I will start with a review of KSB/A stability, especially their local version and then discuss joint work with Janos Kollar, showing that it is enough to check these conditions, including flatness, up to codimension 2. This implies that we have a very good understanding of this stability condition in general, because local KSB-stability is trivial at codimension 1 points, and quite well understood at codimension 2 points, since we have a complete classification of 2-dimensional slc singularities.

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The mapping class group is an important algebraic invariant of a topological space and an important object in low-dimensional geometry and topology, especially in the geometry and topology of 3-manifolds. In this lecture, we will first introduce and review some material about mapping class groups. Then, we will review theorems that explain the structure of these groups and their relations to the Teichmüller spaces. After that, we will discuss some applications of this material in the theory of Riemann surfaces and number theory, especially Grothendieck-Teichmüller theory.

Kahler Calabi-Yau threefolds can be deformed into a non-Kahler object by degeneration and resolution. We will discuss various aspects of the geometrization of this process by special non-Kahler metrics.
This is joint work with T. Collins and S.-T. Yau.

google meet's link:
https://meet.google.com/rtm-twkv-ury

Let $X$ be a smooth connected projective $K3$ surface over the complex numbers and let $Spl(r; c_1, c_2)$ be the moduli space of simple sheaves on $X$ of fixed rank $r$ and Chern classes $c_1$ and $c_2$. In 1984, Mukai proved that $Spl(r; c_1, c_2)$ is a smooth algebraic space of dimension $2rc_2-(r-1)c_1^2-2r^2+2$ with a natural symplectic sstricture, i.e., it has a non-degenerate closed holomorphic 2-form. In my talk, I will present a useful method to construct isotropic and Lagrangian subspaces of $Spl(r; c_1, c_2)$. This is joint work with Barbara Fantechi.

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This lecture is intended as an introduction to some of the metaphysical and epistemological issues that are usually raised in connection with mathematical thought. These issues include, among other things, the nature of abstract objects like numbers and sets, some of the well-known arguments for the existence of such objects as well as the epistemological problems that the existence of those objects incur. There will also be some brief discussions of certain anti-realist approaches to mathematical ontology.

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I introduce Fano varieties and the classification problem. I explain the conjectural framework of Fano/Landau�Ginzburg correspondence and its consequences for the classification of Fano varieties. I intend this to be an accessible colloquium-style presentation.

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Seen through the lens of the Minimal Model Program, the classification of algebraic varieties can be summarised into two main steps: firstly, the algorithm of the MMP allows to decompose a variety with mild singularities, birationally, into a tower of fibrations whose general fibres have ample, anti-ample, or numerically trivial canonical divisor. In view of this decomposition, the natural second step to take is to study these 3 classes of algebraic varieties in detail, for example, studying their moduli theory and/or any other property that could shed light on ways to understand all possible elements that belong to such classes. It turns out a very important property to understand in this process is boundedness: a collection of varieties is bounded when the elements of the given collection can be parametrised using a finite type geometric space. The property of boundedness plays a crucial role in the construction of proper moduli spaces of finite type. Moreover, if a given collection of algebraic varieties is bounded (in char 0), then the topological types of their underlying analytic spaces belong to only finitely many homeomorphism classes: hence, all of their topological invariants come in just finitely many possible different versions. While over the past 15 years, several breakthroughs have completely settled the question of boundedness (and the subsequent construction of moduli spaces) in the case of log canonical models (varieties/pairs with ample canonical divisor) and Fano varieties (those with anti-ample canonical divisor), the situation is still quite unclear in the case of trivial numerical divisor. In this seminar, I will try to explain what is known, or not, and what those challenges are that make the situation quite more complicated than the other 2 cases. I will moreover explain how we can overcome most of the issues if we assume that a K-trivial variety is endowed with a fibration structure of relative dimenesion one. The seminar includes results from various works I developed over the past 10 years with G. Di Cerbo, C. Birkar, S. Filipazzi, and C. Hacon. Moreover, I will talk about current work in progress with P. Engel, S. Filipazzi, F. Greer, M. Mauri were we show various new boundedness results for K-trvial varieties fibered in K3 surfaces or abelian varieties.

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The concept of hypergeometric functions was coined by John Wallis in the middle of 17th century as a generalization of many elementary functions. Since then, these functions have played a key role in various branches of mathematics. Great mathematicians such as Euler, Gauss, Riemann and Klein have investigated the theory of hypergeometric functions from various aspects and have shown the role of these functions in the connection of differential equations, geometry and number theory. In the 20th century, with the generalizations of these functions, unexpected applications of them in theoretical physics (string theory) were found, which led to the solution of conjectures in enumerative geometry. In this lecture, we will have a general overview of these efforts.

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We discuss construction of a derived Lagrangian intersection theory of moduli spaces of perfect complexes, with support on divisors on compact Calabi Yau threefolds. Our goal is to compute deformation invariants associated to a fixed linear system of divisors in CY3. We apply a Tyurin degeneration of the CY3 into a normal-crossing singular variety composed of Fano threefolds meeting along their anti-canonical divisor. We show that the moduli space over the Fano 4 fold given by total space of degeneration family satisfies a relative Lagrangian foliation structure which leads to realizing the moduli space as derived critical locus of a global (-1)-shifted potential function. We construct a flat Gauss-Manin connection to relate the periodic cyclic homology induced by matrix factorization category of such function to the derived Lagrangian intersection of the corresponding “Fano moduli spaces”. The later provides one with categorification of DT invariants over the special fiber (of degenerating family). The alternating sum of dimensions of the categorical DT invariants of the special fiber induces numerical DT invariants. If there is time, we show how in terms of “non-derived” virtual intersection theory, these numerical DT invariants relate to counts of D4-D2-D0 branes which are expected to have modularity property by the S-duality conjecture. This talk is based on joint work with Ludmil Katzarkov and Maxim Kontsevich, recent work with Jacob Krykzca, and former work with Vladimir Baranovsky.

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The pants graph of a compact orientable surface S, introduced by Hatcher and Thurston, is a simplicial graph whose vertices are maximal essential multicurves on S and whose edges correspond to certain local moves on multicurves. The importance of the pants graph comes from a theorem of Brock showing that the pants graph is a good combinatorial model for the Teichmuller space with the Wiel-Petersson metric, which is the space of hyperbolic metrics on S up to isotopy. In joint work with Marc Lackenby, we give upper bounds for distance in the pants graph and derive applications for distance in the Teichmuller space and also for volumes of hyperbolic 3-manifolds. I will give an overview of these results, no prior familiarity with the Teichmuller is needed for the talk.

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This lecture is a continuation of the lecture of about a year ago with the same title. No familiarity with the last lecture will be assumed and the emphasis will be on topics that were not discussed in the previous lecture and have some novelty.

Kahler manifolds appear naturally in both Algebraic and Differential Geometry: projective complex varieties from Algebraic Geometry, and smooth manifolds with Riemannian metrics with holonomy U(n). Understanding when a manifold may admit a Kahler structure and how far it is from that is a key problem in geometry. Classically, tools from global analysis (harmonic forms and Hodge theory) provide striking topological properties of Kahler manifolds, linking topology and analysis. We shall review this circle of ideas, and recent results on the construction of manifolds admitting complex structures (and even both complex and symplectic structures simultaneously) but not admitting Kahler structures.

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Meeting ID: 894 4268 0102
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Actions of algebraic groups appear in many problems of Algebraic Geometry, encoding certain symmetries. Often one is interested in factoring out these symmetries and producing a new geometric object, a so-called quotient that parametrises the orbits of the action. In this framework, Geometric Invariant Theory (GIT) developed by David Mumford in 1965 deals with the problem of the existence of quotients, with an algebro-geometric structure, of an algebraic variety by the action of a reductive group. This method is extremely useful which also has important and surprising connections with symplectic geometry. In this lecture we shall give a gentle introduction to GIT construction, mainly focusing on the affine case. The algebraic side of this story is about finding the subalgebra of invariants (for an algebra with a group action) which dates back to Hilbert�s 14th problem.

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Thanks to Faltings, Arakelov and Parshin's solution to Mordell's conjecture we know that smooth complex projective curves of genus at least equal to 2 have finite number of rational points. A key input in the proof of this fundamental result is the boundedness of families of smooth projective curves of a fixed genus (greater than 1) over a fixed base scheme. The latter was generalized by the combined spectacular results of Kovacs-Lieblich and Bedulev-Viehweg to higher dimensional analogues of such curves; the so-called canonically polarized projective manifolds. In this talk I will discuss our recent extension of this boundedness result to the case of families of varieties with semiample canonical bundle (for example Calabi-Yaus). This is based on joint work with Kenneth Ascher (UC Irvine).

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This will be an expository talk on how algebraic geometry and more specifically Hodge theory have helped to prove some longstanding conjectures in combinatorics and the theory of matroids. These are the results of June Huh, Karim Adiprasito, Eric Katz, Petter Branden, and many more. These results earned June Huh his 2022 Fields Medal.

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Meeting ID: 824 7830 2122
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In this talk, we are going to uncover a paper written by E. Artin one hundred years ago, in 1924, entitled (in German) "Ein mechanisches System mit quasi-ergodischen Bahnen". "A Mechanical System with Quasi-Ergodic Trajectories " The connection between continued fractions and geodesics in the hyperbolic plane goes back to Humbert (1916) and Smith (1877) and it is used in an ingenious way by E. Artin in the above paper provided the first example of a dense geodesic trajectory on a special Riemann surface, the so-called "modular surface". In the first part of the talk, we will try to cover some preliminaries and generalities about discrete subgroups of the group of isometries of the hyperbolic plane and their relations to the various tessellations of the hyperbolic plane / Poincare disc. In the second part, we will try to talk about a special tessellation called Farey tessellation of the hyperbolic plane and its relation to continued fractions. One of the applications of this connection is the existence of a dense geodesic on the modular surface. If time permits, we will also try to discuss the relationship between the Diophantine approximation of an irrational number and the behavior of the corresponding geodesic on the modular surface.

