# 2010-2011 Colloquium

This page contains information about the Mathematics Colloquium at the University of Toronto. The colloquium meets regularly on Wednesdays, 4:10-5pm, 6183 Bahen Centre. Refreshments are served before the colloquium in the Math Lounge at 3:30pm. If you will be speaking, details on how to use the equipment in BA 6183 can be found here.

Colloquium announcements are made here: Colloquium announcements

# Fall Term

## September 29, Braverman, 16:10-17:00 @BA6183

 Mark Braverman (University of Toronto) 2010-2011 Colloquium Wednesday September 29 16:10-17:00 BA6183 Title: On bounded-depth Boolean circuits Abstract: A Boolean circuit of depth d is a circuit comprised of AND, OR and NOT gates arranged in at most d layers. This class of circuits is one of the few complexity classes where unconditional lower bounds, i.e. computational impossibility results exist. Many of the bounds follow from a deep connection between bounded-depth circuits and low-degree multivariate polynomials. In this talk we will discuss some of these connections. We will then present a recent proof of the 1990 Linial-Nisan conjecture on the computational power of bounded-depth circuits. The conjecture stated that bounded-depth Boolean circuits of size poly(n) cannot distinguish inputs drawn from a k-wise independent distributions from uniform inputs, where k=poly(log n). The talk will be almost completely self-contained, relying only on elementary probability theory. arXiv 2010_09_29_Braverman_Notes 2010_09_29

## October 06, Lewis, 16:10-17:00 @BA6183

 James Lewis (University of Alberta) 2010-2011 Colloquium Wednesday October 06 16:10-17:00 BA6183 Title: New Invariants on Algebraic Cycles Abstract: I will explain the intertwining role of Hodge theory and algebraic cycles, beginning from the classical constructions in the 1960's to the more recent developments using arithmetical normal functions. arXiv 2010_10_06_Lewis_Notes 2010_10_06

## October 13, Gluck, 16:10-17:00 @BA6183

 Herman Gluck (UPenn) 2010-2011 Colloquium Wednesday October 13 16:10-17:00 BA6183 Title: Optimality of Hopf fibrations and Hopf vector fields Abstract: The Hopf fibration of a round 3-sphere by parallel great circles was introduced by Heinz Hopf in 1931. It provided the first example of a homotopically nontrivial map from one sphere to another of lower dimension, spurring the development of both homotopy theory and fibre spaces in their infancy. Several years later, Hopf presented three families of fibrations of round spheres by parallel great subspheres of dimensions 1, 3 and 7, which were later seen to encompass all possible instances of this phenomenon. We will prove optimality of the above in the following sense: (1) Given a Hopf fibration of a round sphere by parallel great subspheres, the projection map to the base space is, up to isometries of domain and range, the unique Lipschitz constant minimizer in its homotopy class. (2) Given a Hopf fibration of a round sphere by parallel great circles, view a unit vector field tangent to the fibres as a cross-section of the unit tangent bundle of the sphere. Then this vector field is, up to isometries of domain and range, the unique Lipschitz constant minimizer in its homotopy class. Previous attempts to find a mathematical sense in which Hopf fibrations and vector fields are optimal (typically in terms of volume or energy minimization) have met with limited success. All of this represents joint work with Dennis DeTurck and Pete Storm. The talk is intended to be accessible to graduate students. arXiv 2010_10_13_Gluck_Notes 2010_10_13

## October 27, Morgan, 16:10-17:00 @BA6183

 Frank Morgan (Williams College) 2010-2011 Colloquium Wednesday October 27 16:10-17:00 BA6183 Title: Pentagonal Tilings and Other Partition Problems Abstract: Although the regular pentagon does not tile the plane, there are many tilings by unit-area pentagons, largely not understood. We identify two such pentagonal tilings as ones which minimize perimeter. We discuss a number of other open tiling and partitioning problems in the plane, in R^3, and in other spaces. No prerequisites, undergraduates welcome. arXiv 2010_10_27_Morgan_Notes 2010_10_27

