Seminars are listed in reverse chronological order. The
top of the list is subject to change, since more seminars
are still being planned. All seminars are held at
3:10p.m.
in 1432 Stevenson Center
unless otherwise noted. For further
information on events in the department, you may also
consult the
colloquia schedule, the
weekly
calendar and past
calendars.
Wednesday, April 27, 4:10 pm, SC 1432
Martin Siegwart, Department of Mathematics, Vanderbilt University.
Title: Asymptotic behavior of some nonlocal parabolic problems.
Abstract: We consider the asymptotic behavior of a nonlocal quasilinear parabolic
equation. We study the case where the associated elliptic problem has a unique
equilibrium. It is shown that under certain assumptions this equilibrium is a global
attractor.
Wednesday, April 27 , 3:10, SC 1432.
Okihiro Sawada, Waseda University Tokyo, Japan.
Title:
On the incompressible Navier-Stokes flow
with unbounded initial data.
Abstract:
In this talk we consider the nonstationary incompressible
Navier-Stokes equations (NS) in the whole space for
nondecaying initial velocity at space-infinity.
By simple observations of the 1-dimensional problem (that is
the Burgers equation), or by several known exact solutions
(explicit given functions that solve (NS) in
classical sense), we know that linearly growing data
represent the border case between time-local well-posedness and
ill-posedness. Especially, we mainly discuss the initial velocity
$U_0(x):=Mx+u_0(x)$, where $M$ is a constant matrix, and
$u_0$ is a function in $L^p$. The main result is to establish
a local existence theorem of mild solutions. Our main
tool is the Ornstein-Uhlenbeck semigroup theory. Note that
our semigroup is not analytic. Therefore, we cannot expect
to control time derivatives of solutions, in general.
However, calculating its spatial derivatives shows that our
solutions are regular in spatial variables. Moreover, the estimates
of higher order derivatives imply that our solutions are
analytic in spatial-variables, provided $M$ is skew-symmetric.
This can be seen by estimates of the convergence
rate of the Taylor expansion.
Wednesday, April 20 , 2005.
No meeting.
Wednesday, April 13 , 2005.
Ralph Showalter, Department of Mathematics, Oregon State University.
Title: Diffusion in deforming porous media.
Abstract: We report on some recent progress in the mathematical theory
of nonlinear fluid transport and poro-mechanics, specifically, the
design, analysis and application of mathematical models for the flow of
fluids driven by the coupled pressure and stress distributions within a
deforming heterogeneous porous structure.
The goal of this work is to develop a set of mathematical models of
coupled flow and deformation processes as a basis for fundamental
research on the theoretical and numerical modeling and simulation of
flow in deforming heterogeneous porous media.
Wednesday, April 6, 3:00-4:00pm,
SC 1432
Karatzyna Anna Rejniak,
Mathematical Biosciences
Institute, Ohio State University.
Title: From individual cells to complex tissues: a
cell-based model of growing tumors.
Abstract: Most tumors in vivo become highly non-homogeneous even at very
early stages of their growth. In order to address different aspects of
tumor formation and development on the level of single cells, we propose
a two-dimensional time-dependent mathematical model taking explicitly
into account individually regulated biomechanical processes of tumor
cells and communication between cells and their microenvironment. A
mathematical framework of this model is constituted by the immersed
boundary method and couples the dynamics of separate elastic cells with
the continuous description of a viscous incompressible cytoplasm inside
the cells and the extracellular matrix outside the tissue. I will
present numerical simulations addressing the self-organized formation of
tumor microregions; the formation of micro-architecture of ductal
carcinoma in situ; and the directional sensing and migration of invading
tumor cells.
Friday, April 1, 4:10pm,
SC 1431
Ioannis Kevrekidis,
Chemical Engineering, PACM and Mathematics,
Princeton University.
Title: Equation-free modeling for complex/multiscale systems.
Abstract: OIn current modeling , the best available descriptions of a system often
come at a fine level (atomistic, stochastic, microscopic,
individual-based) while the questions asked and the tasks required by
the
modeler (prediction, parametric analysis, optimization and control) are
at
a much coarser, averaged, macroscopic level. Traditional modeling
approaches start by first deriving macroscopic evolution equations from
the microscopic models, and then bringing our arsenal of mathematical
and
algorithmic tools to bear on these macroscopic descriptions.
