During the last decades lots of efforts have been invested in developing A. Weil's discourse on the analogy in mathematics. This method essentially aims to bring number theory into the light of geometry, through a bridge called the arithmetic of function fields. In this context, and as it may be observed through Langlands philosophy, moduli spaces (/stacks) for global G-shtukas appear as analogs for Shimura varieties, and consequently, they play a crucial role in Langlands program over function fields. They possess local counterparts which are called Rapoport-Zink spaces for local P-shtukas (i.e. function fields analogs for Rapoport-Zink spaces for p-divisible groups). In this talk we overview the landscape of this analogy, by translating Deligne's conception of Shimura varieties (as a moduli for motives) to the realm of function fields. We explain some results in this direction and some further results related to the deformation theory and local model theory for the moduli of G-shtukas. If time permits we also discuss some of the applications.

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We use the method of atomic decomposition to study the action of transfer operators associated to piecewise expanding maps. It turns out that these transfer operators are quasi-compact even when the associated potential, the dynamics and the underlying phase space have very low regularity. In particular it is often possible to obtain exponential decay of correlations, the Central Limit Theorem and almost sure invariance principle for fairly general observables, including unbounded ones. Indeed the class of observables for which we obtain such results often coincides with certain Besov spaces. Joint work with Alexander Arbieto (UFRJ-Brazil).

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[ICMC-IMPA-IPM Joint Mathematics Seminars]

The counting problem for closed geodesics over negatively curved manifolds, and more generally, for closed orbits of Anosov flows is studied extensively in the literature. However, when the metric is not negatively curved, or when the flow of a vector field is not Anosov, closed geodesics/orbits are not isolated and there are some obstacles for defining a well-behaved count function. We discuss some of these obstructions. In particular, we introduce a function which assigns an integer weight to every compact and open subset of the space of closed geodesics for arbitrary Riemannian metrics over closed manifolds.

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[ICMC-IMPA-IPM Joint Mathematics Seminars]

In this presentation, we start with some results about closed geodesics' growth rate in various cases (including geodesics on manifolds with negative sectional curvatures) and the similarity of some of them with prime number theorem. After that, we introduce the marked length spectrum function. An important conjecture says this function identifies the metric of a closed negatively curved manifold.

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[ICMC-IMPA-IPM Joint Mathematics Seminars]

Ledrappier proved that the invariant measures of linear cocycles having zero Lyapunov exponents have certain extra invariance. This was generalized by Avila and Viana for smooth cocycles, in particular they proved that the invariant measures for partially hyperbolic skew products have a disintegration invariant by holonomies, this is known as ''invariance principle''.
This has several applications, such as obtaining genericity of non-uniformly hyperbolic systems, finding physical measures, and classifying the measures of maximal entropy.
In this presentation we will generalize the invariance principle to partially hyperbolic non-skew products (without compact center leaves) which allows us to extend several of the previous applications to more general partially hyperbolic ones. In particular we will give an application to classify measures of maximal entropy of the perturbation of the time one map of Anosov flows.

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In the first half of this talk we are going to give an overview of the basic theory of Lyapunov exponents of linear cocycles focusing on the relationship between them and the invariant skew-product measures. The idea is to understand how these measures can be used to give nice properties of the Lyapunov exponents such as positivity and continuity. In the second half, we are going to use these ideas to study the behavior of the Lyapunov exponents of linear cocycles when the base dynamics is a model of a partially hyperbolic skew-product.

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by the day before the event.

The concept of Lyapunov exponent goes back to Lyapunov's 1892 thesis on the stability of differential equations, and has numerous applications in various branches of mathematics and science.
Starting from the 1960s, it found its proper mathematical framework in ergodic theory, where it has had a prominent role ever since. In this colloquium lecture I will review a few recent developments, especially about the way Lyapunov exponents depend on the underlying dynamical system.

The stability theorem of Lyapunov asserts that, under an additional regularity condition, the stability of the trivial solution of a linear equation remains valid for nonlinear perturbations. The regularity condition of Lyapunov essentially means that a certain limits exist. Such limits are called characteristic exponents or Lyapunov exponents. In this talk we are going to introduce the concept of linear cocycles, and present the theorem of Furstenberg-Kesten which guarantees the existence of (extremal) Lyapunov exponents. Moreover, we are going to present the multiplicative ergodic theorem of Oseledets, and introduce some results regarding the invariant principle and continuity problem.

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by the day before the event.

