1 | Date | Speaker | Affiliation | Title |
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2 | 17/10/2013 | |||

3 | 24/10/2013 | Yves Benoist | ORSAY/IIAS | Stationary measures on projective spaces |

4 | 31/10/2013 | Boaz Klartag | Tel Aviv Univesity | Moment Measures |

5 | 07/11/2013 | Jake Solomon | Hebrew University | Geometry of the space of Lagrangian submanifolds |

6 | 14/11/2013 | Itai Benjamini | Weizmann | Euclidean vs. graph Metrics |

7 | 21/11/2013 | Ilijas Farah | York University | Necessary applications of set theory to operator algebras |

8 | 28/11/2013 | Yiannis Sakellaridis | Rutgers (Newark) | Spherical varieties and the "relative" Langlands program |

9 | 05/12/2013 | Leonid Polterovich | Tel Aviv University | Inside the Poisson bracket |

10 | 12/12/2013 | Jean Bourgain | IAS | Toral eigenfunctions and their nodal sets |

11 | 19/12/2013 | (no lecture; conference at the IIAS) | ||

12 | 26/12/2013 | Yuval Peres | Microsoft research | Random walks on groups and the Kaimanovich-Vershik 1983 conjecture for lamplighter groups |

13 | 02/01/2014 | Jordan Ellenberg | University of Wisconsin | Configurations, arithmetic groups, cohomology, and stability |

14 | 09/01/2014 | Cy Maor | Hebrew University | Dimension-reduction (and other limit models) in non-Euclidean elasticity (Perelman Prize) |

15 | 16/01/2014 | Peter Sarnak | Princeton\IAS\IIAS | Topologies of nodal sets of random band limited functions [DELAYED TO TUESDAY 21/1/2014 at 2pm!!!] |

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18 | Semester Break | |||

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20 | 20/02/2014 | Gil Kalai | Hebrew University | Some old and new problems in combinatorics and geometry |

21 | 27/02/2014 | Itay Kaplan | Hebrew University | Groups and fields in model theory |

22 | 06/03/2014 | Shai Evra | Hebrew University | Simplicial complexes with large 'girth' and large chromatic number (Tzafriri Prize) |

23 | 13/03/2014 | Yaron Ostrover | Tel Aviv Univeristy | From symplectic measurements to Mahler conjecture via billiard dynamics |

24 | 20/03/2014 | Steve Zelditch | Northwestern | Shapes and Sizes of eigenfunctions (Zabrodsky lecture) |

25 | 27/03/2014 | Roman Sauer | Karlsruher Institut für Technologie | Volume and the growth of homology |

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27 | 01/05/2014 | James Maynard | Small gaps between primes (Openning lecture for Joram Seminar, May 1-2) | |

28 | 08/05/2014 | Amit Solomon | Hebrew University | Topology of gradient flow near a critical point (Zochovitzky lecture) |

29 | 15/05/2014 | Ursula Hamenstaedt | Bonn | Flat surfaces, dynamics and trace fields |

30 | 22/05/2014 | Daniel Spielman | Yale | Ramanujan Graphs of Every Degree (Erdos Lecture) |

31 | 29/05/2014 | Nicolas de Saxce | Hebrew University | Expansion in simple lie groups |

32 | 02/06/2014 | Percy Deift | New York University | Universality in numerical computations with random data. Case studies. **Please note special time & place [Monday 4pm at IAS lecture hall]** |

33 | 12/06/2014 | Alex Sodin | Princeton | Airy processes from the theory of random matrices |

34 | Joint AMS-IMU meeting in Tel Aviv\Bar Ilan | |||

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38 | ABSTRACTS | |||

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40 | Yves Benoist | Let V be a real finite dimensional vector space and m a probability measure on GL(V). A classical theorem of Furstenberg tells us that, when the action of the group spanned by the support of m is irreducible and proximal, then there exists a unique m-stationary probability measure on the projective space P(V). We will explain the importance of this result and how to extend it when this action is only supposed to besemisimple. This is a joint work with JF. Quint. | ||

41 | Boaz Klartag | We describe a certain bijection between convex functions in a finite-dimensional linear space X (modulo translations), and finite Borel measures on the dual space X* with barycenter at the origin. The construction is related to toric Kahler-Einstein metrics in complex geometry, to the Prekopa-Leindler inequality, and to the Minkowski problem in convex geometry. Joint work with D. Cordero-Erausquin. | ||

