# 2009-2010 Colloquium

## February 24, 2010: Alex Nabutovsky, TBA

Speaker: Alex Nabutovsky (Toronto)
Title: TBA

## January 27, 2010: Jeremy Quastel, TBA

Speaker: Jeremy Quastel
Title: TBA

## January 20, 2010: Askold Khovanskii, Bernstein-Kushnirenko theorem and its generalizations

Title: Bernstein-Kushnirenko theorem and its generalizations
Abstract: The Bernstein-Kushnirenko theorem computes the number of solutions in $(\mathbb{C}^*)^n$ of a system of equations $P_1 = 0, \dots = P_n = 0$, where

$P_1, \dots, P_n$ are generic functions with given Newton polyhedra. The answer is given in terms of the mixed volumes of these Newton polyhedra. Recently Kiumars Kaveh and myself have found far-reaching generalizations of this theorem. Among these generalizations there are: an extension of intersection theory of divisors, a new version of Hodge index inequality, elementary proofs of Alexandrov-Fenchel inequality in convex geometry and its analogues in algebraic geometry, a version of Bernstein{Kushnirenko theorem for varieties equipped with an action of a reductive algebraic group. I will try to explain these results.

Abstract.

## December 9, 2009; Dan Knopf; TBA

Speaker: Dan Knopf (UT Austin)
Title:TBA
Abstract: TBa

## December 2, 2009, 16:10; Laszlo Erdös, The local relaxation flow approach to universality for random matrices

Speaker: Laszlo Erdös (Munich, Harvard)
Title: The local relaxation flow approach to universality for random matrices
Abstract: The local eigenvalue statistics of the Gaussian Unitary Ensemble (GUE) is given by Dyson's celebrated sine kernel. The universality conjecture states that this law also holds for a much more general class of random matrices. In this talk I present a new method to approach this problem via a localized stochastic relaxation flow for the eigenvalues. The core of the method is general and is able to establish the universality of local spectral statistics of a broad class of large random matrices. We show that the local distribution of the eigenvalues coincides with the local statistics of the corresponding Gaussian ensemble provided the eigenvalues are close to their classical location determined by the limiting density of eigenvalues. This information can be obtained by well established methods for various matrix ensembles. We demonstrate the method by proving local spectral universality for Wigner and Wishart matrices.

## November 25, 2009; Mircea Mustata; Log canonical thresholds: an overview

Speaker: Mircea Mustata (Michigan)
Title: Log canonical thresholds: an overview
Abstract: The log canonical threshold is an invariant of singularities that comes up in many different contexts: it is related to integrability of functions, to solving equations modulo pm, and to spaces of arcs. After an overview of characterizations in various settings, I will discuss some recent results and open problems.

## November 18

16:10

Morwen Thistlethwaite (Tennessee): The relative merits of hyperbolic and algebraic knot invariants When Wed, November 18, 16:10 – 17:00 Where Department of Mathematics, 40 St. George Str., Toronto, ON M5S 2E4, Canada (map) Description The relative merits of hyperbolic and algebraic knot invariants The canonical cell decomposition of the complement of a hyperbolic knot is a complete invariant of the knot type, and is computed with great rapidity by the program SnapPea. So what is the catch? Why is it sometimes preferable to forego hyperbolic geometry and instead resort to representations of the knot group onto non-solvable finite groups? These questions will be discussed in relation to the classification of knots in the tables and the determination of their symmetries. As an aside, a uniform method will be described for computing hyperbolic structures on knot complements, by attaching labels to the crossings and edges of a knot diagram and then solving some equations.

## November 11

16:10

Felipe Cucker (CUHK): On a problem posed by Steve Smale When Wed, November 11, 16:10 – 17:00 Where BA6183 (map) Description Wednesday November 11 Felipe Cucker (City University of Hong Kong) On a problem posed by Steve Smale

## November 4

16:10

Vitali Milman (Tel Aviv): Duality and Rigidity for Family of Convex Functions in R^n When Wed, November 4, 16:10 – 17:00 Where BA6183 (map) Description Wednesday November 4 Vitali Milman (University of Tel Aviv) Duality and Rigidity for Family of Convex Functions in R^n

## October 28

16:10

Benoit Collins (Ottawa): Free probability theory and quantum information theory When Wed, October 28, 16:10 – 17:00 Where BA6183 (map) Description Wednesday October 28 Benoit Collins (University of Ottawa) Free probability theory and quantum information theory.

## October 21

16:10

Thomas Spencer (IAS): Statistical mechanics, random band matrices and hyperbolic symmetry When Wed, October 21, 16:10 – 17:00 Where BA6183 (map) Description Wednesday October 21 Thomas Spencer (Institute for Advanced Study) Statistical mechanics, random band matrices and hyperbolic symmetry.

## October 14

16:10

Larry Guth (Toronto): Using algebraic topology to prove inequalities about surface areas When Wed, October 14, 16:10 – 17:00 Where BA6183 (map) Description Wednesday October 14 Larry Guth (Toronto) Using algebraic topology to prove inequalities about surface areas.

## October 7

16:10

Thomas Zink (Bielefeld): Overconvergent Witt differentials and rigid cohomology When Wed, October 7, 16:10 – 17:00 Where BA6183 (map) Description Wednesday October 7 Thomas Zink (Universität Bielefeld) Overconvergent Witt differentials and rigid cohomology.

## September 30

16:10

Colloquium: Departmental Meeting When Wed, September 30, 16:10 – 17:00 Where BA6183 (map) Description Wednesday September 30 Departmental meeting

## September 23

16:10

Vikraman Balaji (Chennai): Vector bundles and non-abelian mathematics When Wed, September 23, 16:10 – 17:00 Where BA6183 (map) Description Wednesday September 23 Vikraman Balaji (Chennai Mathematical Institute) Vector bundles and non-abelian mathematics

## September 16 2009 (15:30@Fields230); Hendrik Lenstra; Modelling finite fields

Speaker: Hendrik Lenstra (Leiden):

Title:Modelling finite fields
Abstract: (Distinguished Lecture Series September 16-18, 2009 Fields Institute, Room 230 3:30 - 4:30 pm) Finite fields made their first explicit appearance in the group-theoretic investigations of the French mathematician Evariste Galois, in 1830. Nowadays they play an important role in many parts of pure and applied mathematics. Concrete computations in a finite field require the availability of an explicit model for the field. The present lecture series addresses a number of fundamental issues that arise in the context of designing such a model. What should, in the first place, be meant by an "explicit model" for a finite field? Can such a model be constructed efficiently? And can it be recognized? Further issues arise if different models for the "same" finite field are encountered. Can an identification between two such models be found efficiently? And if there are more than two, how can one guarantee the consistency of the several pairwise identifications found? Between any two finite fields of the same cardinality there is an isomorphism, but that isomorphism is not in general canonically determined. In the algorithmic world the situation turns out to be better: between any two explicit models for finite fields of the same cardinality one can efficiently construct an isomorphism that may for all practical purposes be called canonical. This surprising result, which may well have practical implications, was recently proved in collaboration with Bart de Smit. It depends on the good algorithmic properties of suitably defined "standard" models for finite fields. The lectures address a general mathematical audience, and they do not presuppose any specialized knowledge. A precise formulation of the key results requires the language of theoretical computer science, but the proof techniques are all taken from algebra and number theory.

## Colloquium Organizer

• Marco Gualtieri (Send comments and questions to mgualt@math.toronto.edu)