1 | Speaker | Affiliation | Title | |
---|---|---|---|---|
2 | 30/10/2014 | Ori Gurel Gurevich | Hebrew University | Recurrence of planar graph limits |
3 | 06/11/2014 | Martin Bridson | Oxford | Non-positive curvature and finite shadows of infinite groups |
4 | 13/11/2014 | Gang Tian | Princeton | Landau Lecture I : Einstein metrics and linear group actions |
5 | 20/11/2014 | Karim Adiprasito | Hebrew University | Combinatorial methods for complex hyperplane arrangements |
6 | 27/11/2014 | Amnon Yekutieli | BGU | Nonabelian Multiplicative Integration on Surfaces |
7 | 04/12/2014 | Tali Kaufman | Bar Ilan | High dimensional expanders |
8 | 11/12/2014 | Wojtek Samotij | Tel Aviv | Counting independent sets in hypergraphs with applications |
9 | 18/12/2014 | Mladen Bestvina | Utah | Asymptotic dimension of a curve complex |
10 | 25/12/2014 | David Fisher | Indiana | Groups acting on manifolds: around the Zimmer program |
11 | 01/01/2015 | Sefi Ladkani | BGU | From groups to clusters |
12 | 08/01/2015 | Sam Grushevsky | Stony Brook | Meromorphic differentials with real periods, and the geometry of the moduli space of curves |
13 | 15/01/2015 | Manor Mendel | Open University | Expanders in metric spaces |
14 | 22/01/2015 | Peter Keevash | Oxford | Hypergraph matchings |
15 | 29/01/2015 | Alex Postnikov | MIT | |
16 | ||||
17 | ||||
18 | Semester Break | |||
19 | ||||
20 | 05/03/2015 | Purim | ||
21 | 12/03/2015 | Chloe Perin | Hebrew University | First-order logic on the free group |
22 | 19/03/2015 | Terence Tao | UCLA | Dvoreztky lectures |
23 | 26/03/2015 | Passover | ||
24 | 02/04/2015 | Passover | ||
25 | 09/04/2015 | Passover | ||
26 | 16/04/2015 | Jonathan Breuer | Hebrew University | Spectral Theory and Orthogonal Polynomial Ensembles |
27 | 23/04/2015 | Independence Day | ||
28 | 30/04/2015 | Boris Dubrovin | SISSA | Zabrodsky lecture: Integrable systems and moduli spaces. |
29 | 07/05/2015 | Eran Asaf | Hebrew University | Tzafriri Prize: The p-adic Local Langlands Programme and Integral Models |
30 | 14/05/2015 | Barry Simon | Caltech | Singular Eigenvalue Perturbation Theory |
31 | 21/05/2015 | Emanuel Milman | Technion | Curvature-Dimension Condition for Non-Conventional Dimensions |
32 | 28/05/2015 | IMU meeting | ||
33 | 04/06/2015 | Hebrew University | The isomorphism problem in ergodic theory | |
34 | 11/06/2015 | Konstantin Golubev | Hebrew University | Zohovitzki prize: On the chromatic number of a simplicial complex |
35 | 18/06/2015 | Richard Schwartz | Brown | Outer billiards and the plaid model |
36 | 25/06/2015 | Barak Weiss | Tel Aviv | Dynamics of the horocycle flow on the eigenform locus |
37 | ||||
38 | ABSTRACTS | |||
39 | Ori Gurel Gurevich | 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? When n goes to infinity, the resulting object is a random rooted graph called the Uniform Infinite Planar Triangulation (UIPT). Angel, Benjamini and Schramm conjectured that the UIPT and similar object are recurrent, that is, a simple random walk on the UIPT returns to its starting vertex almost surely. In a joint work with Asaf Nachmias, 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. | ||
40 | Martin Bridson | There are many situations in geometry and group theory where it is natural or convenient to study infinite groups via their finite quotients. If a group G is residually finite (ie every element survives in some finite quotient) then one might hope to understand G via its finite quotients -- equivalently, its pro-finite completion. But how hard is it find finite quotients, and to what extent do they determine G? In this lecture I'll describe recent advances in this area, emphasizing how an enhanced understanding of spaces of non-positive curvature has underpinned progress on algebraic questions. I'll outline why there is no algorithm that, given a finitely presented group, can determine if the group has a non-trivial finite quotient (a theorem of Bridson and Wilton), and I shall describe applications of this theorem to different areas of mathematics. The lecture will be accessible to a broad mathematical audience. | ||
41 | Gang Tian | The study of Kähler-Einstein metrics was initiated by E. Calabi in the 50's. In the 70's, Yau's celebrated solution of the Calabi conjecture yielded the existence of Kähler-Einstein metrics with vanishing first Chern class. Aubin and Yau solved the existence problem for the case of negative first Chern class. Since then, it has been a challenging problem to study the existence of Kähler-Einstein metrics on Fano manifolds. A Fano manifold is a compact Kähler manifold with positive first Chern class. There are obstructions to the existence of Kähler-Einstein metrics on Fano manifolds. Recently, the problem has been solved by relating existence to K-stability, a geometric stability generalizing substantially the stability from classical geometric invariant theory. In the first lecture, I will first recall some known facts about Riemann surfaces. Then we give a brief tour on the study of Kähler-Einstein metrics in the last two decades. Next I will explain how those Einstein metrics are related to the study of algebraic group actions and geometric stability. | ||
