We revisit the old construction of Gromov and Lawson that yields a Riemannian metric of positivescalar curvature on a connected sum of manifolds admitting such metrics. This is joint workwith C. Sormani and J. Our refinement is to show that the "tunnel" constructed betweenthe two summands can be made to have arbitrarily small length and volume. This talk will investigate a certain class of continuous time Markov processes using machinery from algebraic topology. To each such process, we will associate a homological observable, the average current, which is a measurement of the net flow of probability of the system.

We show that the average current quantizes in the low temperature limit. We also explain how the quantized version admits a topological description. Accessibility is an important concept in the study of groups and manifolds as it helps decomposing the object in question into simpler pieces.

In my talk I will survey some accessibility results of groups and manifolds, and explain how to relate the two. I will then discuss a joint work with Benjamin Beeker on a higher dimensional version of these ideas using CAT 0 cube complexes. We will survey recent developments in the symplectic topology that lead to various notions of distance on the category of Lagrangian submanifolds of a symplectic manifold. We will explain both the algebraic as well as geometric sides of the story and outline some applications. In tame geometry, a cell or cylinder is defined as follows.

A onedimensional cell is an interval; a two-dimensional cell is the domainbounded between the graphs of two functions on a one-dimensional cell;and so on. Cellular decomposition covering or subdiving a set intocells and preparation theorems decomposing the domain of a functioninto cells where the function has a simple form are two of the keytechnical tools in semialgebraic, subanalytic and o-minimal geometry.

Using these tools we show that thereal cells of the subanalytic cellular decomposition and preparationtheorems can be analytically continued to complex cells. Complex cells are closely related to uniformization and resolution ofsingularities constructions in local complex analytic geometry. Inparticular we will see that using complex cells, these constructionscan be carried out uniformly over families which is impossible in theclassical setting. The Choquet order on measures is used to establish that states on a function system always have a representing measure supported on the set of extreme points of the state space in a technical sense.

We introduce a new operator-theoretic order on measures, and prove that it is equivalent to the Choquet order. This leads to some improvements in the classical theory, but more importantly it leads to some new operator-theoretic consequences. This yields a significant strengthening of the classical approximation theorems of Korovkin and Saskin. This is joint work with Matthew Kennedy. The talk explains how one can decide whether a Lipschitz selection exists. The result is joint work with P. Legendre duality is prominent in mathematics, physics, and elsewhere.

In recent joint work with Berndtsson, Cordero-Erausquin, and Klartag, we introduce a complex analogue of the classical Legendre transform. Let G be a group and let r n,G denote the number of isomorphism classes of n-dimensional complex irreducible representations of G. Representation growth is a branch of asymptotic group theory that studies the asymptotic and arithmetic properties of the sequence r n,G.

In Larsen and Lubotzky conjectured that all irreducible lattices in a high rank semisimple Lie group have the same polynomial growth rate. In this talk I will explain the conjecture and describe the ideas around the proof of a variant of the conjecture: if the lattices have polynomial representation growth which is known to be true in most cases then they have the same polynomial growth rate. The Hilbert scheme of points on the plane is one of the central objects of modern geometry. We will review some of the interesting connections of this space with representation theory and the theory of symmetric functions, and we will present some recent geometric results motivated by knot theory.

Diffeology, introduced around by Jean-Marie Souriau following earlier work of Kuo-Tsai Chen, gives a way to generalize differential calculus beyond Euclidean spaces.

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Examples include possibly non-Hausdorff quotients of manifolds and spaces of smooth mappings between possibly non-compact manifolds. A diffeology on a set declares which maps from open subsets of Euclidean spaces to the set are "smooth". In spite of its simplicity, diffeology often captures surprisingly rich information. I will present the subject through a sample of examples, results, and questions. By Roth's theorem, any irrational algebraic number x also satisfies this property, so that from that point of view, algebraic numbers and random numbers behave similarly.

The sloshing problem is a Steklov type eigenvalue problem describing small oscillations of an ideal fluid. We will give an overview of some latest advances in the study of Steklov and sloshing spectral asymptotics, highlighting the effects arising from corners, which appear naturally in the context of sloshing.

