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ACTIVITIES IN 2019

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Lecture

Speakers: Sam Nariman  (University of Copenhagen) Title: On Obstructions to Extending Group actions to Bordisms Description: Motivated by a question of Ghys, we talk about cohomological obstructions to extending group actions on the boundary $\partial M$ of a $3$-manifold to a $C^0$-action on $M$. Among other results, we show that for a $3$-manifold $M$, the $\mathbb{S}^1 \times \mathbb{S}^1$ action on the boundary does not extend to a $C^0$-action of $\mathbb{S}^1 \times \mathbb{S}^1$ as a discrete group on $M$, except in the trivial case $M \cong \mathbb{D}^2 \times \mathbb{S}^1$. Using additional techniques from 3-manifold topology, homotopy theory, and low-dimensional dynamics, we find group actions on a torus and a sphere that are not nullbordant, i.e. they admit no extension to an action by diffeomorphisms on any manifold $M$ with $\partial M \cong \mathbb{T}^2$ or $\mathbb{S}^2$. This is a joint work with K. Mann. Date & Time: 2019/12/31, 10:30--12:00 Location:  Lecture Hall 2, IPM Niavaran Building, Niavaran Square, Tehran

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Lecture

Speakers: Eaman Eftekhary  (IPM) Title: Counting Closed Geodesics on Riemannian Manifolds Description: Associated with every closed oriented smooth manifold $M$, let $R_M$ denote the space of all pair $(L,g)$, where $g$ is a Riemannian metric on $M$ and $L$ is a real number which is not the length of any closed $g$-geodesics. A locally constant geodesic count function $\pi_M:R_M\rightarrow \mathbb{Z}$ is defined which virtually counts the number of closed $g$-geodesics of length less than $L$ at $(L,g)\in R_M$. In particular, when $g$ is negatively curved, $\pi_M(L,g)$ is precisely the number of prime closed $g$-geodesics which have length smaller than $L$. The asymptotic growth of the number of closed $g$-geodesics may subsequently be studied. Date & Time: 2019/12/31, 09:00--10:15 Location:  Lecture Hall 2, IPM Niavaran Building, Niavaran Square, Tehran

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Graduate Course

Lecturers: M. Hosseini, M. Nassiri, M. Teymuri (School of Mathematics, IPM) Title: Introduction to Group Actions Date & Time: Thursdays, Fall 2019 8:30-10:00 & 10:30-12:00 (First lecture: Thursday, October 3, 2019, 8:30 a.m.) Location:  Lecture Hall 2, Niavaran Building, IPM, Sh. Bahonar Square, Tehran Description: This course is a part of the Thematic Program on Dynamical Systems. امکان ثبت نام رسمی درس به عنوان دانشجوی مهمان در پژوهشکده ریاضیاتِ پژوهشگاه دانشهای بنیادی وجود دارد. Website: math.ipm.ir/gt/GA2019.html

Lecture

Speakers: Nahid Ghodratipour  (Sharif University of Technology) Title: Conformally Invariant Observables in the Double-dimer Model Description: We are going to talk about a two-dimensional model in statistical mechanics, the double-dimer model, which is simply a straightforward generalization of the dimer model. The latter is well-known in both physics and mathematics because of its particular features and substantial relations to many contexts, known in combinatorics as a perfect-matching problem. Here the emphasis is on the critical behavior of the loop ensembles of the double-dimer model, conjectured to converge to the level sets of the Gaussian free field and Schramm-Loewnerevolution or conformal loop ensembles with parameter 4. We use the Grassmannian representation of the dimer model to compute some loop-related observables in rectangular domains on the square lattice. The asymptotic behavior of the results in the continuum turns out to be consistent with previously known ones or with the aforementioned conjecture. Date & Time: 2019/10/16, 15:30--16:30 Location:  Lecture Hall 1, IPM Niavaran Building, Niavaran Square, Tehran

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Mathematics Colloquium

Speakers: Péter Komjáth  (Eötvös Loránd University, Hungary) Title: Constructions in Euclidean spaces using the Axiom of Choice. Description: We consider some applications of AC as the coloring of R^n with countably many colors with no two points in rational distance getting the same color. Date & Time: 2019/10/9, 16:00--17:00 Location:  Lecture Hall 1, IPM Niavaran Building, Niavaran Square, Tehran