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Meeting ID: 897 3971 8132
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This talk is based on a project started since 2016 on representations theory of affine Lie superalgebras. The study of representations over affine Lie (super)algebras has a long history; the first attempt in this regard dates back to 1974 when a certain class of modules over affine Lie algebras were studied and classified by the eminent mathematician, Victor Kac. In this talk, we will discuss different classes of modules over an affine Lie (super)algebra and state the difficulties appearing to obtain a classification and that how we can overcome these difficulties.

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A Tevelev degree is a type of Gromov-Witten invariant where the domain curve is fixed in the moduli. After reviewing the basic definitions and previously known results, I will report on joint work with Lehmann, Lian, Riedl, Starr, and Tanimoto, where we improve the Lian-Pandharipande bound on asymptotic enumerativity of Tevelev degrees of hypersurfaces and provide counterexamples to asymptotic enumerativity for certain other Fano varieties.

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We study the spectrum of the Laplacian on finite-area hyperbolic surfaces of large volume, focusing on small eigenvalues i.e. those below 1/4. I will discuss some recent results and open problems in this area. Based on joint works with Michael Magee and with Joe Thomas.

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The known results about the rationality vs irrationality of smooth Fano complete intersections $X^n\subset\mathbb P^{n+c}$ of dimension $n=3,4,5$ and fixed type $(d_1,\ldots, d_c)$ suggest an uniform approach to treat several open cases: index one; index two; quartic fourfolds and fivefolds; etc. From one hand one would like to decide the rationality/irrationality of every element in the numerous cases where the stable irrationality of the very general element is known (e.g. quartic fourfolds and fivefolds, quintic fivefolds, etc); from the other hand one hopes to put some further light on several longstanding conjectures (e.g. the irrationality of the very general cubic fourfold). After an introduction of the general problem and after recalling the state of the art, we shall present some of our recent results on these topics.

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Over the last 20 years, the Rasmussen invariant of knots in S^3 has had a number of interesting applications to questions about surfaces in B^4. In this talk I will survey some recent extensions of the invariant to knots in other three-manifolds: in connected sums of S^1 x S^2 (joint work with Marengon, Sarkar, and Willis), in RP^3 (joint work with Willis, and also separate work of Chen), and in a general setting (work by Morrison, Walker and Wedrich; and independently by Ren-Willis). I will describe how these invariants give bounds on the genus of smooth surfaces in 4-manifolds, and can even detect exotic 4-manifolds with boundary.

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Black Box principles, introduced by Saharon Shelah, are prediction principles provable within standard mathematics (ZFC). In this talk, we review the history of their development and explore some of their applications.

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A set of points Z in $\mathbb{P}^3$ is an (a, b)-geproci set (for GEneral PROjection is a Complete Intersection) if its projection from a general point to a plane is a complete intersection of curves of degrees a and b. We will report on some results in order to pursue classification of geproci sets. Specifically, we will show how to classify (a, b)-geproci sets Z which consist of a points on each of b skew lines, assuming the skew lines have two transversals in common. We will show in this case that b ≤ 6. Moreover we will show that all geproci sets of this type and with no points on the transversals are contained in the $F_4$ configuration. We conjecture that a similar result is true for an arbitrary number a of points on each skew line, replacing containment in $F_4$ by containment in a half grid obtained by the so-called standard construction.

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The study of knots in the $3$-space, has a long story. A mathematical framework for this study formed in 19th century and some interesting theorems were proved in the first half of 20th century. Nevertheless, perhaps the more important developments occurred towards the end of 20th century, as well as the past years of the new millennium. We will review some of the major steps, including some of the new tools developed for the study of knots and links, as well as some open questions in this area with relatively simple statements.

In this talk, I will review several key concepts in Tropical Geometry, highlighting the naturality and numerous applications that arise when integrating the Theory of Positive Closed Currents into this framework. This talk is based on previous works with June Huh, Karim Adiprasito and ongoing joint work with Tien Cuong Dinh.

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Secret sharing is a cryptographic concept introduced in 1979. In a secret sharing scheme, a secret is distributed among a set of participants by giving each one a share. The shares are computed by applying a public rule to the secret and some randomness. Only certain pre-specified subsets of participants are qualified to recover the secret, while the secret remains hidden from all other subsets. Secret sharing is significant from both theoretical and practical perspectives. In this talk, I will survey our results from 2018 to 2024 and introduce several open problems.

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Tropical geometry is a combinatorial counterpart of algebraic geometry, transforming polynomials into piecewise linear functions and their solutions (varieties) into polyhedral fans. This transformation is intricately linked to the concept of Grobner bases, which provide a powerful tool in computational algebra. Specifically, all possible Grobner bases of an ideal are encoded within a polyhedral fan, with the tropical variety appearing as a subfan. Despite its significance, the computational complexity of tropical varieties often limits computations to small-scale instances. In this talk, we introduce a geometric approach that enables the effective computation of various points within tropical varieties. One application of this method is the computation of toric degenerations, which are important objects in algebraic geometry. These degenerations can be modeled on polytopes, and there exists a dictionary between their geometric properties and the combinatorial invariants of the corresponding polytopes. This dictionary can be extended from toric varieties to arbitrary varieties through toric degenerations.

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The second order elliptic partial differential equations have been studied for more than a century. The most important and basic example of such equations is the Laplace equation. It is a linear equation related to many topics in mathematics and Physics. The main property of such an equation is that solutions of Laplace equations are regular. Another important example of elliptic equations is the Monge-Ampere equation which is fully nonlinear. In contrast to the Laplace equation, the solutions of the Monge-Ampere equation may fail to be smooth. In this talk, we go over basic properties of Monge-Ampere equations. At last we will go over the celebrated results of Caffarelli on interior regularity of solutions of the Monge-Ampere equation.

حدس الیوت در سال 1990 مطرح شد و پیش بینی می کرد که کلاس سی استار جبرهای جدایی پذیر یکدار ساده میانگین پذیر با استفاده از پایاهای K-نظریه قابل طبقه بندی هستند. این حدس بر پایه نتایجی بود که برای کلاسهای کوچکتر اثبات شده بود. این حدس برای کلاسهای بزرگی ثابت شده است و مثالهای نقضی در سالهای 2002 و 2008 برای آن پیدا شد. برنامه طبقه بندی الیوت که از زمینه های پژوهشی پویا و گسترده در جبر عملگرها می باشد، به بررسی این حدس و کاربردهای آن در جبر عملگرها، سیستم های دینامیکی و سایر شاخه های ریاضی می پردازد. در این سخنرانی، ضمن بیان این برنامه و نتایج بدست آمده، ارتباط و کاربردهای آن در برخی از شاخه های ریاضی را شرح می دهیم.

This talk focuses on pointwise ergodic theorems and their connection to Bourgain's path to his Fields Medal. We aim to cover four key milestones:
Part One: We investigate the norm convergence of ergodic averages, concluding with an explanation of the Baby Spectral Theorem.
Part Two: We examine the pointwise ergodic theorem established by Birkhoff in 1931. Depending on the time available, we will cover at least one proof, exploring Calderon's transference in the process.
Part Three: This section delves into the circle method and its comparison to the continuous Fourier Transform. We will also discuss several properties of prime numbers and provide an approximation for averages along primes.
Part Four: We introduce the concept of oscillation and present a proof of the pointwise ergodic theorem along primes. Time permitting, we will explore further developments post-1990.

The references are uploaded in the following link:
https://drive.google.com/drive/folders/1_zIDuDybfkfYuHhrlLtwe-rd6SdpraPZ?usp=drive_link

This talk focuses on pointwise ergodic theorems and their connection to Bourgain's path to his Fields Medal. We aim to cover four key milestones:
Part One: We investigate the norm convergence of ergodic averages, concluding with an explanation of the Baby Spectral Theorem.
Part Two: We examine the pointwise ergodic theorem established by Birkhoff in 1931. Depending on the time available, we will cover at least one proof, exploring Calderon's transference in the process.
Part Three: This section delves into the circle method and its comparison to the continuous Fourier Transform. We will also discuss several properties of prime numbers and provide an approximation for averages along primes.
Part Four: We introduce the concept of oscillation and present a proof of the pointwise ergodic theorem along primes. Time permitting, we will explore further developments post-1990.

The references are uploaded in the following link:
https://drive.google.com/drive/folders/1_zIDuDybfkfYuHhrlLtwe-rd6SdpraPZ?usp=drive_link

One of the first steps in studying three-dimensional manifolds is the Poincaré conjecture. This conjecture states that every simply connected, closed, and orientable three-dimensional manifold is homeomorphic to the 3-sphere. In the 1980s, Thurston formulated a stronger conjecture than the Poincaré conjecture. Thurston's Geometrization Conjecture states that every closed, orientable, and prime three-dimensional manifold can be cut along a number of tori such that the interior of each resulting manifold has one of Thurston's eight geometric structures with finite volume. In dimension 2, the Geometrization Conjecture reduces to the uniformization theorem. Our goal is to study the decomposition theorems for three-dimensional manifolds (the Prime Decomposition and JSJ Decomposition) and then examine Thurston's eight geometries to precisely state the Geometrization Conjecture. Following this, we will explore some consequences of this conjecture, such as the fact that the fundamental group is almost a complete invariant for closed three-dimensional manifolds.

Some references:
Three-Dimensional Geometry and Topology, William P. Thurston and Silvio Levy
An Introduction to Geometric Topology, Bruni Martelli

One of the first steps in studying three-dimensional manifolds is the Poincaré conjecture. This conjecture states that every simply connected, closed, and orientable three-dimensional manifold is homeomorphic to the 3-sphere. In the 1980s, Thurston formulated a stronger conjecture than the Poincaré conjecture. Thurston's Geometrization Conjecture states that every closed, orientable, and prime three-dimensional manifold can be cut along a number of tori such that the interior of each resulting manifold has one of Thurston's eight geometric structures with finite volume. In dimension 2, the Geometrization Conjecture reduces to the uniformization theorem. Our goal is to study the decomposition theorems for three-dimensional manifolds (the Prime Decomposition and JSJ Decomposition) and then examine Thurston's eight geometries to precisely state the Geometrization Conjecture. Following this, we will explore some consequences of this conjecture, such as the fact that the fundamental group is almost a complete invariant for closed three-dimensional manifolds.