42 | Karim Adiprasito | One of the most notorious and mystifying subjects in combinatorics and discrete geometry is the influence and the necessity of underlying algebraic and geometric structures. We still do not understand what makes, for instance, complex hyperplane arrangements so special for them to exhibit some so wonderful, unexpected properties, and we sometimes feel divided into two opposing camps concerning the future of the field: -- a pessimist wants us to believe that most of the strong and deep algebraic methods are necessary, and that beyond the nice world of algebraic geometry, dragons have their lair. -- an optimist, starry-eyed as he might be, could be inclined to believe that the very flexible methods of combinatorics are all that he needs. Be that as it may, most methods and tools for the study complex hyperplane arrangements rely, in one way or the other, on complex algebraic methods, which often come with inherent difficulties. I will present the history and current questions concerning complex hyperplane arrangements, focusing in particular on line arrangements. I proceed to show that, within some limitations, some newly developed combinatorial methods are surprisingly powerful, offering more general theorems, more flexibility and surprising new insights at often much lower cost. The lecture will be accessible to a broad mathematical audience. | ||
43 | Amnon Yekutieli | Nonabelian multiplicative integration on curves is a classical theory, going back to Volterra in the 19-th century. In differential geometry this operation can be interpreted as the holonomy of a connection along a curve. In probability theory this is a continuous-time Markov process. This talk is about the 2-dimensional case. A rudimentary nonabelian multiplicative integration on surfaces was introduced in the 1920's by Schlesinger, but it is not widely known. I will present a more sophisticated construction, in which there is a Lie group H, together with an action on it by another Lie group G. The multiplicative integral is an element of H, and it is the limit of Riemann products. Each Riemann product involves a fractal decomposition of the surface into kites (triangles with strings). There a twisting of the integrand, that comes from a 1-dimensional multiplicative integral along the strings, with values in the group G. My main result is a 3-dimensional nonabelian Stokes Theorem. This result is new; only a special case of it was predicted (without proof) in papers in mathematical physics. The motivation for my work was a problem in twisted deformation quantization. It is related to algebraic geometry (the structure of gerbes), algebraic topology (nerves of 2-groupoids), and mathematical physics (nonabelian gauge theory). I will say a few words about these relations at the end of the talk. The talk itself is a computer presentation with numerous color pictures. I recommend printing a copy of the notes before the talk, from the link below. The talk should be accessible to a wide mathematical audience. Lecture notes at http://www.math.bgu.ac.il/~amyekut/lectures/multi-integ/notes.pdf | ||
44 | Tali Kaufman | Expander graphs have been intensively studied in the last four decades. In recent years a high dimensional theory of expanders has emerged. In this talk I will introduce the notion of high dimensional expanders and some of the motivations for studying them. As opposed to (1-dimensional) expanders, where a random bounded degree graph is an expander; a probabilistic construction of a bounded degree high dimensional expander in not known. A major open problem, formulated by Gromov, is whether *bounded degree* high dimensional expanders could exist for dimension d >= 2. I will discuss a recent construction of explicit bounded degree 2-dimensional expanders, that answer Gromov question in the affirmative. Joint work with David Kazhdan and Alexander Lubotzky. | ||
