Departmental Colloquium

	Date	Speaker	Talk Information
View details	September 6	János Kollár Princeton University, currently visiting the University of Utah	Let M be a subset of C n defined as the common zero set of some holomorphic functions. What can one say about the local structure of M ? It turns out that M is a smooth manifold almost everywhere. Our main interest is in the question: How complicated can M be at special points? As an exercise, you can try to see what happens with M 1 ∈ C 3 given by x 2 + y 3 + z 6 =0 and with M 2 ∈ C 5 given by x 2 + y 2 + z 2 + t 3 + u 5 =0.
View details	September 27	Mark Reeder Boston College	Geometric Invariant Theory (GIT) is the study of orbits of algebraic groups acting on vector spaces. Such actions arise in the "epipelagic zones" of a reductive p-adic group. Besides explaining the previous sentence, I will show how the GIT of epipelagic zones leads to new constructions of representations of p-adic algebraic groups. Such constructions lead, via the conjectural Local Langlands correspondence, to verifiable predictions relating p-adic Galois theory to complex simple Lie algebras, which otherwise appear to be completely different areas of mathematics.
View details	October 18	Tyler Jarvis Brigham Young University, currently visiting the University of Utah	I will talk about connections between binomial coefficients, exterior products, symmetric polynomials, and power sums. Although some of these ideas and relations predate Newton, they provide tools that are surprisingly useful for studying "modern" objects, including manifolds or varieties with group actions (orbifolds), representation theory, and resolutions of singularities.
View details	November 8	Anna Vainchtein University of Pittsburgh	Propagation of phase boundaries, cracks and dislocations in crystal lattices is associated with energy dissipation that takes place in atomically sharp transition zones. Classical continuum theories represent these lattice defects as singularities, and the information about their kinetics is lost. This can lead to non-uniqueness of solutions of the associated initial value problem unless an additional constitutive function that relates the velocity of a moving defect to the driving force is specified. This kinetic relation can be extracted from the underlying discrete model by considering a traveling wave solution representing the moving defect. In this talk, I will illustrate this by considering some prototypical discrete systems with nonconvex interactions. I will then introduce a more general concept of kinetic equations that are nonlocal in time.
View details	November 15	Dieter Kotschick Ludwig-Maximilians-Universität München	The Hirzebruch-Riemann-Roch theorem implies certain relations between the Hodge and Chern numbers of a complex projective variety. After proving his theorem in 1953, Hirzebruch formulated several natural problems about these characteristic numbers. For example, he asked to what extent the Hodge and Chern numbers are topological invariants of the underlying manifold. I will explain the answer to this question obtained in recent joint work with S. Schreieder. Along the way we proved that the Hirzebruch-Riemann-Roch relations are the only universal relations between Hodge and Chern numbers, and we gained some understanding of the collections of Hodge numbers for arbitrary compact Kähler manifolds.
View details	November 29	Skip Garibaldi Emory University, visiting UC San Diego	Are matrix groups determined by their maximal tori? Over number fields, this is an old question attributed to Shimura. Translated into a question about rings, it asks: Are division algebras with the same subfields isomorphic or anti-isomorphic? The answers to these two questions are in many cases "no", but are "yes" in some interesting cases that have applications to analogues of the question "Can you hear the shape of a drum?" for locally symmetric spaces.
View details	December 6	Tom Alberts California Institute of Technology	A multiplicative cascade is a randomization of any measure on the unit interval, constructed from an iid collection of random variables indexed by the dyadic intervals. Given an arbitrary initial measure I will describe a method for constructing a continuous time, measure valued process whose value at each time is a cascade of the initial one. The process also has the Markov property, namely at any given time it is a cascade of the process at any earlier time. It has the further advantage of being a martingale and, under certain extra conditions, it is also continuous. I will discuss applications of this process to models of tree polymers and one-dimensional random geometry. Joint work with Ben Rifkind (University of Toronto).

