# Faculty Activities

*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:*

**דר' יובל פילמוס**

הפקולטה למדעי המחשב

טכניון

**Dr. Yuval Filmus**

The Faculty of Computer Science

Technion

**MATH CLUB 19.12.18**

**לאחר ההרצאה יתקיים טקס הענקת הפרסים של התחרות ע"ש גרוסמן**

**כמה מסובך לצבוע גרפים?**

בהינתן גרף, כמה קל לבדוק האם ניתן לצבוע את קדקודיו ב-* c *צבעים כך שכל קשת מחברת בין שני קדקודים בעלי צבעים שונים?

איך אפשר להשתכנע שצביעה כזו קיימת? ואיך אפשר להשתכנע שצביעה כזו אינה קיימת?

נדון בשאלות אלה כצוהר לעולם של סיבוכיות חישובית.

**Complexity of coloring graphs**

Given an undirected graph, how easy is it to determine whether we can color its vertices with *c* colors, so that no edge is monochromatic?

How easy is it to demonstrate that such a coloring exists? How can we prove that no such coloring exists?

We use these questions as a window to the world of computational complexity.

**ההרצאה תהיה בעברית**

**The lecture will be in Hebrew**

*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.

*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:*

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.

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