## November 03, Villani, 16:00-17:00 [4pm sharp!!] @Room 230, Fields Institute

 Cédric Villani [1] (Université Claude Bernard Lyon I) 2010-2011 Colloquium Wednesday November 03 16:00-17:00 [4pm sharp!!] Room 230, Fields Institute Title: Particle systems and Landau damping (Lecture 2 of 3 distinguished lectures) Abstract: One of the fundamental and initially controversial theories of classical physics is Boltzmann's kinetic theory of gases. Instead of tracking the individual motions of billions of atoms, it describes the evolution of the probability that a particle has a certain position and velocity. The equilibrium probability distributions have been known for more than a hundred years, but it has been very difficult to understand whether and how fast convergence to equilibrium occurs. Villani (in collaboration with Desvillettes) obtained the first results on the convergence rate for initial data not close to equilibrium. In joint work with Mouhot, he established nonlinear Landau damping for the kinetic equations of plasma physics, settling a long-standing debate. He is one of the pioneers in applications of optimal transport theory to geometric and functional inequalities. arXiv 2010_11_03_Villani_Notes 2010_11_03

## November 10, Schedler, 16:10-17:00 @BA6183

 Travis Schedler (MIT) 2010-2011 Colloquium Wednesday November 10 16:10-17:00 BA6183 Title: Poisson varieties and D-modules Abstract: Poisson varieties are natural generalizations of symplectic varieties which are allowed to admit singularities. Motivated by questions concerning their quantization, deformation, and resolution, I will discuss and apply Poisson traces on these varieties, which are linear functionals invariant under Hamiltonian vector fields. I will explain an approach based on an elementary application of D-modules and the algebraic theory of differential equations. I will discuss examples of Poisson varieties such as Kleinian and other isolated surface singularities and their symmetric powers, and nilpotent cones of semisimple Lie algebras, using the Springer resolution. As applications, we deduce bounds on numbers of irreducible representations of spherical symplectic reflection algebras (or on zero-dimensional symplectic leaves in the commutative case), and similarly for universal enveloping and W-algebras modulo a central character. No background on Poisson geometry or D-modules will be assumed. This is joint work with Pavel Etingof. arXiv 2010_11_10_Schedler_Notes 2010_11_10

## November 17, Batyrev, 16:10-17:00 @BA6183

 Victor Batyrev (University of Tubingen) 2010-2011 Colloquium Wednesday November 17 16:10-17:00 BA6183 Title: Non-archimedean integrals over algebraic varieties and their applications Abstract: We gives a general overview of ideas behind the non-archimedean (motivic) integration over algebraic varieties and show how non-archimedean integrals help to produce new interesting invariants inspired by birational geometry. arXiv 2010_11_17_Batyrev_Notes 2010_11_17

## November 24, Tomczak-Jaegermann, 16:10-17:00 @BA6183

 Nicole Tomczak-Jaegermann (University of Alberta) 2010-2011 Colloquium Wednesday November 24 16:10-17:00 BA6183 Title: Singular numbers of random matrices; the asymptotic non-limiting theory Abstract: Random matrix theory considers matrices of a finite size whose entries are random variables from a certain class, and typically is interested in the limiting behaviour of various numerical characteristics when the size tends to infinity. Recent years mark an appearance of a new contrasting point of view, when one seeks quantitative results, valid for matrices of an arbitrary size, in form of inequalities for numerical characteristics, depending on parameters, but not on the size of the matrix. In this approach the size is arbitrary and fixed through the argument, although it may be required to be sufficiently large. Although no limiting behaviour of numerical characteristics is expected to be proved in general, in many problems actually the rate of convergence can be established. In this talk we shall present a survey of results of the asymptotic non-limiting theory of singular numbers of random matrices developed in approximately last six years. arXiv 2010_11_24_Tomczak-Jaegermann_Notes 2010_11_24

