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