Date

Speaker

Talk Information

September 6

János Kollár
Princeton University, currently visiting the University of Utah

Let M be a subset of C n defined as the common zero set of some holomorphic functions. What can one say about the local structure of M ? It turns out that M is a smooth manifold almost everywhere. Our main interest is in the question: How complicated can M be at special points? As an exercise, you can try to see what happens with M 1 ∈ C 3 given by x 2 + y 3 + z 6 =0 and with M 2 ∈ C 5 given by x 2 + y 2 + z 2 + t 3 + u 5 =0.

View details

September 27

Mark Reeder
Boston College

Geometric Invariant Theory (GIT) is the study of orbits of algebraic groups acting on vector spaces. Such actions arise in the "epipelagic zones" of a reductive p-adic group. Besides explaining the previous sentence, I will show how the GIT of epipelagic zones leads to new constructions of representations of p-adic algebraic groups. Such constructions lead, via the conjectural Local Langlands correspondence, to verifiable predictions relating p-adic Galois theory to complex simple Lie algebras, which otherwise appear to be completely different areas of mathematics.

View details

October 18

Tyler Jarvis
Brigham Young University, currently visiting the University of Utah

I will talk about connections between binomial coefficients, exterior products, symmetric polynomials, and power sums. Although some of these ideas and relations predate Newton, they provide tools that are surprisingly useful for studying "modern" objects, including manifolds or varieties with group actions (orbifolds), representation theory, and resolutions of singularities.

View details

November 8

Anna Vainchtein
University of Pittsburgh

Propagation of phase boundaries, cracks and dislocations in crystal lattices is associated with energy dissipation that takes place in atomically sharp transition zones. Classical continuum theories represent these lattice defects as singularities, and the information about their kinetics is lost. This can lead to non-uniqueness of solutions of the associated initial value problem unless an additional constitutive function that relates the velocity of a moving defect to the driving force is specified. This kinetic relation can be extracted from the underlying discrete model by considering a traveling wave solution representing the moving defect. In this talk, I will illustrate this by considering some prototypical discrete systems with nonconvex interactions. I will then introduce a more general concept of kinetic equations that are nonlocal in time.

View details

November 15

Dieter Kotschick
Ludwig-Maximilians-Universität München

The Hirzebruch-Riemann-Roch theorem implies certain relations between the Hodge and Chern numbers of a complex projective variety. After proving his theorem in 1953, Hirzebruch formulated several natural problems about these characteristic numbers. For example, he asked to what extent the Hodge and Chern numbers are topological invariants of the underlying manifold. I will explain the answer to this question obtained in recent joint work with S. Schreieder. Along the way we proved that the Hirzebruch-Riemann-Roch relations are the only universal relations between Hodge and Chern numbers, and we gained some understanding of the collections of Hodge numbers for arbitrary compact Kähler manifolds.

View details

November 29

Skip Garibaldi
Emory University, visiting UC San Diego

Are matrix groups determined by their maximal tori? Over number fields, this is an old question attributed to Shimura. Translated into a question about rings, it asks: Are division algebras with the same subfields isomorphic or anti-isomorphic? The answers to these two questions are in many cases "no", but are "yes" in some interesting cases that have applications to analogues of the question "Can you hear the shape of a drum?" for locally symmetric spaces.

View details

December 6

Tom Alberts
California Institute of Technology

A multiplicative cascade is a randomization of any measure on the unit interval, constructed from an iid collection of random variables indexed by the dyadic intervals. Given an arbitrary initial measure I will describe a method for constructing a continuous time, measure valued process whose value at each time is a cascade of the initial one. The process also has the Markov property, namely at any given time it is a cascade of the process at any earlier time. It has the further advantage of being a martingale and, under certain extra conditions, it is also continuous. I will discuss applications of this process to models of tree polymers and one-dimensional random geometry. Joint work with Ben Rifkind (University of Toronto).

Department of Mathematics

College of Science

Fall 2012