## December 01, Hayden, 16:10-17:00 @BA6183

 Patrick Hayden (McGill University and Perimeter Institute) 2010-2011 Colloquium Wednesday December 01 16:10-17:00 BA6183 Title: Quantum information as asymptotic geometry Abstract: Quantum states are represented as vectors in an inner product space. Because the dimension of that state space grows exponentially with the number of its constituents, quantum information theory is in large part the asymptotic theory of finite dimensional inner product spaces, a field with its own long history. I’ll highlight some examples of how abstract mathematical results from that area, such Dvoretzky’s theorem, manifest themselves in quantum information theory as improvements on the famous "teleportation" procedure and as the raw material for counterexamples to the area's best-known conjecture. More recently, this perspective has led to methods for encrypting arbitrarily long messages using constant-sized secret keys. arXiv 2010_12_01_Hayden_Notes 2010_12_01

## December 08, Klainerman, 16:10-17:00 @BA6183

 Sergiu Klainerman http://www.math.princeton.edu/~seri/ (Princeton University) 2010-2011 Colloquium Wednesday December 08 16:10-17:00 BA6183 Title: Mis-facts and dreams in General Relativity Abstract: arXiv 2010_12_08_Klainerman_Notes 2010_12_08

# Spring Term:

## January 19, Stuart, 16:10-17:00 @BA6183

 David Stuart (DAMPT) 2010-2011 Colloquium Wednesday January 19 16:10-17:00 BA6183 Title: Topological solitons and dispersion Abstract: We will discuss topological solitons and issues which arise in attempts to understand their asymptotic stability and large time dynamics. Examples include monopoles in Yang-Mills theories and magnetic Ginzburg-Landau vortices. Dispersive analysis in the presence of such objects involves Schrodinger operators whose coefficients decay more slowly than usually assumed in perturbative analysis. For example in the case of Ginzburg-Landau vortices it is necessary to understand dispersive estimates for two dimensional magnetic Schrodinger operators in the presence of non-zero magnetic fluxes. This latter fact implies directly that the gauge potential necessarily has slow (critical) decay which means the problem cannot be treated as a perturbation of the free Schrodinger evolution. We consider model problems in which it is possible to develop an analytical understanding of this situation. arXiv 2011_01_19_Stuart_Notes 2011_01_19

## January 26, Saulina, 16:10-17:00 @BA6183

 Natalia Saulina (Perimeter Institute) 2010-2011 Colloquium Wednesday January 26 16:10-17:00 BA6183 Title: Surface operators in 3d topological field theory Abstract: Surface operators in a 3d topological field theory form a 2-category. I will give an explicit example of this structure in abelian Chern-Simons gauge theory and in Rozansky-Witten topological sigma-model. In particular, I will describe properties of line defects(1-morphisms in the 2-category) sitting at the junction of two surface operators. In Rozansky-Witten model, connection with monoidal deformations of the derived category of coherent sheaves will be discussed. In Chern-Simons theory, surface operators will be used to explain the algebraic construction of 2d rational conformal field theory. arXiv 2011_01_26_Saulina_Notes 2011_01_26

## February 2, Dembo, 16:10-17:00 @BA6183

 Amir Dembo (Stanford University) 2010-2011 Colloquium Wednesday February 2 16:10-17:00 BA6183 Title: Statistical Mechanics on Sparse Random Graphs Abstract: Theoretical models of disordered materials lead to challenging mathematical problems with applications to random combinatorial problems and coding theory. The underlying structure is that of many discrete variables that are strongly interacting according to a mean field model determined by a random sparse graph. Focusing on random finite graphs that converge locally to trees we review recent progress in validating the cavity prediction for the limiting free energy per spin and the approximation of local marginals by the belief propagation algorithm. This talk is based on joint works with Andrea Montanari and Nike Sun. arXiv 2011_02_2_Dembo_Notes 2011_02_2