Some references:
Three-Dimensional Geometry and Topology, William P. Thurston and Silvio Levy
An Introduction to Geometric Topology, Bruni Martelli

The problem of counting closed geodesics in Riemannian Manifolds of con- stant negative curvature has been of interest for many years. A method for counting geodesics in manifolds with arbitrary curvature is introduced by Eftekhary. Minimal surfaces in a Riemannian manifold are higher dimensional gener- alizations of closed geodesics. A natural question is how to count these types of submanifolds. We introduce a function which counts minimal tori in a Riemannian manifold (M, g) with dim M > 4. Moreover, we show that this count function is invariant under perturbations of the metric. Looking forward to seeing you organizers

Quantum mechanics emerged as a revolutionary theory that classical physics could not fully explain, particularly at the submicroscopic scale. The shift to quantum mechanics challenged classical beliefs, notably with Heisenberg’s Uncertainty Principle, which established an inherent limit on the precision of measuring operators like position and momentum. Quantum mechanics also introduced the idea of superposition, where particles can exist in multiple states simultaneously until measured, at which point they collapse into a single state. The Heisenberg groups H^n are a significant and simple example of a noncommutative graded Lie group, illustrating the abstract nature of commutation relations between position and momentum operators in quantum mechanics. In this presentation, we initially discuss the representation of the Laplacian in the Heisenberg groups H^n, and subsequently extend it to the graded Lie group. This extension introduces some open problems and conjectures.

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We will discuss the index and spectral flow associated with Fredholm systems with time-reversal symmetry. We connect these to a K-theoretic perspective.

Artinian Gorenstein algebras (AG algebras for short) can be viewed as algebraic analogues of the cohomology rings of smooth projective varieties. The Strong and Weak Lefschetz properties for graded AG algebras take origin from the hard Lefschetz theorem. The properties of an AG quotient $A _F$ of a polynomial ring are related to its Macaulay dual generator $F$, and in particular $A_F$ fails the Strong Lefschetz property if and only if the hessian of $F$ of order $t$ vanishes for some $1\leq t\leq d/2$, where $d=\deg F$ and the usual hessian is obtained for $t=1$. Perazzo polynomials are a large class of polynomials with vanishing hessian so their algebras $A_F$ always fail the SLP. I will report on some recent results concerning the question if the WLP holds for these algebras. Joint work with N. Abdallah, N. Altafi, P. De Poi, L. Fiorindo, A. Iarrobino, P. Macias Marques, R.M. Mir ́o-Roig, L. Nicklasson.

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It is well known that the study of Liouville geometry, in the case of gradient-like Liouville dynamics, can be reduced to a Morse theoretical description in terms of symplectic handle decompositions. Such examples are called Weinstein. On the other hand, the construction and properties of non-Weinstein Liouville structures are far less understood. The first examples of non-Weinstein Liouville manifolds were constructed by McDuff (1991) and Geiges (1995), which were later on generalized by Mitsumatsu (1995), hinting towards further interactions with hyperbolic dynamics. More specifically, Mitsumatsu proved that given an arbitrary closed 3-manifold equipped with a uniformly hyperbolic (i.e. Anosov) flow, one can construct a non-Weinstein Liouville structure on its 4-dimensional thickenning. The purpose of this talk is to show that Mitsumatsu�s construction in fact provides a framework for a contact/symplectic geometric theory of Anosov flows. Furthermore, we discuss how deeper phenomena from the regularity and stability theory of Anosov flows is inherited in the resulting Liouville structure, portraying strong dynamical and geometric rigidity. In particular, we show that our geometric model gives a characterization of non-vanishing Liouville 4-manifolds with C^1-persistent 3-dimensional skeleton in terms of Anosov dynamics.

Starting from scratch, I will define the Monge-Ampere operator and its relation to several fields of mathematics, such as complex analysis, potential theory and complex geometry. Then I shall focus on specific properties of solutions to the Monge-Ampere equation, with an emphasis on estimates and regularity issues. Time permitting I will also discuss the main open problems in the field.

P.N. Here is the link for the webpage of our speaker:
https://apacz.matinf.uj.edu.pl/users/269-slawomir-dinew

Location: https://meet.google.com/you-qymk-ybu

In the 50s, Hirzebruch asked which linear combinations of Hodge and Chern numbers are topological invariants of compact complex manifolds. Building on ideas of Schreieder and Kotschick, who solved the K�hler case, I will present a general answer to this question (and some related ones). Furthermore, I will outline a program how to tackle similar questions when incorporating more cohomological invariants, eg the dimensions of the Bott Chern cohomology groups. This will naturally lead to an algebraic study of the structure of bicomplexes, as well as a number of challenging geometric construction problems.

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In this lecture, we shall talk about Arithmetic from a Logical point view and introduce a related scheme of first-order axioms in the language of Arithmetic, called Open Induction (OI) and show how the problem of characterising the logical hierarchy of subsystems of OI, naturally leads to Galois Theory.

I will briefly recall the notion of stable surfaces and of the corresponding moduli space. Then I will outline a partial description of the boundary points in the case of surfaces with $K^2=1$, $p_g=2$ (joint work with Stephen Coughlan, Marco Franciosi, Julie Rana and Soenke Rollenske, in various combinations) and, time permitting, in the case of Campedelli and Burniat surfaces (joint work with Valery Alexeev).

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A special case of Hironaka's QUESTION F, named F', asks about the strong factorization of birational maps between reduced nonsingular algebraic schemes, which is still open. Suppose that $\varphi : X\dashrightarrow Y$ is such a map, and let $U\subset X$ be the open subset where $\varphi$ is an isomorphism. This problem asks if there exists a diagram $$\xymatrix{ & Z \ar[dl]_{\varphi_{1}}\ar[dr]^{\varphi_{2}}\X \ar[rr]^{\varphi} & & Y}$$ where the morphisms $\varphi_{1}$ and $\varphi_{2}$ are sequences of blow-ups of non-singular centers disjoint from $U$. In this talk, we will discuss how strong factorization can be simplified by providing a complete answer to the problem of toroidalization of morphisms, while we introduce the strong Oda conjecture.

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The interplay of the geometric and the spectral properties of Riemannian manifolds are highly influential in essentially all areas of mathematics, but are far from being fully understood. Some of the recent advancements in understanding this relation could be achieved by means of transfer operators. I shall overview some recent developments in this area with a focus on hyperbolic surfaces, automorphic functions, resonances and the dynamics of the geodesic flow and with an emphasis on insights and heuristics. In particular, I will, at the example of Hecke triangle groups, discuss how an outer symmetry in the dynamics translates into a parity for automorphic functions.

Location: https://meet.google.com/you-qymk-ybu

A polynomial vector field on a complex plane C^2 defines a foliation of the plane and one would like to know when the leaves of this foliation are algebraic, i.e. when the analytic integral curves are algebraic. This is a hard unsolved problem. I will suggest a conjectural approach to this problem which uses the reduction of the vector field modulo primes p. It turns out that the corresponding problem in characteristic p is easy to solve. We then conjecture that the original vector field is algebraic if and only if its reduction modulo a prime p is algebraic for almost all p. We have some partial results towards proving this conjecture. This is a work in progress with D. Leshchiner.


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The geometry of moduli spaces and stacks of vector bundles on curves have been intensively studied from different perspectives; for example, via point counting over finite fields by Harder and Narasimhan, and gauge theoretically by Atiyah and Bott over the complex numbers. Following Grothendieck's vision that a motive of an algebraic variety should capture many of its cohomological invariants, Voevodsky introduced a triangulated category of motives which partially realises this idea. After describing some properties of this category, I will present a formula for the motive of the moduli stack of vector bundles on a smooth projective curve; this formula is compatible with classical computations of invariants of this stack due to Harder, Atiyah--Bott and Behrend--Dhillon. The proof involves rigidifying this stack using Flag-Quot schemes parametrising Hecke modifications as well as a motivic version of an argument of Laumon and Heinloth on the cohomology of small maps, which is closely related to the Grothendieck-Springer resolution. I will explain how to extend this to a formula for the stack of coherent sheaves and, if there is time, I will give an overview of other motivic descriptions of closely related moduli spaces. This is joint work with Simon Pepin Lehalleur.

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Higgs bundles are vector bundles equipped with an additional "twisted endomorphism". Introduced by Nigel Hitchin in a context of mathematical physics, they have turned to be central objects in differential and algebraic geometry. In particular, moduli spaces of Higgs bundles have a very rich geometry that is both related to the geometry of moduli of vector bundles but also has additional symplectic features. I will introduce these moduli spaces and discuss some of what is known about their cohomology and their motivic invariants. There has been a lot of recent progress in this direction and I will try to describe the main threads. I will conclude with a discussion of my joint work with Victoria Hoskins on a motivic version of the "cohomological mirror symmetry" conjecture of Hausel and Thaddeus for SL_n and PGL_n Higgs bundles.

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Asymptotic separation of variables is a tool that Chris Judge and I introduced to prove the generic simplicity of Euclidean triangles. I will explain the ideas behind this approach and how it can be applied to different simple shapes.


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Finite fields are indispensable to modern cryptography and coding theory. It is well known that the implementation of the arithmetic of finite fields greatly depends on their representation, which in turn depends on the existence of irreducible polynomials of special forms. So, studying irreducible polynomials over finite fields besides having a natural theoretical appeal, has an applied appeal, too. There are an abundant number of known results and open problems about the distribution of irreducible polynomials over finite fields. In this talk, we briefly survey what is known and what remains to be explored.

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Like its complex counterpart, non-Archimedean pluripotential theory studies plurisubharmonic functions and Monge-Ampere equations on non-Archimedean analytic spaces in the sense of Berkovich. It provides a useful tool to analyze degenerations of complex analytic objects in general, and the notion of K-stability entering in the Yau-Tian-Donaldson conjecture in particular. This talk will propose a gentle introduction to this circle of ideas.

Location: https://meet.google.com/you-qymk-ybu

Reciprocity Laws have played an important role in the development of Number Theory. The expository article by Jared Weinstein in the Bulletin of the American Mathematical Society (2016) gives a technically sophisticated and commendable survey including fairly recent developments. The purpose of this lecture is to explain to a mathematician with little knowledge of algebraic number theory the origin of the problem, why reciprocity laws are interesting, and what some of the methods of the investigation and accomplishments are. Accordingly, the emphasis in the lecture will be somewhat different from those in standard accounts.

