Abstracts
Plenary Talks
Nathan Keller (Bar Ilan University) - Erdos Talk
From Analysis of Boolean Functions to Social Choice and Extremal Combinatorics
Analysis of Boolean functions (AoBF) applies tools from mathematical analysis to study functions defined on the hypercube {0,1}^n. A central tool is using properties of the discrete Fourier transform of the function under consideration to derive information on the function. Initiated by Kahn, Kalai, and Linial in 1988, AoBF has grown into a prolific field, with connections to various areas of analysis and algebra, and a wide variety of applications, ranging from Machine learning and Social choice to Mathematical physics.
In this talk we shall review the basic notions and theorems of AoBF, and discuss applications to Social choice theory, Extremal combinatorics, and Quantum computation.
The results in extremal combinatorics are based on joint works with Noam Lifshitz.
Doron Puder (Tel Aviv University) - Erdos Talk
Word Measures on Groups: Structure, Applications and Conjectures
Fix a formal word in a free group, such as xyxy^{-2}. It induces a probability measure on every finite or even compact group by substituting x and y with random elements from the group.
In the talk I will survey some of the phenomena and structure unveiled in recent years about these measures, demonstrate the variety of mathematics that goes into this theory, mention applications and state conjectures.
This is based on joint works with Liam Hanany, Michael Magee, Frédéric Naud, Gal Ordo, Ori Parzanchevski and Danielle West.
Lightning Talks
Michael Bersudsky (Technion)
On the image in the torus of sparse points on expanding analytic curves
It is known that the projection to the 2-torus of the normalized parameter measure on a circle in the plane becomes uniformly distributed as the radius grows to infinity. We consider the following sparse version of this phenomenon. Starting from an angle and a sequence of radii {$r_n$} which diverges to infinity, we consider the projection to the 2-torus of the n'th roots of unity rotated by this angle and dilated by a factor of $r_n$. We found that if $r_n$ grows polynomially in n, then the image of these sparse collections becomes equidistributed, independently of the chosen rotation. Interestingly, we proved that the equidistribution might fail dramatically for every initial rotation if one allows the radii grow faster then any polynomial. Nevertheless, we showed that independently of the growth rate of the radii, the equidistribution still holds for almost all rotations. In greater generality, we will discuss this three type of results for the projection to the d-torus of dilations of varying analytic curves in d-space.
Uri Gabor (Hebrew University)
Classification of stationary stochastic processes by finitary coding
One of the remarkable results classifying stationary processes is Ornstein’s isomorphism theorem (1970), asserting that i.i.d. processes with the same entropy are isomorphic. This was strengthened in 1979 by Keane and Smorodinsky who replaced the notion of isomorphism with the more strict notion of finitary isomorphism. Despite their success in lifting the Isomorphism theorem to the finitary category, the question of whether other classification theorems have finitary counterpart remained open. We introduce a new quantity assigned to a process, which is invariant under finitary isomorphism, and use it to show for three classification theorems the invalidity of their finitary counterpart: the preservation of being a Bernoulli shift through factors, Sinai’s factor theorem, and the weak Pinsker property. This gives negative answers to an old conjecture and a recent open problem.
Amitai Kamber (Hebrew University)
Optimal lifting for SLn (Z)
The strong approximation theorem for SLn(Z) states that the modulu q map from SLn(Z) to SLn(Z/qZ) is onto. We want to make this statement quantitative, and in particular to lift a typical element of SLn(Z/qZ) to a small element in SLn(Z), where we measure the size of an integral matrix by the maximal absolute value of its coordinates.
Recently, Sarnak proved that for every epsilon>0, as q grows, almost every element of SL2(Z/qZ) can be lifted to an element of SL2(Z) with coordinates bounded by q^(3/2+epsilon). The exponent 3/2 is optimal by simple counting arguments.
Conjecturally, a similar statement should hold for every n. We will discuss this conjecture and some progress towards it.
Based on joint works with Konstantin Golubev and Hagai Lavner.
Yoni Kasten (Weizmann Institute)
Algebraic characterization of relational camera pose measurements in multiple images
My thesis addresses the problem of \emph{averaging bifocal tensors} in multiview settings. Solutions to this problem form an important step in solving the Structure from Motion (SfM) problem, i.e., the problem of recovering camera parameters and 3D scene structure from collections of 2D images. For a collection of $n$ images, \emph{bifocal tensors} (commonly referred to as \emph{essential and fundamental matrices}), denoted $F_{ij}$ ($i,j \in [n]$), capture the relative position, orientation, and internal camera parameters between cameras $i$ and $j$. These tensors are typically measured (up to unknown scale factors) from noisy point correspondences. Averaging bifocal tensors is a process of removing the noise from these measurements and recovering the underlying camera parameters. My thesis makes the following contributions: 1. It provides a complete algebraic characterization of the manifold of bifocal tensors for $n$ cameras. 2. It provides an optimization framework to project measured bifocal tensors to the nearest manifold. 3. It uses techniques analogous to \emph{parallel graph rigidity} in projective spaces to design an efficient projection algorithm. 4. It introduces a method for handling collinear camera settings, which create singular bifocal tensor structures, by introducing a novel concept of \emph{virtual cameras}. 5. Our implementation demonstrates the efficiency of these algorithms and achieves state-of-the-art results.
Elad Kosloff (Hebrew University)
Counting curves and the appearance of noncommutative graded algebra
Enumerative geometry is the branch of mathematics concerned with counting numbers of possible solutions to geometric questions. Typically this task is carried out by means of intersection theory. In our work we address J holomorphic disks which are fundamental objects in symplectic topology. Enumerating them is a natural question in symplectic topology with applications in hamiltonian dynamics and algebraic geometry. In my talk I want to outline a method, employing a generating function that defines enumerative invariants on the symplectic manifold. Intuitively, the generating function counts J-holomorphic disks with constraints in the interior and boundary of the disk.