In particular, we will outline an approach towards proving the conjectures posed by Fox and Kuttler back in on the asymptotics of sloshing frequencies in two dimensions. The talk is based on a joint work in progress with M. Levitin, L. Parnovski and D. Abstract The Graph Isomorphism problem is the algorithmic problem to decide whether or not two given finite graphs are isomorphic. In the first talk we state a core group theoretic lemma and sketch its role in the algorithm: the construction of global automorphisms out of local information.

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These highly regular structures, going back to Schur , are a common generalization of strongly regular graphs and the more general distance-regular graphs and association schemes arising in the study of block designs on the one hand and the orbital structure of permutation groups on the other hand. Johnson graphs are examples of distance-regular graphs with a very high degree of symmetry.

The purpose of this talk is to introduce a new concept, the "radius" of elements in arbitrary finite-dimensional power-associative algebras over the field of real or complex numbers. It is an extension of the well known notion of spectral radius. As examples, we shall discuss this new kind of radius in the setting of matrix algebras, where it indeed reduces to the spectral radius, and then in the Cayley-Dickson algebras, where it is something quite different.

I will discuss isoperimetric problems and their generalizations and applications. The generalization will involve more global notions of boundary as well as partitions into more than 2 parts.

Metallic nanoparticles are optically extraordinary in that they support resonances at wavelengths that greatly exceed their own size. In creating the dichroic glass of the Lycurgus cup, the ancient Romans had exploited the phenomenon, probably unknowingly, already in the 4th Century. Nowadays, surface-plasmon resonance is fundamental to the field of nanophotonics, where the goal is to manipulate light on small scales below the so-called diffraction limit.

Numerous emerging applications rely on the ability to design and realise compound nanostructures that support tunable and strongly localised resonances. In this talk I will focus on the misleadingly simple-looking eigenvalue problem governing the colours of plasmonic nanostructures. The plasmonic spectrum of these structures can be quite rich. For example, the spectrum of a sphere dimer is compound of three families of modes, each behaving differently in the near-contact limit; moreover, these asymptotic trends mutate at moderately high mode numbers and again at yet larger mode numbers. Craster, Vincenzo Giannini and Stefan A.

I will describe a new approach to chaotic flows in dimension three, using knot theory. The flow can then be related to the geodesic flow on the modular surface. When changing the parameters, we find other knots for the Lorenz system and so this uncovers certain topological phases in the Lorenz system. While the topic of geometric incidences has existed for several decades, in recent years it has been experiencing a renaissance due to the introduction of new polynomial methods. This progress involves a variety of new results and techniques, and also interactions with fields such as algebraic geometry and harmonic analysis.

Studying incidence problems often involves the uncovering of hidden structure and symmetries. In this talk we introduce and survey the topic of geometric incidences, focusing on the recent polynomial techniques and results some by the speaker. We will see how various algebraic and analysis tools can be used to solve such combinatorial problems. I will give a very personal overview of the evolution of mainstream applied mathematics from the early 60's onwards.

This era started pre computer with mostly analytic techniques, followed by linear stability analysis for finite difference approximations, to shock waves, to image processing, to the motion of fronts and interfaces, to compressive sensing and the associated optimization challenges, to the use of sparsity in Schrodinger's equation and other PDE's, to overcoming the curse of dimensionality in parts of control theory and in solving the associated high dimensional Hamilton-Jacobi equations. Studying the regular part and shock curves for the entropy solutions for scalar conservation laws is a major research in this field.

Assume that the initial date is constant in the connected components outside a compact interval, T. P Liu and Dafermos-Shearer obtained an interesting criterion when the solution admits one shock after a finite time for the uniformly convex flux. In this talk I will talk about the same phenomena for the flux which has finitely many inflection points. The proof relies on the structure theorem for entropy solutions for convex flux.

A surprisingly rich variety of techniques from different areas of mathematics have been used successfully in this area: combinatorial, probabilistic, analytic and algebraic. For example, Fourier analysis and representation theory have recently proved useful. I will discuss some results and open problems in the area, some of the techniques used, and some links with other areas. The concept of measurable entropy goes back to Kolmogorov and Sinai who in the late 50ies defined an isomorphism invariant for measure preserving Z-actions.

