Abstract:
To study interesting arithmetic properties of binary forms, we often need a way to order them. I will discuss some natural ways to order binary forms $F(x , y)$ with integer coefficients, especially those of degree $3$ and $4$. I will show some of my recent works as examples of the importance of understanding the invariant theory of integral binary forms in order to count these important arithmetic objects.

Abstract:
It is expected that the global Langlands correspondence will be realized in the cohomology of Shimura varieties. This relies on the idea that these varieties are moduli for motives (and therefore in particular are sources for Galois representations), and in addition, they carry Hecke symmetries. A central point of Langlands' philosophy is that the zeta functions attached to Shimura varieties should be among the L-functions attached to automorphic representations of reductive groups. According to this, the conjecture of Langlands and Rapoport was born (~1987) to describe the structure of the points on the reduction of a Shimura variety. Recall that there is an analogues picture over function fields. Here, the moduli stacks for global G-shtukas replace Shimura varieties. In this talk, we first survey the panorama over function fields by translating the Deligne's conception of Shimura varieties (as a moduli for motives). Then we review the local theory of global G-shtukas, and continue towards the function fields analog of the conjecture of Langlands and Rapoport ...

Abstract:
Embezzlement, first introduced in the context of entanglement theory in quantum information theory, is the phenomenon of starting with a resource and producing a target object while leaving the resource almost untouched. In this talk, an embezzlement lemma is introduced that in some sense states that permutation matrices are dense in the space of unitary operators. Some applications of this lemme, particularly in the theory of nonlocal correlations are also discussed. This talk is based on a joint work with Marc-Olivier Renou.

Abstract:
TBA

Abstract:
We study the BO-ZK operator in the two-dimensional case. This operator contains both local and nonlocal terms and appears in a model describing the electromigration in thin nanoconductors on a dielectric substrate. It is also introduced in the half-wave-Schr\"{o}dinger equation. We first prove the associated anisotropic Gagliardo-Nirenberg inequality and its best constant. Next, as an application, by considering the related evolution equations and their Cauchy problems, we show the uniform bound of solutions in the energy space.

Abstract:
The well-known Cartan-Hadamard conjecture states that the classical isoperimetric inequality for domains in Euclidean space can be generalized to manifolds of nonpositive curvature. It is also known that this conjecture would follow from an estimate for total curvature of convex hypersurfaces. Motivated by these problems, we develop in this talk a comparison formula for total curvature of level sets of functions in Riemannian manifolds, and discuss some of its applications. This is joint work with Joel Spruck.

Abstract:
Using Scholze-Weinstein description of the Rapoport-Zink spaces in terms of p-adic Hodge theory and the vector bundles on the Fargues-Fontaine curve, we show that a certain diagram constructed via the exterior power morphism from the Lubin-Tate tower to the Rapoport-Zink tower is Cartesian. The special case of the determinant morphism is a theorem of Scholze-Weinstein. This result helps us understand the behavior of the exterior power morphism via-à-vis the period morphisms on the Lubin-Tate and Rapoport-Zink towers.

Abstract:
Abelian varieties are algebraizable complex tori. As such they carry polarizations, i.e., special codimension 1 subvarieties) that can be used to embed them in projective space. Theta divisors are minimal among such divisors, in the sense that there is only one holomorphic function (up to a scalar) with poles of order 1 solely along the theta divisor. This property has some interesting consequences for the cohomology of the theta divisor: the primal (or vanishing) cohomology of the theta divisor behaves like a Hodge structure of lower weight. I will survey some results about interesting geometric consequences of this fact for abelian varieties of low dimensions.

Abstract:
This talk aims to study a method for computing the discrepancy of any given distribution of points on the sphere. Studying distributions of points on the sphere has a long history and Thomson’s Problem, inspired by early atomic theory dating back to 1904, was a landmark. While Thomson’s Problem is based on the Coulomb potential, the discrepancy measures the deviation of the number of points in a set from the expected value.