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Lecture

Speakers: Younes Nikdelan  (‎Universidade do Estado do Rio de Janeiro (UERJ), Brazil) Title: Modular Vector Fields and Calabi-Yau Modular Forms Description: In this lecture we introduce a spacial moduli space $\sf T$ of the pairs formed by definite Calabi-Yau $n$-folds (arising from the Dwork family) along with $n+ 1$ differential $n$-forms. We observe that there exists a unique vector field $\textsf{R}$ on $\sf T$, called modular vector field, satisfying a certain equation involving the Gauss-Manin connection. It turns out that the $q$-expansion (Fourier series) of the components of a solution of $\sf R$, which are called Calabi-Yau modular forms, has integer coefficients, up to multiplying by a constant rational number. In particular, in the case of elliptic curves and $K3$-surfaces, where $n=1,2$, the components of a solution can be written in terms of (quasi-)modular forms satisfying certain enumerative properties. A very useful result of these works is that the modular vector field ${\sf R}$ together with the radial vector field and a degree zero vector field generates a copy of the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$. We will finish this talk by endowing the space of Calabi-Yau modular forms with an algebraic structure called Rankin-Cohen algebra. To get the vector field $\sf R$ we use an algebraic method called Gauss-Manin connection in disguise introduced by Hossein Movasati. Date & Time: 2019/7/10, 15:30--16:30 Location:  Lecture Hall 1, IPM Niavaran Building, Niavaran Square, Tehran

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Mathematics Colloquium

Speakers: Fraydoun Rezakhanlou   (University of California, Berkeley) Title: Metastability and Condensation Description: Dynamical systems that are perturbed by small random noises are known to exhibit metastable behavior. Analogously, random walks with tendency towards a finite collection of sites may exhibit metastability. Zero Range Process is a random walk on a simplex with metastable states residing at the vertices. Interpreting this process as a particle system on a one dimensional lattice, the metastable states correspond to the condensates. In this talk I give an overview of some known results in both the continuous and discrete settings, and discuss some open questions. Date & Time: 2019/7/3, 16:00--17:00 Location:  Lecture Hall 1, IPM Niavaran Building, Niavaran Square, Tehran

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Lecture

Speakers: Omid Hatami  (IPM) Title: Combinatorial Model for the Emergence of Local Equilibria Description: Let $G$ be the symmetric group $\mathfrak{S}(N)$. Start with the identity member $e$ in $\mathfrak{S}(N)$ and in each step multiply a random transposition. In step $T$, we have a word $$\pi_T = \tau_1 \tau_2\dots \tau_T$$ where $\tau_i$ are transposition. $\pi_T$ has a cycle decomposition \begin{equation*} \sigma_T~=~\sigma_1\sigma_2\ldots, \end{equation*} where $\sigma_i$'s are arranged in decreasing order of length so that $\sigma_1$ (of length $\ell_1$) is the longest cycle. The basic question is, ``What can we say about $\ell_1$ at time $T$? We will study the answer to this question. Date & Time: 2019/6/19, 15:30--17:00 Location:  Lecture Hall 1, IPM Niavaran Building, Niavaran Square, Tehran

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Mathematics Colloquium

Speakers: Freydoon Shahidi  (Purdue University, USA) Title: Reciprocity and Functoriality Description: I will discuss some number theoretic background to explain classical reciprocity laws of number theory. This will be an introduction to non-abelian reciprocity which generalizes Artin reciprocity map. I will do this in terms of complex representations of local Galois groups which are to parametrize irreducible admissible representations of GL(n,F), where F is a local field. This is called the Local Langlands correspondence (LLC) for GL(n) which has been established by Harris-Taylor, Henniart and Scholze in various forms. I will mention recent results in matching of different arithmetic (Artin) root numbers and L-functions with analytic ones through (LLC). A global Langlands correspondence for number fields remains illusive and very much out of reach. But one can still discuss one of its major consequences, Langlands functoriality principle, for which there has been some striking progress in the past twenty years Date & Time: 2019/6/12, 16:00--17:15 Location:  Lecture Hall 1, IPM Niavaran Building, Niavaran Square, Tehran

