# Techmath - Math Seminars in Israel

*Abstract:*

Abstract: Where extremal combinatorialists wish to optimise a discrete parameter over a family of large objects, probabilistic combinatorialists study the statistical behaviour of a randomly chosen object in such a family. In the context of representable matroids (i.e. the columns of a matrix) over $\mathbb{F}_2$, one well-studied distribution is to fix a small $k$ and large $m$ and randomly generate $m$ columns with $k$ 1’s. Indeed, when $k = 2$, this is the graphic matroid of the Erdos-Renyi random graph $G_{n,m}$. We turn back to the simplest corresponding extremal question in this setting. What is the maximum number of weight-$k$ columns a matrix of rank $\leq n$ can have? We show that, once $n \geq N_k$, one cannot do much better than taking only $n$ rows and all available weight-$k$ columns. This partially confirms a conjecture of Ahlswede, Aydinian and Khachatrian, who believe one can take $N_k=2k$. This is joint work with Wesley Pegden.

*Announcement:*

**Title of talks: Geometric structures in nonholonomic mechanics**

**Lecture 1: **Monday, December 17, 2018 at 15:30

**Lecture 2: **Wednesday, December 19, 2018 at 15:30

**Lecture 2: **Thursday, December 20, 2018 at 15:30

**Abstract**: Nonholonomic mechanics concerns with mechanical systems whose velocity is constrained. If these velocity constraints are linear, they define k-planes at every point of the configuration space of the system. In more complex situations further constraints appear: the movement of the system not only has to be tangent to these k-planes, but must obey conditions in which tangent vectors to the trajectories have constant length, or satisfy other, in general nonlinear, relations. This equips kinematics of nonholonomic mechanical systems with various geometric structures. These are: vector distributions on manifolds, their symmetry groups, differential invariants, associated exterior differential systems, Cartan connections, etc.

In the lectures we will discuss these geometric structures in simple examples of existing (or possible to exist) mechanical systems. We will concentrate on systems whose kinematics is described by a low dimensional parabolic geometry i.e. a geometry modeled on a homogeneous space G/P, with G being a simple Lie group, and P being its parabolic subgroup. The considered systems will include a movement of ice skaters on an ice rink, rolling without slipping or twisting of rigid bodies, movements of snakes and ants, and even movements of flying saucers. Geometric relations between these exemplary nonholonomic systems will be revealed. An appearance of the simple exceptional Lie group G2 will be a repetitive geometric phenomenon in these examples.

*Abstract:*

In ordinary algebra, characteristic zero behaves differently from characteristic p>0 partially due to the possibility to symmetrize finite group actions. In particular, given a finite dimensional group G acting on a rational vector space V, the "norm map" from the co-invariants V_G to the invariants V^G is an isomorphism (in a marked contrast to the positive characteristic case). In the chromatic world, the Morava K-theories provide an interpolation between the zero characteristic represented by rational cohomology and positive characteristic represented by F_p cohomology. A classical result of Hovey-Sadofsky-Greenlees shows that the norm map is still an ismorphism in these "intermediate characteristics". A subsequent work of Hopkins and Lurie vastly generalises this result and puts it in the context of a new formalism of "higher semiadditivity" (a.k.a. "amidexterity"). I will describe a joint work with Tomer Schlank and Shachar Carmeli in which we generalize the results of Hopkins-Lurie and extend them to the telescopic localizations and draw some consequences (along the way, we obtain a new and more conceptual proof for their original result).

*Abstract:*

We will start by discussing a special class of automorphisms of a Poisson point process on an infinite measure space called Poisson suspensions and explain that the space of Poisson suspensions is a Polish group. After this we will explain an if and only if criteria for existence of an absolutely continuous invariant measure and show that a group has Kazhdan's property T if and only if all of its actions as Poisson suspensions are not properly nonsingular. If time permits we will show how one can use the previous construction to obtain a simple proof of a result of Bowen, Hartman and Tamuz that a group does not have Kazhdan property T if and only if it does not have a Furstenberg entropy gap in the sense of Nevo.