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In her thesis, Mirzakhani proved that the number of simple closed geodesics (that is, closed geodesics without self-intersection) of length at most $L$ in a fixed hyperbolic surface $X$ is asymptotic to a constant times $L^{6g - 6}$, as $L \to \infty$ (here, $g \geq 2$ is the genus of $X$). More recently, Eskin-Mirzakhani-Mohammadi presented an alternative proof of Mirzakhani's theorem by relating the count in question to lattice point counts in Teichmuller balls of large radius, a count that was previously studied by Athreya-Bufetov-Eskin-Mirzakhani. Their approach enabled them to obtain a more precise result, namely, to obtain a power-saving error term for the count in the question.
In this short course, we aim to highlight the connection between lattice point counts in Teichmuller space and simple closed geodesic counts on a fixed hyperbolic surface. Our exposition follows in part Arana-Herrera's 2021 paper which obtains power-saving error terms for a closely related count (that is, mapping class group orbits of a fixed filling curve), and follows in part a work in progress of mine.

More information: https://drive.google.com/drive/folders/1_0qR8OtN737UFGdYwt3jkhl86Nz7zmUE?usp=sharing

In her thesis, Mirzakhani proved that the number of simple closed geodesics (that is, closed geodesics without self-intersection) of length at most $L$ in a fixed hyperbolic surface $X$ is asymptotic to a constant times $L^{6g - 6}$, as $L \to \infty$ (here, $g \geq 2$ is the genus of $X$). More recently, Eskin-Mirzakhani-Mohammadi presented an alternative proof of Mirzakhani's theorem by relating the count in question to lattice point counts in Teichmuller balls of large radius, a count that was previously studied by Athreya-Bufetov-Eskin-Mirzakhani. Their approach enabled them to obtain a more precise result, namely, to obtain a power-saving error term for the count in the question.
In this short course, we aim to highlight the connection between lattice point counts in Teichmuller space and simple closed geodesic counts on a fixed hyperbolic surface. Our exposition follows in part Arana-Herrera's 2021 paper which obtains power-saving error terms for a closely related count (that is, mapping class group orbits of a fixed filling curve), and follows in part a work in progress of mine.

More information: https://drive.google.com/drive/folders/1_0qR8OtN737UFGdYwt3jkhl86Nz7zmUE?usp=sharing

We consider boundary value problems on manifolds with boundary where the boundary exhibits singularities of fibred cusp type. The simplest (unfibred) cusp is what is sometimes called an incomplete cusp, e.g. the complement of two touching circles in the plane. The fibred version includes the complement of two touching balls in $\mathbb{R}^n$. Our results also extend to geometries which are conformal to these incomplete cusps, for example fundamental domains of Fuchsian groups or uniformly fattened infinite cones in $\mathbb{R}^n$. For the Laplacian on such spaces, or more general elliptic operators $P$ whose structure relates well to the geometry, we study one of the basic objects of the theory of boundary value problems: the Calderon projector, which is essentially the projection to the set of boundary values (Cauchy data) of the homogeneous equation $Pu=0$. For a smooth compact manifold with boundary, it is classical that the Calderon projector is a pseudodifferential operator (PsiDO). In the case of the Laplacian, one can deduce from this that the Dirichlet-to-Neumann operator, which is fundamental to many spectral theoretic questions studied currently (like the Steklov spectrum), and also of interest in inverse problems, is a PsiDO also. We extend these results to the case where the boundary has fibred cusp singularities: both the Calderon projector and the Dirichlet-Neumann operator are in a PsiDO calculus adapted to the geometry, the so-called phi-calculus. This yields a precise description of their integral kernels near the singularities. In the talk, I will introduce the necessary background on the phi-calculus. This is joint work with K. Fritzsch and E. Schrohe.

Location: https://meet.google.com/you-qymk-ybu

In her thesis, Mirzakhani proved that the number of simple closed geodesics (that is, closed geodesics without self-intersection) of length at most $L$ in a fixed hyperbolic surface $X$ is asymptotic to a constant times $L^{6g - 6}$, as $L \to \infty$ (here, $g \geq 2$ is the genus of $X$). More recently, Eskin-Mirzakhani-Mohammadi presented an alternative proof of Mirzakhani's theorem by relating the count in question to lattice point counts in Teichmuller balls of large radius, a count that was previously studied by Athreya-Bufetov-Eskin-Mirzakhani. Their approach enabled them to obtain a more precise result, namely, to obtain a power-saving error term for the count in the question.
In this short course, we aim to highlight the connection between lattice point counts in Teichmuller space and simple closed geodesic counts on a fixed hyperbolic surface. Our exposition follows in part Arana-Herrera's 2021 paper which obtains power-saving error terms for a closely related count (that is, mapping class group orbits of a fixed filling curve), and follows in part a work in progress of mine.

More information: https://drive.google.com/drive/folders/1_0qR8OtN737UFGdYwt3jkhl86Nz7zmUE?usp=sharing

In her thesis, Mirzakhani proved that the number of simple closed geodesics (that is, closed geodesics without self-intersection) of length at most $L$ in a fixed hyperbolic surface $X$ is asymptotic to a constant times $L^{6g - 6}$, as $L \to \infty$ (here, $g \geq 2$ is the genus of $X$). More recently, Eskin-Mirzakhani-Mohammadi presented an alternative proof of Mirzakhani's theorem by relating the count in question to lattice point counts in Teichmuller balls of large radius, a count that was previously studied by Athreya-Bufetov-Eskin-Mirzakhani. Their approach enabled them to obtain a more precise result, namely, to obtain a power-saving error term for the count in the question.
In this short course, we aim to highlight the connection between lattice point counts in Teichmuller space and simple closed geodesic counts on a fixed hyperbolic surface. Our exposition follows in part Arana-Herrera's 2021 paper which obtains power-saving error terms for a closely related count (that is, mapping class group orbits of a fixed filling curve), and follows in part a work in progress of mine.

More information: https://drive.google.com/drive/folders/1_0qR8OtN737UFGdYwt3jkhl86Nz7zmUE?usp=sharing

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In her thesis, Mirzakhani proved that the number of simple closed geodesics (that is, closed geodesics without self-intersection) of length at most $L$ in a fixed hyperbolic surface $X$ is asymptotic to a constant times $L^{6g - 6}$, as $L \to \infty$ (here, $g \geq 2$ is the genus of $X$). More recently, Eskin-Mirzakhani-Mohammadi presented an alternative proof of Mirzakhani's theorem by relating the count in question to lattice point counts in Teichmuller balls of large radius, a count that was previously studied by Athreya-Bufetov-Eskin-Mirzakhani. Their approach enabled them to obtain a more precise result, namely, to obtain a power-saving error term for the count in the question.
In this short course, we aim to highlight the connection between lattice point counts in Teichmuller space and simple closed geodesic counts on a fixed hyperbolic surface. Our exposition follows in part Arana-Herrera's 2021 paper which obtains power-saving error terms for a closely related count (that is, mapping class group orbits of a fixed filling curve), and follows in part a work in progress of mine.

More information: https://drive.google.com/drive/folders/1_0qR8OtN737UFGdYwt3jkhl86Nz7zmUE?usp=sharing

In this talk I will discuss two important model-theoretic constructions, namely ultra-product and Fraisse-Hrushovski constructions and overview some of their applications in various fields of mathematics.

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In this talk, I will survey classification results for vector bundles on smooth threefolds over an algebraically closed field. I will start with classical results in the affine case, and then show how to complete the classification in that case. Then, I will pass to quasi-projective threefolds, focusing on the case of complex varieties.

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The Nonvanishing Conjecture and the Abundance Conjecture are longstanding open problems in the Minimal Model Program. I am going to present some unexpected generalizations which appeared in the literature in the last few years and to discuss a few variants of them.

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The (one-phase) Bernoulli problem deeply influenced the modern free boundary regularity theory and is still an object of intensive research. Free boundary problems are often PDE-described problems with a priori unknown (free) interfaces or boundaries. Such types of problems appear in Physics, Geometry, Probability, Biology, or Finance, and the study of solutions and free boundaries uses methods from PDE, Calculus of Variations, and Geometric Measure Theory. We first review the very classical theory, which is about how to understand the regularity of free boundaries. Then, we present some recent results for the Bernoulli problem with the p-Laplace operator.

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I will report on a joint work with Yuri Tschinkel (Simons Foundation) and Zhijia Zhang (New York University) on linearizability of actions of finite groups on singular cubic threefolds.

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An interval exchange transformation (IET) is a map from the unit interval to itself for which the interval admits a finite partition by subintervals over each of which, the map acts as translation in such a way that the image of the interior of the intervals is disjoint from one another. This family of maps arise naturally as the return maps of directional flows on translation surfaces. The ergodic theory of IETs has been studied extensively by many authors. In this talk, we will review some ergodic properties of this family. Then, we introduce linear cocycles over the space of IETs and study their spectral properties. A well-known cocycle is the Kontsevich-Zorich cocycle. The study of the Lyapunov spectra of this cocycle has implications for asymptotic growth of Birkhoff sums and understanding of the dynamics of typical IETs. One can show that this cocycle preserves a symplectic structure, thereby having symmetric spectra with respect to zero. The Kontsevich-Zorich conjecture proved by Forni, Avila-Viana asserts that the Lyapunov spectra of the restricted Kontsevich-Zorich cocycle, and consequently that of the restricted Zorich cocycle, are simple and nonzero. Our main focus in this talk, will be on the study of the spectrum of another linear cocycle introduced by Bufetov and Solomyak, known as the twisted (or spectral) cocycle. This cocycle is designed so that its iterations control the exponential Birkhoff sums of locally constant functions, analogous to the fact that the iterations of the Zorich cocycle govern the behavior of ordinary Birkhoff sums of such functions. This cocycle and its cousins have appeared in several papers studying rates of weak mixing and the spectral measures associated with IETs and substitution dynamical systems. We will show that the spectrum of this cocycle is non-degenerate if the genus of the associated translation surface is bigger than one.
This talk is based on a recent joint work with Pedram Safaee (UZH).

