Colloquia
Department of Mathematics and Statistics
Texas Tech University
This talk is concerned with the Uniformization problem: which intrinsic qualities of a metric space allow a good parameterization by a
Euclidean space? In the first part of the talk, we consider geometrically good (conformal, quasiconformal) parameterizations. While
including a wide range of fractal examples, spaces as such enjoy geometric and analytic properties and a great amount of first-order
calculus can be performed on them. In the second part of the talk, we discuss measure-theoretically good (Lipschitz, Holder)
parameterizations. The problem of classifying spaces admitting such parameterizations is is one of the most important problems
in geometric measure theory and it is associated to the famous Traveling Salesman Problem.
Geometric PDEs are concerned with utilizing PDE techniques to study geometric problems. A general theme in my research is the investigation of equilibrium configurations with respect to natural quantities modelling the energy or entropy of geometric objects. Those equilibria enjoy several extremal properties that are usually described by elliptic PDEs. Consequently, understanding these equations would advance our knowledge about the associated geometries. In this talk, I'll describe my contribution to fundamental conjectures in the following concrete directions. First, we'll discuss PDEs on manifolds, exploiting elliptic equations that arise in special manifolds particularly in dimension four. The second direction focuses on geometric eigenvalue problems. Here the geometry of an object is examined through studying appropriate elliptic operators and their eigenvalues. Third, we'll talk about geometric flows and applications in which PDEs arise as a mechanism to change the shape of a manifold.
Capacity of sets and condensers in Euclidean spaces has been an object of study by researchers working in several areas of Analysis and Geometry due to its applications and the important role it plays in physics.
In the first part of this talk, we will present some basic facts about conformal and analytic capacity and its connection and applications
to holomorphic functions. A distortion theorem will be examined and the asymptotic behavior of capacity under covering maps.
In the second part we will introduce the modulus metric and examine a conjecture of J. Ferrand, G. Martin and M. Vuorinen from 1991 that every isometry in the modulus metric is a conformal mapping. We will discuss the tools we used and the method we followed to solve the conjecture in the recent papers [1] and [2].
References:
[1] Dimitrios Betsakos and Stamatis Pouliasis, ''Isometries for the modulus metric are quasiconformal mappings'', Trans. Amer. Math. Soc. Published electronically: November 21, 2018.
[2] Stamatis Pouliasis and Alexander Yu. Solynin, ''Infinitesimally small spheres and conformally invariant metrics'', submitted, 2018. https://arxiv.org/abs/1812.04651
In this talk, we will overview the recent developments in the theory of minimal surfaces, and constant mean curvature surfaces in 3-manifolds.
In particular, we will discuss the asymptotic Plateau problem in $ H^3 $ and $ H^2 \times R $, the Calabi-Yau conjecture, and minimal surfaces in hyperbolic 3-manifolds.

Conformal geometry is the study of spaces where we can define the angles of crossing curves, but not their lengths. One of the most successful programs for studying conformal geometry has been the holographic program initiated by Fefferman and Graham, which proceeds by realizing the conformal space as the "boundary at infinity" of a complete Riemannian Einstein manifold of one higher dimension. I will describe this program, and then discuss my project to extend it to conformal spaces which themselves have boundaries. This entails relating the conformal space to an Einstein manifold with a "corner at infinity" of codimension two, and the presence of the corner leads to several interesting analytic complications arising from doubly singular PDEs.
In this talk, we present a method to analyze certain slow-fast
motions in dynamical systems. For singular perturbed dynamical
systems, the well-known Geometric Singular Perturbation Method
(GSPM) is usually applied to find the special limit cycles -- slowfast periodic solutions. However, many practical problems might
be not able or very difficult to be put in the form of singular
perturbed equations, but they still exhibit slow-fast motions. For
such cases, based on dynamical system theory, we developed a
method to identify and analyze certain slow-fast motions. We will
use several biological examples to illustrate our method, and give
a comparison between the GSPM and our method.
Min Wang - Title: An Introduction to Bayesian Hypothesis Testing in High
Dimensions
Abstract: In this talk, we briefly discuss how to implement Bayes
factor-based procedures for the hypothesis testing problems and
study their asymptotic properties when the model dimension grows
with the sample size. We first consider the Bayes factor for
ANOVA designs as an intermediate step toward developing the
Bayes factors for general linear hypothesis testing that has been
well studied in the frequentist literature. Finally, we generalize the
Bayes factor method for testing equality of means of two normal
populations in the "large-p, small-n" setting.
Chunmei Wang - Title: Weak Galerkin (WG) Finite Element Methods for PDEs
Abstract: The speaker will present the basic principles of weak
Galerkin (WG) finite element methods for second order elliptic
problems. An abstract framework for WG will be presented and
discussed for its potential in the scientific applications.
