Speakers

Abstract:
The question of whether the world in which we live is discrete or continuous has fascinated philosophers, physicists and mathematicians since antiquity, and from the point of view of the latter, has given birth to two cultures of mathematics. After having motivated this question by praising the scopes of these two practices of mathematics, I will explain how these two worlds, sometimes so distant, are found so harmoniously together at infinity.

Abstract:
In 2005, Nualart-Peccati discovered a significant result in the realm of probabilistic limit theorems on the Wiener space known as the fourth moment theorem stating that for a sequence $F_n$ of random variables living in a fixed Wiener chaos with variance one, $$F_n \stackrel{\text{law}}{\longrightarrow} N \sim \mathscr{N}(0,1) \quad \text{ if and only if} \quad E(F^4_n) \to 3 \, (= E(N^4)).$$ Recently, these sort of mathematical statements have been extensively got attention and it is culminating in the so-called Malliavin-Stein approach. The talk provides an introduction on the aforementioned approach with emphasis on the breakthrough technique of the Markov generators, spectral properties, and the Gamma calculus introduced by Michael Ledoux.

Abstract:
While the model theory of locally compact fields are well understood (starting from pioneering decidability works of Tarski and Ax-Kochen), very little is known regarding the model theory of number fields due to their Godelian undecidability where a direct approach is not possible. In this talk, I will give some model-theoretic results for number fields that are established from local theories. Such a local to global transition is made via motivic and $p$-adic integration to establish a result on meromorphic continuation of Dirichlet series that are Euler products of local integrals defined by means of model-theoretic data, and to understand model theory and measure theory of adeles. This result will then give solution to a number of open problems in number theory and algebra.

Abstract:
Several fundamental problems that arise in optimization and computer science can be cast as follows: Given vectors $v_1,\ldots,v_m \in \mathbb{R}^d$ and a constraint family $\mathcal{B} \subseteq 2^{[m]}$, find a set $S \in \mathcal{B}$ that maximizes the squared volume of the simplex spanned by the vectors in $S$. A motivating example is the ubiquitous data-summarization problem in machine learning and information retrieval where one is given a collection of feature vectors that represent data such as documents or images. The volume of a collection of vectors is used as a measure of their diversity , and partition or matroid constraints over $[m]$ are imposed in order to ensure resource or fairness constraints. Even with a simple cardinality constraint $(\mathcal{B}={[m] \choose r})$, the problem becomes NP-hard and has received much attention starting with a result by Khachiyan who gave an $r^{O(r)}$ approximation algorithm for this problem. Recently, Nikolov and Singh presented a convex program and showed how it can be used to estimate the value of the most diverse set when there are multiple cardinality constraints (i.e., when $\mathcal{B}$ corresponds to a partition matroid). Their proof of the integrality gap of the convex program relied on an inequality by Gurvits and was recently extended to regular matroids. The question of whether these estimation algorithms can be converted into the more useful approximation algorithms -- that also output a set -- remained open.
The main contribution of this paper is to give the first approximation algorithms for both partition and regular matroids. We present novel formulations for the sub-determinant maximization problem for these matroids; this reduces them to the problem of finding a point that maximizes the absolute value of a non-convex function over a Cartesian product of probability simplices. The technical core of our results is a new anti-concentration inequality for dependent random variables that arise from these functions which allows us to relate the optimal value of these non-convex functions to their value at a random point. Unlike prior work on the constrained sub-determinant maximization problem, our proofs do not rely on real-stability or convexity and could be of independent interest both in algorithms and complexity where anti-concentration phenomena have recently been deployed.

Abstract:
I will briefly discuss work in progress on studying the exponential memory loss for piecewise expanding $C^{1+}$ maps with countably many branches on metric spaces. I will mostly discuss the sufficient conditions for proving such a result including the conditions on the complexity growth of the partition elements. We will not assume or use the existence of a Markov structure for such dynamical systems and give explicit estimates on the constants involved in the exponential decay.

Abstract:
The analog of the Riemann curvature tensor for noncommutative tori manifests itself in the term $a_4$ appearing in the heat kernel expansion of the Laplacian of curved metrics. This talk presents a joint work with Alain Connes, in which we obtain an explicit formula for the $a_4$ associated with a general metric in the canonical conformal structure on noncommutative two-tori. Our final formula has a complicated dependence on the modular automorphism of the state or volume form of the metric, namely in terms of several variable functions with lengthy expressions. We verify the accuracy of the functions by checking that they satisfy a family of conceptually predicted functional relations. By studying the latter abstractly we find a partial differential system which involves a natural flow and action of cyclic groups of order two, three and four, and we discover symmetries of the calculated expressions with respect to the action of these groups. At the end, I will illustrate the application of our results to certain noncommutative four-tori equipped with non-conformally flat metrics and higher dimensional modular structures.