45 | Wojtek Samotij | Many important theorems and conjectures in combinatorics, such as the theorem of Szemeredi on arithmetic progressions and the Erdos-Stone theorem in extremal graph theory, can be phrased as statements about families of independent sets in certain uniform hypergraphs. In recent years, an important trend in extremal and probabilistic combinatorics has been to extend such classical results to the so-called `sparse random setting'. This line of research has recently culminated in the breakthroughs of Conlon and Gowers and of Schacht, who developed general tools for solving problems of this type. Although these two approaches solved very similar sets of longstanding open problems, the methods used are very different from one another and have different strengths and weaknesses. In the talk, we describe a third, completely different approach to proving extremal and structural results in sparse random sets that also yields their natural `counting' counterparts. We give a structural characterization of the independent sets in a large class of uniform hypergraphs by showing that every independent set is almost contained in one of a small number of relatively sparse sets. We then show how to derive many interesting results as fairly straightforward consequences of this abstract theorem. Based on joint work with Jozsef Balogh and Robert Morris. | ||
46 | Mladen Bestvina | Asymptotic dimension, introduced by Gromov, is an important large scale invariant of metric spaces. E.g. groups of finite asymptotic dimension satisfy the Novikov conjecture, by a theorem of Yu. It was recently proved (Bestvina-Bromberg-Fujiwara) that mapping class groups have finite asymptotic dimension. A crucial ingredient is the fact, due to Bell-Fujiwara, that a curve complex has finite asymptotic dimension. An attempt to follow this program and apply it to Out(F_n) runs into several difficulties, one of which is that it is unknown that a suitable analog of the curve complex has finite asymptotic dimension. In this joint work with Ken Bromberg we offer an alternative proof of the Bell-Fujiwara theorem, which also gives linear bounds on the asymptotic dimension. In this approach, one must estimate the capacity dimension of the Gromov boundary. In the talk, I will start with a review of dimension theory and of the basic objects involved in the argument, and will outline the proof. | ||
47 | David Fisher | I will give a survey of various results concerning when ”large groups” can act smoothly on compact manifolds. This field emerges out of some conjectures of Zimmer's concerning when lattices in semisimple Lie groups can act on compact manifolds. I will try to mention a number of open problems. | ||
48 | Sefi Ladkani | I will present a new combinatorial construction of finite-dimensional algebras with some interesting representation theoretic properties: they are of tame representation type, symmetric and have periodic modules. The quivers we consider are dual to ribbon graphs and they naturally arise from triangulations of oriented surfaces with marked points. The class of algebras that we get contains in particular the algebras of quaternion type introduced and studied by Erdmann with relation to certain blocks of group algebras. On the other hand, it contains also the Jacobian algebras of the quivers with potentials associated by Fomin-Shapiro-Thurston and Labardini to triangulations of closed surfaces with punctures. Hence our construction may serve as a bridge between modular representation theory of finite groups and cluster algebras. All notions will be explained during the talk. | ||
49 | Sam Grushevsky | We will discuss some results on the geometry of the moduli space of Riemann surfaces, and applications, obtained using meromorphic differentials with all periods real. These constructions are motivated by Whitham perturbation theory of integrable systems, and somewhat parallel to those used in Teichmuller dynamics. Based on joint work with Igor Krichever. | ||
50 | Manor Mendel | An expander is a graph G=(V,E) of small degree such that all of its subsets have many edges leaving them. This assertion is equivalent to the following approximate equality: For any mapping f:V-->H of the vertices into Hilbert space, the average of ||f(u)-f(v)||^2 over ALL PAIRS u,v in V is within a constant factor of the average of ||f(x)-f(y)||^2 over ALL EDGES {x,y} in E. This geometric reformulation of expansion is important in metric geometry, leading to the question whether it holds true when the Hilbert space is replaced by other metric spaces. This talk will survey some of the current known results in this direction, as well as challenging open questions. In particular, we will discuss the existence of expanders with respect to certain Banach spaces, simply connected spaces of non-positive curvature in the sense of Aleksandrov, and random regular graphs. We will concentrate on the following natural question: Is it the case that if one family of expanders satisfies the above approximate equality with respect to a metric space X then is it necessarily the case that every family of expanders satisfies it with respect to X? While the answer to this question is "Yes" when X is a Hilbert space (or even an L_p space), we will show that in general the answer is "No": There exists a metric space X and a sequence of expanders that satisfies the above approximate equality with respect to X, while random regular graphs (which are known to be expanders) fail to satisfy it with respect to X. The construction uses tools from Combinatroics, Probability, Geometry, and Functional Analysis. The talk is based on the following papers: http://arxiv.org/abs/1207.4705, http://arxiv.org/abs/1301.3963, http://arxiv.org/abs/1306.5434 coauthored jointly with A. Naor. | ||