Suppose that $(K,v_0)$ is a valued field, $f(x)\in K[x]$ is a monic and irreducible polynomial and $(L,v)$ is an extension of valued fields, where $L=K[x]/(f(x))$. Let $A$ be a local domain with quotient field $K$ dominated by the valuation ring of $v_0$ and such that $f(x)$ is in $A[x]$. The study of these extensions is a classical subject. In this talk, we discuss the history of this subject, connections with resolution of singularities, and recent progress. We will discuss our recent work with Razieh Ahmadian on the problem of describing the structure of the associated graded ring ${\rm gr}_v A[x]/(f(x))$ of $A[x]/(f(x))$ for the filtration defined by $v$ as an extension of the associated graded ring of $A$ for the filtration defined by $v_0$. We give a complete simple description of this algebra when there is unique extension of $v_0$ to $L$ and the residue characteristic of $A$ does not divide the degree of $f$. To do this, we show that the sequence of key polynomials constructed by MacLane's algorithm can be taken to lie inside $A[x]$. This result was proven using a different method in the more restrictive case that the residue fields of $A$ and of the valuation ring of $v$ are equal and algebraically closed in a recent paper by Cutkosky, Mourtada and Teissier.

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Long time existence of Calabi flow on Riemann surfaces was first proved by Chrusciel and later by Chen. In this talk, We give a different proof for the long time existence.

In this talk I aim to discuss asymptotically almost sure (aka a.a.s.) expansion lower bounds of the uniform ensemble of d-regular graphs and show that random π-lifts may be used to obtain some improvements. In this regard, after introducing the model, I briefly go through some techniques already used in this area of research and based on a recent joint contribution with MH. Shojaedin, I will introduce a general reduction method that provides a.a.s. lower bounds when a.a.s. upper bounds are known, which is based on the analysis of a contiguous ensemble constructed through random π-lifts. This model gives rise to a dual approximation as a conditional optimization problem that can be handled using Bernstein and normal approximation schemes. In particular, I will report some consequences of this approach as improvements of existing a.a.s. lower bounds for the case of small degrees and the case of asymptotically large degrees. Subject to my time limitations I may also talk about the iterated π-lift model and its spectral properties as well as some connections of the subject to statistical mechanics and physics.

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We discuss the algebraic geometry behind coupled cluster (CC) theory of quantum many-body systems. The high-dimensional eigenvalue problems that encode the electronic Schroedinger equation are approximated by a hierarchy of polynomial systems at various levels of truncation. The exponential parametrization of the eigenstates gives rise to truncation varieties. These generalize Grassmannians in their Pluecker embedding. We explain how to derive Hamiltonians, we offer a detailed study of truncation varieties and their CC degrees, and we present the state of the art in solving the CC equations. This is joint work with Fabian Faulstich and Svala Sverrisdóttir.

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This is mostly an introduction to and short survey of hyperkahler manifolds and Lagrangian fibrations, including some known results and some open problems.

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I will start by displaying a model of a free boundary problem which by now belongs to the folklore. Then, I will go over some well-known facts in regard to the Poisson problem with zero Dirichlet boundary conditions. Finally, I will introduce the optimization problems which will lead us to the aforementioned free boundary problems. Some open problems and intriguing questions will be shared as well.

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Strongly-correlated quantum systems are often extremely fragile and notoriously hard to control, which poses challenges for possible technological applications. That is why a certain subclass of quantum states, the so-called topological phases of matter, recently attracted much attention. These are characterised by a certain degree of stability and robustness under perturbations, rooted in their special mathematical properties. Apriori, it is not always clear whether a given quantum state of matter is topological or not. We propose a mathematical criterion, which we call ''the geometric test", to tell whether a state of matter is in a topological phase. We then apply our test to strongly-interacting states of matter in Quantum Hall effect, observed in certain 2d materials (Gallium-arsenide, graphene, ...) at low temperatures and in strong magnetic fields. I will explain the idea of the test (which works pretty well) and the results, based on recent and upcoming works with Dimitri Zvonkine (Paris) and with Igor Burban (Paderborn).

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A nodal domain refers to a connected region where the eigenfunction is nonzero. The simplest topological invariant of nodal domains is the nodal count. The celebrated Pleijel nodal domain theorem gives an upper bound on the asymptotic behaviour of the nodal count of the Laplace eigenfunctions with Dirichlet boundary condition. There have been advancements in this direction including its extension to the Neumann problem and more generally to the Robin problem with a non-negative parameter as well as improvements of the Pleijel upper bound for the Dirichlet problem. In this talk, we discuss an implication of these results. In particular, we focus on the extension of the Pleijel-type nodal domain theorem to the Robin problem without restriction on the sign of the Robin parameter. This is joint work with David Sher.

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Here you can find more information about our speaker:
https://research-information.bris.ac.uk/en/persons/asma-hassannezhad/publications/

This is a general overview of some of the most prominent Programs in Mathematics. We give a very short (and mainly historical) account of programs proposed by Riemann (1854) Klein (1872) Poincarè (1892) Hilbert (1900/1921) Weil (1949) Langlands (1967) Grothendieck (1984) and Mori (1988). We also review some of the most famous classification programs, such as ’classification of von Neumann algebras’ (1936) ’classification of finite simple groups’ (1972) and ’classification of separable simple nuclear unital C*-algebras’ (1993). We give a short account of two programs proposed for the current century by Smale (1999) and Simon (2000). This is not a technical review and we hope to give a glimpse, or if successful, an essence of all these.

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In this talk, I will start with some basic facts on harmonic maps in the smooth setting. I will then touch upon various definitions of harmonic maps with singular targets or domains or both and will discuss some recent and new results in this direction.

You can find a short scientific biography of the speaker here:
https://sajjadlakzian.iut.ac.ir/biography

The topology of character varieties of surface groups is well known to be of fundamental importance, in part through its relationship, via the non-abelian Hodge theory to the moduli spaces of Higgs bundles. The calculation of Serre (also known as E-) polynomials of these varieties received an important impetus with works of Hausel and Rodriguez-Villegas (2008). These arithmetic methods, as well as new geometric techniques have been successfully developed and applied to other cases, like moduli of quiver representations, and character varieties of free or free abelian groups. In this talk we will introduce another point of view in the computations of the E-polynomials of character varieties for arbitrary finitely presented groups, based on a natural stratification coming from affine GIT and the combinatorics of partitions.

We discuss two notions of percolation on graphs. The first one is $r$-neighbor bootstrap percolation in which an activation process of the vertices of a given graph is carried out. The process starts with some initially activated vertices and then, in each round, any inactive vertex with at least $r$ active neighbors becomes activated. The second one is the so called graph bootstrap percolation in which we activate the edges of a graph instead of vertices. Starting with some initially activated edges, we activate any inactive edge which creates a new copy of $H$, the pattern or base graph. Two problems, one of combinatorial nature and the other of probabilistic type are reviewed. The minimum size of a set of initially activated vertices leading to the activation of all vertices of a graph $G$ in the $r$-neighbor bootstrap percolation is denoted by $m(G, r)$. Computing $m(G, r)$ is a combinatorial problem. In the second problem, each vertex of $G$ is initially activated with a probability $p$ and the question is to find the probability threshold of activation of all vertices of the graph $G$. We talk about the latest developments on both problems and also the connections between $r$-neighbor bootstrap percolation and graph bootstrap percolation.

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I will survey some recent developments regarding the minimal model program for foliations defined over a complex algebraic variety, together with some applications towards the study of fibrations in birational geometry.

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For a full exceptional sequence of vector bundles on the projective plane there is a remarkable equation, the so-called Markov equation, in terms of the ranks of the three vector bundles. This equation, slightly modified, has been used in a joint work with Beineke and Brustle for cluster mutations for quivers with three vertices. The aim of this talk is to define the natural generalization for full exceptional sequences with n members. This leads to the notion of a polynomial invariant, that is a polynomial in indeterminants x(i,j) for i < j between 1 and n. This allows to evaluate such a polynomial at any full exceptional sequence. We define a polynomial invariant to be a polynomial whose value does not depend on the full exceptional sequence, it only depends on the underlying category. In the talk we define polynomial invariants, present several examples and relate them to the natural braid group action. Eventually, we classify all polynomial invariants.

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The Calabi Problem is a formidable problem in the confluence of differential and algebraic geometry. It asks which compact complex manifolds admit a Kahler-Einstein metric. A necessary condition for the existence of such a metric is that the canonical class of the manifold has a definite sign. For manifolds with zero or positive canonical class, the Calabi problem was solved by Yau and Aubin/Yau in the 1970s. They confirmed Calabi's prediction, showing that these manifolds always admit a Kahler-Einstein metric. On the other hand, for projective manifolds with negative canonical class, called ''Fano manifolds'', the problem is much more subtle: Fano manifolds may or may not admit a Kahler-Einstein metric. The Calabi problem for Fano manifolds has attracted much attention in the last decades, resulting in the famous Yau-Tian-Donaldson conjecture. The conjecture, which is now a theorem, states that a Fano manifold admits a Kahler-Einstein metric if and only if it satisfies a sophisticated algebro-geometric condition, called ''K-polystability''. In the last few years, tools from birational geometry have been used with great success to investigate K-polystability. In this talk, I will present an overview of the Calabi problem, the recent connections with birational geometry, and the current state of the art in dimension 3.

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For over a century de Rham cohomology has played a central role in geometry, analysis and arithmetic. In positive characteristic, it is still a mysterious object. I will explain recent discoveries about it, due to Bhatt-Lurie, Drinfeld, and Petrov, in the wake of the theory of prismatic stacks.

Link: https://vroom.ui.ac.ir/b/jav-qqx-92n-2wr

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Grobner bases are a powerful tool in polynomial ideal theory with many applications in various areas of science and engineering. A Grobner basis is a particular generating set for a given ideal which represents many useful properties of the ideal. The general theory of Grobner bases along with the first algorithm for constructing them were introduced by Buchberger in 1965 in his Ph.D. thesis. An staggered linear basis is indeed a linear basis containing a Grobner basis for a given ideal. This notion was first introduced by Gebauer and Moller in 1988, however the algorithm that they described for computing these bases was not complete. In this talk, we first give a brief overview on the theory of Grobner bases (as well as of staggered linear bases) and then present a simple algorithm for computing staggered linear bases.