Abstract:
The idea of using pade approximations for approximating irrational numbers is originally due to Thue and Sigel. Hypergeomtric method is related to approximating the special values of hypergeometric series normally by using pade approximations and it’s main concern is obtaining rational approximation to irrational numbers. This makes hypergeometric method one of the main effective methods for solving Diophantine equations. In this talk We will discuss some classic and recent results concerning solving Diophantine equation by using hypergeometric method.

Abstract:
A major trend in non-commutative harmonic analysis is to investigate function spaces related to Fourier analysis (and representation theory) of non-abelian groups. The Fourier algebra, Rajchman algebra and Fourier-Steiltjes algebra, which are associated with the regular representation, the $C_0$ universal representation and the universal representation of the ambient group respectively, are important examples of such function spaces. These function algebras encode the properties of the group in various ways; for instance the non-existence of derivations on such algebras indicates their lack of analytic properties, which in turn translates into forms of either commutativity or discreteness for the group itself. In this talk, we study some Banach algebra properties of these function algebras. In particular, we present explicit constructions of continuous derivations on the Fourier algebras of two important matrix groups, namely the group of ${\mathbb R}$-affine transformations and the Heisenberg group. Using the structure theory of Lie groups, we extend our results to semisimple Lie groups and nilpotent Lie groups. If time permits, we will discuss weighted versions of the Fourier algebra, called the Beurling-Fourier algebra, and some of their properties.

Abstract:
The quadratic value theorem provides a dichotomy principle for an anisotropic quadratic form $q$ over a given field $F$ and an irreducible polynomial $p$ in a polynomial ring $A$ in finitely many variables over $F$:
(1) either $q$ remains anisotropic over the residue field of $A$ at $p$,
(2) or a scalar multiple of $p$ lies in the group generated by the nonzero values represented by $q$ over $A$.
In this talk, we examine the validity of this principle in a wider context.

Abstract:
An approximate unitary representation of a group $G$ is a map $f : G \longrightarrow U \left( n \right)$ where $n$ is a positive integer and $f \left( g \right) f \left( h \right) \approx f \left( g h \right)$ for all $g , h \in G$, where the approximation is in a common norm. Ulam asked in 1940 whether such $f$ can be approximated by an exact representation. We prove that the answer to the Ulam’s question is affirmative when the norm in question is any Schatten $p$-norm.

Abstract:
In this talk I will talk about local Langlands correspondence. I’m going to introduce the main ingredients needed to state the correspondence. Time permitting, I will also say few words about the proof(s) of the correspondence in the $GL_n$ case.

Abstract:
In the past decades, "weighted inequalities" have been a very attractive realm in singular Integral theory. One basic problem concerning them consists in determining conditions for a given operator (e. g. Hilbert transform) to be bounded in $L^p(w)$ with an appropriate weight $w$. To solve such problems, probabilistic methods or the dyadic technique --- which is a game of cubes --- has been used. In this talk, we will present some recent results about weighted bounds for "multilinear square functions" and certain singular integral operators such as linear and multilinear Fourier multipliers and the Riesz transforms associated to Schrödinger operators on $\mathbb{R}^n$.
References:
[1] T. A. Bui, M. Hormozi, Weighted bounds for multilinear square functions, Potential Analysis 46 (1), 135-148
[2] T. A. Bui, J. M. Conde-Alonso, X. T. Duong, M. Hormozi, Weighted bounds for multilinear operators with non-smooth kernels, Studia Mathematica 236, 245-269
[3] W. Damian, M. Hormozi, K. Li, New bounds for bilinear Calderon-Zygmund operators and applications, 35 pages, Accepted to Revista Matematica Iberoamericana

Abstract:
Since he nineteenth century English poet Edward Fitzgerald, produced a small booklet of translation of the Ruba'iayyat of Umar Kháyyam, it almost immediately elevated Kháyyam's name on a par with the giants of Classical Persian poetry. By our time Khayyam's name enjoy wide spread recognition as a brilliant poet who produced The Quatrains . On a secondary level, however, he is mentioned as a leading mathematician and a philosopher of his time. This phenomenon is not typical of pre-modern mathematicians or poets, as the two tracks appear unrelated in frst sight.
In this talk, I will present a historical investigation into the circumstances of Khayyam's life and career, based on the extant evidence, to put the two seemingly divergent persona in context, and seek answers to questions regarding his mathematical and literary legacy.

