# Math Colloquium

Math colloquium meets on Mondays at 12:15 in Schreiber 006, Tel Aviv University.

# Fall 2005

### 21.11.2005, 12:15 Yakov Eliashberg, Stanford, USA Applications of symplectic geometry to low-dimensional topology. (This is the first lecture in the framework of DISTINGUISHED LECTURES IN TOPOLOGY supported through the Michael Bruno Memorial Awards.)

ABSTRACT: Symplectic geometry enters low-dimensional topology in several ways. First of all, there is a number of canonical constructions which associate with smooth manifolds and their submanifolds some symplectic or contact manifolds and their Lagrangian and Legendrian submanifolds. Then symplectic and contact invariants of the constructed objects become smooth invariants of the original manifolds. This approach was successfully applied by L. Ng for his study classical knots in the 3-space. I will briefly review his results and also discuss some ongoing attempts to use this approach for defining new invariants of 4-manifolds. Gromov's theory of holomorphic curves is another door through which symplectic geometry enters low-dimensional topology, and I will discuss some of the constructions and results in this direction.

### 28.11.2005, 12:15 Pierre Lochak, Universite Paris 6, France Thurston with Grothendieck? From diffeomorphisms of surfaces to Grothendieck-Teichmueller theory.

ABSTRACT. Using concrete examples I will explain how Thurston's viewpoint on the diffeomorphisms of surfaces and its subsequent ramifications should be quite relevant for exploring the landscape delineated in Grothendieck's Esquisse d'un programme'. This includes introducing origamis' which are a priori of a purely topological nature and can also be viewed as a higher dimensional analog of `dessins d'enfants'; in particular the Galois group of the rational numbers acts faithfully on them.

### 05.12.2005, 12:15 Michael Polyak, Technion, Haifa Counting lines and other geometric shapes (or where rigid algebraic geometry meets smooth topology).

ABSTRACT. In complex enumerative geometry one counts algebraic-geometric objects with certain properties, e.g. a number of rational curves of degree d passing through 3d-1 points. I will discuss a real counterpart of such problems, starting from some simple examples and relating them to the theory of finite type invariants. I will also discuss a general setting to produce such invariants using maps of configuration spaces and homology intersections.

### 12.12.2005, 12:15 Victor Palamodov, Tel Aviv University Rigidity of Riemannian manifolds and the Inverse Kinematic problem

ABSTRACT: Whether a Riemannian manifold $M$ is uniquely determined by lengths of its closed geodesics? The (affirmative) answer is known in few special cases: Michel, Gromov, Croke, Uhlmann,... If $M$ is conformal to a bounded domain in Euclidean space, the problem is to recover the conformal coefficient as function of Euclidean coordinates from knowledge of lengths of all geodesics connecting boundary points. The particular case is known as the inverse kinematic problem in geophysics: to reconstruct the velocity of elastic waves in the globe from day surface measurements of travel times. This problem has a long history: Herglotz, V. Markushevich, M. Gerver, V. Romanov, R. Muhometov, G. Beylkin, J. Bernstein,... The newest progress in the problems will be described in the talk. The basic notions will be explained as well as the main arguments, which are quite elementary.

### 19.12.2005, 12:15 Yuval Peres , University of California, Berkeley Point processes, the stable marriage algorithm, and Gaussian power series.

ABSTRACT: We consider invariant point processes, i.e., random collections of points with distribution invariant under isometries: the simplest example is the Poisson point process. Given a point process M in the plane, the Voronoi tesselation assigns a polygon (of different area) to each point of M. The geometry of "fair" allocations is much richer: There is a unique "fair" allocation that is "stable" in the sense of the Gale-Shapley stable marriage problem. Zeros of power series with Gaussian coefficients are a different source of point processes, where the isometry invariance is connected to classical complex analysis. In the case of independent coefficients with equal variance, the zeros form a determinantal process in the hyperbolic plane, with conformally invariant dynamics. Surprisingly, in this case the number of zeros in a disk has a coin-tossing interpretation. (Talk based on joint works with C. Hoffman, A. Holroyd and B. Virag).

### 09.01.2006, 12:15 Dan Romik , University of California, Berkeley Random Young tableaux, Young diagrams, permutations and sorting networks.