Amin Rahman - Title: Mathematically tractable models in cancer biophysics and walking droplets
Abstract:Attacking problems through models of varying complexity often
provides a broad picture of the phenomenon of interest. If a model is burdened
by the baggage of assumptions, it will not capture the pertinent dynamics. On
the other hand, a model that does not obviate negligible contributions falls prey
to computational constraints. Sometimes, if we are fortunate, we may discover
reduction techniques that eliminate undesirable computational cost. If these
simpler models replicate the qualitative behavior of the phenomena, it may allow
for fast numerical experiments, which can often reveal insight missed by more
complex models. In this talk, I present models arising from cancer biophysics
and walking droplets, and discuss insights from fast numerical experiments.
Cezar Lupu - Title: Riemann zeta and multiple zeta values
Abstract: In this talk, we bring into perspective the infamous Riemann zeta
function and its natural generalization, the multiple zeta function. We focus more
on the evaluations of such objects at positive integers. The techniques used in
these evaluations rely on the properties of some special functions.Although they
look rather simple, it turns out that the Riemann zeta and multiple zeta values
play a very important role at the interface of analysis, number theory, geometry
and physics with applications ranging from periods of mixed Tate motives to
evaluating Feynman integrals in quantum field theory.The talk should be
accessible to non specialists and graduate students.
We develop and analyze a 2x2 dynamical system describing flow through a single pore to study the
dynamics of the appearance and dissolution of gas bubbles during two-component (CO2, H2O), two-phase
(gas, liquid) flow. Our analysis indicates that three regimes occur at conditions pertinent to petroleum
reservoirs. These regimes correspond to a critical point changing type from an unstable node to an unstable
spiral and then to a stable spiral as flow rates increase. Only in the stable spiral case do gas bubbles achieve
a steady-state finite size. Otherwise, all gas bubbles that form undergo, possibly oscillatory, growth and then
dissolve completely. Under steady flow conditions, this formation and dissolution repeats cyclically.
Background Introduction: Compositional flow involving a dissolved gas is of importance in many areas,
including oil reservoir production, pipeline transport, CO2 sequestration, and the disposal of radioactive
waste. Such flow involves the inherent possibility of creation of a gas phase and its subsequent transport.
Excluding specific tertiary recovery practices such as CO2 foam flooding, keeping potential gas components
dissolved in fluid phases is important for efficient extraction in reservoirs; the presence of gas bubbles and the
resultant fluid-gas menisci complicates flow and can compete with fluid movement. The ability to control
formation pressure or flow rates to prevent bubble formation is important for extraction efficiency. A challenge
to the numerical simulation of compositional flow in porous media is the change in the system of equations
that accompanies the appearance or disappearance of the gas phase. This difficulty has been addressed by
several approaches. In our computations of two-phase, two-component (H2O, CO2) flow in a 3D pore
network, we noted the periodic appearance and dissolution of the gas phase in certain pores. Intensive
evaluation of our algorithms led us to conclude that the phenomenon was not numerical in origin. In reviewing
the literature on gas transport in porous media at reservoir scales, in micromodel studies, specific studies on
gas bubble formation, and mathematical studies of gas phase disappearance in water-hydrogen systems, we
have been unable to find any mention of this periodic phenomenon. We have therefore pursued a
mathematical investigation. In this article, we extract a 2x2 dynamical system from the mathematical model
upon which our computations were based in order to study the mechanics of this phase-cycling phenomenon.
In this talk I will:
- present the mathematical model and derive the dynamical system,
- summarize the direction fields, critical points and solution trajectories,
- report the results of numerical solutions to the dynamical system
- provide summary critique of the work
Particle and mesh-free methods offer significant computational advantages in settings where quality mesh
generation required for many compatible PDE discretizations may be expensive or even intractable. At
the same time, the lack of underlying geometric grid structure makes it more difficult to construct meshfree methods mirroring the discrete vector calculus properties of mesh-based compatible and mimetic
discretization methods. In this talk we survey ongoing efforts at Sandia National Laboratories to develop
new classes of locally and globally compatible meshfree methods that attempt to recover some of the key
properties of mimetic discretization methods.
We will present two examples of recently developed ``mimetic’’-like meshfree methods. The first one is
motivated by classical staggered discretization methods. We use the local connectivity graph of a
discretization particle to define locally compatible discrete operators. In particular, the edge-to-vertex
connectivity matrix of the local graph provides a topological gradient, whereas a generalized moving leastsquares (GMLS) reconstruction from the edge midpoints defines a divergence operator.
The second method can be viewed as a meshfree analogue of a finite volume type scheme. In this
method, the metric information that would be normally provided by the mesh, such as cell volumes and
face areas, is reconstructed algebraically, without a mesh. This reconstruction process effectively creates
virtual cells having virtual faces and ensures a local conservation property matching that of mesh-based
finite volumes. In contrast to similar recent efforts our approach does not involve a solution of a global
optimization problem to find the virtual cell volumes and faces areas. Instead, we determine the
necessary metric information by solving a graph Laplacian problem that can be effectively preconditioned
by algebraic multigrid.
Several numerical examples will illustrate the mimetic properties of the new meshfree schemes. The talk
will also review some of the ongoing work to build a modern software toolkit for mesh-free and particle
discretizations that leverages Sandia’s Trillinos library and performance tools such as Kokkos.