Abstract:
Let $\Omega \subset {\Bbb R}^2$ be a bounded domain with a regular boundary $\partial\Omega$, and let $\Gamma$ be a part of $\partial \Omega$ with nonempty relative interior.
Assume that a subdomain $\omega \subset \subset \Omega$ is given such that its boundary $\gamma := \partial\omega$ is a Jordan curve and let us denote by ${\bf n}$ the exterior normal to the boundary of $\Omega \setminus \overline{\omega}$. The question we address in this talk is the following: given a target function $h$ defined on $\gamma$, can one find a control function $v$ defined on $\partial\Omega$ having its support ${\rm supp}(v)\subset \Gamma$, and such that the solution $\Psi$ of
$\Delta\Psi = 0\quad\mbox{in }\, \Omega, \qquad {\partial\Psi \over \partial{\bf n}} = v\quad\mbox{on }\, \partial\Omega,$
satisfies
${\partial\Psi \over \partial{\bf n} } = h\quad \mbox{on }\, \gamma\qquad\mbox{or}\qquad \left\|{\partial\Psi \over \partial{\bf n} } - h\right\|_{H^{-1/2}(\gamma)} \leq \epsilon\mbox{ ?}$
(Here $H^{-1/2}(\gamma)$ denotes the Sobolev space of order $-1/2$ on $\gamma$). The motivation of this question lies in its application to the approximate Lagrangian control of Euler equation. The same question is studied when $\Omega \subset {\Bbb R}^N$ is a sooth bounded domain and $N \geq 2$.
The talk will also address the following ill-posed problem: let $\Omega$ be the rectangular domain $\Omega := (0,\pi )\times (0,\ell) \subset {\Bbb R}^2$ for some $\ell >0$ and set $\Gamma_{0} := [0,\pi]\times\{0\}$ and $\Gamma := \{0\}\times[0,\ell] \cup \{\pi\}\times[0,\ell]$. We give necessary and sufficient conditions on the Cauchy data $f_{0},g_{0}$ so that there exists $u \in H^1(\Omega)$ satisfying $\Delta u = 0$ in $\Omega$ and
${\partial u\over \partial{\bf n} } = 0 \quad \mbox{on }\, \Gamma,\qquad u = f_{0} \quad\mbox{on }\, \Gamma_{0}\quad \mbox{and }\, {\partial u \over\partial{\bf n} } = g_{0}\quad \mbox{on }\, \Gamma_{0}.$

Abstract:
We introduce a logical theory of differentiation for a real-valued function on a nite dimensional real Euclidean space. A real-valued continuous function is represented by a localic approximable mapping between two semi-strong proximity lattices, representing the two stably locally compact Euclidean spaces for the domain and the range of the function. Similarly, the Clarke subgradient, equivalently the $L$-derivative, of a locally Lipschitz map, which is non-empty, compact and convex valued, is represented by an approximable mapping. Approximable mappings of the latter type form a bounded complete domain isomorphic with the function space of Scott continuous functions of a real variable into the domain of non-empty compact and convex subsets of the nite dimensional Euclidean space partially ordered with reverse inclusion. Corresponding to the notion of a single-tie of a locally Lipschitz function, used to derive the domain-theoretic $L$-derivative of the function, we introduce the dual notion of a single-knot of approximable mappings which gives rise to Lipschitzian approximable mappings. We then develop the notion of a strong single-tie and that of a strong knot leading to a Stone duality result for locally Lipschitz maps and Lipschitzian approximable mappings. The strong single-knots, in which a Lipschitzian approximable mapping belongs, are employed to define the Lipschitzian derivative of the approximable mapping. The latter is dual to the Clarke subgradient of the corresponding locally Lipschitz map de ned domain-theoretically using strong single-ties. A stricter notion of strong single-knots is subsequently developed which captures approximable mappings of continuously di erentiable maps providing a gradient Stone duality for these maps. Finally, we derive a calculus for Lipschitzian derivative of approximable mapping for some basic constructors and show that it is dual to the calculus satis ed by the Clarke subgradient. This is a joint work with Prof. Abbas Edalat.

