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Scheduled Colloquia (2008--2009) Colloquia are usually
held in Room 250, Parker Hall from 4:00 to 4:50 pm on Fridays with
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Past Colloquia (2007--2008) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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Title: Positive and free
boundary solutions for some nonlinear
Title: Partial latin
squares, partial gerechte designs, list colouring and
Title: The Horn conjecture
and other related problems
Title: Finite groups with
permutably and S-permutably embedded
Title: Homogeneous
continua, mutual aposyndesis and products of
Title: One horse racing
story in ancient China, two types of card
Title: Iterated Brownian
motion and a related class of processes
Title: Phase field
models for biofilm growth, expansion, and biofilm- flow interaction
(abstract below)
Title: Performance and
staffing of many server queues under heavy
Title: About the stability
of rotating gas balls (abstract below)
Title: $\mathcal A$-stability of global attractors of competition Past Colloquia (2007)
Title: Analysis and stability of RDEs on continuously deforming
domains
(abstract below)
Faculty Host: Georg Hetzer
Title: Deviation
inequalities in probability and geometry and an application
to longest convex chains (abstract below)
Faculty Host: Andras Bezdek
Title: The incomparable
Leonhard Euler at 300: Two reflections
(abstract below) Faculty Host: Joint AU-AUM
Mathematical Colloquium
Title: Stationary solutions
for a quasilinear model for phase transitions Faculty Host: Georg Hetzer
University, Budapest, Hungary)
Title: Packing convex bodies into a minimal convex polytope of given
shape (abstract below)
Faculty Host: Andras Bezdek
Technology, Changsha, China) Title:
Attractors of random parabolic equations (abstract
below)
Faculty Host: Wenxian Shen
Kunming University of Science and Technology,
China) Title:
Dynamical understanding of loop soliton solution for several
nonlinear wave equations
Faculty Host: Wenxian Shen Abstracts
(2008) Title: Inverse limits, economics, and backward dynamics
Abstract: Some economic models, such as the cash-in-advance model of money,
have the property that the dynamics are ill defined going forward in time,
but are well defined going backward in time. We apply the theory of inverse
limits to characterize topologically possible solutions to a dynamic
economic model with this property. We show that such techniques are
particularly well suited for analyzing the dynamics going forward in time
even though the original map is ill defined in this direction. We analyze
the inverse limit of the cash-in-advance model of money and illustrate how
information about the inverse limit is useful for detecting or ruling out
complicated dynamics.
Title: Length bias in the
measurements of carbon nanotubes
Abstract: To measure carbon nanotube lengths, atomic force microscopy and
special software are used to identify and measure nanotubes on a square
grid. Current practice does not include nanotubes that cross the grid, and,
as a result, the sample is length-biased. The selection bias model can be
demonstrated through Buffon's needle problem, extended to general curves
that more realistically represent the shape of nanotubes observed on a grid.
In this article, the nonparametric maximum likelihood estimator is
constructed for the length distribution of the nano-tubes, and the
consequences of the length bias are examined. Probability plots reveal that
the corrected length distribution estimate provides a better fit to the
Weibull distribution than the original selection-biased observations, thus
reinforcing a previous claim about the underlying distribution of
synthesized nanotube lengths.
Title: Longitudinal data analysis using sufficient
dimension reduction
Abstract: There have been an increasing number of applications where the
data are repeatedly measured at a sequence of time points. In this talk we
investigate a sufficient dimension reduction approach for analyzing such
longitudinal data. The proposed method can effectively reduce the dimension
of the predictors while retaining full regression mean information. Variable
selection for longitudinal data is studied, and graphical diagnosis and
model fitting after dimension reduction are discussed. The proposed method
is flexible enough to encompass a variety of commonly used longitudinal
models. Effectiveness of the method is demonstrated through an extensive
numerical study.
Title: Geometry-based kernel selection for classification
dimension
Abstract: Kernel-based classification methods such as support vector machine
use the data embedding idea which maps data into a higher dimensional
feature space via a kernel function. In practice, selecting the
hyperparameter in a kernel function is a critical step in order to obtain a
good performance. We derive a novel method of choosing the hyperparameter in
the Gaussian kernel by considering the geometry of the embedded feature
space. The proposed method is independent of the specific classification
algorithm and empirically shown to give competitive performance, while
having much better computational efficiency than cross-validation. We also
demonstrate with data examples that the proposed method is robust to the
sampling variability, unlike cross-validation.