Over the last few years, and with several collaborators, we have
developed
and validated a mathematically inspired, computational enabling
technology
that allows the modeler to perform macroscopic tasks acting on the
microscopic models directly. We call this the ``equation-free" approach,
since it circumvents the step of obtaining accurate macroscopic
descriptions.
We will argue that the backbone of this approach is the design of
(computational) experiments. In traditional numerical analysis, the main
code "pings" a subroutine containing the model, and uses the returned
information (time derivatives, function evaluations, functional
derivatives) to perform computer-assisted analysis. In our approach the
same main code "pings" a subroutine that sets up a short ensemble of
appropriately initialized computational experiments from which the same
quantities are estimated (rather than evaluated). Traditional continuum
numerical algorithms can thus be viewed as protocols for experimental
design (where "experiment" means a computational experiment set up and
performed with a model at a different level of description).
Ultimately, what makes it all possible is the ability to initialize
computational experiments at will. Short bursts of appropriately
initialized computational experimentation -through matrix-free numerical
analysis and systems theory tools like variance reduction and
estimation-
bridges microscopic simulation with macroscopic modeling. Some
connections
of this approach with data analysis techniques will also be discussed.
Wednesday, March 30 , 2005.
Vincenzo Vespri,
Department of Mathematics, University of Florence, Italy.
Tilte: regularity results for a calss of degenerate equations arising
from Financial mathematics.
Abstract:
We consider a class of degenerate equations arising from finance (in
particular from European and
American options). The degeneration is of polynomial type. We prove for
these equations some regularity
properties such as estimates in L^p and C^\alpha, generation of
analytic semigroups, characterization
of the domain of the operator.
Wednesday, March 23 , 2005.
Andras Czirok, Department of Anatomy & Cell Biology, University
Kansas Med Center.
Title: Pattern Formation During Embryonic Development.
Abstract:
Living organisms, from bacteria to vertebrates, generate sophisticated
patterns. The formation and regulation of these structures, as well as
the emergence of associated functions, is understood only in a handful
of cases, partially due to the lack of sufficiently detailed
experimental data. Recent advances in automatized microscopy allowed
the time-resolved tracking of embryonic development at cellular
resolution. Our studies in bird embryos revealed large-scale, traveling
wave-like and vortex-like tissue motions. These velocity patterns are
likely to be driven by mechanical stress fields present in the embryo.
Superimposed upon the global movements are cellular rearrangements --
typical examples are seen during the formation of the primordial
vascular bed. Vasculogenesis is a relatively simple morphogenetic
process requiring only one cell type that can be readily observed and
manipulated in avian embryos. The resulting "polygonal" vascular
structure is shown to be resulted by contact cell-cell interactions:
adhesion and protrusive activity.
Friday, March 18, 4:10pm,
SC 1432
Jan Prüss,
Department of Mathematics,
University of Halle, Germany.
Title: Well-posedness of the Navier-Stokes equations for a class of
non-Newtonian fluids.
Abstract:
In this talk we present a result on well-posedness of the Navier-Stokes
equations for so-called generalized Newtonian fluids. The stress-strain
relation is similar to that of a Newtonian fluid, with the exception
that the viscosity is a function of the second invariant of the rate of
strain tensor. A typical law covered by our approach is the power law of
Oswald-de Waele. We are able to consider the complete physical
reasonable range of exponents, in contrast to results available in the
literature so far.
Wednesday, March 16 , 2005.
Jui-Ling Yu,
Mathematics
Department, Michigan State
University.
Title: A fully explicit two stage method for solving
reaction-diffusion-chemotaxis system.
Abstract: Reaction-diffusion-chemotaxis systems have been proved to be
fairly accurate mathematical models for many pattern formations problems
in chemistry and biology. These systems are important for computer
simulations of the patterns, parameter estimations as well as analysis
of the biological properties. In order to solve
reaction-diffusion-chemotaxis systems, efficient and reliable numerical
algorithms are essential for the pattern generations.
In this talk, a general reaction-diffusion-chemotaxissystem is
considered and specific numerical issues are discussed. I propose a
fully explicit discretization combined with a variable optimal time step
strategy for solving the reaction-diffusion-chemotaxis system. Theorems
about stability and convergence of the algorithm are given to show that
the algorithm is highly stable and efficient. Numerical experiment
results are given for one testing problem and one real experimental
problem.