Shir Peleg (Tel Aviv University)
A generalized Sylvester-Gallai type theorem for quadratic polynomials
In this work we prove a version of the Sylvester-Gallai theorem for quadratic polynomials
that takes us one step closer to obtaining a deterministic polynomial time algorithm for testing
zeroness of \Sigma^{[3]}\Pi\Sigma\Pi^{[2]} circuits. Specifically, we prove that if a finite set of irreducible quadratic polynomials Q satisfy that for every two polynomials Q1, Q2 in Q there is a subset K ⊂ Q, such that Q1, Q2 \notin K \and whenever Q1 and Q2 vanish then also ∏i_{i\in K}Qi vanishes, then the linear span of the polynomials in Q has dimension O(1). This extends the earlier result [Shp19] that showed a similar conclusion when |K| = 1.
An important technical step in our proof is a theorem classifying all the possible cases in
which a product of quadratic polynomials can vanish when two other quadratic polynomials
vanish. I.e., when the product is in the radical of the ideal generated by the two quadratics.
This step extends a result from [Shp19] that studied the case when one quadratic polynomial
is in the radical of two other quadratics.
Parallel Sessions
Algebra
Time: 14:15
First talk: Nadya Gurevich (Ben-Gurion University)
Title: Fourier transforms on the basic affine space of a quasi-split group
Abstract: For a quasi-split group G over a local field F, with Borel subgroup B = T U and Weyl group W, there is a natural geometric action of G×T on L2(X), where X = G/U is the basic affine space of G. For split groups, Gelfand and Graev have extended it to an action of G×(T semidirect product W) by generalized Fourier transforms Φ_w, that can be viewed as normalized intertwining operators. It is based on the definition for the group SL(V), dim(V) = 2 of the operator Φ_w : L2(V) → L2(V), which happens to be the usual Fourier transform. In this case G × (T semidirect product W) acts on L2(V) by the minimal representation Π of Sp(4).
We shall extend this result for quasi-split groups. The main challenge is to define the operator Φ_w for the quasi-split group G = SU(3) of rank one. Similarly to the split case we shall consider the action G×(T semidirect product W) on L2(X) using the minimal representation Π of Spin(8).
This is joint work with David Kazhdan.
Time: 14:55
Second talk: Alexei Entin (Tel Aviv University)
Title: The minimal ramification problem in inverse Galois theory
Abstract: For a number field K and a finite group G the Boston-Markin Conjecture predicts the minimal number of ramified places (of K) in a Galois extension L/K with Galois group G. The conjecture is wide open even for the symmetric and alternating groups S_n, A_n over the field of rational numbers Q.
We formulate a function field version of this conjecture, settle it for the rational function field K=F_q(T) and G=S_n with a mild restriction on q,n and make significant progress towards the G=A_n case.
We also discuss some other groups and the connection between the minimal ramification problem and the Abhyankar conjectures on the etale fundamental group of the affine line in positive characteristic.
Time: 15:35
Third talk: Beeri Greenfeld (Bar-Ilan University)
Title: How do algebras grow?
Abstract: The most important invariant of a finite dimensional algebra is its dimension. Let $A$ be a finitely generated, infinite dimensional associative (or Lie) algebra over some base field $F$. A major way to 'measure its infinitude' is to study its growth rate, namely, the asymptotic behavior of the dimensions of the spaces spanned by (at most n)-fold products of some fixed generators. Up to a natural asymptotic equivalence relation, this becomes a well-defined invariant of the algebra itself, independent of the specification of generators.
The question of `how do algebras grow?', or, which functions can be realized as growth rates of algebras (perhaps with additional algebraic properties, as grading, simplicity etc.) plays an important role in classifying infinite dimensional algebras of certain classes, and is thus connected to ring theory, noncommutative projective geometry, combinatorics of infinite words, symbolic dynamics and more.
We present new results on possible and impossible growth rates of important classes of associative and Lie algebras, by combining novel techniques and constructions from noncommutative algebra, combinatorics of (infinite trees of) infinite words and convolution algebras of étale groupoids attached to them.
As applications, we answer two open questions of Petrogradsky (from 1997 and 2000) on infinite dimensional Lie algebras, extend earlier constructions of Bartholdi and Smoktunowicz (2014) on monomial algebras to settings they left open, and derive the first constructive counterexamples to a question of Bergman (1988) on various dimensions of noncommutative rings.
Examples are given to prove that a certain polynomial error factor appearing along the journey plays a crucial role, even for growth types arbitrarily close to exponential, yielding the first non-trivial gaps in the space of intermediate growth functions of algebras.
Time: 16:15
Fourth talk: Ari Shnidman (Hebrew University)
Title: Derivatives of automorphic L-functions and representation theory
Abstract: I'll present a formula relating the central derivative of an automorphic L-function over a function field to the intersection number of a pair of algebraic cycles in the moduli space of PGL_2-shtukas. The proof requires giving a representation theoretic interpretation for "differentiating" the L-function. After stating the formula, I will describe the pretty representation theory that underlies the proof.
Analysis and Geometry
Time: 14:15
First Talk: Yanir Rubinstein (University of Maryland / Weizmann Institute)
Title: Basis divisors, balanced metrics, and Kahler-Einstein metrics
Abstract: Using log canonical thresholds and basis divisors Fujita--Odaka introduced purely algebro-geometric invariants $\delta_m$ whose limit in $m$ is now known to characterize uniform K-stability on a Fano variety. A basic question since Fujita--Odaka's work has been to find an analytic interpretation of these invariants.
In joint work with G. Tian and K. Zhang we show that each $\delta_m$ is the coercivity threshold of a quantized Ding functional on the $m$-th Bergman space and thus characterizes the existence of balanced metrics. This approach has a number of applications. The most basic one is that it provides an alternative way to compute these invariants, which is new even for projective space. Second, it allows us to introduce algebraically defined invariants that characterize the existence of the otherwise transcendental Kahler-Ricci solitons. Third, it leads to approximation results involving balanced metrics in the presence of automorphisms that extend some results of Donaldson.