42 | Jake Solomon | The roots of symplectic geometry lie in classical mechanics. A symplectic manifold is the space of all possible positions and momenta of a given mechanical system. A Lagrangian submanifold is a subspace roughly corresponding to a semi-classical state in quantum mechanics; for example, it may be the subspace where position is fixed but momentum is undetermined as required by Heisenberg's uncertainty principle. A central theme in symplectic geometry is to determine when one Lagrangian submanifold can evolve to another according to the laws of classical mechanics. If so, the two Lagrangians are called exact isotopic. On the other hand, Lagrangian submanifolds arise in high energy physics as boundary conditions for open strings. However, only the special Lagrangians, or those which minimize volume, are supersymmetric. Much research has aimed to resolve the question of when an exact isotopy class of Lagrangian submanifolds contains a special Lagrangian representative, and if so, whether it is unique. Currently, very little is known. In this talk, I will discuss a functional on each exact isotopy class of Lagrangian submanifolds, which has special Lagrangians for critical points. It is convex along the geodesics of a negatively curved metric. Thus questions of existence and uniqueness for special Lagrangians are related to the existence of geodesics. The definitions are motivated by mirror symmetry and a notion of stability familiar from algebraic geometry. No background in geometry or physics will be assumed. | ||

43 | Itai Benjamini | We will discuss global and local graph limits. Global: how well the graph metric on bounded degree graphs can approximate the metric of homogeneous manifoldsequipped with some invariant length metric. Local: We will comment on local limits of some random graphs. | ||

44 | Ilijas Farah | Connections between set theory and operator algebras in the past century were few and sparse, although von Neumann was an important figure in the early history of both areas and some key mid-century developments in descriptive set theory occurred within an operator-algebraic context in the work of Mackey, Glimm, and Effros. Recently, there has been great progress at the interface of operator algebras, set theory and model theory. Long-standing open problems on the structure of C*-algebras were solved using methods from set theory. I will survey some of these results, with particular emphasis to ones related to the rigidity of quotient structures. | ||

45 | Yiannis Sakellaridis | In this talk I will survey the Langlands program from the point of view of harmonic analysis on homogeneous spaces. When these spaces are spherical varieties (such as symmetric spaces) we are led to a generalization of several aspects of the program. Among other things, this sheds some light on the very mysterious relationship between periods of automorphic forms and L-functions. | ||

46 | Leonid Polterovich | We discuss constraints on the Poisson brackets coming from symplectic topology, applications to symplectic intersections and Lagrangian knots, as well as links to quantum mechanics. | ||

47 | Jean Bourgain | In studying spectral aspects of smooth manifolds,the flat torus is perhaps the simplest case because eigenfunctions are completely explicit.There is also a mysterious analogy with the largely unproven phenomena conjectured in the hyperbolic case. Although there is a better analytic grip in the torus case,the most basic problems around the distribution of eigenfunctions and nodal sets turn out to be very hard and the partial progress made relies on diverse areas,including incidence geometry, diophantine analysis,the theory of elliptic curves and of course harmonic analysis.The purpose of the talk will be to give some impression of the various issues one runs into. | ||

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49 | Yuval Peres | Let G be an infinite group with a finite symmetric generating set S. The corresponding Cayley graph on G has an edge between x,y in G if y is in xS. Kaimanovich-Vershik (1983), building on fundamental results of Furstenberg, Derrienic and Avez, showed that G admits non-constant bounded harmonic functions iff the entropy of simple random walk on G grows linearly in time; Varopoulos (1985) showed that this is equivalent to the random walk escaping with a positive asymptotic speed. Kaimanovich and Vershik (1983) also described the lamplighter groups (groups of exponential growth consisting of finite lattice configurations) where (in dimension at least 3) the simple random walk has positive speed, yet the probability of returning to the starting point does not decay exponentially. They conjectured a complete description of the bounded harmonic functions on these groups; In dimension 5 and above, their conjecture was proved by Anna Erschler (2011). In the talk, I will discuss the background and present a proof of the Kaimanovich-Vershik conjecture for all dimensions, obtained in joint work with Russ Lyons; the case of dimension 3 is the most delicate. | ||

