Iowa State University Mathematics
Colloquium
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Time:
4:10-5p.m. 290 Carver Hall
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Date |
Speaker |
Title (Click on the title of a talk for the abstract if available). |
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Jan. 12,Thursday |
Milton Jara IMPA-Rio de Janeiro and
CIMS- |
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Jan. 19,Thursday |
Yongtao Zhang University of California - Irvine |
Computational analysis of morphogen gradients during embryo development
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Jan. 24,Tuesday |
Xiaoqiang Wang |
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Jan. 26,Thursday |
Jue Yan |
Discontinuous Galerkin Method: High Order PDEs, Interface Capturing and Hamilton-Jacobi Equations |
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Feb. 7, Tuesday |
Changfeng Gui IMA & University of |
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Feb. 14, Tuesday |
Michael Klibanov UNC Charlotte |
Some theoretical and numerical topics in inverse problems |
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Feb. 15 Wed. 3:10-4p.m. at 290 Carver |
Boumediene Hamzi UC Davis |
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Feb. 23, Thursday |
Alberto Bressan Pennsylvania State University |
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Feb. 28, Tuesday |
Shouhong Wang |
A new
bifurcation theory for nonlinear partial differential equations |
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Mar. 7, Tuesday |
Luen-Chau Li |
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Mar. 14, Tuesday |
Spring break |
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Mar. 21, Tuesday 305 Carver |
University or |
Ramanujan's Lost Notebook
(Joint
colloquium with computer science department) |
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Mar. 23, Thursday |
Wilfrid Gangbo |
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Mar. 27, Monday 001 Carver |
Fan Chung, UCSD |
Random graphs and Internet graphs
(Joint
colloquium with computer science department) |
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Mar. 30 Thursday |
Ken Davidson |
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Apr. 4, Tuesday |
Rostyslav O. Hryniv, IAPMM, Lviv, Ukraine |
What spectra can nonselfadjoint
Sturm--Liouville operators have? |
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Apr. 11, Tuesday |
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Critical Thresholds in the Fokker-Planck Equation for Polymers |
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Apr. 18, Tuesday |
Dhruv Mubayi, UIC |
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April 20, Thursday 3:10-4p.m. |
Kathrin Bringmann, |
Freeman Dyson's "Challenge for the Future": The mock theta functions. |
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Apr. 25, Tuesday |
Marton Balazs, University of Wisconsin |
The four outfits and the fluctuations
of the simple exclusion process |
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Apr. 27, Thursday |
Jayadev Athreya , |
Abstracts:
Jan 19, Thursday Dr. Yongtao Zhang, University of California - Irvine
Title: Computational analysis of morphogen gradients during embryo development
Abstract: A morphogen is a substance whose nonuniform distribution in a field of cells differentially determines the fate and phenotype of those cells. During the embryo development of both vertebrates and invertebrates, the bone morphogenetic protein (BMP) binding with cell receptors acts as a morphogen to induce the dorsal-ventral patterning. Using experimental and computational analysis, we investigate how morphogens and other ligands cooperate to produce the desired pattern and dynamics in the Drosophila embryo. In particular, we find that the morphogen activity is much less robust than previously claimed. Then we consider the extension of the one-dimensional model to a more realistic three-dimensional reaction-diffusion system for the Zebrafish embryo development. The complex geometrical shape of the Zebrafish embryo during 30%-epiboly ~ shield stage is approximated by an open spherical ring. Computational analysis on the model reveals that two synergistic feedback loops in the zygotic control cooperate with the maternal control to regulate the complex gene-network and drive a stable BMP morphogen gradient pattern in the Zebrafish embryo.
One of the major computational challenges in this study is the severe stability constraint on the time step due to the stiffness of reactions and diffusions. To overcome this difficulty, we have designed a new class of efficient semi-implicit numerical schemes which treat the linear diffusions exactly and explicitly, and the nonlinear reactions implicitly. A novel decoupling technique results in that the size of the nonlinear system arising from the implicit treatment of the reactions is independent of the number of spatial grid points; it only depends on the number of original equations. The stability region for this new class of schemes is much larger than existing methods, and its second order version is unconditionally stable with respect to both diffusion and reaction.