51 | Peter Keevash | Perfect matchings are fundamental objects of study in graph theory. There is a substantial classical theory, which cannot be directly generalised to hypergraphs unless P=NP, as it is NP-complete to determine whether a hypergraph has a perfect matching. On the other hand, the generalisation to hypergraphs is well-motivated, as many important problems can be recast in this framework, such as Ryser's conjecture on transversals in latin squares, and the Existence Conjecture for combinatorial designs (the subject of my lecture series in the 18th Midrasha Mathematicae). In this talk we will discuss a characterisation of the perfect matching problem for uniform hypergraphs that satisfy certain density conditions (joint work with Richard Mycroft), and a polynomial time algorithm for determining whether such hypergraphs have a perfect matching (joint work with Fiachra Knox and Richard Mycroft). | ||
52 | Alex Postnikov | |||
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57 | Purim | |||
58 | Chloe Perin | We will give an overview of questions one might ask about the first-order theory of free groups and related groups: how much information can first-order formulas convey about these groups or their elements, what algebraic interpretation can be given for model theoretic notions such as forking independence, etc. Some of these questions have been answered, others are still open - our aim is to give a feel for the techniques and directions of this field. We will assume no special knowledge of model theory. | ||
59 | Terence Tao | Small and large gaps between the primes There are many questions about the gaps between consecutive prime numbers which are not completely solved, even after decades of effort. For instance, the twin prime conjecture, which asserts that the gap between primes can equal 2 infinitely often, remains open. However, there has been recent progress in understanding both very small and very large gaps between primes. Last year, in a breakthrough work of Yitang Zhang, it was shown that there were infinitely many gaps between primes of bounded size; Zhang's original bound here was 70 million, but it has since been cut down to 246 thanks to the efforts of James Maynard and an online collaborative "Polymath" project. In the opposite direction, recent work of Kevin Ford, Ben Green, Sergei Konyagin, and myself have shown the existence of prime gaps significantly larger than their average spacing, improving upon earlier work of Rankin, Pintz, and others. The two proofs are somewhat different in nature; the first argument relies on sieve theory methods, whereas the second argument uses probabilistic arguments combined with the results of Ben Green and myself in arithmetic progressions in the primes. We present the main ideas of both of these results in this lecture. | ||
60 | Passover | |||
61 | Passover | |||
62 | Jonathan Breuer | The notion of an orthogonal polynomial ensemble generalizes many important point processes arising in random matrix theory, probability and combinatorics. The most famous example perhaps is that of the eigenvalue distributions of unitary invariant ensembles of random matrix theory (such as GUE). Remarkably, the study of fluctuations of these point processes is intimately connected to the study of one dimensional discrete Schroedinger operators. This talk will review recent work elucidating and exploiting this connection in the context of microscopic universality, laws of large numbers and central limit theorems. | ||
63 | Independence day | |||
64 | Boris Dubrovin | The study of interconnections between integrable systems and topology of Deligne - Mumford moduli spaces of stable algebraic curves and their generalisations has been initiated by E.Witten and M.Kontsevich at the beginning of 90s. In the talk I will explain some basic ideas about the conjectural correspondence between the class of smooth projective varieties with semisimple quantum cohomology and a class of hierarchies of integrable PDEs. | ||