ABSTRACT. Plancherel measure is an important probability model that encodes the probabilistic behavior of lengths of increasing subsequences in random permutations. It has been shown in recent years to be in many ways a discrete analogue of the GUE (Gaussian Unitary Ensemble) random matrix model. I will introduce a new model, uniform random square Young tableaux, which turns out to be a natural deformation of Plancherel measure that can be analyzed using similar techniques. I will survey some new results on this model and some applications to the combinatorics of random permutations and random "sorting networks", which are ways to sort a list of N distinct numbers from increasing to decreasing order by applying a minimal-length sequence of adjacent transpositions. The talk is based on joint works with Boris Pittel, Omer Angel, Alexander Holroyd, Balint Virag, Scott Sheffield and Rick Kenyon.

### 16.01.2006, 12:15 Leonid Polterovich , Tel Aviv University Title: Symplectic maps: algebra, geometry, dynamics.

ABSTRACT: Symplectic maps appear as a natural generalization of area-preserving diffeomorphisms of surfaces. They play a central role in the mathematical model of classical mechanics. We will focus on the following topics: (a) growth rate of symplectic maps and the trichotomy hyperbolic/parabolic/elliptic in the context of diffeomorphisms; (b) obstructions to symplectic actions of finitely generated groups, including a symplectic version of the Zimmer program on actions of lattices. We discuss results in these directions based on modern methods of symplectic topology

### 30.01.2006, 12:15 Steven Schochet , Tel Aviv University The Incompressible Limit

ABSTRACT: The incompressible limit is a singular limit of a system of partial differential equations that describes fluid flow. Because there are several variants of the equations for fluid flow and a variety of classes of initial data and boundary conditions that may be considered, there are in fact a large number of incompressible limits. After its introduction almost a hundred years ago, interest in the incompressible limit was renewed about thirty years ago, and it has become one of the prototypical problems in the theory of singular limits for partial differential equations. In this talk, which is aimed at a general mathematical audience, I will describe the relevant equations and their physical background, the nature of singular limits including in particular the relation to the theory of averaging for ODEs, some basic theory and examples of partial differential equations, a variety of results that have been obtained over the years concerning the incompressible limit, key ideas from some of their proofs, and some recent results and open problems.

# Spring 2006

### 06.03.2006, 12:15 Sergey Fomin , University of Michigan, USA Cluster algebras.

ABSTRACT: The talk will survey the basic definitions and results of the emerging theory of cluster algebras, viewed from a combinatorial perspective. Joint work with Andrei Zelevinsky.

### 13.03.2006, 12:15 Emmanuel Giroux , ENS Lyon, France Open books in contact geometry.

ABSTRACT: we will first describe the notion of an open book and how it appears under distinct names in different areas of mathematics---dynamical systems, complex algebraic geometry, algebraic and geometric topology. Then we will see that these various aspects of open books are facets of their general relations with contact geometry. Finally, we will discuss a number of questions raised by these relations.

### 27.03.2006, 12:15 Alfred Inselberg , Tel Aviv University, Israel & San Diego SuperComputing Center, USA Visualizing R^N and some applications.

ABSTRACT. The desire to understand the underlying geometry of multidimensional problems motivated several visualization methodologies to augment our limited 3-dimensional perception. Parallel Coordinates are rigorously developed obtaining a 1-1 mapping between subsets of N-space and subsets of 2-space. It leads to representations for lines, flats, curves, hypersurfaces and constructions algorithms in N-space involving intersections, proximity, interior point construction and topologies of flats useful in approximations. This is a VISUAL Multidimensional Coordinate System. It is illustrated on some applications to Air Traffic Control, Data Mining on multivariate datasets and Decision Support systems (based on hypersurfaces) capable of doing Feasibility, Trade-Off and Sensitivity Analyses for complex multivariate processes. There will be several interactive demonstrations.

### 24.04.2006, 12:15 Viatcheslav Kharlamov , Universite Louis Pasteur, Strasbourg On Dif=Def problem in real algebraic geometry.