At last, I will talk a little bit about our new work on an efficient
iterative numerical method (called fast sweeping method) for static
Hamilton-Jacobi equations, which have potential applications on tissue growth.
We constructed high order fast sweeping methods on rectangular meshes and
extended original fast sweeping methods to unstructured meshes (triangular
meshes).
Jan 24, Tuesday Dr. Xiaoqiang WANG, Institute of Mathematics and
Applications, University of Minnesota
Title: Phase Field Models and Simulations of Vesicle Bio-Membranes
Abstract:
Recently, we began to systematically model and simulate the shape deformation
of vesicle membranes using a unified energetic variational phase field method
based on the minimization of elastic bending energy with volume and surface
area constraints. Mathematical theory and numerical algorithms are developed to
for the phase field models. Rigorous convergence theories of the numerical
methods are investigated. Many simulations are carried out in static and
dynamic, axis-symmetric and full 3D, one component and multi-component cases.
The new phase field modeling approach has the advantage of avoiding tracking
the free interfaces, and it can easily handle topological changes. Meanwhile, a
series of formulae for retrieving the Euler number of the vesicles have been
given and discussed which may be useful for detection and control purposes.
The 3D codes developed for the equilibrium shape deformations and the
deformations and interactions with fluid fields allow us to conduct extensive
computational studies. Both known and new equilibrium configurations have been
discovered in our numerical simulations. A detailed examination of the energetic
bifurcation landscape has been carried out. We have further studied the effect
of the spontaneous curvature and have conducted simulations of vesicle
transformations in fluids. The further development of the phase field approach
for multicomponent vesicles gives us more tools to understand new and complex
phenomena recently being experimentally studied by biologists.
Jan 26, Thursday Dr. Jue YAN, University of
California at Los Angelos
Title: Discontinuous Galerkin Method: High Order PDEs, Interface
Capturing and Hamilton-Jacobi Equations
Abstract:
We will have three parts in this talk.
First, we discuss local discontinuous Galerkin methods for high order partial
differential equations, including KdV type equations, forth order Bi-harmonic
equations and other nonlinear dispersive equations.
Second, We will discuss some applications of discontinuous Galerkin method to
ncompressible two-phase flow problems, in which level set method is used for
the interface tracking.
Finally we will present a new discontinuous Galerkin scheme for Hamilton-Jacobi
equations and the coupling with sweeping method for time-independent
Hamilton-Jacobi equations like Eikonal equation.
Feb. 7 Tuesday, Prof. Changfeng
Gui IMA &
Title: Entire Solutions in Phase Transition
Abstract:
Entire solutions often play
an important role in the study of partial differential equations since they
arise naturally in the blow-up analysis of singularities. In this talk, I will
survey some existence and symmetry
results on various entire solutions
related to phase transition, including the Allen-Cahn model and multi-phase
model.
Feb. 15,
Wednesday (3:10-4p.m. 290 Carver) Dr. Boumediene Hamzi,
Title:
"The Controlled Center Dynamics"
Abstract: In this talk we present the ``Controlled Center
Dynamics'' which is the control theory analog of the center manifold theory of
dynamical systems.
The center manifold theorem can be viewed as a model
reduction technique for a nonlinear dynamics around an equilibrium where
one or more eigenvalues of its linear part are on the imaginary axis.
If the rest of the eigenvalues are in the open left
half plane then the local asymptotic stability of the equilibrium is decided by
the local asymptotic stability of the dynamics on the center manifold.
This leads to a reduction of the dimension of the dynamics that needs to be
analyzed to determine local asymptotic stability of the equilibrium.
For a nonlinear control system around an
equilibrium, the local asymptotic stability of the linear controllable
directions can be easily achieved by linear feedback. Therefore the
stabilizability of the whole system should depend on a reduced order model that
corresponds to the stabilizability of the linearly uncontrollable directions.
The controlled center dynamics technique formalizes this intuition.