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The term moving interfaces and free boundary refers to a class of problems in which the domain of the problem itself is also an a priori unknown, as well as the basic unknown solution to governing equations. Hence, finding the domain is part of the problem. In this lecture I shall present some basic models in moving interfaces and free boundary problems. These includes obstacle problem, Stefan problem, Hele-Shaw flow and Muskat problem. I will also review the state of art for regularity results in obstacle problem.

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It is an open problem to describe the shape of the effective cone of an algebraic surface. Nagata conjecture predicts part of this shape when the surface is the blow-up of the projective plane at general points. More recently Ciliberto and Kouvidakis proved that Nagata conjecture implies that the two-dimensional effective cone of the symmetric product C_2 of a general, genus g > 9, curve C is open on one side whenever g is not a square. In this talk I will show that the effective cone of the blow-up of C_2 at a general point is non-polyhedral for a general positive genus curve C. This result generalizes previous statements of J.F. Garcia and G. McGrat about the genus 1 case. To prove the statement we first show that having polyhedral effective cone is a closed property for families of surfaces having the same Picard group and then we prove it in the hyperelliptic case. This is joint work with Luca Ugaglia.

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This is an interdisciplinary study and research program related to the areas of differential geometry, algebraic and differential topology, complex (K�ahler) geometry, complex algebraic geometry, dynamical systems, quantum computation as well as classical and quantum mechanics. We will first study the Special Theory of Parity which is the study of the manifold R^N\times R^N. In General Theory of Parity, we study a general manifold of even-dimensional from the differential and dynamical point of view. Our motivation is the following pioneering paper: �Geometrical Formulation of Quantum Mechanics by A. Ashtekar and T.A. Schilling�. We invite all students and researchers working in one of the above mentioned areas to attend in this program.

I will give a survey of some problems and recent results concerning the link between the monodromies of structure connections on certain Frobenius manifolds with the numerology of the derived categories of Fano varieties.

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We construct the heat kernel on the setting of manifolds with fibred boundary metrics. In physics, these metrics arise as gravitational instantons or complete hyperkahler four-manifolds. For such manifolds, we analyze the asymptotics of heat kernels in finite and long time. The construction approach is the $\phi$-calculus of Melrose and Mazzeo combined with the resolvent construction of Guillarmou and Hassell. We present the analytic torsion of such manifolds as an application of these constructions and impose the natural question of how to relate this to the topological invariant, i.e. Reidemeister Torsion in this setting.

Keywords: Heat kernel, Resolvent at low energy, fibred boundary manifolds, Analytic torsion

Reference: http://oops.uni-oldenburg.de/4772/1/talspe20.pdf

Bell non-locality and its experimental realizations are celebrated results in quantum physics. Non-locality asserts that certain correlations in nature cannot be explained classically, i.e., by the local hidden variable model. In Bell's setting we study correlations between two or more distant parties who share a single source in common. Recently, by the development of quantum communication networks, the more general setup of non-locality in networks is also studied. In this setup, the parties have several sources in common that are shared through a network, so we expect richer non-locality in networks comparing to the standard Bell's setup. In this talk after an introduction to non-locality, I will explain some examples of non-locality in networks.

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This talk briefly explains how to find closed simply connected Sasaki-Einstein 5-manifolds from K-stable log del Pezzo surfaces. It then lists closed simply connected 5-manifolds that are known so far to admit Sasaki-Einstein metrics. It also presents possible candidates for Sasaki-Einstein 5- manifolds to complete the classification of closed simply connected Sasaki-Einstein 5-manifolds.

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Passcode: 362880

lll

In recent years, a novel attitude to the classical problem of identifying and classifying special linear systems in projective n space, has been emerged. For a subvariety Z of the projective n space with defining ideal I, let P_1,�,P_s be general distinct points in this space and let m_1,�,m_s be positive integers which at least one of them is greater than one. On the subspace of those elements of degree d part of the homogeneous ideal I which vanish at P_i with multiplicity at least m_i, each fat point m_iP_i defines a specific number of linear relations on this subspace. For a given set of points P_i with multiplicity m_i, it is expected that these linear equations to be linearly independent. If it is not the case, then one says that the variety Z admits an unexpected hypersurface with respect to the fat point subscheme defined by these fat points, and this linear subspace is called a special linear system on the variety Z. Each element of this subspace defines a hypersurface, known as an unexpected hypersurface. In this talk, we review some interesting examples which brought into the scene with this new approach and explain some existing methods to construct unexpected hypersurfaces.

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I will discuss the Riemann zeta function and the significance of its zeros to prime numbers. Also, I will look at the distribution of zeta zeros and mention some of my related works on the subject.

To get more information about the colloquium, join to the following google group: https://groups.google.com/g/ipm-math-colloquium

We discuss some stricking properties of stable vector bundles over curves, which are frequently used in moduli and Brill-Noether arguments of these bundles. Then, after a quick historical surf in the topic, we give an upper bound for dimensions of Brill-Noether schemes of rank 2 stable vector bundles.

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In this talk I will try to address some problems related to cohomology of dynamical systems and its relation to rigidity problems as well as applications to cohomology of groups.


[Zoom: https://us02web.zoom.us/j/83021642552?pwd=dmM4SFdxRmI0MWZaTmFNNWphR3RFdz09]


[BeST Dynamics Seminars, A joint Beijing-Shenzhen-Tehran mathematical activity]

We discuss the smoothness of toric quiver varieties. When a quiver Q is defined with the identity dimension vector, the corresponding quiver variety is also a toric variety. So it has a fan representation and a quiver representation. I consider only quivers with canonical weight and we classify smooth such toric quiver varieties. I show that a variety corresponding to a quiver with the identity dimension vector and the canonical weight is smooth if and only if it is a product of projective spaces or their blowups.

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Zoom Meeting link: https://us06web.zoom.us/j/81660172318

Suppose n algorithms adaptively choose n pieces of an input (x1,...xn)=X one by one from some predefined distributions, and that Pr[f(X)=1]=p for a (known) Boolean function f. This defines a coin-tossing protocol with coin bias p. Now suppose an adversary A who observes the protocol, as it proceeds, can replace up to k of the inputs. How much can such adversary A increase Pr[f(X)=1] in the two natural settings below. 1.The tampered messages are chosen at random. 2. A can choose the tampered messages at wish. I will survey what is known about the question above, with the focus on (2) while aiming for polynomial-time attacks. I will also briefly mention the connections between the problem above and randomness-tampering attacks on encryption, data poisoning attacks on machine learning algorithms, as well as a new algorithmic approach to measure concentration in product spaces. Based on a sequence of joint works with Omid Etesami (IPM), Ji Gao (UVA), and Saeed Mahloujifar (Princeton) published at TCC'17, ALT'18, ALT'19, ICML'19, SODA'20, TCC'21.
Joining info: https://groups.google.com/g/ipm-math-colloquium

Since introducing Calabi-Yau varieties, a vast number of works in mathematics and theoretical physics have been dedicated to the study of differential equations which are related to these varieties. The solutions of these differential equations, or system of differential equations, provide us with innumerous infinite series or q-expansions (Fourier series) with integer coefficients which are generating functions of certain quantities. In lower dimensions, let us say in dimensions 1 and 2 which are elliptic curves and K3 surfaces, usually these encountered q-expansions are (quasi-)modular forms. But in higher dimensions we can not relate them with the classical quasi-modular forms and call them Calabi-Yau modular forms. This minicourse aims to present these concepts and related facts in two consecutive lectures.

Online: https://meet.google.com/zoi-mdaj-ghe

Rapoport-Zink spaces for p-divisible groups are local counterparts for Shimura varieties. According to the dictionary between function fields and number fields, they correspond to the RZ-spaces for local P-shtukas. We review the construction of these moduli spaces and then discuss our approach for computing the semi-simple trace of Frobenius on their (nearby-cycles) cohomology.

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Meeting ID: 9086116889
Passcode: 362880

Since introducing Calabi-Yau varieties, a vast number of works in mathematics and theoretical physics have been dedicated to the study of differential equations which are related to these varieties. The solutions of these differential equations, or system of differential equations, provide us with innumerous infinite series or q-expansions (Fourier series) with integer coefficients which are generating functions of certain quantities. In lower dimensions, let us say in dimensions 1 and 2 which are elliptic curves and K3 surfaces, usually these encountered q-expansions are (quasi-)modular forms. But in higher dimensions we can not relate them with the classical quasi-modular forms and call them Calabi-Yau modular forms. This minicourse aims to present these concepts and related facts in two consecutive lectures.

Online: https://meet.google.com/zoi-mdaj-ghe

A well-known theorem by Hartshorne and Hirschowitz states that a generic configuration of lines has good postulation. So what about non-reduced configurations? Can adding a multiple line to the configuration still preserve it’s good postulation? This is the question we mainly deal with in this talk. In the first part of the talk we introduce the postulation problem for projective schemes, then we discuss the problem for the family of schemes supported on generic linear configurations, which are the ones of particular interest. In the second part of the talk we focus on the postulation of generic lines and one multiple line in projective space. We give our main theorem providing a complete description to the case of lines and a double line, then we propose a conjecture to the general case, finally we discuss what is known about the conjecture and more recent results on it.

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In this talk, we will introduce the notion of relation type of formal and analytic algebras and show that it is well defined by using of André-Quillen homology. In particular, the relation type is an invariant of an affine algebraic variety and a complex space germ. We will discuss and essay to explain the relation type of singular subscheme of isolated hypersurface singularities. This talk is based on joint ongoing work with Maryam Akhavin.

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I will discuss a 2-dimensional stable pair theory of nonsingular complex 4-folds that is parallel to Pandharipande-Thomas' 1-dimensional stable pair theory of 3-folds. The stable pairs of a 4-fold are related to its 2-dimensional subschemes via wall-crossings in the space of polynomial stability conditions. In Calabi-Yau case, Oh-Thomas theory is applied to define invariants counting these stable pairs under some restrains. This is a joint work with Yunfeng Jiang and Jason Lo.

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This is going to be a report of my ongoing joint research project with Mr. Moghaddamzadeh on finite projective geometries. However, a good portion of the talk will be spent on explaining Grothendieck's generalizations of Galois theory.

