קולוקוויום Ical Atom

A family of problems in diophantine geometry has the following form: We fix a collection of ”special“ algebraic varieties where the 0-dimensional are called ”special points“. In general, if V is a sepcial variety then the special points are Zariski dense in V, and one would like to prove the converse: If V is an irreducible algebraic variety and the special points are Zariski dense in V then V itself is special. Particular cases of the above are the Manin-Mumford conjecture, the Mordell-Lang conjecture, and others. In 1990‘s Hrushovski showed how model theoretic techniques could be applied to solve certain such problems. In 2008 Pila and Zannier developed a different framework. which allows to apply model theory and especially the theory of o-minimal structures, in order to tackle questions of this nature over the complex numbers. Pila himself used these methods to prove some open cases of the Andre-Oort conjecture and since then there was an influx of articles which use similar techniques. At the heart of the Pila-Zannier method lies a theorem of Pila and Wilkie on rational points on definable sets in o-minimal structures. In this survey-like talk I will describe the basic ingredients of the Pila-Zannier method and its applications, in one or two simple cases.

It has been suggested that every good mathematical pattern ought to be manifest in nature. This talk surveys the speaker‘s work with A.Y. Carmi on a manifestation of low-dimensional topology‘s diagrammatic calculus of tangles in the theory of information fusion networks e.g. sensor networks and complex systems. It turns out that various sets of information-theoretic quantities such as sets of entropies naturally admit an algebraic structure called a quandle. The topological toolbox is completely different from the toolbox of statistical inference. Our diagrams are similar but not identical to classical tangle diagrams, and our work involves studying their diagrammatic algebra. The talk is planned to be accessible to a general mathematical audience.

Is there a point set $Y$ in $R^d$, and $C>0$, such that every convex set of volume 1 contains at least one point of $Y$ and at most $C$? This discrete geometry problem was posed by Gowers in 2000, and it is a special case of an open problem posed by Danzer in 1965. I will present two proofs that answers Gowers‘ question with a NO. The first approach is dynamical; we introduce a dynamical system and classify its minimal subsystems. This classification in particular yields the negative answer to Gowers‘ question. The second proof is direct and it has nice applications in combinatorics. [This is a joint work with Omri Solan and Barak Weiss].

Within the field of dynamics, entropy is a real number which measures the amount of chaos or complexity in a dynamical system. Entropy was first introduced for actions of the integers by Kolmogorov in 1958, and it led to huge advances in the field. During the 1970‘s and 80‘s entropy theory was largely extended to actions of amenable groups (such as abelian groups and solvable groups). In 2008, Lewis Bowen made a dramatic breakthrough by extending the notion of entropy to actions of sofic groups (such as linear groups and free groups). A new chapter in entropy theory is now unfolding. In this talk, I will discuss some of the history of classical entropy theory and then discuss some recent breakthroughs.

The category of spectra is one of the most important constructions in modern algebraic topology. It appears naturally in the study of cobordism classes of manifolds, aa the classification of generalized cohomology theories and also can be thought of as a homotopical analog of abelian groups. In the last years J. Lurie and other authors began redeveloping algebra with Spectra taking the role of abelian groups. Analogs of commutative and non-commutative rings , modules, lie-algebras and many others developed, and many theorems where proved that are analogs of the classical case. I’ll describe some of the tools and the ideas appearing in this constructions and sketch some applications. The same way one can do algebra in different characteristics (a prime $p$ or zero) which appear as points of the scheme $Spec(\mathbb{Z})$, One can find all possible “characteristics” of Spectra. Those are classified by a pair $(p, n)$ where $p$ is a prime and $n$ is a natural number called a height. Working in a given characteristic $(n, p)$ one obtains what is called the $K(n)$-local category at height $n$ and prime $p$. It is a well known observation that for a given height $n$ certain “special” phenomena happen only for small enough primes. Further, in some sense, the categories $C_{p,n}$ become more regular and algebraic as $p$ goes to infinity for a fixed $n$. The goal of this talk is to make this intuition precise.

Given an infinite sequence of mathematical structures, logicians have a method to construct a limiting one by using ultra-product. We shall define a notion of “ultraproduct of categories” and then describe a collection of categories $D_{n,p}$ which will serve as algebro-geometric analogs of the $K(n)$-local category at the prime $p$.

