# Techmath - Math Seminars in Israel

*Abstract:*

It is known that the Fourier transform of a measure which vanishes onIt is known that the Fourier transform of a measure which vanishes on [-a,a] must have asymptotically at least a/pi zeroes per unit interval. One way to quantify this further is using a probabilistic model: Let f be a Gaussian stationary process on R whose spectral measure vanishes on [-a,a]. What is the probability that it has no zeroes on an interval of length L? Our main result shows that this probability is at most e^{-c a^2 L^2}, where c>0 is an absolute constant. This settles a question which was open for a while in the theory of Gaussian processes.I will explain how to translate the probabilistic problem to a problem of minimizing weighted L^2 norms of polynomials against the spectral measure, and how we solve it using tools from harmonic and complex analysis. Time permitting, I will discuss lower bounds. Based on a joint work with Ohad Feldheim, Benjamin Jaye, Fedor Nazarov and Shahaf Nitzan (arXiv:1801.10392).

*Abstract:*

I will discuss some recent results on minimal actions of general countable groups. In particular I will describe a new property of such minimal actions called the DJ property which is defined in terms of the notion of disjointness of actions and explain how it is related to an old question of Furstenberg on the algebra spanned by the minimal functions on a group. All concepts above will be explained.

*Abstract:*

The octopus lemma states that certain operators on the symmetric group are positive semi-definite. Its original application was to resolve a long-standing conjecture of Aldous related to the spectral gap of interacting particle systems. Since then it has found other applications. We will survey this new topic, perhaps some proofs will be involved.

*Abstract:*

The octopus lemma states that certain operators on the symmetric group are positive semi-definite. Its original application was to resolve a long-standing conjecture of Aldous related to the spectral gap of interacting particle systems. Since then it has found other applications. We will survey this new topic, perhaps some proofs will be involved.

*Abstract:*

We will discuss convolution semigroups of states on locally compact quantum groups. They generalize the families of distributions of L\'evy processes from probability. We are particularly interested in semigroups that are symmetric in a suitable sense. These are proved to be in one-to-one correspondence with KMS-symmetric Markov semigroups on the $L^\infty$ algebra that satisfy a natural commutation condition, as well as with non-commutative Dirichlet forms on the $L^2$ space that satisfy a natural translation invariance condition. This Dirichlet forms machinery turns out to be a powerful tool for analyzing convolution semigroups as well as proving their existence. We will use it to derive geometric characterizations of the Haagerup Property and of Property (T) for locally compact quantum groups, unifying and extending earlier partial results. Based on joint work with Adam Skalski.

*Abstract:*

Mathematical epidemiology uses modelling to study the spread of contagious diseases in a population, in order to understand the underlying mechanisms and aid public health planning. In recent years there is growing interest in applying similar models to the study of `social contagion': the spread of ideas and behaviors. It is of great interest is to consider the ways in which social contagion differs from biological contagion at the individual level, and to use mathematical modelling to understand the population-level consequences of these differences. In this talk I will present simple `two-stage' contagion models motivated by social-science literature, and study their dynamics. It turns out that these models give rise to some interesting and non-intuitive nonlinear phenomena which do not arise in the `classical' models of mathematical epidemiology, and which might have relevance to understanding some real-world observations.

*Abstract:*

A main goal of geometric group theory is to understand finitely generated groups up to a coarse equivalence (quasi-isometry) of their Cayley graphs. Right-angled Coxeter groups, in particular, are important classical objects that have been unexpectedly linked to the theory of hyperbolic 3-manifolds through recent results, including those of Agol and Wise. I will give a brief background of what is currently known regarding the quasi-isometric classification of right-angled Coxeter groups. I will then describe a new computable quasi-isometry invariant, the hypergraph index, and its relation to other invariants such as divergence and thickness.

*Abstract:*

T.B.A.

*Abstract:*

**Advisor: **Emanuel Milman

**Abstract: **We establish new sharp inequalities of Poincare or log-Sobolev type, on weighted Riemannian manifolds whose (generalized) Ricci curvature is bounded from below. To this end we establish a general method which complements the 'localization' theorem which has recently been established by B. Klartag. Klartag's theorem is based on optimal transport techniques, leading to a disintegration of the manifold measure into marginal measures supported on geodesics of the manifold. This leads to a reduction of the problem of proving a n-dimensional inequality into an optimization problem over a class of measures with 1-dimensional supports. Our method is based on functional analytic techniques, and leads to a further reduction of the optimization problem into a simpler problem over a sub-class of model-space measures. By employing ad-hoc analytical techniques we solve the optimization problems associated with the Poincare and the log-Sobolev inequalities. Quiet unexpectedly the solution to the problem of characterizing the sharp Poincare constant reveals anomalous behavior within a certain domain of the generalized-dimension parameter, hinting on a new phenomena.

*Announcement:*

Title of lectures: **The** **Virtual fibering theorem**.