*Announcement:*

**Title of talks: Geometric structures in nonholonomic mechanics**

**Lecture 1: **Monday, December 17, 2018 at 15:30

**Lecture 2: **Wednesday, December 19, 2018 at 15:30

**Lecture 2: **Thursday, December 20, 2018 at 15:30

**Abstract**: Nonholonomic mechanics concerns with mechanical systems whose velocity is constrained. If these velocity constraints are linear, they define k-planes at every point of the configuration space of the system. In more complex situations further constraints appear: the movement of the system not only has to be tangent to these k-planes, but must obey conditions in which tangent vectors to the trajectories have constant length, or satisfy other, in general nonlinear, relations. This equips kinematics of nonholonomic mechanical systems with various geometric structures. These are: vector distributions on manifolds, their symmetry groups, differential invariants, associated exterior differential systems, Cartan connections, etc.

In the lectures we will discuss these geometric structures in simple examples of existing (or possible to exist) mechanical systems. We will concentrate on systems whose kinematics is described by a low dimensional parabolic geometry i.e. a geometry modeled on a homogeneous space G/P, with G being a simple Lie group, and P being its parabolic subgroup. The considered systems will include a movement of ice skaters on an ice rink, rolling without slipping or twisting of rigid bodies, movements of snakes and ants, and even movements of flying saucers. Geometric relations between these exemplary nonholonomic systems will be revealed. An appearance of the simple exceptional Lie group G2 will be a repetitive geometric phenomenon in these examples.

*Abstract:*

Abstract: An important and challenging class of two-stage linear optimization problems are those without relative-complete recourse, wherein there exist first-stage decisions and realizations of the uncertainty for which there are no feasible second-stage decisions. Previous data-driven methods for these problems, such as the sample average approximation (SAA), are asymptotically optimal but have prohibitively poor performance with respect to out-of-sample feasibility. In this talk, we present a data-driven method for two-stage linear optimization problems without relative-complete recourse which combines (i) strong out-of-sample feasibility guarantees and (ii) general asymptotic optimality. Our method employs a simple robustification of the data combined with a scenario-wise approximation. A key contribution of this work is the development of novel geometric insights, which we use to show that the proposed approximation is asymptotically optimal. We demonstrate the benefit of using this method in practice through numerical experiments.

*Abstract:*

We will present a combinatorial gadget satisfying certain symmetry properties and show how it allows us to study the normal subgroup growth of a rather large class of nilpotent groups, including base extensions of free nilpotent groups of class two and of amalgamations of base extensions of Heisenberg groups. These are results of joint work with Angela Carnevale and Christopher Voll. The talk will assume no prior knowledge of the subject.

*Abstract:*

Two important results in Boolean analysis highlight the role of majorityTwo important results in Boolean analysis highlight the role of majority functions in the theory of noise stability. Benjamini, Kalai, and Schramm (1999) showed that a boolean monotone function is noise-stable if and only if it is correlated with a weighted majority. Mossel, O’Donnell, and Oleszkiewicz (2010) showed that simple majorities asymptotically maximize noise stability among low influence functions. In the talk, we will discuss and review progress from the last decade in our understanding of the interplay between Majorities and noise-stability. In particular, we will discuss a generalization of the BKS theorem to non-monotone functions, stronger and more robust versions of Majority is Stablest and the Plurality is Stablest conjecture. We will also discuss what these results imply for voting.

*Abstract:*

The hydrogen atom system is one of the most thoroughly studied examples of a quantum mechanical system. It can be fully solved, and the main reason why is its (hidden) symmetry. In this talk I shall explain how the symmetries of the Schrödinger equation for the hydrogen atom, both visible and hidden, give rise to an example in the recently developed theory of algebraic families of Harish-Chandra modules. I will show how the algebraic structure of these symmetries completely determines the spectrum of the Schrödinger operator and sheds new light on the quantum nature of the system.

