Colloquium

Research seminars

VIGRE Working Groups

Clifford lectures

David M Bressoud, Macalester College

"Exploiting Symmetries: Alternating Sign Matrices and the Weyl Character Formulas"

Abstract:

This will be an overview of some of the points of interaction between symmetric functions and representation theory on the one hand, and questions in number theory and combinatorics on the other, culminating in recent work of Okada enumerating alternating sign matrices (aka the six-vertex model of statistical mechanics) through evaluations of Weyl character formulas.

Alessandro Conflitti , DMUC Universidade de Coimbra, Portugal

"Highly generalized Eulerian calculus on Coxeter systems"

Abstract:

Classical Eulerian calculus deals with the distribution of the descent and inversion statistics on the symmetric group, the archetypal example of Coxeter system.

We define a family of statistics over a generic finite Coxeter system indexed by subsets of its reflections set, thus highly generalizing the above--mentioned ones. We study the corresponding generating functions, proving that they have a lot of interesting combinatorial properties.

Specifically, we prove equidistribution results, namely we investigate some conditions in order to have that different subsets have the same associated generating function. In particular, considering the symmetric group $S_{n}$ we prove that any poset of size $n$ corresponds to a subset of traspositions of $S_{n}$, and that the generating function of the corresponding statistic includes partial linear extensions of such a poset. Finally, we explicitily compute the associated generating functions for several classes of subsets.

Jason Behrstock, Columbia University

"Quasi-isometric classification of 3-manifold groups"

Any finitely generated group can be endowed with a metric which is unique up to maps of bounded distortion (quasi-isometries). A fundamental question is to classify finitely generated groups up to quasi-isometry. Considered from this point of view, fundamental groups of 3-manifolds provide a rich source of examples. We will survey the world of 3-manifold groups from classical results to the recent resolution of some long standing questions. This talk is intended for a broad audience of mathematicians.

Matthew Hedden , MIT

" Symplectic geometry and invariants for low-dimensional topology "

Abstract:

Over the past few years, ideas from symplectic geometry have had a major impact on low-dimensional topology.  Some of the most impressive results stem from a set of invariants developed by Ozsvath and Szabo.  Though defined using symplectic geometry, they turn out to be surprisingly powerful invariants of low-dimensional objects e.g. knots, and three- and four-manifolds.  In this talk, I will survey these invariants and discuss how I have used them to prove results related to knot theory, complex curves,  surgery theory in dimension three, and the theory of foliations and contact structures on three-manifolds.

Friday

February 8

3:30pm

G310

Max Wakefield,Hokkaido University- Japan

"The characteristic polynomial of a multiarrangement"

Abstract:

One of the most fundamental invariants of a hyperplane arrangement is its characteristic polynomial. For a hyperplane arrangement this polynomial encodes combinatorial, topological, and even algebraic information. In this talk we will review this characteristic polynomial and then discuss a generalization to multiarrangements of hyperplanes.

Louiza Fouli, UT Austin

"The core of ideals"

Abstract:
Let R be a Noetherian local ring with infinite residue field k and I an R-ideal. The ideal J is a reduction of I if J I and I^r+1 =JI^r for some positive integer r. A reduction can be thought of as a simplification of the ideal I. The notion of a reduction for an ideal was introduced by D.
Northcott and D. Rees in order to study multiplicities. Reductions are connected to the study of blowup algebras such as the Rees ring R(I)= R[It] of I, and the associated graded ring gr_{I} (R)=R[It]/IR[It] of I.

In general minimal reductions are not unique. To remedy this lack of uniqueness, one considers the intersection of all reductions, namely the core of the ideal, core(I). This object, that appears naturally in the context of the Brian\c con-Skoda theorem, encodes information about all possible reductions. We present some recent work on the shape of the core of ideals.

