# Techmath - Math Seminars in Israel

*Abstract:*

We study the existence of zeroes of mappings defined in Banach spaces. We obtain, in particular, an extension of the well-known Bolzano-Poincar\'{e}-Miranda theorem to infinite dimensional Banach spaces. We also establish a result regarding the existence of periodic solutions to differential equations posed in an arbitrary Banach space.

This is joint work with David Ariza Ruiz (Sevilla) and Jes\'{u}s Garcia Falset (Valencia).

*Abstract:*

Announcement We are happy to announce the Conference "Harmonic Analysis and PDE" in honor of Professor Vladimir Maz'ya to be held in Holon Institute of Technology, May 26-31, 2019. The meeting will provide a forum for discussions and exchange of new ideas, perspectives and recent developments in the broad field of Modern Analysis. The topics to be addressed include (but not restricted to): Complex and Harmonic Analysis, Partial Differential Equations, Operator Theory and Nonlinear Analysis and Quasiconformal Mappings and Geometry. The conference is organized by Holon Institute of Technology (Holon, Israel), Bar-Ilan University (Ramat Gan, Israel) and RUDN (Moscow, Russia). The registration fee is 150 Euro and covers the conference materials, lunches, coffee breaks, get together party and social program. For details please browse: http://golberga.faculty.hit.ac.il/HAPDE19/HAPDE19.html .

*Abstract:*

Reiamnnian manifolds of constant curvature -1 are called "hyperbolic manifolds". Their theory is rich and interesting.Covering theory tells us that their fundamental groups could be seen as subgroups of the group of isometries of "the hyperbolic space" - the unique (up to isometry) simply connected hyperbolic manifold. The latter group happens to coincide with a certain group of matrices, denoted O(n,1). It so happens that a compact (or finite volume) hyperbolic manifolds could be recovered from its fundamental group - this is Mostow's theorem. In fact, such manifolds are rather rare - there are only countably many of them, and surprisingly often their existence is related to some phenomena of arithmeticity: the matrices involved have only integer values (or values in an integer ring of a number field). When this is the case, the corresponding manifold is said to be "arithmetic". In my talk I will give a gentle introduction to the subject and present a recent theorem that we obtained together with Fisher, Miller and Stover.

*Abstract:*

We construct and study a stationary version of the Hastings-Levitov(0) model. We prove that unlike the classical model, in the stationary case particle sizes are constant in expectation, yielding that this model can be seen as a tractable off-lattice Diffusion Limited Aggregation. The stationary setting together with a geometric interpretations of the harmonic measure yields new geometric results such as stabilization, finiteness of arms and unbounded width in mean of arms. Moreover we can present an exact conjecture for the fractal dimension. Joint work with Noam Berger and Amanda Turner.

*Abstract:*

After revisiting the efficient determination of approximations of invariant manifolds, this talk will address the closure problem of nonlinear systems subject to an autonomous forcing and placed in parameter regimes for which no slaving principle holds. In particular, solutions which do not lie on any invariant manifold and for which we seek for a reduced parameterization, will be of primary interest. Adopting a variational framework, we will show that efficient parameterizations can be explicitly determined as parametric deformations of invariant manifolds; deformations that are themselves optimized by minimization of cost functionals naturally associated with the dynamics.

Rigorous results will be derived that show that - given a cutoff dimension - the best manifolds that can be obtained through our variational approach, are manifolds which are in general no longer invariant. The minimizers are objects, called the optimal parameterizing manifolds (PMs), that are intimately tied to the conditional expectation of the original system, i.e. the best vector field of the reduced state space resulting from averaging of the unresolved variables with respect to a probability measure conditioned on the resolved variables.

Applications to the closure of low-order models of Atmospheric Primitive Equations and Rayleigh-Benard convection will be then discussed. The approach will be finally illustrated - in the context of the Kuramoto-Sivashinsky turbulence with many unstable modes - as providing efficient closures without slaving for a cutoff scale k_c placed within the inertial range and the reduced state space just spanned by the unstable modes, without inclusion of any stable modes whatsoever. The underlying optimal PMs obtained by our variational approach are far from slaving and allow for remedying the excessive backscatter transfer of energy to the low modes encountered by known parameterizations in their standard forms, when they are used at this cutoff wavelength. In other words, our variational approach will be shown to fix the inverse error cascade, i.e. errors in the modeling of the parameterized (small) scales that contaminate gradually the larger scales, and may spoil severely the closure skills for the resolved variables. Depending on time extension to stochastic systems will be discussed. This talk is based on a joint work with Honghu Liu and James McWilliams.

*Abstract:*

Machine learning and information theory tasks are in some sense equivalent since both involve identifying patterns and regularities in data. To recognize an elephant, a child (or a neural network) observes the repeating pattern of big ears, a trunk, and grey skin. To compress a book, a compression algorithm searches for highly repeating letters or words. So the high-level question that guided this research is: When is learning equivalent to compression? We use the quantity $I(input; output)$, the mutual information between the training data and the output hypothesis of the learning algorithm, to measure the compression of the algorithm. Under this information theoretic setting, these two notions are indeed equivalent. a) Compression implies learning. We will show that learning algorithms that retain a small amount of information from their input sample generalize. b) Learning implies compression. We will show that under an average-case analysis, every hypothesis class of finite VC dimension (a characterization of learnable classes) has empirical risk minimizers (ERM) that do not reveal a lot of information. If time permits, we will discuss a worst-case lower bound we proved by presenting a simple concept class for which every empirical risk minimizer (also randomized) must reveal a lot of information.