Time: 14:55
Second Talk: Shira Tanny (Tel Aviv University)
Title: The Poisson bracket invariant: soft and hard approaches.
Abstract: The Poisson bracket is a basic operation on functions on a symplectic manifold playing a fundamental role in geometry and dynamics. A theorem by Entov-Polterovich states that functions forming a partition of unity with "symplectically small" supports cannot commute with respect to the Poisson bracket. In 2012 Polterovich conjectured a quantitative version of this theorem. I will discuss three related topics: a solution of this conjecture in dimension two (with Lev Buhovsky and Alexander Logunov), a link between this problem and Grothendieck's theorem from functional analysis (with Efim Gluskin), and a Floer-theoretical approach to partitions of unity (with Yaniv Ganor). All symplectic preliminaries will be explained.
Time:15:35
Third Talk: Aron Wennman (Tel Aviv University)
Title: The hole event for Gaussian complex zeros and the emergence of quadrature domains
Abstract: The Gaussian Entire Function (GEF) is a distinguished random Taylor series with independent Gaussian coefficients, whose zero set is invariant in distribution with respect to isometries of the plane. We consider the zero distribution of the GEF, conditioned on the rare event that no zero lies in a given (large) region. For circular holes, Ghosh and Nishry observed that as the radius of the hole tends to infinity, the density of zeros vanishes not only on the given hole, but also on an annulus beyond the (rescaled) hole — a 'forbidden region' emerges.
In this talk we are concerned with the shape of the forbidden region for non-circular holes. I plan to discuss how one can study this problem through a type of constrained obstacle problem, and indicate why the solution is given in terms of a special class of algebraic domains -- the (subharmonic) quadrature domains.
This reports on ongoing joint work with Alon Nishry.
Time: 16:15
Fourth Talk: Adi Glucksam (University of Toronto)
Title: Stationary random entire functions and related questions
Abstract: Let T be the action of the complex plane on the space of entire functions defined by translations, i.e T_w takes the entire function f(z) to the entire function f(z+w). B.Weiss showed in `97 that there exists a probability measure defined on the space of entire functions, which is invariant under this action. In this talk I will present optimal bounds on the minimal possible growth of functions in the support of such measures, and discuss other growth related problems inspired by this work. In particular, I will focus on the question of minimal possible growth of frequently oscillating subharmonic functions.
The talk is partly based on a joint work with L. Buhovsky, A.Logunov, and M. Sodin.
Geometry and Groups
Time: 14:15
First Talk: Gili Golan (Ben Gurion University)
Title: On maximal subgroups of Thompson group F
Abstract: We study subgroups of Thompson group F by means of a 2-automaton associated with them. We prove that a maximal subgroup of infinite index of F must be closed (i.e., it coincides with the subgroup of F accepted by the 2-automaton associated with it). We also prove that F has infinitely many non-isomorphic maximal subgroups.
Time: 14:55
Second Talk: Alex Margolis (Technion)
Title: Discretisable quasi-actions
Abstract: If a group G acts isometrically on a metric space X and Y is any metric space that is quasi-isometric to X, then G quasi-acts on Y. A fundamental problem in geometric group theory is to straighten (or quasi-conjugate) a quasi-action to an isometric action on a nice space. We will introduce and investigate discretisable spaces, those for which every cobounded quasi-action can be quasi-conjugated to an isometric action of a locally finite graph. Work of Mosher-Sageev-Whyte shows that free groups have this property, but it holds much more generally. For instance, every hyperbolic group is either commensurable to a cocompact lattice in rank one Lie group, or it is discretisable.
We give several applications and indicate possible future directions of this ongoing work, particularly in showing that normal and almost normal subgroups are often preserved by quasi-isometries. For instance, we show that any finitely generated group quasi-isometric to a Z-by-hyperbolic group is Z-by-hyperbolic. We also show that within the class of residually finite groups, the class of central extensions of finitely generated abelian groups by hyperbolic groups is closed under quasi-isometries.
Time: 15:35
Third Talk: Gil Goffer (Weizmann Institute)
Title: Invariable generation and Wiegold's conjecture
Abstract: I will discuss the notion of invariably generated groups, and present a construction of an invariably generated group that admits an index two subgroup that is not invariably generated. The construction answers questions of Wiegold and of Kantor-Lubotzky-Shalev. This is a joint work with Nir Lazarovich.
Time: 16:15
Fourth Talk: Maria Gerasimova (Bar Ilan University)
Title: Isoperimetry, Littlewood functions, and unitarisability of groups
Abstract: Let G be a discrete group. A group G is called unitarisable if any uniformly bounded representation on any Hilbert space H can be conjugated to a unitary representation. It is well known that amenable groups are unitarisable. It has been open ever since whether amenability characterises unitarisability of groups.
Dixmier: Are all unitarisable groups amenable?
We define the Littlewood exponent Lit(G) of a group G:
We will show that, on the one hand, Lit(G) is related to unitarisability and amenability and, on the other hand, it is related to some geometry of G.
We will discuss some examples and applications of this connection.
Set Theory and Logic
Time: 14:15
First Talk: Eran Alouf (Hebrew University)
Title: Dp-minimal expansions of (Z,+)
Abstract: I will review the research about dp-minimal expansions of (Z,+), and then present a recent result classifying (Z,+,<) as the unique dp-minimal expansion of (Z,+) defining an infinite subset of N.
All the relevant notions will be defined in the talk.
Time: 14:55
Second Talk: Eliana Barriga (Ben Gurion University)
Title: O-minimal universal covers of semialgebraic groups over real closed fields
Abstract: Semialgebraic groups over a real closed field can be seen as a generalization of the semialgebraic groups over the real field, and also as a particular case of the groups definable in an o-minimal structure.