50 | Jordan Ellenberg | Consider the following two objects: * The congruence subgroup of level p in SL_n(Z); that is, the group of integral matrices congruent to 1 mod p; * The ordered configuration space of n points on a manifold M, which is to say, the space parametrizing ordered n-tuples of distinct points on M; Each of these objects carries a natural action of the symmetric group S_n on n letters. (In the first case, this is by permuting the elements of the standard basis; in the second case, by permuting the points in the n-tuple.) What's more, each one is naturally described by cohomology groups H^i, which inherit the action and thus become representations of S_n. Although these examples are quite different, it turns out there is a general notion of stability which applies to both of these cases (and many other examples in representation theory, algebraic geometry, and combinatorics.) In some sense, each H^i is "the same" representation of S_n for all sufficiently large n. This implies, for instance, that the dimensions of these cohomology groups are (for sufficiently large n) polynomials in n. In the congruence subgroup context, our results refine a 2012 theorem of Putman. In the configuration space context, the result here refines a 2011 theorem of Church. The main ingredient is the theory of FI-modules, developed by myself, Tom Church, and Benson Farb, together with a Noetherianness theorem proved by the three of us and Rohit Nagpal: http://arxiv.org/abs/1204.4533 http://arxiv.org/abs/1210.1854 | ||

51 | Cy Maor | Elasticity theory deals with finding the configuration of elastic bodies in the Euclidean space under various boundary conditions. Mathematically speaking, this reduces to finding an embedding of a 3D Riemannian manifold (the elastic body) in the 3D Euclidean space, such that the embedding minimizes an elastic energy, which, roughly speaking, measures the distance of the embedding from being rigid. Many of the interesting elastic bodies are "thin", i.e. have one or more slender dimensions. Therefore, a natural question arises - is there an elastic energy on embeddings of 2D or 1D manifolds that encompasses the behavior of a "thin" 3D manifold? This question dates back to Euler and Bernoulli, but only since the mid-1990's, rigorous derivations of such dimensionally-reduced models have been proved. In this talk I will discuss some recent work (joint with Raz Kupferman and Jake Solomon) on dimensionally-reduced models for elastic bodies with arbitrary smooth metric (a.k.a. non-Euclidean elasticity). If time permits, I will discuss other types of limit models in elasticity theory. Some familiarity with basic notions in Riemannian geometry will be assumed, but no prior knowledge in the calculus of variations. | ||

52 | Peter Sarnak | We discuss various Gaussian ensembles for real homogeneous polynomials in several variables and the question of the distribution of the topologies of the connected components of the zero sets of a typical such random real hypersurface. For the "real-Fubini-Study ensemble" and at other end the "monochromatic wave ensemble", one can show that these have universal laws. Some qualitative features of these laws are also established. Joint work with I.Wigman | ||

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57 | Gil Kalai | Using this familiar title by Erdos, I will describe some old and new problems in combinatorics and geometry. I. Questions around Borsuk's problem. II) Questions that involve combinatorics and topology. Solving puzzles based on triangulations of spheres, Embedding of 2-dimensional complexes in four-dimensional space. III) Questions about convex polytopes. E.g., Ramsey type and Erdos-Ko-Rado typeproblems for polytopes. IV) Questions around Tverberg theorem. | ||

58 | Itay Kaplan | In the model theory of groups and fields, we ask questions such as: to what extent does the first order theory of the structure in question determine its algebraic properties, and vice versa. In the late 60's and the 70's, Shelah had developed his Classification Theory. He was motivated by purely model theoretic questions such as "how many models does a first order theory have?". He defined classes of theories, usually by some combinatorial property that they have (or lack) and developed a vast machinery to analyze them. Now one may ask what can be said about the algebraic properties of groups or fields whose first order theory belong to a certain class. I plan to survey some results around this question. I will define every notion I will need for this talk (including what is a formula). | ||

59 | Shai Evra | In 1959 Erdos proved by random methods that there exist graphs with arbitrary large girth and arbitrary large chromatic number. Explicit constructions were given in 1988 by Lubotzky, Philips & Sarnak : the Ramanujan graphs. In this talk we will study the high dimensional analogous question, i.e., for simplical complexes instead of graphs. Here one should explain first what is "girth" and what is "chromatic number". After doing this, we will show how representation theory (and in particular a quantitative form of Kazhdan property T, due to Hee Oh) leads to a proof that the Ramanujan complexes constructed in 2005 by Lubotzky-Samuels-Vishne give simplical complexes of large girth and large chromatic number. This is a joint work with Konstantin Golubev and Alex Lubotzky. | ||