Abstract:
Erdős, Faudree, and Rousseau (1992) showed that a graph on $n$ vertices and at least $\lfloor n^2/4\rfloor + 1$ edges has at least $2\lfloor n/2\rfloor + 1$ edges in triangles. To see that this result is sharp, consider the graph obtained by adding one edge to the larger side of the complete bipartite graph $K_{\lceil n/2 \rceil, \lfloor n/2\rfloor}$. In this talk, we give an asymptotic formula for $h(n, e, K_3)$, the minimum number of edges contained in triangles in a graph having $n$ vertices and $e$ edges, where $e > n^2/4$ arbitrary. The main tool of the proof is a generalization of Zykov's symmetrization method that can be applied for several graphs simultaneously. We apply our weighted symmetrization method to tackle Erdős' conjecture concerning the minimum number of edges on $5$-cycles. We further extend our results to give an asymptotic formula for $h(n, e, F)$, the minimum number of $F$-edges in an $(n, e)$-graph when $n \rightarrow \infty$ and $F$ is a given $3$-chromatic graph. This is a joint work with Zóltan Füredi.

Abstract:
We live today the second quantum revolution, where the quantum properties such as coherent superposition and entanglement are used in a controlled and reproducible manner to develop new technological tools in computation, communication and high-precision measurement. After a brief discussion on particular features of quantum systems with respect to classical ones (composite systems modeled by tensor products, and the irreversible and unpredictable nature of quantum measurements), I will overview the main mathematical models behind these systems and some of their properties. These models include the discrete Markov chain models for quantum systems under measurement, the continuous-time stochastic master equations, and the Lindblad-type master equation modeling the average evolution. I will also briefly discuss some control and stabilization problems which are at the center of the current developments for instance in the area of quantum error correction.

Abstract:
We will talk about the diophantine equations from a logical point of view and explain that for a subsystem $T$ of the Peano Arithmetic how diophantine equations can be studied within $T$. For a special $T$ called Open Induction, this study leads us, through Wilkie's theorem, to an algebro-geometric situation wherein we consider a kind of "arithmetic" solutions of diophantine equations in the space of all solutions. This latter space is the real spectrum of a commutative ring. In modern real algebraic geometry, the real spectrum has the importance of the Zariski spectrum in the usual algebraic geometry. We will also briefly discuss a recent result of the speaker on the distribution of the "arithmetic" points in the real spectrum.

Abstract:
A multivariate polynomial $p \left( z_1 , \ldots , z_n \right)$ is stable if $p \left( z_1 , \ldots , z_n \right) \neq 0$ whenever $\Im(z_i) > 0$ for all $i$. Strongly Rayleigh distributions are probability distributions on $0-1$ random variables whose generating polynomial is stable. They can be seen as a natural generalization of determinantal distributions. Borcea, Branden and Liggett used the geometry of stable polynomials to prove numerous properties of strongly Rayleigh distributions, including negative association, and closure under conditioning and truncation.
In this talk I will go over basic properties of stable polynomials and strongly Rayleigh distributions; then, I will describe algorithmic applications in counting, sampling and optimization.
Based on joint works with Nima Anari, Alireza Rezaei, Amin Saberi, Mohit Singh.

Abstract:
Since Hilbert's theorem $90$, many Galois cohomological methods have been developed and applied to some interesting questions in algebraic number theory. In this talk we are concerned with questions related to interplay of Galois cohomology and the ring of integer valued polynomials. Our main focus will be on some concrete examples involving some low degree number fields.

Abstract:
According to a classical result of Bertoin (1998), if the initial data for Burgers equation is a Levy Process with no positive jump, then the same is true at later times and there is an explicit equation for the evolution of the associated Levy measures. In 2010, Menon and Srinivasan published a conjecture for the statistical structure of solutions to scalar conservation laws with certain Markov initial conditions, proposing a kinetic equation that should suffice to describe the solution as a stochastic process in $x$ with $t$ fixed (or in $t$ with $x$ fixed). In a joint work with Dave Kaspar, we have been able to establish this conjecture. Our argument uses a particle system representation of solutions.

Abstract:
We consider the action of invariant differential operators on the symmetric algebra of a mutiplicity-free representation of a basic classical Lie superalgebra. We show that for the general class of examples that arise from the TKK construction (where the representation space is indeed a Jordan superalgebra), there is a distinguished basis of the algebra of invariant differential operators (known as the Capelli basis) whose spectrum is given by suitable specialisations of super analogues of Macdonald polynomials defined by Sergeev and Veselov.