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The Multiplicative Ergodic Theorem (MET) is a powerful tool with various applications in different fields of mathematics, including analysis, probability theory, and geometry, and a cornerstone in smooth ergodic theory. Oseledets first proved it for matrix cocycles; since then, the theorem attracted many researchers to present new proofs and formulations with increasing generality.
This talk intends to provide a new version of MET for stationary compositions on a (possibly random) field of (potentially distinct) Banach spaces, depending on the random sample. MET has two versions, and in the first talk, I will concentrate on the one-sided form of this theorem. The primary motivation of this work is to implement a dynamical approach for stochastic delay equations. Analyzing the long-time behavior of this type of equation is a challenging task; since their corresponding solutions often fail to admit the flow property. Our MET, in particular, can be applied to this family of equations to prove the existence of the Lyapunov exponents. In the second part of the talk, which is supposed to be given in the week after, I will talk about the semi-invertible version of MET.
This work is based mainly on the speaker's P.h.D thesis.


[IPM Youth Seminars on Topology and Dynamics]

First, we talk about the topology of orbits of points under hyperbolic diffeomorphisms and the important theorem: Smale spectral decomposition. After that, we will talk about the structure of pre-images, pre-orbits, and limit sets, under Anosov endomorphisms, and some important topological properties of those sets under endomorphisms. Especially linear Anosov endomorphisms over tori. In this regard, we consider a fixed point and its pre-images under a linear endomorphism over as a lattice in. This talk is based on the works of Feliks Przytycki; My Ph.D. thesis, and the book ...


[IPM Youth Seminars on Topology and Dynamics]

In this talk we review construction of toric varieties and classification of (torus equivariant) line bundles and vector bundles on them (after Klyachko). We interpret Klyachko's data of a vector bundle as a "piecewise linear map" into the Tits building of the general linear group GL(r). This "building" perspective helps to approach many questions about vector bundles on toric varieties in a new light. As an application of this idea, we obtain a classification of (torus equivariant) vector bundles on toric schemes in terms of "piecewise affine maps" to the Bruhat-Tits building of GL(r). This is work in progress with Chris Manon and Boris Tsvelikhovsky. I try to cover most of the background material.

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Passcode: 362880

Foliations are well-behaved decompositions of a manifold into submanifolds with certain properties, called leaves. Here we focus on the more recent definition of singular foliations after I. Androulidakis and G. Skandalis, which carry more data than just a leaf decomposition, and we define singular Riemannian foliations (SRF) over this family.
Finally, we present a notion of equivalence relation deciding when two SRFs have the same transverse geometry, and we find an invariant through Poisson geometry. This is a joint work in progress with T. Strobl.


[IPM Youth Seminars on Topology and Dynamics]

Bounded cohomology for groups and spaces was originally defined by Gromov in the '80s and it is intimately related to the geometric and dynamical properties of the groups. For example, Ghys used the bounded Euler class to classify certain group actions on the circle up to (semi)conjugacy. However, unlike the group cohomology, it is notoriously difficult to calculate bounded cohomology of groups. And in fact, there is no countably generated group known for which we can completely calculate the bounded cohomology unless it is trivial in all positive degrees like the case of amenable groups. In this talk, I will report on a joint work with Nicolas Monod on the bounded cohomology of certain homeomorphism groups.
In particular, we show that the bounded cohomology of $ \rm{Homeo}(\mathbb{S}^1)$ and $\rm{Homeo}(\mathbb{D}^2)$ are polynomial rings generated by the Euler class.


[IPM Youth Seminars on Topology and Dynamics]

In the first session, we will introduce interval exchange transformations and discuss some of their typical properties; a typical IET is uniquely ergodic and weakly mixing, it is never strongly mixing. We will postpone the proofs to the second talk. Recall that a dynamical system is weakly mixing if the Cesaro averages of correlations tend to zero. We will show that a dichotomy holds regarding the speed of decay of Cesaro averages of sufficiently smooth observables. The rate is polynomial for IETs that can be suspended to give flows on surfaces of genus at least two (non-rotation class IETs) and is logarithmic for rotation type IETs (the ones whose corresponding suspension surface is always a Torus). We also show a logarithmic lower bound for the decay of Cesaro averages of correlations for rotation-type IETs thereby ruling out the possibility of having a polynomial rate of decay in this case.


[IPM Youth Seminars on Topology and Dynamics]

One of the main goals of global analysis is to understand the structure of meaningful subsets of the group of diffeomorphisms of a manifold M. In this lecture we will consider the set of partially hyperbolic diffeomorphisms on M, where M is a three dimensional compact orientable manifold. Up to algebraic and geometric construction there are some classic different examples:
• Hyperbolic linear automorphisms in T3,
• Circle extensions of Anosov surface maps,
• time-one maps of Anosov flows that are either suspensions of hyperbolic surface maps or mixing flows.
In this lecture we add some hypotheses to smooth partially hyperbolic maps to show that the above examples are all possible types of partially hyperbolic under these hypotheses (up to isotopy classes).

REFERENCES:
[1] J. Franks; Anosov diffeomorphisms. In Amer. Math. Soc., editor, Global Analysis. Proc. Sympos. Pure Math 14 (1968), pages 61–93.
[2] A. Hammerlindl and R. Potrie; Pointwise partial hyperbolicity in three dimensional nilmanifolds. Journal of the London Mathematical Society 89 (2014), no. 3, 853–875.
[3] P. Carrasco, E. PUJALS, and F. Hertz; Classification of partially hyperbolic diffeomorphisms under some rigid conditions. Ergodic Theory and Dynamical Systems (2020), 1–12.
[4] R. Sagin and J. Yang; Lyapunov exponents and rigidity of Anosov automorphisms and skew products. Advances in Mathematics 355 (2019).
[5] P. D. Carrasco, F. Rodriguez-Hertz, J. Rodriguez-Hertz, and R. Ures; Partially hyperbolic dynamics in dimension three. Ergodic Theory and Dynamical Systems 38 (2017) no. 3, 2801–2837.
[6] A. Gogolev; Surgery for partially hyperbolic dynamical systems, I: Blow-ups of invariant submanifolds. Geometry and Topology 22 (2018), no. 4, 2219–2252.

[IPM Youth Seminars on Topology and Dynamics]

References:
R. Abraham, Bumpy metrics, Global analysis (Berkeley, Calif., 1968), Proc. Sympos. Pure Math., 14, (1970).
C. Robinson, Generic properties of conservative systems I. Am. Journ. Maths. 92 (1970),
Elismar R. Oliveira. “Generic properties of Lagrangians on surfaces: the Kupka-Smale theorem”. In: Discrete Contin. Dyn. Syst.21.2 (2008).
L. Rifford, R. Ruggiero, Generic properties of closed orbits of Hamiltonian flows from Mañé’s viewpoint. Int. Math. Res. Not., 22 (2012).
A. Figalli and L. Rifford. “Closing Aubry sets II”. In: Comm. Pure Appl. Math.68.3 (2015).
Shahriar Aslani and Patrick Bernard. “Normal Form Near Orbit Segments of Convex Hamiltonian Systems”. In: International Mathematics Research Notices (Jan. 2021).

[IPM Youth Seminars on Topology and Dynamics]
Venue: meet.google.com/qix-iqiy-txt

We study attracting invariant graphs and attracting multi-graphs for skew product systems. First, we focus on skew product systems driven by a baker map having an n-dimensional closed ball $B_n \subseteq \mathbb{R}^n$ as the fiber and investigate the geometrical structures of invariant graphs and multi-graphs for these systems. We introduce an n-porcupine attractor which is a generalization of porcupine horseshoes discovered by Diaz and Gelfert. We construct an open set in the space of all such skew products such that any skew product belonging to this set admits a non-uniformly hyperbolic maximal attractor. Morover this attractor either is an n-porcupine attractor or an attracting continuous invariant graph. Then, we provide some related results on the ergodic properties of attracting graphs and investigate stability results for such graphs under deterministic perturbations. In our context the rates of contraction are non-uniform thus we have non-uniformly hyperbolic attractors that are the support of ergodic SRB measures. Additionally, we prove that the SRB measure varies continuously with the skew product in the Hutchinson metric. Furthermore, we construct robust attracting multi-graphs or porcupine multi-graphs for skew products driven by a baker map. Finally, for a certain class of skew products driven by expanding circle maps we show that the following dichotomy is ascertained: the non-uniformly maximal attractor is either a massive attractor or a thick attractor.

[IPM Youth Seminars on Topology and Dynamics]
Venue: meet.google.com/qix-iqiy-txt

We study properties and applications of the mean orbital pseudo-metric $\bar{\rho}$ on a topological dynamical system $(X,T)$ defined by \[ \bar{\rho}(x,y)= \limsup_{n\to \infty} \min_{\sigma \in S_n} \frac{1}{n}\sum_{k=0}^{n-1} d(T^k(x), T^{\sigma(k)}(y)), \] where $x,y\in X$, $d$ is a metric for $X$, and $S_n$ is the permutation group of the set $\{0,1,\ldots,n-1\}$. Writing $\hat{\omega}(x)$ for the set of $T$-invariant measure generated by the orbit of a point $x\in X$, we prove that the function $x\mapsto \hat{\omega}(x)$ is $\bar{\rho}$ uniformly continuous. This allows us to characterise equicontinuity with respect to the mean orbital pseudo-metric ($\bar{\rho}$-equicontinuity) and connect it to such notions as uniform or continuously pointwise ergodic systems studied recently by Downarowicz and Weiss. This is joint work with F. Cai, D. Kwietniak, and J. Li.

[IPM Youth Seminars on Topology and Dynamics]
Venue: meet.google.com/qix-iqiy-txt

In this talk, we will present some new results regarding the spectrum of transfer operators associated to different classes of dynamical systems. Our goal is to obtain precise information on the discrete spectrum. We will describe a general principle which allows us to obtain substantial spectral information. We will then consider several settings where new information can be obtained using this approach, including affine expanding Markov maps, monotone maps, hyperbolic diffeomorphisms. (Joint work with Oliver Butterley and Carlangelo Liverani.)

Reference:
O. Butterley, N. Kiamari, and C. Liverani, Locating Ruelle resonances, arXiv: 2012.13145 (2020).