Title: Probability maps for brain activity via fMRI
Abstract: We build a probability map of the brain based on fMRI data from
multiple subjects. The model used is taken in part from particle
interactions models in physics and in part from social network models. The
neighborhood system for the social network is built using a principal
components breakdown of the image. The fit is achieved creatively, using
Markov chain Monte Carlo methods as well as pseudo-likelihood methods.
Preliminary results are pleasing!
Title: Positive and free boundary solution for some
nonlinear
Abstract: We give an overview of some recent work concerning existence and
multiplicity of positive solutions for some non-Lipschitz and even singular
nonlinearities. Solutions with a free boundary (the boundary of the subset
where a non-negative solution is zero) are also considered.
Title: Partial latin squares, partial gerechte designs,
list colouring and
Abstract: click here
Title: The Horn conjecture and other related problems
Abstract: click here
Title: Finite groups with permutably and $S$-permutably
embedded
Abstract: click
here
Title: Homogeneous continua, mutual aposysdesis and
products of
Abstract: First, I will give a brief introduction to homogeneous continua
with some examples, motivation and history of the subject. The second part
of this talk will be devoted to a new study involving the mutual aposyndesis
and semi-indecomposability of continua. In particular, I will present a
characterization of mutually aposyndetic products of solenoids. Title: Dualities and decomposition numbers
Abstract: Motivated by a duality of finite groups of Lie type, we will
present a general construction of dualities for any Lie type situation,
e.g., Lie algebras, quantum groups, category O,
character sheaves, etc. In the case of the general linear group (and the
Iwahori-Hecke algebra), we will show how this duality relates to the elusive
decomposition numbers arising in the modular representation theory of these
objects. Title: Solenoids, bihomogeneity, embedding and foliations
Abstract: We will first survey some of the basic properties of solenoids.
Then we shall address the questions of when solenoids are bihomogeneous and
when they embed in Euclidean space of comparatively small dimension.
Finally, we shall consider how and when solenoids occur as minimal sets of
smooth foliations of closed manifolds. Title: One horse racing story in ancient China, two types
of card
Abstract: Motivated by a horse racing story of ancient China, we consider
two types of card games, whose outcomes are related to the inertia of
Hermitian matrices with prescribed eigenvalues. The study has interesting
connections and implications to other areas, such as probability,
statistics, and algebraic combinatorics. Title: Iterated Brownian motion and a related class of
processes
Abstract: Suppose a solid has a crack filled with a gas. If the crack
reaches the surrounding medium, how long will it take the gas to diffuse out
of the crack? Iterated Brownian motion serves as a physical model for
diffusions in a crack. Although this process is not a Markov process (it
does not satisfy the Chapman-Kolmogorov equations), it does have connections
with a parabolic operator. We study the lifetime asymptotics of iterated
Brownian motion in bounded and unbounded domains. We also extend generalized
isoperimetric-type inequalities to iterated Brownian motion in several
domains. I will also talk about Large deviations results for a class of
processes related to iterated Brownian motion. Title: Phase field models for biofilm growth, expansion,
and biofilm- flow interaction
Abstract: We derive a set of phase field models for biofilms using the
one-fluid two-component formulation in which the combination of
extracellular polymeric substances (EPS) and the bacteria are effectively
modeled as one fluid component while the collective ensemble of nutrient and
the solvent are modeled as the other. The biofilm is assumed an
incompressible continuum. The growth modes are identified in linearized
analysis. Numerical simulations are carried out in one and two space
dimensions using a velocity-corrected projection method for incompressible
flows. Biofilm growth, expansion, streaming, rippling, and detachment are
simulated in shear cells numerically. Viscoelastic properties of the biofilm
are investigated as well. Title: Performance and staffing of many server queues under
heavy load
Abstract: Motivated by applications in call centers, we analyze a heavily
loaded queueing model of many servers. We improve the classical result of
Halfin and Whitt (1981) by deriving refinements of their limit theorem for
the probability of delay. The key idea behind our approach is a relation
between the Poisson and Normal distribution which dates back to Szego
(1912). We apply our results to investigate the performance of the square
root staffing rule. Title:
About the stability of rotating gas balls
Abstract: We consider the question of nonlinear stability of the equilibrium
states of barotropic, self-gravitating viscous fluids which are slowly
rotating like a rigid body. These equilibrium states as well as the
non-stationary solutions occupy part of space, and a constant pressure is
assumed on the free surface, but no surface tension. Although the rotation
is slow, the stability of these equilibria cannot be obtained by a simple
perturbation argument from the case of a non-rotating configuration, as the
disturbances of the surfaces even in the non-rotating case do not
necessarily decay exponentially. Title: $\mathcal A$-stability of global attractors of
competition
Abstract: We study the structural stability of the attractor ($\mathcal
A$-stability) for two-species competition-diffusion systems with the
Morse-Smale property. Such systems generate semiflows on positive cones of
certain infinite-dimensional Banach spaces (e.g., fractional order spaces).
