Departmental Colloquium 2012-2013
Thursdays, 4:15 PM, JWB 335
Spring 2013
January 24: SPECIAL COLLOQUIUM
Speaker: Srikanth Iyengar, University of Nebraska,
Lincoln
Title: Group algebras and commutative rings
Abstract: The goal of this talk will be to describe a bridge between the modular representation
theory of finite groups and modules over polynomial rings. This has given us new insights and results
concerning modular representations, and has also lead to unexpected results in commutative algebra. The
talk with be based on joint work with Avramov, Benson, Carlson, Buchweitz, Krause, and Claudia Miller.
January 31:
Speaker: Peter Trapa, University of Utah
Title: Unitary representations of reductive Lie groups
Abstract: Unitary representations of Lie groups appear in many places in mathematics: in harmonic analysis (as generalizations of
the sines and cosines appearing in classical Fourier analysis); in number theory (as spaces of modular and automorphic forms); in quantum
mechanics (as "quantizations" of classical mechanical systems); and in many other places. They have been the subject of intense study for
decades, but their classification has only recently recently emerged. Perhaps surprisingly, the classification has inspired connections
with interesting geometric objects (equivariant mixed Hodge modules on flag varieties). These connections have made it possible to
extend the classification scheme to other related settings.
The purpose of this talk is to explain a little bit about the history and motivation behind the study of unitary representations and
offer a few hints about the algebraic and geometric ideas which enter into their study. This is based on a recent preprint with Adams,
van Leeuwen, and Vogan.
February 7: SPECIAL COLLOQUIUM
Speaker: Sébastien Motsch, Center for Scientific
Computation and Mathematical Modeling, University of Maryland
Title: Mathematical modeling of self-organized dynamics
Abstract: In many biological systems, we observe the emergence of
self-organized dynamics (e.g. school of fish, ant colonies, pedestrian
traffic). Modeling is an essential tool to better understand their
behavior. Based on experimental data, we introduce some recent models
which aim to explain such dynamics.
Since biological systems can reach up to millions of individuals, we
discuss in a second part how we can derive “macroscopic models” from
several microscopic models. In contrast with particle systems in
physics, models of self-organized dynamics do not conserve momentum or
energy. This lack of conservation requires to introduce new tools to
derive and analyze their macroscopic limits.
February 14: SPECIAL COLLOQUIUM
Speaker: Jun Allard, University of California, Davis
Title: Traveling waves in crawling cells
Abstract: Crawling cells, including the white blood cells that patrol
your body in search of infections, display several distinct dynamical
patterns driven by both biochemistry (diffusion and reactions between
chemical species) and mechanics (physical forces between the
components inside cells). Our understanding of these spatiotemporal
patterns has been aided by mathematical modeling using techniques
including partial differential equations (PDEs). Recently, traveling
waves have been observed in the protein actin, which powers certain
cells’ ability to crawl. Following experimental observation of one
type of crawling cell, the fish epithelial keratocyte, we hypothesized
that traveling waves are excitable waves arising from interactions of
three components: actin, adhesion sites that attach the cell to its
environment, and VASP, a protein that regulates actin. We developed a
mathematical model formulated as a system of PDEs with a nonlocal
integral term and noise. Numerical solutions lead to a number of
predictions, including that VASP also exhibits a traveling wave out of
phase with the actin wave, later discovered in further experiments.
Our model also reveals a role for tension in the membrane that
surrounds the cell, which would otherwise be difficult to observe
directly by experiment.
March 7:
Speaker:
Clint Dawson,
University of Texas at Austin
Title: Discontinuous Galerkin methods for coastal ocean applications
Abstract: Flow and transport in the coastal ocean are described by coupled
hydrodynamic systems describing currents, water levels, waves and
atmosphere-ocean coupling. In addition, ocean circulation models can be
used to drive transport models of, e.g., hydrocarbons in the coastal
environment. In this talk, we will describe a discontinuous Galerkin
framework for solving coupled systems arising in coastal ocean
applications. DG methods have proven to be accurate at modeling coastal
ocean physics, we will also discuss local time stepping methods aimed at
improving the efficiency of DG methods.
March 28:
Speaker:
Jean-Luc Thiffeault, University of
Wisconsin-Madison
Title: Biomixing: when organisms stir their environment
Abstract: As fish, micro-organisms, or other bodies move through a fluid, they
stir their surroundings. This can be beneficial to some fish, since
the plankton they eat depends on a well-stirred medium to feed on
nutrients. Bacterial colonies also stir their environment, and this
is even more crucial for them since at small scales there is no
turbulence to help mixing. It has even been suggested that the total
biomass in the ocean makes a significant contribution to large-scale
vertical transport, but this is still a contentious issue. We propose
a simple model of the stirring action of moving bodies through both
inviscid and viscous fluids. An attempt will be made to explain
existing data on the displacements of small particles, which exhibits
probability densities with exponential tails. A large-deviation
approach helps to explain some of the data, but mysteries remain.
This is joint work with Steve Childress, George Lin, and Peter
Mueller.
April 4:
Speaker:
Lisa Fauci, Tulane University
Title: A Tale of Waving Tails: Calcium-Driven Dynamics of Undulatory Swimmers
Abstract: Locomotion due to body undulations is observed across the entire spectrum of swimming organisms, from
microorganisms to fish. The internal force generating mechanisms range from the action of dynein molecular motors within a
mammalian sperm to muscle activation in lamprey. We will present recent progress in building multiscale computational
models that couple biochemistry, passive elastic properties and active force generation with a surrounding fluid for these
two swimmers.
April 11:
Speaker:
Mark Andrea de Cataldo , State
University of New York, Stony Brook
Title: A curious symmetry: P=W.
Abstract: In this talk, which is aimed at non-experts, I will introduce two moduli spaces of structures on a compact Riemann
surface, relate them via a result called non-abelian Hodge theorem, and then discuss a recently observed and somewhat mysterious
symmetry relating their cohomology rings.
This is joint work with T. Hausel and L. Migliorini.
Fall 2012
September 6:
Speaker: János
Kollár, Princeton
University,
currently visiting the University of Utah
Title: Local topology of analytic spaces
Abstract: Let M be a subset of Cn 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 M1∈ C3 given by
x2+y3+z6=0 and with M2∈
C5 given by
x2+y2+z2+t3+u5=0.
September 27:
Speaker: Mark Reeder, Boston
College
Title: Geometric Invariant Theory and epipelagic representations of p-adic
groups
Abstract: 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.
October 18:
Speaker:
Tyler Jarvis, Brigham Young University, currently
visiting the University of Utah
Title: Binomial coefficients, exterior products, symmetric polynomials, and lambda rings
in orbifold geometry
Abstract: 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.
November 8:
Speaker:
Anna Vainchtein ,
University of Pittsburgh
Title: From discrete to continuum: kinetic relations and beyond
Abstract: 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.
November 15
Speaker:
Dieter
Kotschick, Ludwig-Maximilians-Universität
München
Title: Characteristic numbers of algebraic varieties
Abstract: 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.
November 29:
Speaker: Skip
Garibaldi, Emory University, visiting UC San Diego
Title: Algebraic groups with the same tori
Abstract: 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.
December 6: SPECIAL COLLOQUIUM
Speaker: Tom Alberts, California Institute of
Technology
Title: Diffusions of Multiplicative Cascades
Abstract: 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).