While a similar theory can be developed in an analogous manner for abelian or even amenable groups, the situation gets far more complicated when dealing with groups which are "very" non-commutative, such as free groups. We start the talk with a warm-up about the classical Kolmogorov-Sinai entropy. Using the free group on two generators as an illustrative example, we show how to define cocycle entropy as a new isomorphism invariant for measure preserving actions of quite general countable groups.

Further, we draw connections to other notions of entropy and to open problems in the field. We conclude the talk by clarifying pointwise almost sure approximation of cocycle entropy values. To this end, we present a first Shannon-McMillan-Breiman theorem for actions of non-amenable groups. Joint work with Amos Nevo. The monodromy groups associated to differential equations with regular singularities have, associated to them a monodromy group.

We give a survey of results proved concerning this questions. The aim of the lecture is to show the importance of the knowledge of the set of decomposed places in a uniform pro-p extension of number fields for the mu-invariant of the Class group along the tower. The talk will be elementary and easily accessible. I will describe a long line of research around the asymptotic density of rational points in transcendental varieties, starting from the work of Bombieri-Pila on analytic curves in the late eighties and on to its vast generalization in the work of Pila-Wilkie about ten years ago.

The latter sits at the crossroads between analysis, logic and diophantine geometry, and has attracted considerable attention in the last decade after playing the key role in new proofs of several conjectures on unlikely arithmetic intersections, including the first proof of the Andre-Oort conjecture by Pila. I will give a taste of the philosophy of these applications in an elementary example. Finally I will discuss one of the main open problems of the area, the Wilkie conjecture, and describe some recent progress obtained in a joint work with Dmitry Novikov.

A serious obstacle in applying these Fourier sums, seems to be a lack of control over the dimensions of representations of G. What does a random planar triangulation on n vertices looks like? More precisely, what does the local neighbourhood of a fixed vertex in such a triangulation looks like?

In a joint work with Ori Gurel-Gurevich we prove this conjecture. The proof uses the electrical network theory of random walks and the celebrated Koebe-Andreev-Thurston circle packing theorem. We will give an outline of the proof and explain the connection between the circle packing of a graph and the behaviour of a random walk on that graph.

The minimal ramification problem may be considered as a quantitative version of the inverse Galois problem. So, the problem is to compute or to bound m G. This leads to novel upper bounds on m G , for finite groups G that are realizable as the Galois group of a branched covering. Our methods are based on sieve theory results, in particular on the Green-Tao-Ziegler theorem on prime values of linear forms in two variables, on the theory of specialization in arithmetic geometry, and on finite group theory.

## Research Publications and Interest

The goal of this talk is to convince you that you have been unknowingly using bounded cohomology all your life and to encourage you to come out and use it more openly. To this end we will explain how the natural desire to count leads to bounded cohomology, and how Eudoxus used bounded cohomology to define the ordered field of real numbers around BC. Slightly more recent developments in bounded cohomology and its interactions with geometry, algebra, probability and combinatorics will also be discussed.

We will also explain the special relationship between bounded cohomology and the Technion, which goes back if not to ancient times then at least to the s. We will state a number of open problems which can be understood by a first year student, but whose solution might be a challenge even for professional researchers. Throughout the talk we will focus on the second bounded cohomology and its combinatorial description through quasimorphisms. Cube complexes have come to play an increasingly central role within geometric group theory, as their connection to right- angled Artin groups provides a powerful combinatorial bridge between geometry and algebra.

I will describe how we can exploit the locality of a maximal independent set MIS to the extreme, by showing how to update an MIS in a dynamic distributed setting within only a single adjustment in expectation. The approach is surprisingly simple and is based on a novel analysis of the sequential random greedy algorithm. No background in distributed computing will be assumed.

The talk is based on joint work with Elad Haramaty and Zohar Karnin. Needle decomposition is a technique in convex geometry, which enables one to prove isoperimetric and spectral gap inequalities, by reducing an n-dimensional problem to a 1-dimensional one. In this lecture we will explain what needles are, what they are good for, and why the technique works under lower bounds on the Ricci curvature. The first part of the talk will be about the relation between the singularities of the fibers of p and the analytic properties of push-forwards of smooth measures by p.