In this talk, the Polar Coordinates method was introduced in the context of Thomson’s problem and the order of the growth of the discrepancy for this method is investigated. Applying tiny modifications to this method leads to the best-known rate. Besides, a new algorithm is introduced that enables one to determine whether the discrepancy of a given distribution is greater than any given number or not, and in the latter case, it specifies the location where a high discrepancy occurs.

Abstract:
Gromov defined various notions of convergence for a sequence of compact or non-compact metric spaces, that formalize the intuition that the scaled lattice $\delta \mathbb{Z}^d$ converges to $\mathbb{R}^d$ as $\delta\to 0$. Such convergence has applications in group theory and geometry. It has also found important applications in probability theory. Most notably, Gromov-Hausdorff convergence is used in the study of limits of random graphs and other types of discrete structures, which is still a very active topic. In various applications, the metric spaces under study are equipped with additional structures (e.g., a measure) and it is required to study their limit as well. Such an additional structure can be a measure, a point or finitely many points, marks on points, a closed subset, a continuous curve, a cadlag curve, etc. Various recent papers considered such additional structures and, to study their limit, generalized the Gromov-Hausdorff metric (or convergence) appropriately. The generalizations use the same idea as Gromov’s, but are not implied by that. Therefore, the mentioned papers either stated and proved all results (being a metric, completeness, separability, and convergence criteria) or mentioned them without proof. The former required a lot of space and was not the main goal of the papers, and the latter had errors in some papers due to various technicalities. In this presentation, after a review of the literature, we present a unified framework for generalizing the Gromov-Hausdorff metric for future use by using the notion of functors. This work is based on the preprint arxiv.org/abs/1812.03760.

Abstract:
I will discuss a recent result on the sum-product problem in finite fields, showing that given a subset $A$ of a finite field of sufficiently large characteristic $p$, if $|A|< p^{26/51}$ then $$\max\{|A+A|, |AA|\}> |A|^{16/13 - o(1)}.$$ This talk is based on a joint work with Sophie Stevens (RICAM).

Abstract:
I will talk about a new geometric inequality: the Sphere Covering Inequality. The inequality states that the total area of two distinct surfaces with Gaussian curvature less than 1, which are also conformal to the Euclidean unit disk with the same conformal factor on the boundary, must be at least 4π. In other words, the areas of these surfaces must cover the whole unit sphere after a proper rearrangement. We apply the Sphere Covering Inequality to solve several open problems about uniqueness and symmetry of solutions of mean field type equations. In particular we apply this inequality to prove an old conjecture of A. Chang and P. Yang about the best constant of a Moser-Trudinger type inequality.

Abstract:
The Montgomery-Dyson conjecture roughly states that the statistical properties of the zeros of the Riemann zeta function on the critical line (after rescaling) are similar to those of the eigenvalues of large random unitary matrices (after rescaling). In the past two decades, inspired by this conjecture, and in connection with number theory problems, many new ideas have been emerged in random matrix theory, new results proven and new spectacular conjectures stated (e.g. the moments conjecture by Keating and Snaith). In this talk I shall introduce a new object that we have introduced in 2017 which helped us solve the ratio conjecture in random matrix theory: it is a random holomorphic function whose zero set is a sine kernel point process. In a recent remarkable work, Valko and Virag have called this object the stochastic zeta function and have shown several chacaterizations and extensions of this object. Yet other possible extensions have been recently proved and studied by Lambert and Paquette. I shall focus here on how the idea of this object emerged and how it helps to give a natural solution to the ratio conjecture. I will also discuss an equivalent probabilistic formulation of the GUE conjecture (this is the Montgomery-Dyson conjecture for all correlation levels) involving the stochastic zeta function. In any case, we shall see that the stochastic zeta function (and its generalisations proposed by others) appears as the scaling limit of random analytic functions which have some spectral interpretation. If time allows, I will briefly discuss ongoing work with J. Najnudel where we are able to provide a very general framework/theorem which contains all known convergence results to these generalised stochastic zeta functions.

