Recent Advances in Orbifold Theory

Speaker: 

Professor Yongbin Ruan

Institution: 

Univ. of Wisconsin, Madison

Time: 

Friday, November 12, 2004 - 3:00pm

Location: 

MSTB 254

In this talk, I will give a survey on
some of recent advances in orbifold theory and focus
on the application. It includes the computation for
cohomology of Hilbert scheme of points of algebraic surface,
symplectic resolution, twisted K-theory and many other stuff.

Mathematical models in tumor growth

Speaker: 

Professor Avner Friedman

Institution: 

Ohio State Univ.

Time: 

Monday, February 7, 2005 - 11:00am

Location: 

1114 Natural Sciences 1

Tumor growth has been modeled at the macroscopic level by using established physical laws coup
led with biological processes which are described in a phenomenological fashion. Such model c
onsists of a system of PDEs in the tumor region; this region is changing in time, and thus its
boundary is a "free boundary." In this talk, I shall introduce basic material on free bounda
ry problems, and then proceed to describe models of tumor growth. I shall state results on exi
stence theorems, the shape of the free boundary, and on its asymptotic behavior as time goes t
o infinity.

Localization and String Duality

Speaker: 

Professor Kefeng Liu

Institution: 

UCLA

Time: 

Thursday, November 4, 2004 - 4:00pm

Location: 

MSTB 254

I will discuss the proofs of some conjectural formulas
about Hodge integrals on moduli spaces of curves.
The generating series for all genera and all marked
points of such integrals are expressed in terms of
finite closed formulas from Chern-Simons knot invariants.
Such conjectures were made by string theorists based
on large N duality in string theory. I will explain
our proofs from localization techniques. Their relation
to toric Calabi-Yau manifolds and equivariant index
theory in gauge theory will also be discussed.
These are joint works with C.-C. Liu, J. Zhou and J. Li.

The Integral Basis Problem of Eichler

Speaker: 

Professor Haruzo Hida

Institution: 

UCLA

Time: 

Thursday, October 14, 2004 - 4:00pm

Location: 

MSTB 254

It is a classical problem to determine the span
of the theta series of a given quadraic space over
a small ring. In such a way, Jacobi proved his
famous formula of the number of ways of expressing
integers as sums of four squares.
For the norm form of a definite quaternion algebra B,
we determine the span integrally over very small ring
(for example, if B only ramifies at one prime p,
we shall determine the span over Z[1/(p-1)]).

ANALYSIS ON THE WORM DOMAIN

Speaker: 

Professor Steven Krantz

Institution: 

Washington Univ at St. Louis and MSRI

Time: 

Tuesday, November 30, 2004 - 4:00pm

Location: 

MSTB 254

The Diederich-Fornaess worm domain has proved to be of fundamental
importance in the understanding of the geometry of pseudoconvex domains in
multidimensional complex space. More recently, the worm has proved to be
an important example for the study of the inhomogeneous Cauchy-Riemann
equations in higher dimensions.

In forthcoming work, Krantz and Marco Peloso have done a complete
analysis of the Bergman kernel on a version of the worm domain.
We produce an asymptotic expansion for the kernel and calculate
its mapping properties. We can recover versions of the results
of Kiselman, Barrett, Christ, and Ligocka on the worm.

Phylogenetic Algebraic Geometry

Speaker: 

Professor Bernd Sturmfels

Institution: 

UC Berkeley

Time: 

Thursday, December 2, 2004 - 4:00pm

Location: 

MSTB 254

Many widely used statistical models of evolution are algebraic varieties, that is, solutions sets of polynomial equations. We discuss this algebraic representation and its implications for the construction of maximum likelihood trees in phylogenetics. The ensuing interaction between combinatorial algebraic geometry and computational biology works as a two-way street: biologists may benefit from new mathematical tools, while mathematicians will find a rich source of open problems concerning objects reminiscent of objects familiar from classical projective geometry.

Optimal decisions: From neural spikes, through stochastic differential equations, to behavior.

Speaker: 

Professor Philip Holmes

Institution: 

Princeton Univ.

Time: 

Thursday, January 13, 2005 - 4:00pm

Location: 

McDonnell Douglas Auditorium

There is increasing evidence from in vivo recordings in monkeys
trained to respond to stimuli by making left- or rightward eye
movements, that firing rates in certain groups of `visual' neurons
mimic drift-diffusion processes, rising to a (fixed) threshold prior
to movement initiation. This supplements earlier observations of
psychologists, that human reaction time and error rate data can be
fitted by random walk and diffusion models, and has renewed interest
in optimal decision-making ideas from information theory and
statistical decision theory as a clue to neural mechanisms.

I will review some results from decision theory and stochastic
ordinary differential equations, and show how they may be extended and
applied to derive explicit parameter dependencies in optimal
performance that may be tested on human and animal subjects. I will
then describe a biophysically-based model of a pool of neurons in a
brainstem organ - locus coeruleus - that is implicated in widespread
norepinephrine release. This neurotransmitter can effect transient
gain and response threshold changes in cortical circuits of the type that
the abstract drift-diffusion analysis requires. I will argue that, in
spite of many gaps and leaps of faith, a rational account of how neural
spikes give rise to simple behaviors is beginning to emerge.

This work is in collaboration with Eric Brown, Rafal Bogacz, Jeff
Moehlis and Jonathan Cohen (Princeton University), and Ed Clayton,
Janusz Rajkowski and Gary Aston-Jones (University of Pennsylvania).
It is supported by the National Institutes of Mental Health.

Transport in celestial and molecular systems

Speaker: 

Professor Jerry Marsden

Institution: 

Caltech

Time: 

Thursday, January 27, 2005 - 4:00pm

Location: 

MSTB 254

This topic is concerned with the application of computational
dynamics tools to the problem of transport in celestial mechanics and
molecular systems. The celestial problems are typified by computing the
probability that a member of the Hilda group of asteroids (which are
just inside the orbit of Jupiter) cross the orbit of Mars in a certain
interval of time. In the molecular context, the computation of the
ionization rate of the Rydberg atom will be used to illustrate the
techniques.

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