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Lecture

Speakers: Muhammad Ali Khan  (Centre of Mathematics, University of Porto (CMUP)) Title: Statistical (In)Stability in Strange Attractors. Description: We will introduce the concept of physical measures and statistical stability for dynamical systems. We will discuss the existence of physical measures for certain families of dynamical systems including the well-known families of geometric and contracting Lorenz flows and the corresponding one dimensional maps. As an ultimate goal of this talk, we will present the results concluding statistical stability and instability for geometric and contracting Lorenz flows, respectively. Date & Time: 2019/5/1, 15:30--16:30 Location:  Lecture Hall 1, IPM Niavaran Building, Niavaran Square, Tehran

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Mathematics Colloquium

Speakers: Siavash Shahshahani   (Sharif University of Technology ) Title: Struggle Between Syntax and Semantics in Mathematics Description: Mathematical practice throughout history has featured a dichotomy that can broadly be characterized as a struggle between syntax and semantics. We survey the history and the roots of this dichotomy and offer some speculation and thinking points. Date & Time: 2019/4/24, 16:00--17:00 Location:  Lecture Hall 1, IPM Niavaran Building, Niavaran Square, Tehran

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Lecture

Speakers: Roman Karasev  (Moscow Institute of Physics and Technology) Title: Gromov's waists for non-radial Gaussian and radial non-Gaussian measures Description: Mikhail Gromov in 2003 proved the waist theorem for a radially symmetric Gaussian measure in $\mathbb R^n$: For any such measure and any continuous map $ f: \mathbb R^n \to\mathbb R^m$, there exists a point $y\in\mathbb R^m$ such that for any $t>0$ the measure of the $t$-neighborhood of the preimage $f^{-1}(y)$ is not less than the measure of the $t$-neighborhood of the standard linear subspace $\mathbb R^{n-m}\subseteq\mathbb R^n$. We were thinking about a version of this theorem for a non-radial Gaussian measure. We managed to establish such with a similar statement, of course, the linear subspace $\mathbb R^{n-m}\subseteq\mathbb R^n$ at the end of the statement needs to be chosen more carefully. It is curious that our proof also gives a simplification of Gromov's proof with the help of Caffarelli's theorem on monotone transportation. We were also thinking about certain radial measures that are not Gaussian, mostly finding counterexamples to the waist theorem's statement. Full exposition is in the paper: https://arxiv.org/abs/1808.07350. This is a joint work with Arseniy Akopyan from IST Austria. Date & Time: 2019/4/16, 14:30--16:00 Location:  Lecture Hall 2, IPM Niavaran Building, Niavaran Square, Tehran

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Lecture

Speakers: Ali Khezeli  (Tarbiat Modares University) Title: On Generalizations of the Gromov-Hausdorff Metric Description: Gromov has defined a metric on the set of all compact metric spaces. This metric is defined for group-theoretic purposes but has found important applications in probability theory as well. Also, there exist several generalizations of this metric. For instance, the Gromov-Hausdorff-Prokhorov metric defines the distance of two measured metric spaces. Other examples consider metric spaces equipped with a distinguished point, a closed subset, a curve, a tuple of such structures, etc. In this talk, a general approach is presented for generalizing the Gromov-Hausdorff metric to consider metric spaces equipped with some additional structure. This abstract framework unifies the existing generalizations in the literature. It is also useful for studying new examples of additional structures which will be needed in the future works of the author. The framework is provided both for compact metric spaces and for boundedly-compact pointed metric spaces. In addition, completeness and separability of the metric are proved under some conditions. This enables us to study random metric spaces equipped with additional structures, which is the main motivation of this work. Date & Time: 2019/4/10, 15:30--16:30 Location:  Lecture Hall 1, IPM Niavaran Building, Niavaran Square, Tehran