In this talk, I will present and discuss some of our main results on definably connected algebraic semialgebraic groups over a real closed field R through the study of their o-minimal universal covering groups and of their relationship with the R-points of some connected R-algebraic group.
We will see that the o-minimal universal covering group of a definably compact definably connected group definable in a sufficiently saturated real closed field R is an open subgroup of the o-minimal universal covering group of the R-points H (R) of some Zariski-connected R-algebraic group H.
This research is part of my PhD thesis at the University of Haifa, Israel and Universidad de los Andes, Colombia.
Time: 15:35
Third Talk: Tom Benhamou (Tel Aviv University)
Title: The Forcing Method and Prikry-Type Forcing
Abstract: In this talk we will start by presenting an overview and give motivation for the forcing method, which is a method to produce models of set theory. Then we will address the problem of preserving cardinals and cofinalities. Finally we will introduce some Prikry type forcing, which produce models in which cradinals are preserved and cofinality of some cardinals changes. More specifically, we will define Prikry forcing and a result by Gitik, Kanovei, Koepke, that every intermediate ZFC model of a Prikry generic extension in again Prikry generic. If time permits, we will present Magidor forcing and present a generalization of Gitik, Kanovei and Koepke's result due to the speaker and Gitik.
The audience is assumed to be familiar with basic concepts of set theory and logic, such as ordinals and cardinals.
Time: 16:15
Fourth Talk: Alejandro Poveda (Hebrew University)
Title: A new framework for iterating Prikry-Type forcings and applications to Stationary Reflection
Abstract: In this talk we will give an overview of the theory of Σ-Prikry forcings and their iterations, recently introduced in a series of papers. We will begin motivating the class of Σ-Prikry forcings and showing that many Prikry-type posets that center on countable cofinalities fall into this framework. Afterwards, we will present a viable iteration scheme for these posets. Finally, and if time permits, we will try to discuss some applications of the framework to the investigation of consistency result in singular combinatorics. Specifically, we shall discuss a recent construction of a model where א_ω is a strong limit cardinal, the Singular Cardinal Hypothesis fails at א_ω and every stationary subset of א_ω+1 reflects. This shows the mutual consistency of two classical results of Magidor.
Joint work with Assaf Rinot and Dima Sinapova.
Ergodic Theory
Time: 14:15
First Talk: Zemer Kozlov (HUJI)
Title: The orbital equivalence of Bernoulli actions and their Sinai factors
Abstract: Recent results on the classification of Bernoulli shifts into orbit equivalence classes show that a nonsingular Bernoulli shift on $2$ symbols can never have an absolutely continuous infinite $\sigma$-finite invariant measure. In addition there is strong evidence that the only possible Krieger types of Bernoulli shifts on $2$ symbols are type $\mathrm{III}_1$, type $I$ (dissipative) and type $\mathrm{II}_1$ (equivalent to stationary product measure). Quite surprisingly,when the state space is infinite, the situation changes dramatically and there are Bernoulli shifts on countably many symbols of arbitrary Krieger type (K-Soo, Berendschot-Vaes).
In this talk I will explain an elementary construction of a type-$\mathrm{III}_{\lambda}$ Bernoulli shift, show that our nonsingular Bernoulli shifts have independent and identically distributed factors and discuss further challenges when one tries to understand whether given two non-stationary Bernoulli shifts, is one a factor of the other. This talk is based on joint work with Terry Soo (UCL).
Time: 14:55
Second Talk: Kevin Bouchner (Weizmann Institute)
Title: From Patterson-Sullivan theory to representation theory of negatively curved groups
Abstract: We will describe how dynamical properties of discrete negatively curved groups can be used to describe their representation theory.
The talk will focus on a recent result characterizing geometrically the existence of non-tempered representations analogous to the complementary series.
Time: 15:35
Third Talk: Erez Nesharim (Hebrew University)
Title: Homogeneous flows and numbers badly approximable by algebraics
Abstract: In their 1998 paper, Kleinbock and Margulis proved a uniform estimate for non-divergence of unipotent flows on homogeneous spaces and applied it in order to show that almost every point on a nondegenerate manifold is not very well approximable by rationals. We use their method and a variant of Schmidt's game in order to prove the existence of badly approximable points on any nondegenerate curve. In combination with an extension of Kleinbock-Margulis' results to ``friendly'' measures by Kleinbock--Lindenstrauss--Weiss, the same argument implies the existence of a real number which is badly approximable by algebraic numbers. This settles a question asked independently by Beresnevich and Bugeaud in 2014. After introducing the results I will focus on the case of uncountably many badly approximable points on the parabola.
This talk is based on a joint work with Victor Beresnevich and Lei Yang.
Time: 16:15
Fourth Talk: Jeremias Epperlein (Ben Gurion University)
Title: Iterated Minkowski sums, horoballs and north-south-dynamics
Abstract: Let $G$ be a countable group generated by a finite set $A$ containing
the identity. This talk will present results about the topological
dynamical system $\varphi_A: 2^G \to 2^G$ defined by $M \mapsto
MA=\{ma : m \in M, a \in A\}$. I will concentrate on properties of the
pair $(G,A)$ which can be recovered from the dynamics of $\varphi_A$.
This is based on joint work with Tom Meyerovitch.
Applied Mathematics
Time: 14:15
First Talk: Anne Le Blanc (Tel Aviv University)
Title: Block Finite Difference Methods and their Relation to Finite Element Methods
Abstract: In the classical FD method the same stencil is used at each grid point and the orders of the global error and the truncation error are the same. Block Finite Difference methods (BFD) are FD methods in which the domain is divided into blocks, or cells, containing two or more grid points with a different scheme used for each grid point, unlike the standard FD method. In this approach, the interaction between the different truncation errors and the dynamics of the scheme may inhibit the error from growing, hence error reduction is obtained, leading to a more efficient method.
