**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.

## Speakers

### Invited Speakers

- Omid Amini (CNRS)
- Ehsan Azmoodeh (University of Helsinki)
- Jamshid Derakhshan (University of Oxford)
- Javad Ebrahimi Boroojeni (EPFL)
- Peyman Eslami (University of Warwick)
- Farzad Fathizadeh (Caltech)
- Amir Ghadermarzi (University of Tehran)
- Mahya Ghandehari (University of Delaware)
- Mohammad Gholamzadeh Mahmoudi (Sharif University of Technology)
- Mohammad Hadi Hedayatzadeh (IPM)
- Omid Hatami (IPM)
- Mahdi Hormozi (Shiraz University)
- Hadi Jorati (Ohio State University)
- Otared Kavian (Université de Versailles)
- Zeinab Maleki (Isfahan University of Technology)
- Mazyar Mirrahimi (INRIA)
- Shahram Mohsenipour (IPM)
- Shayan Oveis Gharan (University of Washington)
- Ali Rajaei (Tarbiat Modares University)
- Fraydoun Rezakhanlou (University of California, Berkeley)
- Hadi Salmasian (University of Ottawa)
- Maryam Sharifzadeh (University of Warwick)
- Mehdi Tavakol (Pohang University of Science and Technology)
- Hamid Usefi (Memorial University of Newfoundland)
- Rashid Zare-Nahandi (IASBS)

### Contributed Speakers

CNRS

Title: Discrete-Continuous Duality, and the Question of the Discrete Limit of Continuous Objects

University of Helsinki

Title: The Malliavin-Stein Approach in Probabilistic Approximations

**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.

University of Oxford

Title: Mathematical Logic and Number Theory

**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.

EPFL

Title: Constrained Sub-Determinant Maximization via Anti-Concentration

**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.

University of Warwick

Title: Exponential Memory Loss for Piecewise Expanding Maps of Metric Spaces

**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.

Caltech

Title: The Term $a_4$ in the Heat Kernel Expansion of Noncommutative Tori

**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.

University of Tehran

Title: Applications of the Hypergeometric Method to Diophantine Equations

**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.

University of Delaware

Title: Derivatives on Fourier-type Algebras of Locally Compact Groups

**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.

Sharif University of Technology

Title: A Dichotomy Principle for Representations of Primes by Quadratic Forms

**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.

IPM

Title: Introduction to the Local Langlands Correspondence

**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.

Shiraz University

Title: Weighted Norm Estimates for Singular Integrals

**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

Ohio State University

Title: The Lost Legacy of Umar Khayyam's Mathematical Works

**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.

Isfahan University of Technology

Title: On Erdős' Conjecture on Pentagonal Edges and Other Type of Edges

**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.

INRIA

Title: Open Quantum Systems, Measurement and Control

**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.

IPM

Title: Logical Arithmetic and Real Algebraic Geometry

**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.

University of Washington

Title: Real Stable Polynomials, Strongly Rayleigh Distributions and Algorithmic Applications

**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.

University of California, Berkeley

Title: Markovian Solutions to Scalar Conservation Laws

**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.

University of Ottawa

Title: The Capelli Eigenvalue Problem for Lie Superalgebras

**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.

University of Warwick

Title: Proof of Komlós Conjecture on Hamiltonian Subsets

**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.

Pohang University of Science and Technology

Title: Tautological Classes with Twisted Coefficients

**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.

Memorial University of Newfoundland

Title: Polynomial Identities, Group Rings and Enveloping Algebras

**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.

IASBS

Title: Linearity of Resolution of Monomial Ideals and the Role of Characteristic of the Base Field

**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.