Thursday, March 10, 4:10pm,
SC 1432
Denis A. Labutin,
Department of Mathematics,
University of California Santa Barbara.
Title: Local regularity for critical equations.
Abstract:
We prove a local regularity result for a class of
nonlinear pseudodifferential equations arising in
conformal geometry. That is, any Sobolev weak solution is
actually smooth. The main part of the equation is the Paneitz
operator. This natural pseudodifferential operator appeared recently in
several branches of geometry.
Wednesday, March 2 , 2005.
Junfeng Liu,
Yale University Center for
Genomics and Proteomics, Yale
University.
Title: Global Bayesian Approach to Identifying
Biomarkers from MALDI-MS Data.
Abstract: Mass spectrometry has been broadly used in biological
research for biomarker discovery and pharmaceutical research and
development. Beyond preliminary peak detection by certain threshold
method, we develop a global simulation-based approach for peak alignment
by constructing a symmetric transition kernel. We interpret diverse
aspects of our sampler by measure theory to justify the achieved
improvement upon first reversible jump Markov chain Monte Carlo example
initiated by Green (1995). Finally we make comparisons among different
models based on sample classifications, and the improved performance is
demonstrated by our smaller error rates from 10 fold cross-validation.
Monday, February 28th, 3:10pm,
SC 1432
Elizabeth Weitzke,
Department of Cell
Biology,
University of Connecticut
Health Center.
Title: Building Cell Models: Mathematical Techniques
for the Simulation of Cellular Dynamics
Abstract:
Abstract: Research efforts in computational cell modeling have
increased significantly in the last decade due to technological advances
in experimental techniques and data collection, advances in computer
capabilities, and a growing number of studies that explain biological
phenomena. New experimental data and insight into the inner workings of
cellsÕ hierarchical functions have enabled computational researchers to
develop mathematical models that aid in further understanding signal,
metabolic, genomic, and proteomic networks, and the temporal and spatial
localizations associated with these networks. In this talk, I will
present the mathematical framework and solution techniques for capturing
the coupling of transcription, translation and metabolism, as well as
initial investigation into cellular motility through examination of
actin dynamics at the leading edge of motile cells.
Since many biological functions and processes exhibit nonlinear
phenomena, effective models of such systems must have the capability to
mimic such behaviors.
Therefore, in order to predict the response of a cell to changes in its
surroundings or to modifications of its genetic code, the dynamics are
modeled using equations for metabolism, transport, transcription, and
translation. The coupling among processes is accounted for and multiple
scale techniques allow for the simulation of processes that occur on a
wide range of time scales. A rate equation formulation is used to
simulate transcription from an input DNA sequence while the resulting
mRNA is used via ribosome-mediated polymerization kinetics to accomplish
translation. Feedback associated with the creation of species necessary
for metabolism by the mRNA and protein synthesis modifies the rates of
production of factors (e.g. nucleotides and amino acids) that, in turn,
affect the dynamics of transcription and translation.
Investigation into the molecular mechanisms controlling assembly and
disassembly of actin filaments at the leading edge of motile cells is
also of great interest. The subunits that compose actin filaments
diffuse quickly throughout the cytoplasm, whereas the filaments cannot.
This results in rapid structural reorganization within the cell so that
the cell can respond to an external stimulus, whereby disassembly of
filaments at one site and reassembly of them at another is rapid.
Accessory proteins associated with cytoskeleton function regulate the
spatial distribution and the dynamic behavior of the filaments. These
accessory proteins bind to the filaments or their monomers to determine
the sites of assembly and to change the kinetics of filament assembly
and disassembly. The varied forms and functions of actin filaments
depend on accessory proteins and are modeled using Virtual Cell
software. The resultant mathematical model is coupled with the
reaction/diffusion/advection simulation engine in Virtual Cell to
predict actin dynamics in geometries representing the cell leading edge.
Wednesday, Feburary 23 , 2005.
Edward Swim,
Department of Mathematics & Statistics,
Texas Tech University.
Title:
A nonconforming finite element method for fluid-structure interaction
problems.