On the other hand, the Finite Element method (FE) consists in finding an approximation of the solution in a certain form, usually a linear combination of a set of chosen trial functions.
The Discontinuous Galerkin (DG) method is a class of FE method using a completely discontinuous polynomial space for the approximation of the solution and trial functions. We first show that our BFD scheme can be viewed as a DG scheme, proving stability during the process. Then, performing a Fourier like analysis, we prove optimal convergence of the BFD scheme.
Time: 14:55
Second Talk: Alona Mokhov (Afeka College of Engineering)
Title: Approximation of Set-Valued Functions - an overview
Abstract: The talk surveys our work on the approximation of set-valued functions (multifunction), functions mapping the points of a closed real interval to general compact sets in R^n. The approach is to adapt approximation methods for real-valued functions to set-valued functions, by replacing operations between numbers by operations between sets.
For multifunctions with compact convex images, adaptation based on Minkowski convex combinations of sets yields approximating operators in the Hausdorff metric. Yet, if the images of set-valued functions are not necessarily convex, then the approximations methods may fail. To avoid covexification, approximation methods for set-valued functions with general compact images are adapted using metric linear combinations of sets. This adaptation method is not restricted to positive operators. As examples we study metric Bernstein operators, metric Schoenberg operators and metric polynomial interpolants. Error estimates are given with respect to Hausdorff distance in terms of the regularity properties of the approximated set-valued function. Using metric linear combinations of sets the metric integral of set-valued functions is defined. The metric integral is free of the convexifcation drawback, which is typical to the commonly used Aumann integral of SVFs. We use the metric integral to smooth a multifunction by defining its metric Steklov set-valued function. The error is measured by the averaged Hausdorf distance.
Joint work with Nira Dyn and Elza Farkhi,
Time: 15:35
Third Talk: Boaz Nadler (Weizmann Institute)
Title: The Trimmed Lasso: Sparse Recovery Guarantees and Practical Optimization
Abstract: Consider the sparse approximation or best subset selection problem: Given a vector y and a matrix A, find a k-sparse vector x that minimizes the residual ||Ax-y||. This sparse linear regression problem, and related variants, plays a key role in high dimensional statistics, compressed sensing, and more.
In this talk we focus on the trimmed lasso penalty, defined as the L_1 norm of x minus the L_1 norm of its top k entries in absolute value. We advocate using this penalty by deriving sparse recovery guarantees for it, and by presenting a practical approach to optimize it. Our computational approach is based on the generalized soft-min penalty, a smooth surrogate that takes into account all possible k-sparse patterns. We derive a polynomial time algorithm to compute it, which in turn yields a novel method for the best subset selection problem. Numerical simulations illustrate its competitive performance compared to current state of the art.
Joint work with Tal Amir and Ronen Basri.
Time: 16:15
Fourth Talk: Jeremy Schiff (Bar-Ilan University)
Title: KP reductions from the KP lattice
Abstract: The Kadomtsev-Petviashvili (KP) equation is a nonlinear partial differential equation in 2+1 dimensions that describes the complex web-like patterns of waves that can appear on the surface of shallow water. The KP equation is the simplest nontrivial equation in the KP hierarchy, a system of equations with diverse applications in mathematics and theoretical physics. Many important nonlinear partial differential equations in 1+1 dimensions, such as the Burgers', Korteweg-de Vries, nonlinear Schrodinger and Boussinesq equations, all arise as reductions of the KP hierarchy.
Among the numerous special properties of KP (both the equation and the hierarchy) is the existence of a Backlund transformation (BT), a way to construct a new solution from a given one by solution of a simpler system. Backlund transformations commute, and this allows the construction of an associated lattice, in which the vertices are solutions of KP (equation or hierarchy), connected by an edge if they are related by a BT. In this talk we show how reductions of KP (some old, some new) arise by imposing translational invariances on this lattice.
Joint work with Sasha Rasin (Ariel).
Time: 14:15-14:20
Opening
Time: 14:20-14:50
First Talk: Anna Hoffman (Weizmann Institute of Science)
Title: What do mathematicians wish to teach secondary school teachers about the discipline of
mathematics?
Abstract: This study investigates what mathematicians wish to teach teachers about the discipline of mathematics. Data source included interviews with five research mathematicians who taught academic mathematics courses to practicing secondary mathematics teachers. Analysis revealed that expanding teachers’ knowledge about the discipline of mathematics was one of the main objectives of the interviewees. They referred to nine topics that can be grouped into three key aspects: (1) the essence of mathematics, (2) doing mathematics, and (3) the worth of mathematics. In this presentation I will characterize and illustrate each aspect.
Time: 14:50-15:20
Second Talk: David Ginat (Tel Aviv University)
Title: רמות הפשטה בחשיבה אלגוריתמית
Abstract: הפשטה הינה מוטיב יסודי במדעי המחשב בכלל ובחשיבה אלגוריתמית בפרט. בבעיה אלגוריתמית נתונות נקודת מוצא ומטרה, ויש לנסח סדרת פעולות חישוב כדי להוביל מנקודת המוצא למטרה. סדרת פעולות החישוב מבטאת נקודת מבט אופרטיבית, של “איך להשיג את המטרה”. נקודת מבט זו הינה פרטנית, מפורשת, ומתקיימת ברמת הפשטה נמוכה יחסית. במקרים רבים חשוב שהיא תהיה קשורה לרמת הפשטה גבוהה יותר, אשר מבטאת נקודת מבט דקלרטיבית, של “מהם המאפיינים של הבעיה לפתרון”. הבחנה במאפיינים מתאימים הינה מפתח לפתרון אלגוריתמי נכון ויעיל. ההרצאה תתמקד באתגר של שילוב בין רמות ההפשטה המתוארות .בפתרון בעיות אלגוריתמיות
Time: 15:20-15:30
Break
Time: 15:30-16:00
Third Talk: Elena Naftaliev (Achva Academic College) and Marita Barabash (Achva Academic College; Ministry of Education)
Title: Experimental mathematics at school – A three-faced coin: Mathematical, didactic and
research aspects
Abstract: Experimental mathematics, always part of the discipline, has been significantly upgraded due to advanced technological tools. Nevertheless, this approach has not yet influenced mathematics teaching, curricula, and teachers’ practices. Our research focuses on organically interweaving formal deductive and TBICMs (Technology-Based Interactive Curriculum Materials)-supported experimental approaches toward mathematics into teaching and learning mathematics in school.