We show, using normal forms under the feedback group, how the stabilizability of the overall system can be reduced to the stabilizability of the dynamics on a controlled center manifold. Part of the feedback is used to stabilize the linearly stabilizable directions and the other part is used to shape the center manifold. The shape of the center manifold determines the dynamics on it and the goal is to shape the center manifold so that its dynamics is locally asymptotically stable. We illustrate this approach by stabilizing systems with a transcontrollable, a fold, and a Hopf control bifurcations.
February 23, Thursday, Prof. Alberto Bressan, Pennsylvania State University
Title: Stability of approximations to hyperbolic
conservation laws
Abstract:
The talk will present a survey of basic techniques and recent results in the
theory of hyperbolic conservation laws. Approximate solutions obtained by the
Glimm scheme, vanishing viscosity, relaxation approximations and semidiscrete
approximations satisfy a uniform bound on the total variation. All these
approximations converge to a unique limit, depending continuously on the
initial data in the L^1 norm. On the other hand, we will show how fully
discrete numerical schemes can produce a large increase the total variation, so
that no a priori bound can hold. For general hyperbolic systems, a
rigorous proof of convergence of these numerical schemes remains an open
problem.
February 28, Tuesday Prof. Shouhong Wang
Title:
Abstrac: In this talk, I shall present a new bifurcation theory for
nonlinear partial differential equations and its applications. The theory is
centered at a new notion of bifurcation called attractor bifurcation, together
with new strategies for Lyapunov and center manifold reductions. Applications
to the Rayleigh-Benard convection and to the Ginzburg-Landau model of superconductivity
will be given in this talk as well. This is joint work with Tian Ma.
March 7, Tuesday, Prof. Luen-Chau
Li, Penn. State University
Title: From Poisson
groups to Poisson groupoids
Abstract: This is a
survey talk in which I will discuss some of the basic notions in the theory of
Poisson Lie groups and Poisson groupoids. Historically, Poisson Lie groups was
introduced by Drinfeld in the eighties as a result of considerations concerning
some work in mathematical physics. Since then, the subject has found
connections with many areas of mathematics. On the other hand, Poisson
groupoids was introduced by Weinstein in an attempt to unify Drinfeld's Poisson
groups and the symplectic groupoids of Karasev-Weinstein. An important class of
Poisson groupoids is the so-called dynamical Poisson groupoids of Etingof and
Varchenko. Towards the end of the
talk, I will discuss some recent activites in understanding the geometry and
applications of this interesting class of Poisson groupoids.
This talk is aimed at a general mathematical audience
and there are essentially no prerequisites.
March 21 Tuesday (305
Carver) Prof. Bruce C. Berndt, UIUC
Title: Ramanujan's Lost Notebook
Abstract: Srinivasa Ramanujan, generally regarded as the greatest
mathematician in Indian history, was born in 1887 and died in 1920 at the age
of 32. Most of his work was recorded without proofs in notebooks. In the spring
of 1976, while searching through papers of the late G. N. Watson at Trinity
College, Cambridge, George Andrews found a sheaf of 138 pages of Ramanujan's
work. In view of the fame of Ramanujan's "ordinary" notebooks,
Andrews naturally called this collection of sheets Ramanujan's "lost
notebook." This work, comprising about 650 results with no proofs, arises
from the last year of Ramanujan's life and represents some of his deepest work.
After a brief history of Ramanujan's life and notebooks, the history and origin
of the lost notebook will be given. The remainder of the lecture will be
devoted to a survey of some of the most interesting entries in the lost
notebook. These include claims in q-series, theta functions, continued
fractions, integrals, partitions, and other infinite series.
March 27, Monday (001
Carver) Prof. Fan Chung, University of California at San Diego
Title: Random graphs and Internet graphs
Abstract:
We will discuss some recent developments on random graphs with given
expected degree distributions.Such ramdom graphs can be used to model various
very large graphs arising in Internet and telecommunications. In turn, these
"massive graphs" shed insights and lead to new directions for random
graph theory. For example, it can be shown that the sizes of connected
components depend primarily on the average degree and the second-order average
degree under certain mild conditions. Furthermore, the spectra of the adjacency
matrices of some random power law graphs obey the power law while the spectra
of the Laplacian follow the semi-circle law. We will mention a number of
related results and problems that a re suggested by various applications of
massive graphs.