Abstract: In this talk I will present a nonconforming finite element methodology
using a three-field formulation to analyze a fluid-structure interaction
problem. The methodology is used to couple a Lagrangian model describing
the structure with the arbitrary Lagrangian-Eulerian strategy used to
describe the fluid in order to simulate a full unsteady physical
phenomenon. Consistency error estimates are obtained which show that the
numerical scheme employed yields a first order approximation for the
solution to the fluid-structure interaction problem. Finally, I will
present a discrete energy estimate to demonstrate the stablity of the
proposed method.
Friday, February 18, 3:10pm,
SC 1206.
Anuj Srivastava,
Center for Applied Vision and Imaging Science,
Department of Statistics,
Florida State University.
Title: Statistical Tools for Registration and Analysis of Shapes.
Abstract: In this talk I will summarize some recent activities of our
group
in the area of shape analysis using images. There are two main
issues in this area: (i) registration of shapes across images, and
(ii) quantification of differences between registered shapes. To
illustrate these issues, I will describe the problem of jointly
registering and comparing shapes of closed, planar curves. The
main idea here is to represent shapes as elements of a shape space
(without using landmarks), and to compare shapes using geodesic
paths on these shape spaces. These geodesic paths are constructed
under one of two models: (i) non-elastic shapes: only bending and
no stretching of curves is allowed, and (ii) elastic shapes: both
bending and stretching of curves are allowed. This framework is
then used to derive a statistical framework for analysis of
shapes. I will also present some recent work on registration of 2D
images using diffeomorphism mappings (Research performed with Washington
Mio,
Eric Klassen, Xiuwen Liu, Shantanu Joshi, and Sanjay Saini).
Thursday, February 10, 4:10pm,
SC 1432.
Ugur Abdulla,
Department of Mathematics,
Florida Institute of Technology.
Title:
Boundary regularity
and irregularity for the diffusion equation and its Brownian motion
counterpart.
Abstract: First, I will present a necessary and sufficient condition for
the regularity of a characteristic top boundary point of an arbitrary
open subset of IRN+1(N _ 2) for the diffusion (or heat) equation. It is
noteworthy that the necessary and sufficient condition expressed in
terms of the local geometry or in terms of the local modulus of the
lower semicontinuity of the boundary manifold. The result implies
asymptotic probability law for the standard Ndimensional Brownian
motion. Then I will introduce a notion of regularity of t = -1 for the
diffusion equation and establish a necessary and sufficient condition
for the existence of a unique solution to the first boundary value
problem for the diffusion equation in a general domain _ IRN+1 which
extends up to t = -1. The probabilistic counterpart of this result is
multidimensional Kolmogorov-Petrovsky test for the asymptotic behaviour
of the Brownian motion trajectory as t " 1. Finally, I will discuss some
open problems and possible impact in Brownian motion theory.
Wednesday, February 9, 4:10pm,
SC 1431.
Ugur Abdulla,
Department of Mathematics,
Florida Institute of Technology.
Title:
Parabolic Problems in Non-smooth Domains with
Applications.
Abstract: Theory of boundary-value problems in non-smooth domains is one
of the most difficult and delicate subjects in the theory of partial
differential equations. In this talk I will discuss some essential
features of the second order parabolic equations in general non-smooth
domains. The principal idea behind my research is that the second order
parabolicity of the equation dictates certain geometry of the boundary
manifold, which may be incorporated into the well established interior
parabolic regularity theory. Although the results are true for quite
more general class of second order degenerate and singular parabolic
equations, for clarity I will address a Dirichlet problem for the
nonlinear diffusion equation in N +1- dimensional noncylindrical domains
with non-smooth and possibly characteristic boundary manifolds. The main
objective of the lecture is to express the criteria for the
well-posedness (existence, uniqueness, comparison and stability) in
terms of the local modulus of lower semicontinuity of the boundary
manifold. The two key problems in this context are:
(1) To find the minimal regularity condition on the lateral boundary
manifold for regularity of the boundary points for weak solutions of the
Dirichlet problem.
(2) To find the minimal regularity condition on the lateral boundary
manifold for the uniqueness of weak solution to the Dirichlet problem.
To clear a question whether any weak solution is at the same time a
Óviscosity solutionÓ.