At the present stage of our research, the teachers who participate in it learn how appropriate
TBICMs usage contributes to the implementation of the experimental-mathematics approach in
school alongside with formal, deductive mathematics, and we intend to present both the
essence of the approach, and our first findings.
Time: 16:00-16:30
Panel discussion: Ruhama Even (Weizmann Institute of Science) – Chair, Uri Bader (Weizmann Institute of Science; Ministry of Education), Michael Fried (Ben Gurion University), Roza Leikin (Haifa University)
Title: The role of mathematics in mathematics education
Time: 16:30-16:45
Open discussion
Discrete Mathematics and Theory of Computer Science
Time: 14:15
First Talk: Tom Gilat (Bar Ilan University)
Title: Decomposition of random walk measures on the one-dimensional torus
Abstract: The main result of this talk is a decomposition theorem for a measure on the
one-dimensional torus. Given a sufficiently large subset $S$ of the positive
integers, an arbitrary measure on the torus is decomposed as the sum of two
measures. The first one $\mu_1$ has the property that the random walk with
initial distribution $\mu_1$ evolved by the action of $S$ equidistributes very
fast. The second measure $\mu_2$ in the decomposition is concentrated on very
small neighborhoods of a small number of points. Work was done as part of PhD studies
under the supervision of Prof. Elon Lindenstrauss.
Time: 14:55
Second Talk: Yelena Yuditsky (Ben Gurion University)
Title: Typical structure of hereditary graph families
Abstract: A family of graphs $\cal F$ is \textit{hereditary} if it is closed under isomorphism and taking induced subgraphs. For example, for a given graph $H$, a hereditary family is the family of all \textit{$H$-free} graphs, that is graphs without an induced copy of $H$.
Alon, Balogh, Bollob\'{a}s and Morris showed that for every hereditary family $\cal F$ there exist $\epsilon >0$ and $l\in \mathbb{N}$ such that the number of graphs in $\cal F$ on $n$ vertices is $2^{(1-1/l)n^2/2+o(n^{2-\epsilon})}$. They showed this bound by deriving various structural properties of almost all graphs in $\cal F$. We study and obtain additional structural properties of almost all graphs in some restricted hereditary families. As an application of our results, we prove the existence of an infinite family of counterexamples for the recent Reed-Scott conjecture about the structure of almost all $H$-free graphs.
Joint work with Sergey Norin.
Time: 15:35
Third Talk: Shoni Gilboa (Open University)
Title: On the maximal number of three-term arithmetic progressions in finite sets in different geometries
Abstract: Green and Sisask observed in 2008 that in any $n$-element set of integers (or real numbers), there are at most $\lceil n^2/2\rceil$ three-term arithmetic progressions. A simple projection argument shows that the same is true in Euclidean spaces of any dimension.
We study this question in general metric spaces, where we consider a triple $(a,b,c)$ of points in a metric space $X$ to be an arithmetic progression if $d_X(a,b)=d_X(b,c)=\textstyle\frac{1}{2}d_X(a,c)$.
In particular, we show that the Green-Sisask result extends to hyperbolic spaces, but does not hold in spherical geometry.
Joint work with Itai Benjamini.
Time: 16:15
Fourth Talk: Elad Aigner-Horev (Ariel University)
Title: Smoothed analysis of Ramsey-type problems
Abstract: Numerous results have been established of late regarding various properties of randomly perturbed/augmented graphs; by that we mean graph distributions of the form G \cup G(n,p), where G is a fixed graph and G(n,p) is “the” binomial random graph of edge density p:=p(n). In our talk, we shall concentrate on Ramsey-type properties of graphs thus sampled with the aim of providing a short survey of recent results in this line of research.
We shall commence with a classical Ramsey properties known for randomly perturbed dense graphs as well as randomly perturbed dense sets of integers demonstrating the connection between the problems at hand to the Kohayakawa-Kruter conjecture (or more specifically its recently established 1-statement) as well as to the so called asymmetric random Rado theorem.
Next, we shall consider anti-Ramsey properties of randomly perturbed dense graphs; in particular the threshold for the emergence of so called rainbow cliques (of fixed size) in such graphs in every proper edge-colouring thereof.
If time permits, we shall conclude, with the emergence of rainbow Hamilton cycles in randomly edge-coloured randomly perturbed dense graphs.
The talk is based on joint works with Oran Danon, Dan Hefetz, Shoham Letzer, and Yury Person
Special Lightning Session
Danil Akhtyamoff and Alon Dogon (Hebrew University)
On uniform Hilbert Schmidt stability of groups
Consider the following question: Given a group G, and a map \phi from G to some unitary group U(n). Knowing that \phi is close to being a homomorphism (i.e. unitary representation), can we find a true unitary representation that is close to it? This question depends a lot on how one measures distances of matrices in U(n). D. Kazhdan proved the answer is affirmative when G is amenable and the operator norm is used to measure distances. In this work, we consider the question when using the normalized Hilbert Schmidt norm on U(n). We prove that virtually abelian groups satisfy this, and under the assumption of finite generation and residual finiteness, these are the only examples.
Tamar Bar-On (Bar-Ilan University)
Profinite completion of free profinite groups
The profinite completion of a free profinite group on infinite set of generators is a profinite group of grater rank. However, it is still unknown whether it is a free profinite group too. I am going to present some partial results regarding to this question.