65 | Eran Asaf | Since the statement of the Taniyama-Shimura-Weil conjecture, it has been known that this is only the tip of the iceberg regarding the deep connection between solving equations in integers, and modular forms. The modularity theorem, proved in 2001, is only a special case of the famous Langlands conjecture for the group GL(2,Q), which to this day remains open. Establishing the local langlands correspondence, in 2001, for GL(n,F), where F is a local field, was not enough. The aim of the p -adic Local Langlands Programme is to establish a similar correspondence of representations with p -adic coefficients. This turns out to be quite involved, and was established for GL(2,Qp) only recently. One of the main tools in establishing this correspondence was the existence of integral models in certain representations of GL(2,Qp). We prove a criterion for the existence of integral models in certain representations of U(3,F), where F is a finite extension of Qp. | ||
66 | Barry Simon | Eigenvalue Perturbation Theory is central to the theory of nonrelativistic quantum mechanics going back to Schrodinger's first papers. This lecture will review what is known about the eigenvalues in physical situations where one doesn't have simple convergence to a new isolated eigenvalue. Included are the anharmonic oscillator and Zeeman effect (divergent series and summability), autoionizing states in atoms (complex scaling and resonances), Stark effect (exponentially small resonances) and double wells (instantons). | ||
67 | Emanuel Milman | Given an n-dimensional Riemannian manifold endowed with a probability density, we are interested in studying its isoperimetric, spectral and concentration properties. To this end, the Curvature-Dimension condition CD(K,N), introduced by Bakry and Emery in the 80's, is a very useful tool. Roughly put, the parameter K serves as a lower bound on the weighted manifold's "generalized Ricci curvature", whereas N serves as an upper bound on its "generalized dimension". Traditionally, the range of admissible values for the generalized dimension N has been confined to [n,\infty]. In this talk, we present some recent developments in extending this range to N < 1, allowing in particular negative (!) generalized dimensions. We will mostly be concerned with obtaining sharp isoperimetric inequalities under the Curvature-Dimension condition, identifying new one-dimensional model-spaces for the isoperimetric problem. Of particular interest is when curvature is strictly positive, yielding a new single model space (besides the previously known N-sphere and Gaussian measure): the sphere of (possibly negative) dimension N<1, which enjoys a spectral-gap and improved exponential concentration. Time permitting, we will also discuss the case when curvature is only assumed non-negative. When N is negative, we confirm that such spaces always satisfy an N-dimensional Cheeger isoperimetric inequality and N-degree polynomial concentration, and establish that these properties are in fact equivalent. In particular, this renders equivalent various weak Sobolev and Nash inequalities for different exponents on such spaces, and implies a stability property for the N-dimensional Cheeger constant. | ||
68 | IMU meeting | |||
69 | Benjy Weiss | I will give a historically oriented survey of what is known as the "isomorphism problem” or the "conjugacy problem" in ergodic theory. After describing the problem and some of the more classical results I will also describe some recent work which points to the inherent difficulties that the problem presents. | ||
70 | Konstantin Golubev | A simplicial complex is a high-dimensional analogue of graph, defined as a family of subsets of a vertex set closed under taking subsets. By analogy for graphs, the chromatic number of a simplicial complex is the least number of colors needed in order to color the vertices in such a way that there is no monochromatic maximal face. In the talk, we will present a generalization of the classical Hoffman lower bound on the chromatic number for graphs to simplicial complexes by means of the spectra of its Laplacian operators. If time permits, we will also discuss Ramanujan Complexes (constructed by Lubotzky, Samuels and Vishne) which serve as an example of simplcial complexes of large girth and large chromatic number. | ||
71 | Richard Schwartz | Outer billiards is a dynamical system which resembles ordinary billiards except that the “ball" bounces around the outside of the shape, so to speak. Even for simple shapes, such as kites (quadrilaterals with bilateral symmetry) the orbits are quite intricate. I will explain a combinatorial model which predicts the structure of the orbits for kites, and explain how this is related to a things such as circle rotations, polytope exchange transformations, and self-similar tilings. |