Our main result states that a Morse-Smale two-species competition system is
structurally $\mathcal A$-stable, which implies that the set of
nonlinearities for which the system possesses the Morse-Smale property is
open in an appropriate space under the topology of $C^2$-convergence on
compacta. Moreover, we provide a sufficient condition under which a system
has the Morse-Smale property. Abstracts
(2007) Title: Analysis and stability of RDEs on continuously
deforming
Abstract: In this talk, I will present an arbitrary Lagragian-Eulerian
formulation applied to reaction-diffusion systems on continuously deforming
domains. Title: Deviation inequalities in probability and geometry
and
an application to longest convex chains
Abstract: The first part of the talk shall give an insight to the fascinating
large deviation principle, which among many other valuable applications have
provided numerous links between probability theory and higher dimensional
geometry. I will discuss some--already--classical results, e.g. the
Azuma-Hoeffding and Talagrand inequalities and the concentration of measure
principle. The second part will be devoted to the following problem: Let $T$
be a triangle in which we choose $n$ uniform independent random points. Fix
two vertices $v_0$ and $v_1$ of $T$, and find the maximal number of points
among the chosen ones which are in convex position together with $v_0$ and
$v_1$. These points form a convex chain between the two vertices, and the
length of this chain, i.e., the number of points involved, is a random
variable. We establish a sharp asymptotic estimate for this quantity, and
with the aim of Talagrand's inequality, prove that the limit shape of the
longest convex chains is the unique parabolic arc connecting $v_0$ and $v_1$
tangent to the sides of $T$. Title: Stationary solutions for a quasilinear model for
phase
transitions in one space dimension
Abstract: We show striking differences in pattern formation produced by the
Cahn-Hilliard model with the p-Laplacian and a C$^1,\alpha$
potential (0<$\alpha$≤1) in place of the regular (linear) Laplace operator
and a C$^2$ potential. The corresponding energy functional exhibits
multi-dimensional continua ("polyhedra") of critical points as opposed to
the classical case with the Laplace operator. Each of these continua is a
finite-dimensional, compact C$^1,1$ manifold with boundary. Some of
the critical points are local minimizers of the energy functional in a
CCC$^1,\alpha$-related topology (0<a≤1), whereas others are only
saddle points. The former are interior points of the corresponding continuum
(viewed as a compact manifold with boundary), while the latter are boundary
points. These facts offer a different explanation of the "slow dynamics" on
the attractor for the dynamical system generated by the corresponding
time-dependent parabolic problem. Title: The incomparable Leonhard Euler at 300: Two
reflections
Abstract: On the 300th anniversary of the birth of Leonhard Euler, the more
one reflects on the sheer volume of his contributions to mathematics and
science, the more one becomes overwhelmed. In a short talk, even accompanied
by the magnificent 12 posters produced for this anniversary by the Swiss
government, we will be able to consider only two facets of Euler's legacy.
We will focus on the trigonometry chapter of his Introducio in Analysin
Infinitorum (Introduction to the Analysis of the Infinite, 2
volumes, 1748; E 101, 102), and we will survey his Lettres Title: Packing convex bodies into a minimal convex
polytope of
given shape
Abstract: We examine some special cases of the following problems: Let n
bounded convex bodies be given and let a convex polytope P be also given in
the d-dimensional Euclidean space. We want to find an algorithm that
determines the exact value of the smallest constant c (as a root of a
polynomial of one variable) for which it holds that there is a packing of
the space with n convex bodies formed by congruent copies of the given n
convex bodies such that the packing fits into the polytope cP, where cP is a
polytope similar to P with coefficient of similarity c. We solve this
problem for n=1 when the convex body is a polytope, and for n=2 when the
convex bodies are Euclidean spheres.
Title: Attractors of random parabolic
equations
Abstract: This talk is concerned with existence and measurability of pullback
attractors for random parabolic equations on non-smooth domain and/or with
"non-smooth" initial data. It first presents some abstract existence theorem
of pullback attractors for random dynamical systems by applying the measure
of non-compactness and $\omega$-limit compactness. It then considers the
pullback attractors for random parabolic equations on smooth domain in
$L_2(D)$ and $H_0^1(D)$, and considers the pullback attractors for random
parabolic equations on non-smooth domain in $L_p(D)$ with $p\gg 1$. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||