Tomer Bauer (Bar Ilan University)
On Ideal Zeta Functions of Free Nilpotent Lie Rings of Class 2
The normal subgroup zeta function of a finitely generated group $G$ is defined to be the Dirichlet series whose $n$-th coefficient is the number of normal subgroups of index $n$.
In a seminal paper, Grunewald, Segal and Smith (1988) established, among other results, properties of the local factors of these zeta functions for nilpotent groups. It is more convenient to compute the ideal zeta function of an associated Lie ring $L$. This ideal zeta function enumerates the two-sided ideals of finite additive index of $L$, and almost all of its local factors are equal to the local factors for $G$.
We compute the local factors for a family of extensions of free nilpotent Lie rings of class $2$. These results support a conjecture that in the cases we study a functional equation holds for all primes, and a SageMath implementation provides an explicit computational evidence for that.
Snir Ben Ovadia (Weizmann Institute)
Hyperbolic SRB measures and the leaf condition
Hyperbolic SRB (after Sinai, Ruelle, and Bowen) measures are an important object in mathematical physics, as a substitute to the Liouville measure for non-closed systems. The question of their existence is still a major open problem. We give a sufficient and necessary condition for their existence in the form of a leaf condition- a geometric condition.
Ido Ben-Shaul (Tel Aviv University)
Solving the functional Eigen-Problem using Neural Networks
In this work, we explore the ability of NN (Neural Networks) to serve as a tool for finding eigenpairs of partial differential equations. The question we aim to address is whether, given a self-adjoint operator, we can learn what are the eigenfunctions, and their corresponding eigenvalues. The topic of solving the eigenproblem is widely discussed in Image Processing, as many image processing algorithms can be thought of as such operators. We suggest an alternative to classical numeric methods of finding eigenpairs, which may potentially be more robust and have the ability to solve more complex problems. In this work, we focus on simple problems for which the analytical solution is known. This way, we are able to make initial steps in discovering the capabilities and shortcomings of DNN (Deep Neural Networks) in the given setting.
Gal Dor (Tel Aviv University)
A Category of Automorphic Representations for GL(2)
Consider the representation theory of a group G. We often want to classify its irreducible representations. Moreover, we would like to understand the decompositions of various spaces of functions on G-spaces X. For the first kind of question, we have a powerful tool: category theory. Studying the category of not necessarily irreducible representations of G tells us a lot about the irreducible ones. For the second kind of question, there is no immediately clear category of representations that are “associated” to a specific G-space X. The goal of this talk is to very briefly showcase some recent constructions regarding the category of representations of G=GL(2). This will be done through explaining how they can be used to construct such categories of representations associated to G-spaces. The methods used appear to be meaningful in the finite field, p-adic and automorphic cases, but this talk will focus on the more elementary settings.
Yoel Grinshpon (Hebrew University)
Spectral fluctuations for Schrodinger operators with a random decaying potential
Linear statistics provide a tool for the analysis of fluctuations of random measures and have been extensively studied for various models in random matrix theory. In this talk we discuss the application of the same philosophy to the analysis of the finite volume eigenvalue counting measure of one dimensional Schroedinger operators with a random decaying potential. In this process, we use methods of path counting as well. This is joint work with Jonathan Breuer and Moshe White.
Ori Sporta Katz (Weizmann Institute)
A phenomenological model for density-driven oceanic flows
In some oceanic systems, the strength of a flow connecting two regions that differ in their average density, scales like the density difference between them, as determined by heat and salinity tracers. Both of these tracers are thus semi-active, their distribution affecting the density-driven component of the overall flow, which in turn affects their distribution.
In this talk, we will present the construction of a novel model for this interplay: we consider a pair of coupled advection-diffusion equations for heat and salinity in a closed 3D basin, that are coupled by averages through their common advection field. The novelty and simplicity of the model comes from the fact that we do not solve the Navier-Stokes equations in order to obtain the advection field. Rather, we consider a flow with a predetermined form that could be anything, e.g. determined from toy models, observations, or numerical simulations, and a strength that is determined dynamically by the average density difference between two regions of the basin. In oceanographic research, the resulting non-linear integral-PDE set can be used to isolate the effect of different tracer sources and boundary conditions, and their variability, on the overall flow strength and its stability in realistic oceanic systems.
In addition to presenting the model’s construction and usefulness, we will briefly discuss our main analytical result: we have proven that a large class of integral-PDEs, to which our model belongs, are well-posed in the sense of Hadamard: given initial and boundary conditions that have sufficient regularity properties, unique solutions exist globally for all times, and exhibit a continuous dependence on initial conditions.
Noam Kimmel (Tel Aviv University)
Covariance of error terms related to the Dirichlet eigenvalue problem
We consider the Dirichlet eigenvalues of a planar domain $\Omega$. These are the eigenvalues of the Laplacian operator on $\Omega$ with a fixed 0 boundary condition. Weyl's conjecture gives an estimation for the number of eigenvalues up to $X$, with some error term $E(X)$. There has been a lot of work dedicated to better understand this error term. For general domains the problem is very difficult, but there are some domains for which $E(X)$ is better understood. For example, when the eigenvalues can be explicitly calculated. These domains include rectangles, special triangles and ellipses. For these domains, the problem of understanding $E(X)$ turns out to be related to another problem - counting lattice points inside expanding ovals. In the talk we investigate the covariance between the error terms of counting lattice points inside different expanding ovals. For ellipses, we give a more precise formula. These cases are of particular interest since these cases come from counting Dirichlet eigenvalues of rectangles and special triangles.
Ohad Klein (Bar Ilan University)
On the distribution of Randomly Signed Sums and Tomaszewski’s Conjecture
A Rademacher sum X is a random variable characterized by real numbers a_1, ..., a_n, and is equal to
X = a_1 x_1 + ... + a_n x_n, where x_1, ..., x_n are independent signs (uniformly selected from {-1, 1}).