60 | Steve Zelditch | Eigenfunctions of the Laplacian on a Riemannian manifold (M, g) represent modes of vibrations of drums and membranes. In quantum mechanics they represent stationary states of atoms. Understanding shapes and sizes of eigenfunctions allows one to visualize these objects. An intriguing problem is to relate the shapes and sizes to the underlying classical mechanics, such as the geodesic flow of (M, g) or the dynamics of billiard trajectories on a billiard table. In this talk we will explain the role of eigenfunctions in quantum mechanics and discuss both classic and new results describing nodal (zero) sets of eigenfunctions. The new results relate nodal sets to classical dynamics. No prior knowledge of quantum mechanics is assumed. | ||

61 | Roman Sauer | Let us consider a tower of finite coverings of a closed manifold. The talk circles around the following question: How does the size of the k-th homology group grow as we go up this tower? This question is linked to questions in group theory and number theory. But in my talk I will explore another connection: to phenomena of largeness in Riemannian geometry. The methods of proof, though, are from algebraic topology and topological dynamics. | ||

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63 | James Maynard | It is believed that there should be infinitely many pairs of primes which differ by 2; this is the famous twin prime conjecture. More generally, it is believed that for every positive integer $m$ there should be infinitely many sets of $m$ primes, with each set contained in an interval of size roughly $m\log{m}$. We will introduce a refinement of the `GPY sieve method' for studying these problems. This refinement will allow us to show (amongst other things) that $\liminf_n(p_{n+m}-p_n)<\infty $ for any integer $m$, and so there are infinitely many bounded length intervals containing $m$ primes. | ||

64 | Amit Solomon | There is an intimate relationship between the critical points of a smooth function on a manifold and the topology of the manifold: As we will see, given a generic function with "non-degenerate" critical points, one can construct a chain complex, known as the Morse complex, whose homology equals the singular homology of the manifold. Unfortunately, when the critical points of the function are degenerate, the Morse complex construction fails miserably. In recent work with J. Solomon, we make the first step towards extending the Morse complex to a class of degenerate functions. Namely, we endow the stable set of a degenerate critical point with a natural stratification generalizing the concept of the stable manifold. The talk will be accessible to graduate students and no prior knowledge in differential geometry or algebraic topology will be assumed (I will define every notion I use). | ||

65 | Ursula Hamenstaedt | A flat surface is a closed oriented surface equipped with an euclidean metric with finitely many singular points, with cone angles 2kpi for k>1. The affine automorphism group of such flat surfaces is naturally a discrete subgroup of PSL(2,R). We explain how to obtain some information on typical such groups and their trace fields using tools from homogeneous dynamics. | ||

66 | Daniel Spielman | We explain what Ramanujan graphs are, and prove that there exist infinite families of bipartite Ramanujan graphs of every degree. Our proof follows a plan suggested by Bilu and Linial, and exploits a proof of a conjecture of theirs about lifts of graphs. Our proof of their conjecture applies the method of interlacing families of polynomials to Mixed Characteristic Polynomials, which we introduce in the first talk. However, we will not assume familiarity with the first talk. In this talk, a bound on the roots of these polynomials will follow from a bound of Heilmann and Lieb on the roots of the matching polynomials of graphs. We also prove that there exist infinite families of irregular bipartite Ramanujan graphs. This is joint work with Adam Marcus and Nikhil Srivastava. | ||

67 | Nicolas de Saxce | Given a set A in a group G, we want to compare the size of A to the size of the product set AAA of elements of G that can be written as products of three elements of A. In the case G is a simple real Lie group, and A is measured by the number N(A,r) of balls of radius r needed to cover it, I will explain that AAA is much larger than A, unless there is an obvious obstruction. I will also present two applications of such a theorem, the first one to the spectral gap property for certain averaging operators on compact simple Lie groups, and the second to the Hausdorff dimension of subgroups of simple Lie groups. | ||

68 | Percy Deift | Universal fluctuations are shown to exist when well-known and widely used numerical algorithms are applied with random data. Similar universal behavior is shown in stochastic algorithms and algorithms that model neural computation. The question of whether universality is present in all, or nearly all, computation is raised. Joint work with G.Menon, S.Olver and T. Trogdon | ||

69 | Alex Sodin | We shall discuss some random processes which were originally found in the context of random matrices, and which conjecturally describe the fluctuations of random interfaces in two dimensions. The conjectures are rather general; however, the rigorous results are mostly limited to integrable models. After introducing the problem, we shall describe an attempt to move slightly beyond the integrable class. No background in either random matrices or random interfaces is assumed. | ||

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