In this talk, we introduce, briefly, the basis of Lorentzian geometry and cohomogeneity one actions. We also, introduce the Einstein universe and consider an interesting cohomogeneity one action on $\mathbb{Ein}^{1,2}$.
A smooth manifold $M$ equipped with a metric tensor $\textbf{g}$ of signature $(1, n)$ is called a Lorentzian manifold. Precisely, the restriction of $\textbf{g}$ to the tangent space $T_pM$ is of the form $−x_1^2+x_2^2+\cdots+x_n^2$.
Let $G$ be a Lie group which acts on a smooth manifold. Then the orbit of $G$ at point $p\in M$ is $G(p) := \{gp : g \in G\}$. If $G$ acts on $M$ smoothly, then every orbit of $G$ is a smooth immersed submanifold of $M$. The action of $G$ on $M$ is called cohomogeneity one if $G$ admits a codimension 1 orbit in $M$. Furthermore, if $M$ is a pseudo-Riemannian manifold and $G$ acts on $M$ isometrically, then every orbit is a pseudo-Riemannian submanifold. In other words, the restriction of the metric on each orbit has constant signature.
Consider the direct product Lorentzian manifold $(M, \textbf{g}) = (\mathbb{S}^1\times \mathbb{S}^n, −d\theta^2+\textbf{g}_{\mathbb{S}^n}$ ) where $(\mathbb{S}^1, d\theta^2)$ is the usual Riemannian circle of radius 1 and $(\mathbb{S}^n, \textbf{g}_{\mathbb{S}^n})$ is the usual Riemannian $n$-dimensional sphere of constant sectional curvature 1. The map $-Id : M \to M$ sending $(x, y)$ to $(-x, -y)$ is an isometry. The $(n + 1)$-dimensional Einstein universe $\mathbb{Ein}^{1,n}$ can be defined as the quotient of $M$ by $\{Id, -Id\}$. The Einstein universe is Lorentzian analogue of the sphere $\mathbb{S}^n$. It compactifies the Minkowski space and is the conformal boundary of Anti de Sitter space.

The main goal of this short course is to present a proof of Calabi_Yau theorem. It was first conjectured by Calabi that any volume form on a compact Kahler manifold can be realized as the volume form associated to a Kahler metric. In a seminal work, Yau proved Calabi's conjecture. A very important consequence of Calabi-Yau theorem is the existence of Kahler-Ricci flat metrics on compact Kahler manifolds with trivial canonical bundle.

Outline of the course:
1) background material on complex manifolds, Hermitian and Kahler metrics, Ricci form
2) Some background on elliptic PDE such as Schauder estimates
3) Yau's original proof on apriori $C^0$ estimate
4) Some advance approach to Calabi-Yau theorem
5) an overview on complex Monge-Ampere via Pluripotential theory (if time permitted)

In this mini-course, we will illustrate some functional analytic approaches to the study of the statistical properties of dynamical systems.

We will present Lasota-Yorke technique for existence of absolutely continuous invariant measures for some classes of dynamical systems. We will then study the spectral properties of Frobenius-Perron operator in order to obtain more information about such invariant measures. If time permits, we will continue to talk about the speed of convergence of the iterates of the transfer operator and the central limit theorems.

References.
A. Boyarsky, P. Gora, Laws of chaos. Invariant measures and dynamical systems in one dimension. Probability and its Applications. Birkhuser Boston, Inc., Boston, MA, 1997.
C. Liverani, Invariant Measures and their Properties. A functional Analytic point of view (available online).

The main goal of this short course is to present a proof of Calabi_Yau theorem. It was first conjectured by Calabi that any volume form on a compact Kahler manifold can be realized as the volume form associated to a Kahler metric. In a seminal work, Yau proved Calabi's conjecture. A very important consequence of Calabi-Yau theorem is the existence of Kahler-Ricci flat metrics on compact Kahler manifolds with trivial canonical bundle.

Outline of the course:
1) background material on complex manifolds, Hermitian and Kahler metrics, Ricci form
2) Some background on elliptic PDE such as Schauder estimates
3) Yau's original proof on apriori $C^0$ estimate
4) Some advance approach to Calabi-Yau theorem
5) an overview on complex Monge-Ampere via Pluripotential theory (if time permitted)

The main goal of this short course is to present a proof of Calabi_Yau theorem. It was first conjectured by Calabi that any volume form on a compact Kahler manifold can be realized as the volume form associated to a Kahler metric. In a seminal work, Yau proved Calabi's conjecture. A very important consequence of Calabi-Yau theorem is the existence of Kahler-Ricci flat metrics on compact Kahler manifolds with trivial canonical bundle.

Outline of the course:
1) background material on complex manifolds, Hermitian and Kahler metrics, Ricci form
2) Some background on elliptic PDE such as Schauder estimates
3) Yau's original proof on apriori $C^0$ estimate
4) Some advance approach to Calabi-Yau theorem
5) an overview on complex Monge-Ampere via Pluripotential theory (if time permitted)

The main goal of this short course is to present a proof of Calabi_Yau theorem. It was first conjectured by Calabi that any volume form on a compact Kahler manifold can be realized as the volume form associated to a Kahler metric. In a seminal work, Yau proved Calabi's conjecture. A very important consequence of Calabi-Yau theorem is the existence of Kahler-Ricci flat metrics on compact Kahler manifolds with trivial canonical bundle.