Special type of Rademacher sums are normally distributed variables X, which satisfy Pr[ |X| <= sqrt Var(X) ] ~ 0.68.
Moreover, various results demonstrate that all Rademacher sums behave "somewhat" like normally distributed variables.
In this framework, Bogusław Tomaszewski asked in 1986:
Is it true that all Rademacher sums X satisfy Pr[ |X| <= sqrt Var(X) ] >= 1/2 ?
We present several tools which allow us to analyze distributions of Rademacher sums, and to answer Tomaszewski's conjecture.
Joint work with Nathan Keller.
Eden Kuperwasser (Tel Aviv University)
A sharp threshold for Schur's theorem in the integers modulo a prime
We say that a subset A of \mathbb{Z}_N is r-Schur if any coloring of the elements of A with r colors admits a monochromatic sum, i.e. elements x,y, and z satisfying x+y=z and all having the same color. Schur's theorem, a basic result in Ramsey Theory, states that whenever N is sufficiently large, the set \mathbb{Z}_N is itself r-Schur. In the talk we will turn to random subsets of \mathbb{Z}_N with density p, that is, the subset that remains when each element is kept independently with probability p and discarded otherwise. We establish the existence of a sharp threshold for the r-Schur property in random subsets of \mathbb{Z}_N, when N goes over the prime numbers, and for any number of colors r. Roughly speaking, this means that as N tends to infinity and p grows from zero to one, the probability that a random subset of \mathbb{Z}_N with density p is r-Schur has two phases: it is either very close to zero or very close to one, with a very small window between those two states.
Our proof relies on Friedgut's criterion for sharp thresholds and the Container Lemma of Saxton and Thomason, and also of Balogh, Morris, and Samotij.
Elyasheev Leibtag (Weizmann Institute)
Homomorphic images of algebraic groups
A topological group with the property that the image of the group under any continuous homomorphism is closed, is sealed (or h-closed). Knowing that a topological group is sealed ensures that its topological properties are not deformed by a continuous homomorphism into another space. Being sealed is a non trivial property on groups, the additive group of the real line for example is not sealed. We show a criterion for sealedness on a the class of algebraic groups over local fields.
Etai Leumi (Tel Aviv University)
LCM Problem For Function Fields
When studying the distribution of prime numbers, Chebychev had the idea of estimating the least common multiple of the first n integers. This was the first important step towards the prime number theorem, and in fact the asymptotic estimate log[lcm(1,...,N)] ~ N is equivalent to the prime number theorem. Later, the l.c.m. problem was generalized to the study of the least common multiple of a polynomial sequence, and in 2011 Cilleruelo conjectured that for any irreducible polynomial f with integer coefficients and degree d > 1, the following estimate holds:
log lcm(f(1),...,f(N)) ~ (d-1)Nlog N , as N tends to infinity.
Cilleruelo proved this estimate for quadratic polynomials, but if deg f = d > 2 the conjecture is still open. Though a good upper bound is known and a lower bound of the correct magnitude, even a single example is yet to be found.
In our work we study the analogous l.c.m. problem for function fields, where for some "special" set of polynomials we proved Cileruello's analogous conjecture.
Alan Lew (Technion)
Representability and boxicity of simplicial complexes
An interval graph is the intersection graph of a family of intervals in the real line. Motivated by problems in ecology, Roberts defined the boxicity of a graph G to be the minimal k such that G can be written as the intersection of k interval graphs.
A natural higher dimensional generalization of interval graphs is the class d-representable complexes. These are simplicial complexes that carry the information on the intersection patterns of a family of convex sets in R^d. We define the d-boxicity of a simplicial complex X to be the minimal k such that X can be written as the intersection of k d-representable complexes.
A classical result of Roberts, later rediscovered by Witsenhausen, asserts that the boxicity of a graph with n vertices is at most n/2. Our main result is the following high dimensional extension of Roberts' theorem: Let X be a simplicial complex on n vertices with minimal non-faces of dimension at most d. Then, the d-boxicity of X is at most (n choose d)/(d+1).
Examples based on Steiner systems show that our result is sharp. The proofs combine geometric and topological ideas.
Matan Seidel (Tel Aviv University)
Random Walks on Circle Packings
A circle packing is a canonical way of representing a planar graph. There is a deep connection between the geometry of the circle packing and the probabilistic property of recurrence/transience of the simple random walk on the underlying graph, as shown in the famous He-Schramm Theorem. The removal of one of the Theorem's assumptions - that of bounded degrees - can cause the theorem to fail. However, by using certain natural weights that arise from the circle packing for a weighted random walk, (at least) one of the directions of the He-Schramm Theorem remains true. Joint work with Ori Gurel-Gurevich.
Michael Simkin (Hebrew University)
A randomized construction of high girth regular graphs
We describe a new random greedy algorithm for generating regular graphs of high girth: Let k > 2 and 0 < c < 1 be fixed. Let n be even and set g = c \log_{k-1}(n). Begin with a Hamilton cycle G on n vertices. As long as the smallest degree \delta (G)<k, choose, uniformly at random, two vertices u,v in V(G) of degree \delta(G) whose distance is at least g-1. If there are no such vertex pairs, abort. Otherwise, add the edge uv to E(G).
We show that with high probability this algorithm yields a k-regular graph with girth at least g. Our analysis also implies that there are (\Omega (n))^{kn/2} k-regular n-vertex graphs with girth at least g.
Raz Slutsky (Weizmann Institute)
On the size of generating sets of lattices in Lie groups
We show that the number of generators of certain discrete groups, namely, lattices, is bounded by their co-volume. To do so, we will introduce some basic notions like locally symmetric spaces and discuss some beautiful results which uncover the surprising link between the volume of certain manifolds and their complexity, be it topological, algebraic or number-theoretic. This is based on joint work with Tsachik Gelander.