## February 9, Lerario, 16:10-17:00 @BA6183

 Antonio Lerario (SISSA, Trieste) 2010-2011 Colloquium Wednesday February 9 16:10-17:00 BA6183 Title: Systems of quadratic inequalities Abstract: Systems of quadratic inequalities are very flexible objects in mathematics, e.g any system of polynomial equations can be reduced to a system of quadratic equations by substitutions. Thus the set X of the solutions of a system of quadratic inequalities can describe a very large class of semi-algebraic sets (the complexity of X is hidden in the number of linearly independent inequalities). To study such a system we focus on the dual object: the convex hull, in the space of all real quadratic forms on $R^n$, of those quadratic forms involved in the system (n is the number of variables in the system). It turns out that the homology of $X$ is determined by the arrangement of this convex hull with respect to the cone of degenerate forms. This approach allows to efficiently compute homology for a very big number of variables n as long as the number of linearly independent inequalities is limited. Moreover, it works also for systems of integral quadratic inequalities, i.e. in the infinite dimension, beyond the semi-algebraic context. The calculations are organized in a spectral sequence whose member $E_2$ and the differential $d_2$ have a simple clear geometric interpretation. This is a joint work with A. Agrachev. arXiv 2011_02_9_Lerario_Notes 2011_02_9

## February 16, ', 16:10-17:00 @BA6183 Reserved by Gualtieri

 () 2010-2011 Colloquium Wednesday February 16 16:10-17:00 BA6183 Title: TBA Abstract: TBA arXiv 2011_02_16__Notes 2011_02_16

## March 2, Krichever, 16:10-17:00 @BA6183

 Igor Krichever (Columbia University) 2010-2011 Colloquium Wednesday March 2 16:10-17:00 BA6183 Title: Integrable systems and geometry of Riemann surfaces Abstract: The remarkable Welter's trisecant conjecture: an indecomposable principally polarized abelian variety $X$ is the Jacobian of a curve if and only if there exists a trisecant of its Kummer variety $K(X)$, was motivated by the celebrated Gunning's theorem and by another famous conjecture: the Jacobians of curves are exactly the indecomposable principally polarized abelian varieties whose theta-functions provide explicit solutions of the so-called KP equation. The latter was proposed earlier by Novikov and was unsettled at the time of the Welter's work. It was proved later by T.Shiota and until recently has remained the most effective solution of the classical Riemann-Schottky problem. The characterization of the Jacobains proposed by the trisecant conjecture is much stronger. The proof of this conjecture based on an notion of integrable linear equations and new type cubic identities for the theta-functions valid for the case of Jacobians on the theta-divisor will be presented. We will also discuss applications of integrable equations of the soliton theory for the characterization problem of Prym varieties. arXiv 2011_03_2_Krichever_Notes 2011_03_2

## March 9, Smets, 16:10-17:00 @BA6183

 Didier Smets (Paris 6 and Ecole Normale Superieure) 2010-2011 Colloquium Wednesday March 9 16:10-17:00 BA6183 Title: On the stability of filament flows and Schrodinger maps Abstract: The study of vortex rings in incompressible 3D fluids dates back to Kelvin and Helmholtz in the mid 1800's. In 1906, Da Rios and Levi-Civita derived a geometric flow for filaments of infinitely small cross section and arbitrary shape. This flow is now widely called the binormal curvature flow or the LIA flow. In the talk, I will first review and then present recent results on stability estimates for the filament flow, and their application to so-called Schrodinger maps. arXiv 2011_03_9_Smets_Notes 2011_03_9

## March 16, ', 16:10-17:00 @BA6183

 () 2010-2011 Colloquium Wednesday March 16 16:10-17:00 BA6183 Title: TBA Abstract: TBA arXiv 2011_03_16__Notes 2011_03_16

