# Techmath - Math Seminars in Israel

*Abstract:*

In Diophantine approximation we are often interested in the Lebesgue and Hausdorff measures of certain lim sup sets. In 2006, Beresnevich and Velani proved a remarkable result — the Mass Transference Principle — which allows for the transference of Lebesgue measure theoretic statements to Hausdorff measure theoretic statements for lim sup sets arising from sequences of balls in R k . Subsequently, they extended this Mass Transference Principle to the more general situation in which the lim sup sets arise from sequences of neighbourhoods of “approximating” planes. In this talk I will discuss a recent strengthening (joint with Victor Beresnevich, York, UK) of this latter result in which some potentially restrictive conditions have been removed from the original statement. This improvement gives rise to some very general statements which allow for the immediate transference of Lebesgue measure Khintchine–Groshev type statements to their Hausdorff measure analogues and, consequently, has some interesting applications in Diophantine approximation

*Abstract:*

The Kobayashi pseudometric on a complex manifold M is the maximal pseudometric such that any holomorphic map from the Poincare disk to M is distance-decreasing. Kobayashi has conjectured that this pseudometric vanishes on Calabi-Yau manifolds. It is still out of reach for general Calabi-Yau manifolds. The proof of Kobayashi conjecture for hyperkahler manifolds is based on ergodic theory. I would explain its proof in application to K3 surfaces.

*Abstract:*

**Advisors: **Dan Garber and Sabach Shoham** **

**Abstract**: Composite convex optimization problems that include a low-rank promoting term have important applications in data and imaging sciences. However, such problems are highly challenging to solve in large-scale: the low-rank promoting term prohibits efficient implementations of proximal based methods and even simple subgradient methods are very limited. On the other hand, methods which are tailored for low-rank optimization, such as conditional gradient-type methods, are usually slow. Motivated by these drawbacks, we present new algorithms and complexity results for some optimization problems in this class. At the heart of our results is the idea of using a low-rank SVD computations in every iteration. This talk is based on joint works with Dan Garber and Shoham Sabach.

*Abstract:*

Composite convex optimization problems that include a low-rank promoting term have important applications in data and imaging sciences. However, such problems are highly challenging to solve in large-scale: the low-rank promoting term prohibits efficient implementations of proximal based methods and even simple subgradient methods are very limited. On the other hand, methods which are tailored for low-rank optimization, such as conditional gradient-type methods, are usually slow. Motivated by these drawbacks, we present new algorithms and complexity results for some optimization problems in this class. At the heart of our results is the idea of using low-rank SVD computations in every iteration. This talk is based on joint works with Dan Garber and Shoham Sabach.

*Abstract:*

The algebra $H^{\infty}(\mathbb{D})$ of bounded analytic functions on the unit disc in the complex plane is a well-studied object. This algebra arises frequently in various areas of mathematics, in particular, function theory, hyperbolic geometry, and operator algebras. The classical Schwarz-Pick lemma tells us that analytic functions bounded by $1$ on the disc are necessarily contractions with respect to the Poincare metric. Furthermore, preserving metric between two points is equivalent to being an isometry and thus a Moebius map. In its other incarnation $H^{\infty}(\mathbb{D})$ is an operator algebra. The connection between the operator algebraic structure and the hyperbolic geometric of the disc was exploited to obtain interpolation and classification results.

However, operator algebras are generally noncommutative, hence it is common to think of them as quantized function algebras. The goal of my talk is to present a noncommutative generalization of this interplay between bounded functions on the disc and its geometry. To this end, I will introduce functions of noncommutative variables and explain how they arise naturally in many (even classical commutative) contexts. The focus of my talk is on bounded nc functions, that turns out to be automatically analytic. We will discuss the generalization of a classical fixed point theorem of Rudin and Herve and give an operator algebraic application.

Only basic familiarity with operators on Hilbert spaces and complex analysis is assumed.

*Abstract:*

In this talk I will describe explicitly the primitive ideals of the classical simple Lie algebras "at infinity". Remarkably, the answer is simpler than in the finite-dimensional case. An unexpected feature is that for sl(infty) and o(infty) all primitive ideals are annihilators of intergrable modules. This is not so for sp(infty) as the Joseph ideal "persists to infinity".

*Abstract:*

In this talk, I will discuss a question which originates in complex analysis but is really a problem in non-linear elliptic PDE. A finite Blaschke product is a proper holomorphic self-map of the unit disk, just like a polynomial is a proper holomorphic self-map of the complex plane. A celebrated theorem of Heins says that up to post-composition with a M\"obius transformation, a finite Blaschke product is uniquely determined by the set of its critical points. Konstantin Dyakonov suggested that it may be interesting to extend this result to infinite degree. However, one must be a little careful since infinite Blaschke products may have identical critical sets. I will show that an infinite Blaschke product is uniquely determined by its "critical structure” and describe all possible critical structures which can occur. By Liouville’s correspondence, this question is equivalent to studying nearly-maximal solutions of the Gauss curvature equation $\Delta u = e^{2u}$. This problem can then be solved using PDE techniques, using the method of sub- and super-solutions.

*Abstract:*

In 1956, Busemann and Petty posed a series of questions aboutIn 1956, Busemann and Petty posed a series of questions about symmetric convex bodies, of which only the first one has been solved. Their fifth problem asks the following. Let K be an origin symmetric convex body in the n-dimensional Euclidean space and let H_x be a hyperplane passing through the origin orthogonal to a unit direction x. Consider a hyperplane G parallel to H_x and supporting to K and let C(K,x)=vol(K\cap H_x)dist (0, G). If there exists a constant C such that for all directions x we have C(K,x)=C, does it follow that K is an ellipsoid? We give an affirmative answer to this problem for bodies sufficiently close to the Euclidean ball in the Banach-Mazur distance. This is a joint work with Maria Alfonseca, Fedor Nazarov and Vlad Yaskin.

*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 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:*

We will discuss a Hamiltonian formalism for cluster mutations using canonical (Darboux) coordinates and piecewise-Hamiltonian flows with Euler dilogarithm playing the role of the Hamiltonian. The Rogers dilogarithm then appears naturally in the dual Lagrangian picture. We will then show how the dilogarithm identity associated with a period of mutations in a cluster algebra arises from Hamiltonian/Lagrangian point of view.(Based on the joint paper with T. Nakanishi and D. Rupel.)

*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.

*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.

*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:*

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:*

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:*