The second part of the talk will be about applications to counting points on varieties, character sums, and random matrices. The first bound is a sharp robust estimate for the Gaussian noise stability inequality of C. Borell which is, in turn, a generalization of the Gaussian isoperimetric inequality. Based in part on a joint work with James Lee. More than two hundred years after Galois, the problem remains largely open. I will review recent results concerning the Ginzburg - Landau equations. They also have found important applications in geometry and topology. These equations have remarkable solutions, localized topological solitons, called the magnetic vortices in the superconductivity and the Nielsen-Olesen or Nambu strings in the particle physics, as well as extended ones, magnetic vortex lattices.

I will review the existence and stability theory of these solutions and how they relate to the modified theta functions appearing in number theory and algebraic geometry. Certain automorphic functions play a key role in the theory described in the talk. This talk has to do with knots invariants which are elements of the Brauer groups and of the Tate Shafarevitch groups of curves over number fields.

This is joint work with Alan Reid and Matt Stover. The two main geometric invariants of a rational function are its monodromy group and ramification type. I will explain the progress made during the past years towards determining all possibilities for these invariants, including contributions by Hurwitz, Zariski, Thom, Guralnick, Thompson, and Aschbacher. I will also present applications to number theory, algebraic geometry, and complex analysis. We will focus on two natural questions which arise in this context.

First, what is the large time behavior of such systems? Second, what is the group behavior of systems which involve a large number of agents? Here one is interested in the qualitative behavior of the group rather than tracing the dynamics of each of its agents. Agent-based models lend themselves to kinetic and hydrodynamic descriptions. We discuss the global regularity of such solutions for sub-critical initial configurations. The Merkurjev-Suslin theorem asserts that the n-torsion part of the Brauer group of a field containing a primitive n-th root of 1 is generated by symbol algebras.

A natural question is what is the minimal number of symbols k for which every n-torsion class is a product of k symbols. We will survey some of the known results and if time permits give the idea behind the proof of a bound on the minimal number of symbols in a geometric situation when the base field contains an algebraically closed field. Quenched invariance principle convergence in law to Brownian motion for random walks on infinite percolation clusters and among i.

The proofs of these results strongly rely on the i. Many important models in probability theory and in statistical mechanics, in particular, models which come from real world phenomena, exhibit long range correlations. An essential ingredient of the proof is a new isoperimetric inequality for correlated percolation models. We do not assume any knowledge of modular forms. Zeros of vibrational modes have been fascinating physicists for several centuries. Courant showed that in higher dimensions only half of this is true, namely zero curves of the n-th eigenfunction of the Laplace operator on a compact domain partition the domain into at most n parts which are called "nodal domains".

It recently transpired first on graphs with a subsequent generalization to manifolds that the difference between this upper bound and the actual value can be interpreted as an index of instability of a certain energy functional with respect to suitably chosen perturbations. We will discuss two examples of this phenomenon: 1 stability of the nodal partitions with respect to a perturbation of the partition boundaries and 2 stability of an eigenvalue with respect to a perturbation by magnetic field.

In both cases, the "nodal defect" of the eigenfunction coincides with the Morse index of the energy functional at the corresponding critical point. Based on joint work with R. Band, P. Kuchment, H. Raz, U. Smilansky and T. In , G. Mapping class groups and their cousins - automorphism groups of free groups are some of the most ubiquitous groups in mathematics.

Their representation theory is known to be very rich, but still remains very mysterious and many very basic questions remain unanswered. We will describe some of what is known about this representation theory, and discuss a recent result answering one such basic question which has been open for some time: given an infinite order mapping class, is there a representation in which its image has infinite order?

We study the stability and instability of equilibrium solutions of a reaction-diffusion equationwith different types of boundary conditions. Stable solutions play an important role in thestudy of pattern formation in biology. Unstable solutions lack of physical significance. The talk is intended to givea flavor of how to apply this formula and does not require any knowledge of geometric analysis.

## Vladimir Steklov (mathematician)

Renormalization is a central idea of contemporary Dynamical Systems Theory, It allows one to control small scale structure of certain classes of systems, which leads to universal features of the phase and parameter spaces. We will review several occurrences of Renormalization in Holomorphic Dynamics: for quadratic-like, Siegel, and parabolic maps that enlighten the structure of many Julia sets and the Mandelbrot set.