## March 23, Rouquier, 16:10-17:00 @BA6183

 Raphael Rouquier (University of Oxford) 2010-2011 Colloquium Wednesday March 23 16:10-17:00 BA6183 Title: Dunkl operators: from analysis to algebra and back Abstract: We will introduce deformations of partial derivatives (Dunkl operators) and discuss their action on polynomial functions. This is encoded in an algebra of deformed differential operators, a Cherednik algebra. Such algebras are related to combinatorics, to singularities and symplectic geometry, and to integrable systems. We will explain how representations of Cherednik algebras can be studied via monodromy representations, leading to a relation with Hecke algebras. This can be lifted through conformal blocks to a more precise relation with affine general linear Lie algebras. On the other hand, the Cherednik algebras can be microlocalized. This microlocalization provides a quantization of the Hilbert schemes of points on the complex plane. arXiv 2011_03_23_Rouquier_Notes 2011_03_23

## March 30, Guth, 16:10-17:00 @BA6183

 Larry Guth (University of Toronto and IAS) 2010-2011 Colloquium Wednesday March 30 16:10-17:00 BA6183 Title: The Erdos distinct distance problem Abstract: Nets Katz and I proved that a set of $N$ points in the plane determines at least $N (\log N)^{-1}$ different distances. This estimate is sharp up to logarithmic factors. It builds on a plan laid out by Elekes and Sharir. In the talk, I will try to explain the main ideas of the proof. The two highlights are an application of basic algebraic geometry such as ruled surfaces, and an application of the ham sandwich theorem. arXiv 2011_03_30_Guth_Notes 2011_03_30

## April 6, Rodnianski, 16:10-17:00 @BA6183

 Igor Rodnianski (Princeton University) 2010-2011 Colloquium Wednesday April 6 16:10-17:00 BA6183 Title: On the formation of singularities in geometric evolution equations Abstract: I will review recent results on singularity formation for the Wave and Schrodinger Map problems. In both cases we uncover a large class of solutions, arising from smooth data, which break down in finite time via concentration of a corresponding harmonic map. The mechanisms for the problems however display substantial qualitative and quantitative differences. I will also discuss the connection between the singularity formation result for the Wave Map and the behavior of geodesics on moduli space, which had been the subject of the "geodesic hypothesis". arXiv 2011_04_6_Rodnianski_Notes 2011_04_6

## April 13, Brudnyi, 16:10-17:00 @BA6183

 Alex Brudnyi (University of Calgary) 2010-2011 Colloquium Wednesday April 13 16:10-17:00 BA6183 Title: Banach-valued Holomorphic Functions on the Maximal Ideal Space of $H^\infty$ and a Problem of Sz. Nagy. Abstract: We present a new direction in the theory of the $H^\infty$ Banach algebra of bounded holomorphic functions on the unit disk with the pointwise multiplication and the supremum norm. Specifically, we study the Banach-valued holomorphic functions defined on open subsets of the maximal ideal space of $H^\infty$. As in the theory of Stein spaces, we establish vanishing of the cohomology for sheaves of germs of such functions, solve the second Cousin problem and prove Runge-type approximation theorems for them. Then we prove that the maximal ideal space of the algebra $H_{comp}^\infty (A)$ of holomorphic holomorphic functions on the unit disk with relatively compact images in a commutative unital complex Banach algebra $A$ is homeomorphic to the direct product of the maximal ideal spaces of $H^\infty$ and $A$. (The problem was posed in the 1960s.) Finally, we establish triviality of some Banach holomorphic vector bundles defined on the maximal ideal space of $H^\infty$ and apply this result to the operator corona problem posed by Sz. Nagy in 1978 solving it for holomorphic operator-valued functions with relatively compact supports. arXiv 2011_04_13_Brudnyi_Notes 2011_04_13

# Colloquium Reservations

• If you wish to reserve a date, email Marco Gualtieri (mgualt@math.toronto.edu).