No prior knowledge on quantum mechanics or representation theory will be assumed.

*Abstract:*

A point scatterer, or the Laplacian perturbed with a delta potential, is a model for studying the transition between chaos and integrability in quantum systems. The eigenfunctions of this operator consist of the Laplace eigenfunctions which vanish at the scatterer, and a set of new, perturbed eigenfunctions. We discuss the mass distribution of the new eigenfunctions of a point scatterer on a flat torus, and present some of our recent results.

*Abstract:*

In this talk I will discuss applications of geometric invariant theory to the study of Hopf algebras. The question which will be considered is the classification of Hopf 2-cocycles on a given finite dimensional Hopf algebra. I will explain why this is in fact a geometric problem, and how geometric invariant theory can helpus here. I will give some examples arising from Bosonizations of nonabelian group algebras and dual group algebras, and present some new family of Hopf algebras arising from such cocycle deformations. If time permits, I will also explain the connection with the universal coefficients theorem, and how some of these invariants relate to surfaces.

*Abstract:*

We all know what is an algebraically closed field and that any field can be embedded into an algebraically closed field. But what happens if multiplication is not commutative? In my talk I'll suggest a definition of an algebraically closed skew field, give an example of such a skew field, and show that not every skew field can be embedded into an algebraically closed one. It is still unknown whether an algebraically closed skew field exists in the finite characteristic case!

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*Announcement:*

Title & abstract of lectures: **TBA**.

Lecture 1: Monday, March 25, 2019 at 15:30

Lecture 2: Wednesday, March 27, 2019 at 15:30

Lecture 3: Thursday, March 28, 2019 at 15:30

*Announcement:*

Title & abstract of lectures: **TBA**.

Lecture 1: Monday, March 25, 2019 at 15:30

Lecture 2: Wednesday, March 27, 2019 at 15:30

Lecture 3: Thursday, March 28, 2019 at 15:30

*Announcement:*

Title & abstract of lectures: **TBA**.

Lecture 1: Monday, March 25, 2019 at 15:30

Lecture 2: Wednesday, March 27, 2019 at 15:30

Lecture 3: Thursday, March 28, 2019 at 15:30

*Announcement:*

**Title**: *The Double Bubble Problem*

**Abstract**: A single round soap bubble provides the least-perimeter way to enclose a given volume of air, as was proved by Schwarz in 1884. The Double Bubble Problem seeks the least-perimeter way to enclose and separate two given volumes of air. Three friends and I solved the problem in Euclidean space in 2000. In the latest chapter, Emanuel Milman and Joe Neeman recently solved the problem in Gauss space (Euclidean space with Gaussian density). The history includes results in various spaces and dimensions, some by undergraduates. Many open questions remain.

*Announcement:*

**Title**: *The Isoperimetric Problem*

**Abstract**: The isoperimetric problem seeks the least-perimeter way to enclose a given volume. Although the answer is well known to be the round sphere in Euclidean and some other spaces, many fascinating open questions remain. Is a geodesic sphere isoperimetric in CP^2? What is the least-perimeter tile of the hyperbolic plane of prescribed area?

*Announcement:*

**Title**: *The Isoperimetric Problem in Spaces with Density*

**Abstract**: Since their appearance in Perelman's 2006 proof of the Poincaré Conjecture, there has been a flood of interest in positive weights or densities on spaces and the corresponding isoperimetric problem. The talk will include recent results and open questions.

*Announcement:*

The aim of the workshop is describing major advancements in bothalgebra and number theory, such as the discovery and advancement ofbounded gaps between twin primes, and the progress on word problems forsimple groups. This workshop day aims at explaining some of theseadvancement to graduate and advanced undergraduate students.

**List of speakers:**

- Eli Aljadeff (Technion)
- Alexei Entin (Tel-Aviv University)
- Danny Krashen (Rutgers University)

**Organizer**: Chen Meiri & Danny Neftin

**More information: TBA...**

*Announcement:*