Abstract:
Komlós conjectured in 1981 that among all graphs with minimum degree at least $d$, the complete graph $K_{d+1}$ minimises the number of Hamiltonian subsets, where a subset of vertices is Hamiltonian if it contains a spanning cycle. We prove this conjecture when $d$ is sufficiently large. In fact we prove a stronger result: for large $d$, any graph $G$ with average degree at least $d$ contains almost twice as many Hamiltonian subsets as $K_{d+1}$, unless $G$ is isomorphic to $K_{d+1}$ or a certain other graph which we specify. This is joint work with Jaehoon Kim, Hong Liu and Katherine Staden.

Abstract:
Let $M_g$ be the moduli space of smooth genus $g$ curves. We define a notion of Chow groups of $M_g$ with coefficients in a representation of $Sp(2g)$, and we define a subgroup of tautological classes in these Chow groups with twisted coefficients. Studying the tautological groups of $M_g$ with twisted coefficients is equivalent to studying the tautological rings of all fibered powers of the universal curve over $M_g$ simultaneously. By taking the direct sum over all irreducible representations of the symplectic group in fixed genus, one obtains the structure of a twisted commutative algebra on the tautological classes. We obtain some structural results for this twisted commutative algebra, and we are able to calculate it explicitly when $g \leq 4$. Thus we determine $R^\bullet(C_g^n)$ completely, for all $n$, in these genera. We also give some applications to the Faber conjecture. This is a joint work with Dan Petersen and Qizhegn Yin.

Abstract:
Rings satisfying a polynomial identity have been proven to enjoy important properties. The pioneering works of Jacobson, Kaplansky and Levitzki resulted in a solution to the bounded Kurosh problem stating that a finitely generated associative algebra over a field in which every element satisfies an algebraic equation of bounded degree is finite dimensional. The Kurosh problem is an analogue of the bounded Burnside problem for groups for which Zelmanov was awarded the Fields medal.
The work of Kemer on $T$-deals and its application to yield a positive answer to the Specht conjecture has been on the spotlight and to some extent under scrutiny in the past few years. I will review some of the recent developments and activities in this regard. Then I will review the results about group rings and enveloping algebras that satisfy a non-matrix polynomial identity.

Abstract:
All of the functionality of the brain is through the coupling of the neurons. In this manuscript, two neurons with different types of excitability, a spiking neuron of type I excitability and an excitable one of type II excitability, couple through a linear form of gap junction. By increasing the coupling strength both neurons tend to tonic spiking, that is the appearance of two-torus in the coupled system. Further increasing in the coupling strength results in the torus break-down through “fold limit cycle bifurcation” on topological torus, then a stable limit cycle appears. This limit cycle corresponds to the emergent bursting oscillations in the neuron of type II excitability while the other neuron is in tonic spiking. Further increasing in the coupling strength makes the coupled system undergo infinitely many cascades of periodic-doubling bifurcations. Which results in bursting Oscillations with different lengths in the neuron of type II excitability. Through these cascades transient and robust chaos of Smale horseshoe type can be observed in the coupled system. Further increasing in the coupling strength leads to the burst synchronization of the neurons. In this manuscript it is approved that burst synchronizations with different lengths can be observed through a sequence of “blue sky catastrophe” bifurcations. This sequence eventually leads to the synchrony.

Abstract:
In this talk we determine the projective unitary representations of finite dimensional Lie supergroups whose underlying Lie superalgebra is $\frak{g} = A \otimes \frak{k}$, where $\frak{k}$ is a compact simple Lie superalgebra and $A$ is a supercommutative associative (super)algebra; the crucial case is when $A = \Lambda_s(\mathbb{R})$ is a Grassmann algebra. Since we are interested in projective representations, the first step consists of determining the cocycles defining the corresponding central extensions. Our second main result asserts that, if $\frak{k}$ is a simple compact Lie superalgebra with $\frak{k}_1\neq \{0\}$, then each (projective) unitary representation of $\Lambda_s(\mathbb{R})\otimes \frak{k}$ factors through a (projective) unitary representation of $\frak{k}$ itself, and these are known by Jakobsen's classification. If $\frak{k}_1 = \{0\}$, then we likewise reduce the classification problem to semidirect products of compact Lie groups $K$ with a Clifford--Lie supergroup which has been studied by Carmeli, Cassinelli, Toigo and Varadarajan.

Abstract:
Let $I$ be an ideal generated by monomials of degree $d$ in the polynomial ring over a field $K$. A clutter $C$ can be corresponded to $I$ in a natural way. The aim of this talk is to present some combinatorial conditions on the clutter $C$ which push the resolution of the corresponding ideal to be linear over any field. Some examples of ideals which have a linear resolution over any field except fields of characteristic $p$ for a given prime $p$ will be presented showing how the characteristic of the base field effects on the linearity of the resolution.