# Faculty Activities

*Abstract:*

The evens and odds form a partition of the integers into arithmetic progressions. It is natural to try to describe in general how the integers can be partitioned into arithmetic progressions. For example, a classic result from the 1950's shows that if a set of arithmetic progressions partitions the integers, there must be two arithmetic progressions with the same difference. Another direction is to try to determine when such a partition is a proper refinements of another non-trivial partition.

In my talk I will give some of the more interesting results on this subject, report some (relatively) new results and present two generalizations of partitioning the integers by arithmetic progressions, namely:

1. Partitions of the integers by Beatty sequences (will be defined).

2. Coset partition of a group.

The main conjecture in thefirst topic is due to A. Fraenkel and describes all the partitionshaving distinct moduli. The main conjecture in the second topic, dueto M. Herzog and J. Schonheim, claims that in every coset partition of a group there must be two cosets of the same index.

Again, we will briefly discuss the history of these conjectures, recall some of the main results and report some new results.

Based on joint projects with Y. Ginosar, L. Margolis and J. Simpson.

*Abstract:*

Let P be a second-order, symmetric, and nonnegative elliptic operator with real coefficients defined on noncompact Riemannian manifold M, and let V be a real valued function which belongs to the class of small perturbation potentials with respect to the heat kernel of P in M. We prove that under some further assumptions (satisfying by a large classes of P and M) the positive minimal heat kernels of P −V and of P on M are equivalent. If time permits we shall show that the parabolic Martin boundary is stable under such perturbations. This is a joint work with Prof. Yehuda Pinchover.

*Abstract:*

See attached file.

*Abstract:*

**Advisor: **Roy Meshulam

**Abstract**: Attached

*Announcement:*

**דר' אריאל לייטנר**

הפקולטה למתמטיקה

טכניון

**Dr. Arielle Leitner**

The Faculty of Mathematics

Technion

**Math Club 30.5.18**

**סיור וירטואלי של גאומטריות**

נחקור איך נראה העולם בגאומטריות אחרות בעזרת משחקים והדמיות במחשב. נראה את הגאומטריות של ת׳רסטון, ונדבר על משפט הגאומטריזציה המפורסם. אם יהיה זמן, נדבר גם על גבולות בין גאומטריות.

**A Virtual Tour of Geometries**

We will explore what it is like to live in different kinds of geometric universes with the aid of computer visualizations and games. We will see some of the Thurston geometries, and discuss the famous Thurston geometrization program. Time permitting, we will discuss how some of these geometries may be deformed to others.

**ההרצאה תהיה בעברית**

**The lecture will be in Hebre**

*Abstract:*

This will be the fourth talk in Adam's lecture series.

*Abstract:*

A K\"ahler group is a group that can be realized as fundamental group of a compact K\"ahler manifold. I shall start by explaining why the question which groups are K\"ahler groups is non-trivial. Then we will address the question which functions can be realized as Dehn functions of K\"ahler groups. After explaining why K\"ahler groups can have linear, quadratic and exponential Dehn function, we show that there is a K\"ahler group with Dehn function bounded below by $n^3$ and bounded above by $n^6$. This is joint work with Romain Tessera.

*Abstract:*

T.B.A.

*Abstract:*

NOTE THE SPECIAL TIME AND PLACE.

This is not a mathematics or a physics talk but it is a talk about mathematicians for mathematicians and physicists. Too often, we think of historical mathematicians as only names assigned to theorems. With vignettes and anecdotes, I'll convince you they were also human beings and that, as the Chinese say, "May you live in interesting times" really is a curse. Among the mathematicians with vignettes are Riemann, Newton, Poincare, von Neumann, Kato, Loewner, Krein and Noether.

*Abstract:*

TBA

*Abstract:*

This talk will be devoted to probabilistic constructions appearing in statistics and geometry. I will introduce the classical notion of VC dimension and discuss how it arises naturally in several problems. One of the questions will be the so-called epsilon-approximation problem. That is, how well what you see in a small random sample approximates the real structure. In the last part of the talk, I will explain how a clever deterministic choice of points may improve standard guarantees provided by the random sampling.