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Lecture

Speakers: Amin Talebi  (Sharif Univ. and Paris XIII) Title: Non-statistical dynamics; Conditions implying the existence, consequences of non-existence Description: For a dynamical system, to be non-statistical means that as the time tends to infinity, the dynamics does not shows convergent statistical behavior in the sense of Birkhoff averages. In this lecture, first we introduce some examples of this kind of behavior in different family of dynamics. Next, we explain the connection between this dynamical behavior and another one called "statistical instability", and show that in any family of dynamical systems, the existence of a "large" set of statistically unstable maps, implies the existence of non-statistical ones. And finally, we present a consequence of this fact for the family of maps that a priori we know they do not display non-statistical behavior, e.g. conservative maps. Date & Time: 2019/3/6, 15:30--17:00 Location:  Lecture Hall 1, IPM Niavaran Building, Niavaran Square, Tehran

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Lecture

Speakers: Pooya Vahidi Ferdowsi  (Caltech) Title: Proximal Actions, Strong Amenability, and Infinite Conjugacy Class Property Description: A continuous action of a countable discrete group on a Hausdorff compact space $X$ is called proximal if for any pair of points $x$, $y$ in $X$, we can simultaneously push them together, i.e. there exists a sequence of group elements $g_n$ with $\lim g_n x = \lim g_n y$. A group is called Strongly Amenable if each of its proximal actions has a fixed point. Glasner introduced these notions and showed Strongly Amenable groups are Amenable. Moreover, he showed that virtually nilpotent groups are Strongly Amenable. In this talk I will present a recent result classifying strongly amenable groups. This is a joint work with Joshua Frisch and Omer Tamuz. Date & Time: 2019/3/3, 10:00--12:00 Location:  Lecture Hall 1, IPM Niavaran Building, Niavaran Square, Tehran

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Mathematics Colloquium

Speakers: Piergiorgio Odifreddi  (University of Turin, Italy) Title: How Godel Became Godel Description: The incompleteness theorem proved by Godel in 1931 is considered one of the jewels of the mathematics of the XX Century, and it has been widely discussed by specialists and crackpots. Usually it is considered as a very subtle statement with a technically very complicated proof, but this is mostly due to the fact that it is usually formulated and proved in the final version published by Godel himself, which is actually the result of a number of successive approximations, each stronger than the previous one. By retracing the intellectual road followed by Godel in the years 1930 and 1931, I will try to show how his incompleteness results are actually very natural and simple answers to questions that were in the air in the Vienna of his times. All in accord with Godel's own view, who repeatedly stressed that the main difficulty of his work was not technical, but ideological. Date & Time: 2019/2/27, 16:00--17:00 Location:  Lecture Hall 1, IPM Niavaran Building, Niavaran Square, Tehran

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Lecture

Speakers: Mohammad Safdari  (Sharif University of Technology) Title: Non-strictly Convex Gradient Constraints in Calculus of Variations Description: We prove the optimal regularity for variational problems with convex gradient constraints. We do not assume any regularity of the constraints; so the constraints can be non-smooth, and they need not be strictly convex. When the domain is smooth enough, we show that the optimal regularity holds up to the boundary. In this process, we also characterize the set of singular points of the viscosity solutions to some Hamilton-Jacobi equations. Furthermore, we obtain an explicit formula for the second derivative of these viscosity solutions; and we show that the second derivatives satisfy a monotonicity property. Date & Time: 2019/2/13, 15:30--17:00 Location:  Lecture Hall 1, IPM Niavaran Building, Niavaran Square, Tehran

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Lecture

Speakers: Mohammad Hadi Mostafid  (Tarbiat Modares University) Title: Ricci Flow and some of its Application Description: Date & Time: 2019/2/6, 15:30--17:00 Location:  Lecture Hall 1, IPM Niavaran Building, Niavaran Square, Tehran

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Mathematics Colloquium

Speakers: Reza Seyyedali  (IPM) Title: Geometric Invariant Theory and Symplectic Quotients Description: Geometric invariant theory (GIT) is an important tool in construction of moduli spaces. By Kempf-Ness theorem, there is a close relationship between GIT and symplectic reduction in the finite dimensional setting. In infinite dimensions, the relationship is more subtle. In this talk, I will go over some examples of such a relationship in finite and infinite dimensions. Date & Time: 2019/1/23, 16:00--17:00 Location:  Lecture Hall 1, IPM Niavaran Building, Niavaran Square, Tehran