Then for a fixed height $n$ we prove:

If time permits we shall describe our ongoing attempts to use these methods to get a version of the $K(n)$-local category corresponding to formal Drinfeld modules (instead of formal groups).

This is a joint project with N. Stapleton and T. Barthel.

I will overview recent developments in harmonic analysis on spherical spaces. This class of spaces includes the class of symmetric spaces and recently became relevant for variations on the Langlands program
due to its relation to periods of automorphic forms.

I will introduce some basic geometric properties of these spaces and then focus on two issues: the decay of generalized matrix coefficients on real spherical spaces and the regularity of generalized (spherical) characters.

After reviewing the necessary background, we will discuss some of our results and elaborate on few applications of these results to problems originating in arithmetic. In particular we will discuss some new results on the problem of counting lattice points in the realm of real spherical spaces.

The main results include quantitative generalizations of Howe-Moore phenomena in the real case and a qualitative generalizations of Howe/Harish-Chandra character expansions in the p-adic case. Our techniques relies on systematic usage of the action of Bernstein center in the p-adic case and the theory of ODE in the real case (using the z-finite action of the center of the universal enveloping algebra).

The lecture is based on recent works with various collaborators (A. Aizenbud, D. Gourevtich, B. Kroetz, F. Knop, H. Schlichtkrull).

Cellular interactions can be modeled as complex dynamical systems represented by weighted graphs. The functionality of such networks, including measures of robustness, reliability, performance, and efficiency, are intrinsically tied to the topology and geometry of the underlying graph. Utilizing recently proposed geometric notions of curvature on weighted graphs, we investigate the features of gene co-expression networks derived from large-scale genomic studies of cancer. We find that the curvature of these networks reliably distinguishes between cancer and normal samples, with cancer networks exhibiting higher curvature than their normal counterparts. We establish a quantitative relationship between our findings and prior investigations of network entropy. Furthermore, we demonstrate how our approach yields additional, non-trivial pair-wise (i.e. gene-gene) interactions which may be disrupted in cancer samples. The mathematical formulation of our approach yields an exact solution to calculating pair-wise changes in curvature which was computationally infeasible using prior methods. As such, our findings lay the foundation for an analytical approach to studying complex biological networks.

The study of generic dynamical systems has lead to some important discoveries in the past. In this talk I will explain the topological notion of a ”generic dynamical system“, survey some historical results and describe a new one, regarding the entropy of symbolic dynamics on amenable groups. Based on joint work with Joshua Frisch.

The algebraic theory of quadratic forms connects fascinating topics, from Hurwitz‘ theorem and Hilbert‘s 17th problem, to the theorems of Voevodsky-Orlov-Vishik and Voevodsky-Rost on the Witt ring and Milnor‘s K-theory, and beyond.

After proving to the audience that this is a beautiful subject, I will try to explain why and how everything is harder in characteristic 2.

I will describe the effects of linkage of quadratic Pfister forms, in particularly in characteristic 2, where one has to distinguish between left- and right-linkage. I will describe a potential invariant (which fails), and construct sets of forms that should be linked, but aren't.

I will talk about the following questions:

1) Given a system of polynomial equations with integer coefficients, how many solutions does it have in the ring Z/N?

​2)​ Given a polynomial map f:R^a–>R^b and a smooth, compactly supported measure m on R^a, does the push-forward of m by f have bounded density?

3) Given a lattice in a higher rank Lie group (say, SL(n,Z) for n>2). How many d-dimensional representations does it ​have?

I will explain how these questions are related to the singularities of certain varieties. Along the way, I'll talk about canonical singularities, random commutators, and the moduli space of local systems.

This is a joint work with Rami Aizenbud

The problem of finding the integral or rational solutions of polynomial equations is one of the oldest in mathematics. The simplest non-trivial case is that of one equation in two variables. While we know that for a sufficiently high degree equation there are only a finite number of solutions, an effective method for finding these solutions in general is not known to exist.

Quite recently, a far reaching program for finding the solutions using fundamental groups and p-adic numbers was invented by Minhyong Kim. I will explain some of the background, and also report on some work with Jennifer Balakrishnan and Steffen Muller using similar ideas to solve a new class of equations. I will give some concrete examples.