A particular motivation for my research arises from the problem about
the evolution of 2 interfaces for the nonlinear degenerate parabolic
equations, formation of singularities in free boundary problems and
asymptotic properties of the brownian motion trajectories.
Wednesday, Feburary 9 , 2005.
Bo Su,
Department of Mathematics,
Iowa State University.
Title:
Existence of Weak Solutions for a Front Propagation
problem in Nonisothermal Polymer Crystallization.
Abstract: Crystallization of polymeric materials is a phase-change
process in strong interaction with heat conduction. In typical example
of moving boundary problems such as Stefan problem, the unknown boundary
is assumed to be isothermal, which leads in a mathematical model to a
homogeneous Dirichlet condition for the (appropriate scaled)
temperature. For polymers, the situation is different, since the phase
change does take place at a fixed temperature (or with kinetic
undercooling close to this temperature), but in a rather large
temperature range between the thermal melting point $T_m$ and the glass
transition temperature $T_g$. The local existence of smooth moving front
in was shown by Friedman- Velazquez under the assumption that initial
interface is small perturbation of a sphere. In this talk, we prove the
global existence of weak solution and show that temperature is Holder
continuous in space and the moving front is Lipschitz graph in
time-space.
Tuesday, February 1, 3:10pm,
120 Wilson Hall.
Ying-Hen Hsiehi,
Department of Applied Mathematics,
National Chung Hsing University, Taiwan.
Title: Mathematical Modeling of SARS Outbreak in Taiwan: Impact of
Intervention Measures.
Abstract: The 2003 world-wide Severe Acute Respiratory Syndrome (SARS)
outbreak provided an opportunity to understand the effectiveness of
quarantine and other interventions measures against newly emerging and
re-emerging infectious diseases. We use Taiwan laboratory confirmed
SARS
case data and quarantine data during the outbreak to formulate a
mathematical model which incorporates quarantine, temporal changes in
infections rate, and public response to the intervention measures and
the
severity of the outbreak. We obtain the SARS fatality ratio, the daily
quarantine rate, and the severity-dependent temporal decrease in
infection
rate resulting from public response, as well as the number of cases the
quarantine was able to prevent. The results show that Level A
quarantine
of potentially exposed contacts prevented 10% of SARS cases and
fatalities,
while the effect of Level B quarantine of travelers from affected areas
was only minor. However, the number of cases prevented through public
response
is noticeable even for a slightly decreased infection rate. Quarantine
in Taiwan had been effective in preventing infections, but not as effective
as that of public response and could have been timelier. The difficulty
and scope involved dictates that, once outbreak occurs, quarantine should be
combined with other intervention measures for swift containment of an
outbreak.
Wednesday, January 26, 2005.
Don Hong,
Visiting Professor, Department of Mathematics & Biostatistics,
Vanderbilt University.
Title: Mass Spectrometry Data Preprocessing in Cancer Study
Abstract:
Mass spectrometry (MS) becomes one of the critical components in cancer
research recently. The matrix-assisted laser desorption ionization
(MALDI) technique allowes the use of MS in applications involving large
molecules. In 2002, the Nobel prize in chemistry recognized MALDI's
ability to analyze intact biological macromolecules.
Though MALDI MS has proven to play a key role in the advancement of
science with the introduction of new fields such as Proteomics, there
are many challenges both in MS data preprocessing and data analysis. In
this talk, we present some recent progress on MS data preprocessing
using wavelets, splines, and also statistical techniques. Some MatLab
implementation results on the data preprocessing will be shown using the
software package developed very recently in Biostatistics Shared
Resource at Vanderbilt Ingram Cancer Center.
Wednesday, January 19, 2005.
Daphne Manoussaki,
Department of Mathematics ,
Vanderbilt University.
Title: Mechanical Forces during Angiogenesis and Vasculogenesis.
Abstract:
During blood vessel formation, endothelial cells follow mechanical as well
as chemical cues in their environment. The theory that describes these
mechanochemical interactions assumes that the extracellular matrix (ECM)
is a viscoelastic material which deforms under cellular traction, and that
cell movement depends on chemoattractant gradients and ECM strain.
Numerical simulations predict cell traction can reorganize the ECM into a
network. I discuss the potential role of chemical vs. mechanical forces
during blood vessel formation.
Previous semesters:
