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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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:
https://groups.google.com/g/ipm-math-colloquium