[IPM Youth Seminars on Topology and Dynamics]
Venue: meet.google.com/qix-iqiy-txt

We give a purely contact and symplectic geometric characterization of Anosov flows in dimension 3 and set up a framework to use tools from contact and symplectic geometry and topology in the study of questions about Anosov dynamics. If time permits, we will discuss a characterization of Anosovity based on Reeb flows and its consequences.

References:
S. Hozoori, Symplectic Geometry of Anosov Flows in Dimension 3 and Bi-Contact Topology. arXiv:2009.02768 (2020).
Y. Mitsumatsu, Anosov flows and non-Stein symplectic manifolds. Annales de l'institut Fourier. Vol. 45. No. 5. 1995. C. Bonatti, J. Bowden, and R. Potrie, Some Remarks on Projective Anosov Flows in Hyperbolic 3-Manifolds. 2018 MATRIX Annals. Springer, Cham, 2020. 359-369.

[IPM Youth Seminars on Topology and Dynamics]
Venue: meet.google.com/qix-iqiy-txt

Many works in the theory of dynamical systems deal with the question of whether a given system is indecomposable or can be decomposed into say `smaller' systems. The concept of indecomposability can be studied from different aspects. In this talk, our focus is on the stable ergodicity of smooth (semi)group actions on manifolds (w.r.t natural volume) which concerns the indecomposability from a measure theoretical point of view. Examples of stably ergodic actions were known in dimension one where the conformality of smooth one-dimensional maps plays a crucial role. Generalizations of one-dimensional arguments work for ergodicity of conformal actions in higher dimensions. However, such generalizations do not provide stably ergodicity, since conformality is not stable in higher dimensions. I will start the talk with a brief introduction of the subject and discuss the ideas of proofs in one-dimensional cases and obstructions for generalization to higher dimensions. Then, I will introduce a local mechanism based on a covering property that guarantees stable `quasi-conformality" for certain higher-dimensional actions and can be used to derive stable ergodicity of such actions. If time permits I will discuss an application of these tools to provide stable ergodic actions on spheres induced by the matrices.

Reference:
A. Fakhari, M. Nassiri and H. Rajabzadeh, Stable local dynamics: expansion, quasi-conformality, and ergodicity, arXiv: 2102.09259 (2021).

[IPM Youth Seminars on Topology and Dynamics]
Venue: meet.google.com/qix-iqiy-txt

The talk will present a broad overview of Abstract Algebra from Galois to our time.


Link: The zoom link will be announced a week prior to the talk at
http://math.ipm.ac.ir/isfahan/
https://researchseminars.org/

In this talk we discuss ergodic optimization and multifractal behavior of Lyapunov exponents for matrix cocycles. We show that the restricted variational principle holds for generic cocycles over mixing subshifts of finite type, and that the Lyapunov spectrum is equal to the closure of the set where the entropy spectrum is positive for such cocycles. Moreover, we show both the continuity of the entropy at the boundary of the Lyapunov spectrum for such cocycles and the continuity of the minimal Lyapunov exponent under the assumption that linear cocycles satisfy a cone condition. We consider a subadditive potential $\Phi.$ We obtain that for $t\rightarrow \infty$ any accumulation point of a family of equilibrium states of $t \Phi$ is a maximizing measure, and that the Lyapunov exponent and entropy of equilibrium states for $t\Phi$ converge in the limit $t\rightarrow \infty$ to the maximum Lyapunov exponent and entropy of maximizing measures.

References:
R. Mohammadpour, Zero temperature limits of equilibrium states for subadditive potentials and approximation of the maximal Lyapunov exponent, Topol. Methods Nonlinear Anal., 55(2), 697–710, 2020.
R. Mohammadpour, Lyapunov spectrum properties and continuity of lower joint spectral radius. arXiv: 2001.03958, (2020).

[IPM Youth Seminars on Topology and Dynamics]
Venue: meet.google.com/qix-iqiy-txt

This talk is divided into two parts. In the first part, after some background on manifolds with negative curvature, I’ll state Margulis’s celebrated counting result proved in his thesis [Mar]. After that, I’ll briefly mention later works inspired by/generalizing his argument. Finally, I’ll sketch an argument proving a counting result due to Paulin and Parkkonen [PP]. This part should be accessible to general audience!
In the second part, which is the more technical part, I’m going to explain how to adapt Paulin and Parkkonen’s argument to prove a similar result for Teichmuller space. My intention is to mainly focus on deeper results that have made this adaptation possible (e.g. [ABEM], [Fr]).

References:
[ABEM] J. Athreya, A. Bufetov, A. Eskin and M. Mirzakhani, Lattice point asymptotics and volume growth on Teichmüller space. Duke Math. J. 161, No. 6, 1055-1111 (2012).
[Fr] I. Frankel, CAT(-1)-Type Properties for Teichmuller Space. arXiv:1808.10022, (2018).
[PP] J. Parkkonen and F. Paulin, Counting common perpendicular arcs in negative curvature. Ergodic Theory Dyn. Syst. 37, No. 3, 900-938 (2017)
[Mar] G. A. Margulis, On some aspects of the theory of Anosov systems. With a survey by Richard Sharp: Periodic orbits of hyperbolic flows. Transl. from the Russian by S. V. Vladimirovna. Berlin: Springer (2004).

[IPM Youth Seminars on Topology and Dynamics]
Venue: meet.google.com/qix-iqiy-txt

The dynamical moduli space of rational maps of degree d, defined as the space of Möbius conjugacy classes of degree d holomorphic self-maps of the Riemann sphere, is a ubiquitous object in complex and arithmetic dynamics. Using the techniques of Geometric Invariant Theory, Silverman constructs this orbit space as an affine variety of dimension 2d-2 which admits a model over the rationals. In the case of degree two, Milnor identifies this space with the affine plane. I will present the results of a joint work with Maxime Bergeron and Sam Nariman regarding the topology of these moduli spaces. We compute the fundamental group of the dynamical moduli space and show that the space is rationally acyclic while its cohomology groups with finite coefficients could be non-trivial. As an application, the ranks of certain rational homotopy groups of the parameter space of rational maps (within the unstable range) will be computed.

Reference:
M. Bergeron, K. Filom and S. Nariman, Topological aspects of the dynamical moduli space of rational maps, arXiv: 1908.10792 (2019).

[IPM Youth Seminars on Topology and Dynamics]
Venue: meet.google.com/qix-iqiy-txt

It is for many years that structural stability of dynamical systems has been taken into consideration by dynamicists and there are various studies, results, and open problems on this topic. This kind of stability cares about the change of the topological behavior of `all' points for the maps nearby the initial map. However, if your aim is to study a dynamical system only from a statistical point of view, this is too restrictive. You can ignore the change of behavior of orbits on a set of zero measure. Moreover, from a statistical point of view, it is not important that for which iterations the orbit of a point meets a subset of the phase space, the only thing which is important is the proportion of times that an orbit meets a given subset. In other words, orbits with different topological behavior may have the same statistical behavior. So it is natural to think about another version of stability while working with statistical properties of your maps. In this talk, using some examples, I would like to present a version of the notion of `statistical stability', and state a few theorems about it.

References:
A. Talebi, Non-statistical rational maps. arXiv:2003.02185, (2020).
A. Talebi, Statistical (in)stability and non-statistical dynamics. arXiv:2012.14462, (2020).

[IPM Youth Seminars on Topology and Dynamics]
Venue: meet.google.com/qix-iqiy-txt

A celebrated theorem of Lind states that a positive real number is equal to the entropy of a shift of finite type, if and only if, it is equal to the logarithm of a Perron algebraic integer divided by some natural number n. Given a Perron number p and a natural number n, we prove that there is a non-negative integral irreducible matrix with spectral radius equal to the nth root of p, and with dimension bounded above in terms of n, the algebraic degree, the spectral ratio, and certain arithmetic information about the ring of integers of its number field. Consequently, there is an irreducible shift of finite type with entropy equal to the logarithm of p divided by n, and with `size' bounded above in terms of the aforementioned data.

Reference:
- M. Yazdi, Non-negative integral matrices with given spectral radius and controlled dimension, arXiv:2101.09268 (2021).

[IPM Youth Seminars on Topology and Dynamics]
Venue: meet.google.com/qix-iqiy-txt

The Eigenvalue Theorem is a basic result in computational algebraic geometry. It says that solving a zero-dimensional system of polynomial equations can be reduced to an eigenvalue problem in linear algebra. The name of Ludwig Stickelberger (1850-1936) is often attached to this theorem, yet papers that use his name never cite any of his papers. My lecture will explore the reasons for this. The answer involves a lovely trace formula in algebraic number theory and an algebra textbook published by Gunter Scheja and Uwe Storch in 1988.
To get more information about the colloquium, join to the following google group:
https://groups.google.com/g/ipm-math-colloquium

A variety $V$ of codimension $r$ in a projective space $\mathbb{P}^n$ is called a set-theoretic complete intersection if $V$, as a set, is the intersection of exactly $r$ hypersurfaces in $\mathbb{P}^n$. I will discuss the history of the general problem, which varieties $V$ are s.t.c.i., with special attention to the still open problem, is every irreducible nonsingular curve in $\mathbb{P}^3$ a set-theoretic complete intersection? In particular I will mention several algebraic criteria, including local cohomology that can in principle be used to show that certain varieties are not s.t.c.i.
To get more information about the colloquium, join to the following google group:
https://groups.google.com/g/ipm-math-colloquium

The ancient Greek word "geometry" is, with its etymological meaning of "measurement of the earth", one of the oldest of mathematics. The discovery of non-euclidean geometries in the 19th century dramatically increased the scope of geometry. This scope was further extended in the 20th century by dividing geometry into branches differing in their objects of study and their methods: topology, differential geometry, Riemannian geometry, symplectic geometry, complex geometry, algebraic geometry, ... Grothendieck is known primarily for having re-founded algebraic geometry on entirely new bases. But he himself considered himself to be a general mathematician, not a specialist. So one may wonder whether the word "geometry", which he used very often without ever defining it, has for him a precise meaning that goes beyond algebraic geometry, and whether certain notions he introduced or to which he gave a central role are likely to apply to everything that one might imagine to be called geometric. The aim of the presentation will be to propose elements of an answer to this question.

To get more information about the colloquium, join to the following google group:
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