In particular, these ideas helped to construct examples of Julia sets of positive area resolving a classical problem in this field. First examples were constructed by Buff and Cheritat about 10 years ago, and more recently a different class, with some interesting new features, was produced by Avila and the author.

In the talk, we will describe these developments. Skip to Menu Skip to Content. Past events Type: Colloquium. Title: Stochastic completeness, uniqueness class and volume growth of graphs. Title: Choquet-Deny groups and the infinite conjugacy class property. Files: Poster. Speaker: Achilles K. Tertikas University of Crete. Particular attention will be on: Point interior singularities in Euclidean Space. Point singularities in Hyperbolic Space. Point boundary singularities in Euclidean Space. It is based on a joint work with G.

Title: Double-samplers and local-to-global list decoding. Title: Quasicrystals and Poisson's summation formula. Title: Brackets and cobrackets in topology of surfaces. Title: An undetermined matrix moment problem and its application to computing zeros of L-functions. Title: On exponential Fermi accelerators and energy Equilibration in Slow-fast systems. Title: Symmetries of the hydrogen atom and algebraic families.

No prior knowledge on quantum mechanics or representation theory will be assumed. Geometric structures in nonholonomic mechanics. Title: Geometric structures in nonholonomic mechanics. Title: Noncommutative analytic functions and their fixed points. Title: Gravitational allocation on the sphere and overhanging blocks. TheElisha Netanyahu Memorial Lectures. Title: Counting holomorphic disks using bounding chains. This is joint work with Jake Solomon. No previous knowledge of any of the objects mentioned above will be assumed. Title: A degenerate isoperimetric problem in the plane. Place: Lidow Rosen Auditorium , Technion.

Title: Persistence barcodes in analysis and geometry.

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Title: How we can partition the integers into arithmetic progressions and related questions. 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. Coset partition of a group. Based on joint projects with Y. Ginosar, L. Margolis and J. Title: Localization of eigenfunctions via an effective potential. Billiard paths on polygons: Where do they lead? Title: Billiard paths on polygons: Where do they lead?

Are Knots rigid?? Title: Invariants and equivalence of ordinary differential equations. From Diophantine equations to Fourier restriction -- and back. Title: From Diophantine equations to Fourier restriction -- and back. What is interesting about 2-D viscous flows? Title: What is interesting about 2-D viscous flows? Infinite Bernoulli convolutions and some generalization: survey of recent results. Title: Infinite Bernoulli convolutions and some generalization: survey of recent results.

Title: Equiangular lines and spherical codes in Euclidean spaces. Joint work with Gershon Wolansky. Description Sobolev spaces become the established and universal language of partial differential equations and mathematical analysis. Among a huge variety of problems where Sobolev spaces are used, the following important topics are the focus of this volume: boundary value problems in domains with singularities, higher order partial differential equations, local polynomial approximations, inequalities in Sobolev-Lorentz spaces, function spaces in cellular domains, the spectrum of a Schrodinger operator with negative potential and other spectral problems, criteria for the complete integration of systems of differential equations with applications to differential geometry, some aspects of differential forms on Riemannian manifolds related to Sobolev inequalities, Brownian motion on a Cartan-Hadamard manifold, etc.

Two short biographical articles on the works of Sobolev in the s and the foundation of Akademgorodok in Siberia, supplied with unique archive photos of S. Sobolev are included. Product details Format Paperback pages Dimensions x x Other books in this series. Add to basket. Different Faces of Geometry Simon K.

Back cover copy Sobolev spaces become the established and universal language of partial differential equations and mathematical analysis. Among a huge variety of problems where Sobolev spaces are used, the following important topics are in the focus of this volume: boundary value problems in domains with singularities, higher order partial differential equations, local polynomial approximations, inequalities in Sobolev-Lorentz spaces, function spaces in cellular domains, the spectrum of a Schrodinger operator with negative potential and other spectral problems, criteria for the complete integrability of systems of differential equations with applications to differential geometry, some aspects of differential forms on Riemannian manifolds related to Sobolev inequalities, Brownian motion on a Cartan-Hadamard manifold, etc.

Two short biographical articles on the works of Sobolev in the 's and foundation of Akademgorodok in Siberia, supplied with unique archive photos of S. Table of contents On the Mathematical Works of S.

Sobolev in the s, V. Burenkov, P. Costea, V.

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