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Lecture

Speakers: Jose Alves  (University of Porto, Portugal) Title: Statistical Stability in Chaotic Dynamics Description: We will consider some classical results on Ergodic Theory and Dynamical Systems motivating the concepts of physical measure and statistical stability. We will address the existence of physical measures and statistical stability for some families of dynamical systems, including the well-known family of Lorenz flows. Date & Time: 2019/1/17, 14:30--15:30 Location:  Lecture Hall 2, IPM Niavaran Building, Niavaran Square, Tehran

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Lecture

Speakers: Ebrahim Akrami  (Semnan University) Title: Quantum Topology, Geometry and Dynamics Description: It is well-known that classical invariants of algebraic and differential topology, including smooth fundamental group, smooth singular homology, de Rham cohomology as well as Morse homology, are not sensitive to smooth structures of manifolds but modern physically-based (TQFT) invariants like Donaldson or Seiberg-Witten invariants are so. Thus it seems there should be a deep relation between quantum theory and manifold theory. In this talk, we investigate this relation. Our starting point is that as classical mechanics is not adequate to describe the structures of atoms and we need quantum mechanics to see atoms' structure, classical algebraic and differential topology is not adequate to describe smooth structures of manifolds. In fact, we are going to say that there is a deep analogy between atomic theory and manifold theory and thus the same techniques used in the former can be used in the later. Based on this idea, in this talk, we construct new invariants of smooth structures of the manifold, namely the quantum version of the smooth fundamental group, smooth singular homology, de Rham cohomology as well as Morse homology. Date & Time: 2019/1/16, 15:30--17:00 Location:  Lecture Hall 2, IPM Niavaran Building, Niavaran Square, Tehran

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Lecture

Speakers: Mohammad Shirazi  (University of Manitoba, Canada) Title: Faber Polynomials, Grunsky Matrix and Period Matrices Description: We extend the idea of Grunsky operator and Faber operator to higher genus compact Riemann surfaces with one boundary curve. In this talk, I will first review the classical concept of Faber Polynomials corresponding to the map $f$, a conformal map on a simply connected domain $\textbf{D}$ of the Riemann sphere with $\Gamma=\partial f (\textbf{D})$. We define the Grunsky coefficients associated to $f$, and review the necessary and sufficient conditions for $f$ to be univalent on $\textbf{D}$. We will define Grunsky and Faber operators and show that the Faber operator is an isomorphism for quasi-circles. I will present some recent work of myself, E. Schippers and W. Staubach generalizing these results to conformal maps into compact Riemann surfaces. Finally, a model which reveals some analogies between the Grunsky operator and the classical period mapping on the Universal Teichmuller space will be discussed. Date & Time: 2019/1/9, 15:30--17:00 Location:  Lecture Hall 2, IPM Niavaran Building, Niavaran Square, Tehran

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Lecture

Speakers: Azizeh Nozad  (IPM) Title: The Moduli Spaces of $G$-Higgs Bundles and the Nilpotent Cone Description: For a non-compact reductive Lie group $G$, the notion of $G$-Higgs bundles over a compact Riemann surface $X$, of genus $g \geq2$, was introduced by Hitchin (80s and 90s). These are appropriate objects for extending the non-abelian Hodge Theorem (the work of Corlette, Donaldson, Hitchin and Simpson) to representations of the fundamental group in a real reductive Lie group $G$. Motivated partially by this identification, the moduli space of $G$-Higgs bundles has been extensively studied. Here we give the obstructions to a deformation retraction from the moduli spaces of $G$-Higgs bundles to the moduli space of semistable principal bundles over $X$, in contrast with the situation when $g = 1$. The existence of those obstructions allows us to deduce the reducibility of the nilpotent cone of the moduli space of $G$-Higgs bundles: that is the pre-image of zero under the Hitchin map. All concepts will be motivated with several examples, and we will give an overview of known results on the moduli spaces of G-Higgs bundles as well as some open problems. Date & Time: 2019/1/2, 15:30--17:00 Location:  Lecture Hall 1, IPM Niavaran Building, Niavaran Square, Tehran

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School of Mathematics, IPM - Institute for Research in Fundamental Sciences Niavaran Building, Niavaran Square, Tehran, Iran Tel: +98 21 222 90 928

Email: gt@ipm.ir