*Abstract:*

I will describe joint work with Stan Alama, Lia Bronsard, Andres Contreras and Jiri Dadok giving criteria for existence and for non-existence of certain isoperimetric planar curves minimizing length with respect to a metric having conformal factor that is degenerate at two points, such that the curve encloses a specified amount of Euclidean area. These curves, appropriately parametrized, emerge as traveling waves for a bi-stable Hamiltonian system that can be viewed as a conservative model for phase transitions.

*Abstract:*

Handwriting comparison and identification, e.g. in the setting of forensics, has been widely addressed over the years. However, even in the case of modern documents, the proposed computerized solutions are quite unsatisfactory. For historical documents, such problems are worsened, due to the inscriptions’ preservation conditions. In the following lecture, we will present an attempt at addressing such a problem in the setting of First Temple Period inscriptions, stemming from the isolated military outpost of Arad (ca. 600 BCE). The solution we propose comprises: (A) Acquiring better imagery of the inscriptions using multispectral techniques; (B) Restoring characters via approximation of their composing strokes, represented as a spline-based structure, and estimated by an optimization procedure; (C) Feature extraction and distance calculation - creation of feature vectors describing various aspects of a specific character based upon its deviation from all other characters; (D) Conducting an experiment to test a null hypothesis that two given inscriptions were written by the same author. Applying this approach to the Arad corpus of inscriptions resulted in: (i) The discovery of a brand new inscription on the back side of a well known inscription (half a century after being unearthed); (ii) Empirical evidence regarding the literacy rates in the Kingdom of Judah on the eve of its destruction by Nebuchadnezzar the Babylonian king.

*Abstract:*

Abstract: Caprace and De Medts discovered that Thompson's V can be written as a group of tree almost automorphisms, allowing to embed it densely into a totally disconnected, locally compact (t.d.l.c.) group. Matui discovered that it can be written as the topological full group of the groupoid associated to a one-sided shift. Combining these, we find countably many different t.d.l.c. groups containing a dense copy of V.

*Abstract:*

In 2003, Welschinger defined invariants of real symplectic manifolds of complex dimensions 2 and 3, which are related to counts of pseudo-holomorphic disks with boundary and interior point constraints (Solomon, 2006). The problem of extending the definition to higher dimensions remained open until recently (Georgieva, 2013, and Solomon-Tukachinsky, 2016-17).

In the talk I will give some background on the problem, and describe a generalization of Welschinger's invariants to higher dimensions, with boundary and interior constraints, a.k.a. open Gromov-Witten invariants. This generalization is constructed in the language of $A_\infty$-algebras and bounding chains, where bounding chains play the role of boundary point constraints. If time permits, we will describe equations, a version of the open WDVV equations, which the resulting invariants satisfy. These equations give rise to recursive formulae that allow the computation of all invariants of $\mathbb{C}P^n$ for odd $n$.

This is joint work with Jake Solomon.

No previous knowledge of any of the objects mentioned above will be assumed.

*Abstract:*

NOTE THE SPECIAL TIME.

TheElisha Netanyahu Memorial Lectures

*Abstract:*

* Abstract: *Given n uniform points on the surface of a two-dimensional sphere, how can we partition the sphere fairly among them?

It turns out that if the given points apply a two-dimensional gravity force to the rest of the sphere, then the basins of attraction for the resulting flow yield such a partition—with exactly equal areas, no matter how the points are distributed. (See http://www.ams.org/publications/journals/notices/201705/rnoti-cvr1.pdf) Our main result is that this partition minimizes, up to a bounded factor, the average distance between points in the same cell. This has an application to almost optimal matching of n uniform blue points to n uniform red points on the sphere. I will also describe open problems regarding greedy and electrostatic matching (Joint work with Nina Holden and Alex Zhai) Another topic where local and global optimization sharply differ appears starts from the classical overhang problem: Given n blocks supported on a table, how far can they be arranged to extend beyond the edge of the table without falling off? With Paterson, Thorup, Winkler and Zwick we showed ten years ago that an overhang of order cube root of n is the best possible; a crucial element in the proof involves an optimal control problem for diffusion on a line segment and I will describe generalizations of this problem to higher dimensions (with Florescu and Racz).