Outline of the course:
1) background material on complex manifolds, Hermitian and Kahler metrics, Ricci form
2) Some background on elliptic PDE such as Schauder estimates
3) Yau's original proof on apriori $C^0$ estimate
4) Some advance approach to Calabi-Yau theorem
5) an overview on complex Monge-Ampere via Pluripotential theory (if time permitted)

The main goal of this short course is to present a proof of Calabi_Yau theorem. It was first conjectured by Calabi that any volume form on a compact Kahler manifold can be realized as the volume form associated to a Kahler metric. In a seminal work, Yau proved Calabi's conjecture. A very important consequence of Calabi-Yau theorem is the existence of Kahler-Ricci flat metrics on compact Kahler manifolds with trivial canonical bundle.

Outline of the course:
1) background material on complex manifolds, Hermitian and Kahler metrics, Ricci form
2) Some background on elliptic PDE such as Schauder estimates
3) Yau's original proof on apriori $C^0$ estimate
4) Some advance approach to Calabi-Yau theorem
5) an overview on complex Monge-Ampere via Pluripotential theory (if time permitted)

Our aim in this course is to review some results and techniques in ergodic theory of action of Lie groups and their discrete subgroups and then to sketch the proof of a profound theorem of G. A. Margulis known as "superrigidity".

Margulis' superrigidity theorem says that under some conditions on Lie groups and their discrete subgroups, any isomorphism between discrete subgroups extends to isomorphism of the ambient groups, and roughly speaking these discrete subgroups determines the Lie groups completely.
To this end, we need to talk about some backgrounds from the structure theory of semisimple Lie groups and Algebraic groups together with some tools from Homogeneous dynamics, for instance Moore's theorem on ergodicity of action of certain closed subgroups of Lie groups on their quotients by lattices.
Finally, we shall briefly discuss some applications of superrigidity in Riemannian geometry and also in characterization of lattices in higher rank simple Lie groups.

The main reference for the course will be the following book:
Zimmer, Robert J. Ergodic theory and semisimple groups. Monographs in Mathematics, 81. Birkhäuser Verlag, Basel, 1984.

In this talk I will make a survey, with historical motivations, about the applications of divergent series in order to treat different problems related with differential equations. We will introduce the theory of summability, developed by Malgrange and Ramis among others, and present some recent progresses in this field, related with the treatment of singularly perturbed differential equations. More precisely, the theory of monomial asymptotic expansions will be presented with some applications and possible future developments.

Devising classification algorithms that are robust to worst-case perturbations has emerged as a challenging problem in theoretical machine learning. In this work, we study whether there is any learning task for which it is possible to design classifiers that are only robust against polynomial-time adversaries; just like how it is done cryptography. Indeed, numerous cryptographic tasks (e.g. encryption of long messages) are only secure against computationally bounded adversaries. We show that computational limitation of attackers can indeed be useful in robust learning by demonstrating a classifier for a learning task in which computational and information theoretic adversaries of bounded perturbations have very different power. Namely, while computationally unbounded adversaries can attack successfully and find adversarial examples with small perturbation, polynomial time adversaries are unable to do so unless they can break standard cryptographic hardness assumptions (in particular digital signatures). No background on machine learning or cryptography will be assumed. Joint work (to appear in ALT 2020) with Sanjam Garg, Somesh Jha, and Saeed Mahloujifar.

A common agreement on the definition of chaos is the sensitive dependence on initial conditions. That means independent of how close two initial conditions are, by letting the system to proceed for a while, the new resulting states of the system are significantly different. In other words, a small error at a starting point will cause a huge difference in the outcome of the system. Since measuring a starting point can not be done accurately, the orbit of states is quite unpredictable. But statistically there is a hope to make a prediction by measuring an observable along orbits of the system. Despite of the alteration of the observable along an orbit, its time average for typical points converges to a constant which is the space average. This is due to the existence of an SRB (Physical) measure. Now, an interesting question is if the space average depends sensitively on system, i.e., if the statistical behavior is stable under the small perturbation of a system? In this talk, we start with some basic definitions and provide some examples of statistical stable chaotic systems: Lorenz-like map and Lorenz attractor. The Lorenz attractor is the first example of robust attractor containing a hyperbolic singularity in dimension three which is called singular hyperbolic attractor. Then we define precisely the singular hyperbolic attractors and discuss their statistical stability. This a joint work with Mohammad Fanaee.