Hailong Dao, University of Utah

"On nice geometric and homological properties of algebraic sets"

Rafe Jones ,University of Wisconsin- Madison

"Arithmetic dynamics and Galois groups of iterated maps"

Abstract:

I'll begin with a brief overview of arithmetic dynamics, which seeks to understand the arithmetic of orbits in discrete dynamical systems. I have considered a question about the set of primes dividing certain orbits, and I will explain how this question reduces to one involving properties of Galois groups of iterated maps. These groups remain mysterious in many cases, although they have many applications, some of which I will discuss. Finally, I'll talk about some of the techniques involved in understanding these groups, and give directions for future projects.

Speaker, Institution

"Title"

Speaker, Institution

"Title"

Nathan Jones , Centre de Recherches Mathématiques Université de Montréal,

"Serre curves and their division fields"

The cyclotomic fields comprise one of the most classical families of number fields.  The N-th cyclotomic field is a Galois extension of the rational numbers whose Galois group is isomorphic to (Z/NZ)^*, the unit group of the integers modulo N.  In this talk, I will discuss a two-dimensional analogue of the N-th cyclotomic field, namely the N-th division field of an elliptic curve, and discuss a theorem that for "almost all" elliptic curves, the N-th division field has Galois group isomorphic to GL_2(Z/NZ), the unit group of the ring of 2 by 2 matrices modulo N.  If time permits, I will discuss applications of this theorem to the problem of averaging constants appearing in various conjectural prime-counting asymptotics attached to elliptic curves.

SPRING BREAK

Mahir Can, University of Pennsylvania

"Recent advances on the rook monoid"

Shellability is a combinatorial property of a cell complex (of a poset) with important topological and algebraic consequences. For example, a shellable complex has the homotopy type of a wedge of r-spheres and its Stanley-Reisner ring is Cohen-Macaulay. Among the interesting classes of shellable posets is the symmetric group with respect to the Bruhat-Chevalley ordering. The rook monoid is a finite (inverse) monoid having symmetric group as its group of invertible elements. There is a natural extension of the Bruhat-Chevalley ordering on the rook monoid (originating from the Bruhat decomposition of the nxn matrices).

In this talk, we shall first give an overview of the symmetric group and the rook monoid,
then we shall work-out some examples showing that the rook monoid is shellable. No prior knowledge is required.

Bangere Purnaprajna, University of Kansas

"Classification of algebraic varieties"

Classification problems are similar to taxonomy in botany or zoology. It is necessary to give a good division and hierarchy to algebraic varieties before we can study them. One wants also to find crucial characteristics to tell us when a given specimen, that is, a given variety, belongs to a certain family.

To separate varieties by dimension of a variety is an obvious first step. The second step is to classify varieties of a given dimension. There is a rough division for varieties of a given dimension based on a quantity called the Kodaira dimension denoted by k, which is a number between -1 and the dimension of the variety. This is a very coarse classification and is like assigning a given vertebrate into fish, amphibians, reptiles, birds and mammals and not being able to say more. For instance, not being able to distinguish between a whale and a bat.

Geometers like their counterparts in zoology want a much finer classification accounting many other complexities. Even for varieties of dimension two (also called surfaces), a finer classification is far from complete. We will concentrate on the finer classification of algebraic surfaces for this talk.

Ian Aberbach , University of Missouri, Columbia

"Uniform bounds in Noetherian Rings"

The talk will try to give a flavor of such results, touching on, for instance, uniform Artin-Rees theorems, uniform annihilation of local cohomology and the connection to uniform annihilators of homology in classes of free complexes, tight closure and its uniform annihilation (i.e., test elements), Briancon-Skoda type theorems, and uniform degrees of nilpotency for parameter ideals.

Mac Hyman , Los Alamos National Laboratory

" New Approaches to Predicting the Spread of Epidemics "

Mathematical models based on the underlying transmission mechanisms of a disease can help the medical/scientific community anticipate the spread of an epidemic and evaluate the potential  effectiveness of different approaches for bringing the epidemic under control. I will describe how these models can aid in understanding of the underlying transmission pathways  of an epidemic and be used to estimate the benefits and the costs of possible interventions. I will describe how the early epidemic models have evolved to account for variations in the infectiousness and transmissibility of different diseases, behavior changes in response to an epidemic, the impact of biased mixing and variations in behavior among people in a susceptible population, and to quantify the uncertainty in the predictions.  The lecture will describe the broad classes of epidemic models, from simple ordinary differential equations to massive stochastic agent-based simulations for understanding the spread of a disease within a major city.  I will describe how these new  models are creating a mathematical foundation  to facilitate collaborations among the biological, public health, behavioral, social, and mathematical science communities.   