*Announcement:*

*Abstract:*

Stable solutions to semilinear elliptic PDEs appear in several problems. It is known since the 1970's that, in dimension $n >9$,there exist singular stable solutions. In this talk I will describe arecent work with Cabré, Ros-Oton, and Serra, where we prove thatstable solutions in dimension $n \leq 9$ are smooth. This answers also to a famous open problem posed by Brezis, concerning the regularity of extremal solutions to the Gelfand problem.

*Abstract:*

Nonstandard analysis was first invented by Abraham Robinson in the early 1960s. It allows to prove theorems of “standard mathematics” taking use of infinite and infinitesimal numbers and many other “nonstandard” mathematical objects. In the mid-1970s Edward Nelson developed an axiomatic approach to nonstandard analysis, with the aim of making nonstandard methods available to the working mathematician. Nelson’s axiomatics is called Internal Set Theory (IST); it is an extension of the “usual” axiomatic Zermelo-Fraenkel set theory, ZFC. As Nelson wrote, “All theorems of conventional mathematics remain valid. No change in terminology is required. What is new in internal set theory is only an addition, not a change.”

In the talk I will describe IST axiomatics and show examples of reasoning in it. I will also discuss the following question. Let H be a proper connected subgroup of the additive group R of real numbers. Is it possible to choose one element in every conjugacy class of R by H? If H consists of all infinitesimals, then the answer is positive. Surprisingly, in general case the answer is negative.

*Abstract:*

Conjugation invariant norms appear in most branches of mathematics. Examples include word norms (autonomous, entropy, fragmentation) and non-discrete norms (Hofer norm) in symplectic geometry. In group theory examples include commutator length and primitive length. After providing some history and motivation, I will focus on subgroups of a group of measure preserving homeomorphisms of a complete Riemannian manifold. I will show that in many cases these norms are unbounded on these groups.

*Abstract:*

During the last 20 years there has been a considerable literature on a collection of related mathematical topics: higher degree versions of Poncelet’s Theorem, certain measures associated to some finite Blaschke products and the numerical range of finite dimensional completely non-unitary contractions with defect index 1. I will explain that without realizing it, the authors of these works were discussing Orthogonal Polynomials on the Unit Circle (OPUC). This will allow us to use OPUC methods to provide illuminating proofs of some of their results and in turn to allow the insights from this literature to tell us something about OPUC. This is joint work with Andrei Martínez-Finkelshtein and Brian Simanek. Background will be provided on the topics discussed.

*Announcement:*

*Abstract:*

A one-day mini conference in memory of Professor Joseph Zaks, 17 June 2019. Schedule: 10:00-10:15 Greetings and some words by the department on Yossi 10:15-11:05 Talk #1: Noga Alon 11:05-11:25 Coffee break 11:25-12:15 Talk #2: Gil Kalai 12:15-14:00 Lunch break (Catering) 14:00-14:50 Talk #3: Nati Linial 14:50-15:10 Coffee break 15:10-16:00 Talk #4: Rom Pinchasi 16:00-17:00 Family and friends talk about Yossi. organizers Raphael Yuster, University of Haifa Anna Melnikov, University of Haifa For updated schedule see: http://sciences.haifa.ac.il/math/wp/?page_id=1382 For more information: please contact the secretary of the mathematics department Michal Ada-Protnov mportnov@univ.haifa.ac.il

*Abstract:*

The workshop will center on Morava's E-theories and the algebra resulting from the Goerss-Hopkins-Miller Theorem, with connections to arithmetic geometry. It should be accessible to graduate students, and will include: 1) The Lubin-Tate moduli problem 2) Hopkins-Kuhn-Ravenel and transchromatic character theory 3) Ambidexterity 4) Power operations on Morava E-theory 5) Actions on the Drinfeld ring 6) The character of the power operation Background talks will be given by other participants of the workshop.

*Abstract:*

The inverse Galois problem over the rationals has inspired several variants over time.

In this talk, following an introduction to the classical problem and the basic notions involved, we will briefly review the variant which asks for the fewest possible ramified primes in a Galois realization of a finite group over the rationals Q, and then spend most of the remaining talk on a recent variant that asks for Galois realizations of a finite group G over Q in which all the nontrivial inertia subgroups have order two. The only groups for which this can happen are those generated by elements of order two, for example finite nonabelian simple groups. If such a group G has a "regular" realization over the rational function field Q(t), as the splitting field of a polynomial f(t,x), then there are computable conditions on the polynomial which guarantee the existence of infinitely many specializations of t into Q which yield realizations over Q with all inertia groups of order two. As examples, three finite simple groups will be given, together with corresponding polynomials. One application of this result is to unramified realizations of these groups over quadratic fields. Another application is to the problem which motivated this result (time permitting). Joint work with Joachim Koenig and Daniel Rabayev.

*Abstract:*

T.B.A.

*Abstract:*

Estimating a manifold from (possibly noisy) samples appears to be a difficult problem. Indeed, even after decades of research, there are no (computationally tractable) manifold learning methods that actually "learn" the manifold. Most of the methods try, instead, to embed the manifold into a low-dimensional Euclidean space. This process inevitably introduces distortions and cannot guarantee a robust estimate of the manifold.

In this talk, we will discuss a new method to estimate a manifold in the ambient space, which is efficient even in the case of an ambient space of high dimension. The method gives a robust estimate to the manifold and its tangent, without introducing distortions. Moreover, we will show statistical convergence guarantees.

It is on a work in progress, joint with Barak Sober.

*Announcement:*

You are invited to the 2019 Summer School:

**Into the forest: group actions on trees and generalizations**

*July 28 – August 1, 2019*

**There will be 5 mini-courses, given by:**

- Indira Chatterji (UNSA)
- Bruno Duchesne (IECL)
- Mark Hagen (UoB)
- Arnaud Hilion (AMU)
- Michah Sageev (Technion)

**Organizer**: Nir Lazarovich