This is joint work with N. Trask, M. Perego, P. Bosler, P. Kuberry, and K. Peterson.
In 1927 Polya proved that the Riemann
Hypothesis is equivalent to the hyperbolicity of Jensen
polynomials for Riemann’s Xi-function. This hyperbolicity
had only been proved for degrees d=1,2,3. We prove
the hyperbolicity of all (but possibly finitely many) the
Jensen polynomials of every degree d. Moreover, we
establish the outright hyperbolicity for all degrees
d<10^26. These results follow from an unconditional
proof of the "derivative aspect" GUE distribution for
zeros. This is joint work with Michael Griffin, Larry
Rolen, and Don Zagier.
Many childhood diseases are vaccine preventable. However,
there continue to be many cases of such vaccine preventable diseases
every year, even in countries with high vaccination coverage. It has
been indicated that waning immunity may be the culprit in such
situations. Mathematical models of waning immunity based on the
SIRS, SIRVS, or SIRWS frameworks have indeed shown that this
could be case. We continue in this venue, by adding new components
to models of waning immunity that allow for further study of this topic,
including immune status dependent susceptibility and transmissibility.
In this talk I will review our models (and their connection to the
underlying biology) and I will provide two case studies of waning
immunity and vaccination using models with and without age structure.
The basic reproduction and control reproduction numbers will be
derived. Strategies for pathogen elimination will be discussed.
The Jones polynomial of knots was one of the
biggest mathematical discoveries of the 1980s. We will
explain how this invariant of knots has led to a new
understanding of the representation theory of Lie groups,
and the geometry of varieties of characters via the tools of
quantization.
Influenza A virus (IAV) has a high mutation rate and large inhost population size, and is thus capable of rapid evolutionary change. While
large-scale antigenic changes have been well-studied at the epidemiological
scale, we tackle the underlying small-scale question: how does the virus
received at the start of an infection in a single individual differ from the
virus transmitted from that individual to others? To address this, we couple a
system of ordinary differential equations, describing the in-host dynamics of
IAV, to a branching process describing the fate of de novo mutant lineages
within the host. This coupled approach is necessary because IAV can reach
extremely high copy numbers in a single infected host, but the transmission
bottleneck may involve as few as one to ten viral particles. We can then
answer the question: how different is the flu you pass on from the flu you
received?
In June 23, 2006's issue of Science magazine,
Pendry et al and Leonhardt independently published their
papers on electromagnetic cloaking with metamaterials. Since
then, there is a growing interest in using metamaterials to
design invisibility cloaks and other interesting devices. In
this talk, I will present some time-domain cloaking models
we studied in recent years. Well-posedness study and
various time-domain finite element methods will be
discussed. I will show numerical simulations of invisibility
cloaks and other interesting simulations such as optical
black holes. I will conclude the talk with some open
issues.
The interaction between viscous fluid flows and elastic objects is common across
many microscale phenomena. I will focus, specifically, on some recent results from and new
research directions for my research group, the Transport: Modeling, Numerics & Theory
laboratory at Purdue. The interaction between an internal flow and a soft boundary presents
an example of a fluid--structure interaction (FSI). This particular type of FSI is relevant to
problems from lab-on-a-chip microdevices for rapid diagnostics to blood pressure
measurement cuffs. Experimentally, a microchannel or a blood vessel is found to deform into
a non-uniform cross-section due to FSIs. Specifically, deformation leads to a non-linear
relationship between the volumetric flow rate and the pressure drop (unlike Poiseuille’s law)
at steady state. We have developed a perturbative approach to deriving these relations.
Specifically, the Stokes equations for vanishing Reynolds number are coupled to the
governing equations of an elastic rectangular plate or axisymmetric cylindrical shell. For
example, the vessel’s deformation can be captured using Kirchhoff--Love plate theory or
Donnell--Sanders shell theory under the assumption of a thin, slender geometry. For the
case of shells, an elegant matched asymptotics problem (with a boundary layer and a corner
layer) provides a closed-form expression for a deformed microtube's radius. Several
mathematical predictions arise from this approach: the flow rate--pressure drop relation, the
cross-sectional deformation profile of the soft conduit, and the scaling of the maximum
displacement with the flow rate. To verify the mathematical predictions, we perform fully 3D,
two-way coupled direct numerical simulations using the commercial software suite ANSYS.
The numerical results are first benchmarked against experimental data in the literature.
Then, the numerical results are compared against the mathematical predictions, showing
excellent agreement. Some extensions to bio/physiological situations (e.g., hyperelastic
conduits and non-Newtonian fluids) and to unsteady flows (e.g., stop-flow lithography in
compliant microchannels) will be discussed.
Severi varieties are roughly parameter
spaces for algebraic plane curves of fixed degree and
geometric genus. More generally, we can use the same
term to describe parameter spaces for curves of a given
homology class and geometric genus on any projective
surface. I will focus on the (open, with some notable
exceptions) problem of determining their irreducible
components. The talk will presume little to no
background in algebraic geometry.