John Lowengrub , University of California, Irvine

"Controlling the shapes of micro- and nano- scale structures"

The ultimate goal of materials design is to start by a specifying a set of desirable properties and then to follow-up by fabricating a material that meets these specifications optimally. By controlling and patterning the micro- and nano- structures, this dream is growing ever closer to reality. Materials that can be uniquely targeted to specific applications have the potential to make an enormous technological impact. In this talk, we present mathematical theory that can be used for controlling the shapes of growing crystals at the microscale and controlling the spatial orientation of nanostructures (quantum dots) during epitaxial growth of thin films.

At the microscale, we demonstrate that there exist critical conditions of growth such that the Mullins-Sekerka instability may be suppressed and instead universal limiting shapes exist. That is, we find that the morphologies of the nonlinearly evolving crystals tend to limiting shapes that evolve self-similarly and depend only on the far-field conditions. We then design protocols by which the compact growth of crystals with desired symmetries can be achieved. We present both 2D and 3D results using adaptive boundary integral methods. Preliminary experimental results are presented that suggest the confirmation of the theory.

The theory at the microscale is then extended to nanoscale studies of monolayer, epitaxially growing islands. Here, the control variables are the deposition flux and a far-field flux that can be manipulated so as to control the shape of the island. We conclude with a study of strained epitaxial thin films. In this case, the relaxation of strain provides a mechanism for influencing the self-organization of quantum dot structures. Using newly-developed, adaptive phase-field methods, we demonstrate that strain patterning, as well as control of the deposition flux, may result in ordered self-organized arrays of nanostructures (quantum dots).

Christo Christov, University of Louisiana at Lafayette

"On the Two-Dimensional Boussinesq Solitons"

The quasi-particle behavior of solitons is very important for physical applications. While the mechanics of soliton interactions is well understood in 1D, the works in multidimension are scant. Even the shapes of the steady propagating, noninteracting solitons are not known for most of the main soliton-supporting models. This is mainly due to the fact that no analytical solutions are available in more than one spatial dimension. For this reason, it is very important to develop numerical approaches that can adequately handle the infinite domain and the bifurcation nature of the problem.

We consider here the steady-propagating single soliton of the so-called Proper Boussinesq equation in two spatial dimensions. We develop three different techniques: a finite difference scheme with nonuniform grid; Galerkin spectral technique base on a special complete orthonormal system of functions on the infinite interval; and a perturbation expansion for small phase speeds, c, of propagation of the soliton up to order , including. We demonstrate that the three techniques are in excellent quantitative agreement, which validates them.

Our results show that there is a dramatic change of the asymptotic behavior of the soliton shape when for phase speeds , i.e. the decay at infinity of the shape of a moving soliton is qualitatively different from the standing soliton. The latter decays super-exponentially with the radial coordinate, while the former decays as . This makes the moving solitons spread wider than the standing soliton. In addition, small depressions (of order of ) appear in the forrunner and the trail of the profile. This appears to be a novel result and can have a profound impact on the way the interaction of two (or more) two-dimensional solitons is modeled. The potential of interaction of two quasi-particles identified with the respective solitons depends predominantly on the asymptotic behavior of soliton's shapes. Thus the algebraically decaying shapes yield an algebraically-decaying attraction potential, which is qualitatively different from the 1D case, where the attraction potential always decay exponentially. The 2D solitons `feel' each other at much farther distances than the 1D ones.

Carlos Julio Moreno, CUNY

"Applications of Hensel's Lemma to the representation theory of SL(2)"

The Representation theory of the p-adic group SL(2) differs significantly in the two cases: p=2 and p odd. We shall examine some of the number theoretic reasons for this from the optic of Hensel's Lemma.