ABSTRACT. Deformation equivalent real algebraic manifolds are equivariantly (respecting the complex conjugation involution) diffeomorphic. The Dif=Def problem asks for what classes of algebraic manifolds diffeomorphic real structure are deformation equivalent. Our goal is to present two recent results. One extends the class of manifolds for which Dif=Def holds to cubic four-folds. The other gives first examples of Dif\ne Def manifolds: it shows that Dif=Def does not hold for Campedelli surfaces.

### 08.05.2006, 12:15 Mikhail Sodin , Tel Aviv University, Israel Random complex zeroes.

ABSTRACT: In the 90-ies, physicists introduced a unique class of Gaussian analytic functions with remarkable unitary invariance of zero points. In the talk, I am going to discuss some recent progress toward understanding the zeroes of these functions. The talk is based on joint works with Fedor Nazarov, Boris Tsirelson, and Alexander Volberg.

### 15.05.2006, 12:15 Alexander Elashvili , Institute of Mathematics of Georgien Academy of Science, Tbilisi, Georgia Lie algebras and singularity theory.

ABSTRACT: Let O_n:=C{x_1,...,x_n} be the algebra of convergent power series in n variable. For f in O_n define the ideal I(f)=(f,d_1f,...,d_nf),where d_if is partial derivation by variable x_i, and consider factor -algebra A(f):=O_n/I(f). We said that f has isolated singularity in 0 if A(f) is finite dimensional vector space over C. Let L(f):=Der(A(f)) be the Lie algebra of all derivations of algebra A(f). In this talk we describe structural properties and some numerical invariants of algebras L(f).

### Counting possible universes in string/M theory.

ABSTRACT: According to string, the vacuum state of our universe is 10 dimensional: the product of the usual 4 dimensional (Minkowski) spactime and a small 3 complex dimensional Calabi-Yau manifold. The problem is to determine how many candidates there are for this CY 3-fold. Physicists often quote the number as 10^{500}. The first purpose of my talk is to explain the problem in mathematical terms. Although the setting is sophisticated, the problem boils down to the combination of a lattice point problem and a problem on critical points of Gaussian random holomorphic functions. I will give a rigorous asymptotic formula for the number of candidate universes and discuss how close it comes to pinning down a specific number in certain string models. The counting formula is reminiscent of counting metastable states of glassy systems as the dimension of the system increases, with the third betti number of the CY 3-fold playing the role of dimension. No prior acquaintance with string theory is assumed.

### Exact Solution of the Six-Vertex Model with Domain Wall Boundary Conditions.

ABSTRACT: The six-vertex model, or the square ice model, with domain wall boundary conditions (DWBC) has been introduced and solved for finite $N$ by Korepin and Izergin. The solution is based on the Yang-Baxter equations, and it represents the free energy in terms of an $N\times N$ Hankel determinant. Paul Zinn-Justin observed that the Izergin-Korepin formula can be re-expressed in terms of the partition function of a random matrix model with a nonpolynomial interaction. We use this observation to obtain the large $N$ asymptotics of the six-vertex model with DWBC in the disordered phase. The solution is based on the Riemann-Hilbert approach and the Deift-Zhou nonlinear steepest descent method. As was noticed by Kuperberg, the problem of enumeration of alternating sign matrices (the ASM problem) is a special case of the the six-vertex model. We compare the obtained exact solution of the six-vertex model with known exact results for the 1, 2, and 3 enumerations of ASMs, and also with the exact solution on the so-called free fermion line. We prove the conjecture of Zinn-Justin that the partition function of the six-vertex model with DWBC has the asymptotics, $Z_N\sim CN^\kappa e^{N^2f}$, as $N\to\infty$, and we find the exact value of the exponent $\kappa$.

### Using the telescope as a microscope: Large scale determination of small scale structure.

ABSTRACT: Just as physicists occasionally use cosmological evidence to reflect on the subatomic world, in geometry, there are a number of situations where one can link the structure at the largest and smallest of scales. The mind does this naturally: watching television we actually see a large number of small pixels, but since we are naturally considering a larger scale, we clump them together, and hypothesize the structure that should be there, were the objects "homogenous" and not pixilated. In this lecture, I will start by giving simple minded examples where large scale structure constrains the small scale and will continue and describe situations where one can actually determine the small scale structure from the large.