Abstract:
We prove a version of a Theorem of Furstenberg in the setting of Mapping class groups. Thurston measure defines a smooth measure class on space of projectivized measured laminations For every measure $\nu$ in this measure class, we produce a measure $\mu$ with finite first moment on the mapping class group such that $\nu$ is the unique $\mu$-stationary measure. In particular, this gives an coding-free proof of the already known result that the Lyapunov spectrum of Kontsevich-Zorich cocycle on the principal stratum of quadratic differentials is simple. This is a joint work with Alex Eskin and Maryam Mirzakhani.

Abstract:
In this talk, first we introduce several statistical physics models that undergo order/disorder phase transition. Then, we explain how techniques from sharp threshold theory answers some of the questions about these models.

Abstract:
In the late 70s, Fathi showed that the group of compactly supported volume-preserving homeomorphisms of the ball is simple in dimensions greater than 2. We present our recent article which proves that the remaining group, that is area-preserving homeomorphisms of the disc, is not simple. This settles what is known as the simplicity conjecture in the affirmative. This is joint work with Dan Cristofaro-Gardiner and Vincent Humilière.

Abstract:
Combining two classical notions in extremal combinatorics, the study of Ramsey-Turán theory seeks to determine, for integers $m\le n$ and $p \leq q$, the number $\mathsf{RT}_p(n,K_q,m)$, which is the maximum size of an $n$-vertex $K_q$-free graph in which every set of at least $m$ vertices contains a $K_p$. Two major open problems in this area from the 80s ask: (1) whether the asymptotic extremal structure for the general case exhibits certain periodic behaviour, resembling that of the special case when $p=2$; (2) constructing analogues of Bollobás-Erdös graphs with densities other than $1/2$.

We refute the first conjecture by witnessing asymptotic extremal structures that are drastically different from the $p=2$ case, and address the second problem by constructing Bollobás-Erdös-type graphs using high dimensional complex spheres with \emph{all rational} densities. Some matching upper bounds are also provided.
This is joint work with Hong Liu, Christian Reiher and Katherine Staden.

Abstract:
Statistical inference --- that is, characterizing the uncertainty of estimators in the form of $p$-values and confidence intervals --- plays a crucial role in scientific research across many disciplines and is at the core of statistical science.
Emerging statistical machine learning and artificial intelligence methods involve models with many parameters, either due to the large number of covariates being modeled, or because of the complexity and flexibility of the approaches. Moreover, many scientific problems involve infinite-dimensional (or function-valued) parameters.
Unfortunately, classical statistical inference procedures fail in such settings.
In this talk, we will discuss recent developments in statistical inference for regularized estimation strategies for high-dimensional models, i.e. models with more parameters than observations. We then discuss the challenges of extending these ideas to the setting of infinite-dimensional (or function-valued) parameters and present a new inference framework that facilitates statistical inference in previously intractable problems.

Abstract:
We introduce the \textit{modular intersection kernel}, and we use it to study how geodesics intersect on the full modular surface $\mathbb{X}=PSL_2\left(\mathbb{Z}\right) \backslash \mathbb{H}$. Let $C_d$ be the union of closed geodesics with discriminant $d$ and let $\beta\subset \mathbb{X}$ be a compact geodesic segment. As an application of Duke's theorem to the modular intersection kernel, we prove that $ \{\left(p,\theta_p\right)~:~p\in \beta \cap C_d\}$ becomes equidistributed with respect to $\sin \theta ds d\theta$ on $\beta \times [0,\pi]$ with a power saving rate as $d \to +\infty$. Here $\theta_p$ is the angle of intersection between $\beta$ and $C_d$ at $p$. This settles the main conjectures introduced by Rickards \cite{rick}.

We prove a similar result for the distribution of angles of intersections between $C_{d_1}$ and $C_{d_2}$ with a power-saving rate in $d_1$ and $d_2$ as $d_1+d_2 \to \infty$. Previous works on the corresponding problem for compact surfaces do not apply to $\mathbb{X}$, because of the singular behavior of the modular intersection kernel near the cusp. We analyze the singular behavior of the modular intersection kernel by approximating it by general (not necessarily spherical) point-pair invariants on $PSL_2\left(\mathbb{Z}\right) \backslash PSL_2\left(\mathbb{R}\right)$ and then by studying their full spectral expansion.