Lecture 1: Monday, November 5, 2018 at 15:30 (about the statement, history and more)

Lecture 2: Wednesday, November 7, 2018 at 15:30

Lecture 3: Thursday, November 8, 2018 at 15:30

The other two lectures will be about the proof, following Prof. Friedl's paper with Takahiro Kitayama.

** Abstract**: In 2008 Agol showed that a 3-manifold with a certain condition on its fundamental group is virtually fibered, i.e. it has a finite covering that is a surface bundle over the circle. A few years later it was shown by Agol and Wise that the fundamental groups of most 3-manifold satisfy Agol's condition, i.e. most 3-manifodls are virtually fibered. We will outline a proof of Agol's theorem following an approach taken by myself and Kitayama.

Light refreshments will be given before the talks in the Faculty lounge on the 8th floor.

*Abstract:*

In 2008 Agol showed that a 3-manifold with a certain condition on its fundamental group is virtually fibered, i.e. it has a finite covering that is a surface bundle over the circle. A few years later it was shown by Agol and Wise that the fundamental groups of most 3-manifold satisfy Agol's condition, i.e. most 3-manifodls are virtually fibered. We will outline a proof of Agol's theorem following an approach taken by myself and Kitayama.

*Abstract:*

T.B.A.

*Abstract:*

The Boltzmann equation without angular cutoff is considered when the initial data is a perturbation of a global Maxwellian with algebraic decay in the velocity variable. Global solution is proved by combining the analysis in moment propagation, spectrum of the linearized operator and the smoothing effect of the linearized operator when initial data in Sobolev space with negative index.

This is a joint work with Ricardo Alonso, Yoshinori Morimoto and Weiran Sun.

*Announcement:*

Title of lectures: **The ****Virtual fibering theorem**.

Lecture 1: Monday, November 2, 2018 at 15:30 (about the statement, history and more)

Lecture 2: Wednesday, November 4, 2018 at 15:30

Lecture 3: Thursday, November 5, 2018 at 15:30

The other two lectures will be about the proof, following Prof. Friedl's paper with Takahiro Kitayama.

**Abstract**: In 2008 Agol showed that a 3-manifold with a certain condition on its fundamental group is virtually fibered, i.e. it has a finite covering that is a surface bundle over the circle. A few years later it was shown by Agol and Wise that the fundamental groups of most 3-manifold satisfy Agol's condition, i.e. most 3-manifodls are virtually fibered. We will outline a proof of Agol's theorem following an approach taken by myself and Kitayama.

Light refreshments will be given before the talks in the Faculty lounge on the 8th floor.

*Abstract:*

The 2018 Newton Conference is designed to illuminate Isaac Newton's legacy, including his scientific doctrine from the perspective of contemporary science, his broad and profound religious outlook and his place in the development of human thought. Some less known aspects of Newton's thought will be presented, such as his interest in chemistry, his connections with the Hebrew language, and his outlook on the historical role of the Jewish people. The speakers represent diverse scientific disciplines, including Physics, Mathematics, History and Philosophy. For registration please browse: https://www.hit.ac.il/events/newton2018 Adir Pridor, Conference Chair

*Announcement:*

Title of lectures: **The ****Virtual fibering theorem**.

Lecture 1: Monday, November 2, 2018 at 15:30 (about the statement, history and more)

Lecture 2: Wednesday, November 4, 2018 at 15:30

Lecture 3: Thursday, November 5, 2018 at 15:30

The other two lectures will be about the proof, following Prof. Friedl's paper with Takahiro Kitayama.

**Abstract**: In 2008 Agol showed that a 3-manifold with a certain condition on its fundamental group is virtually fibered, i.e. it has a finite covering that is a surface bundle over the circle. A few years later it was shown by Agol and Wise that the fundamental groups of most 3-manifold satisfy Agol's condition, i.e. most 3-manifodls are virtually fibered. We will outline a proof of Agol's theorem following an approach taken by myself and Kitayama.

Light refreshments will be given before the talks in the Faculty lounge on the 8th floor.

*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.

*Announcement:*

Title: **Geometric structures in nonholonomic mechanics.**

Lecture 1: Monday, November 12, 2018 at 15:30

Lecture 2: Wednesday, November 14, 2018 at 15:30

Lecture 3: Thursday, November 15, 2018 at 15:30

*Abstract:*

T.B.A.

*Announcement:*

Title: **Geometric structures in nonholonomic mechanics.**

Lecture 1: Monday, November 12, 2018 at 15:30

Lecture 2: Wednesday, November 14, 2018 at 15:30

Lecture 3: Thursday, November 15, 2018 at 15:30

*Announcement:*

Title: **Geometric structures in nonholonomic mechanics.**

Lecture 1: Monday, November 12, 2018 at 15:30

Lecture 2: Wednesday, November 14, 2018 at 15:30

Lecture 3: Thursday, November 15, 2018 at 15:30

*Abstract:*

T.B.A.

*Abstract:*

TBA

*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:*