March 30, Thursday, Prof. Ken Davidson, University of
Waterloo
Title "Operator algebras generated by isometries".
Abstract: I will survey some results on algebras generated by n
isometries with pairwise orthogonal range. I will parallel classical
results for n=1 with more recent work for n >= 2.
April 4,
Tuesday, Prof. Rostyslav Hryniv, Institute for Applied Problems
of Mechanics and Mathematics (IAPMM),
Title: What
spectra can non-selfadjoint Sturm-Liouville operators have?
Abstract:
We address the question, what spectra non-selfadjoint Sturm-Liouville operators on a finite interval can have. Although in the selfadjoint case the question is completely understood, the non-selfadjoint case is more difficult due to possibility of nonsimple and/or nonreal eigenvalues. We solve the inverse spectral problem of reconstructing the complex-valued potential of a Sturm-Liouville operator from two spectra or from a spectrum and the sequence of suitably defined norming constants. We also establish a criterion on solubility of the inverse spectral problem and thus answer the question posed in the title.
April 11, Tuesday, Prof. Hailiang Liu, Iowa State University
Title: Critical Thresholds in the Fokker-Planck Equation for Polymers
Abstract:
In this talk we discuss critical thresholds in the Fokker Planck equation
for polymers. For rigid rod-like molecules of polymers, we investigate the
structure and classification of equilibrium solutions of a 3D nonlinear
Doi-Onsager equation. For the model with the Maier-Saupe potential we provide a
definite answer to the Onsager conjecture (1949):
(1) all equilibrium solutions are uniaxial;
(2) the number and structure of equilibrium solutions hinge on whether the
intensity parameter crosses two critical values: 6.731393 and 7.5.
Furthermore, we present explicit formulas for all stationary solutions. We also
discuss other issues such as their stability, global orientation dynamics as
well as critical threshold phenomena in a dumbbell model for polymeric fluids.
April
18, Tuesday, Prof. Dhruv Mubayi, University of Illinois at
Chicago
Title: Explicit constructions in Graph Ramsey theory
Abstract:
After briefly surveying the major problems of Ramsey theory for graphs, and its
connections to other areas of mathematics, I will present several explicit
constructions of edge-colorings of graphs. One of these provides an
edge-coloring of the complete graph on n vertices such that every copy of K_4
receives at least four colors on its six edges. The number of colors used is at
most cn^{1/2} for some constant c and all n, improving upon the probabilistic
construction of Erdos and Gyarfas. This construction is closely related to the
determination of the multicolor Ramsey number for four-cycles, which is one of
the only cases where we understand the behaviour of these numbers. The tools
used for the construction are a combination of hypergraph methods, and
equations over finite fields.
April
20, Thursday 3:10-4p.m.,
Dr. Kathrin Bringman,
Title: Freeman Dyson's "Challenge for the Future": The mock theta functions.
Abstract:
In his last letter to Hardy, Ramanujan defined 17 peculiar functions which are
now referred to as his mock theta functions. Although these mysterious
functions have been investigated by many mathematicians over the years, many of
their most basic properties remain unknown. This inspired Freeman Dyson to
proclaim
"The mock theta-functions give us tantalizing hints of a grand synthesis
still to be discovered. Somehow it should be possible to build them into a
coherent group-theoretical structure, analogous to the structure of modular
forms which Hecke built around the old theta-functions of Jacobi. This remains
a challenge for the future." -Freeman Dyson 1987, Ramanujan Centenary
Conference
Here we announce a solution to Dyson's "challenge for the future" by
providing the "coherent group-theoretical structure" that Dyson
desired in his plenary address at the 1987 Ramanujan Centenary Conference.
In joint work with Ken Ono, we show that Ramanujan's mock theta functions, as
well a natural generalized infinite class of mock theta functions may be
completed to obtain Maass forms, a special class of modular forms. We then use
these results to prove theorems about Dyson's partition ranks. In particular,
we shall prove the 1966 Andrews-Dragonette Conjecture, whose history dates to
Ramanujan's last letter to Hardy, and we shall also prove that Dyson's ranks
`explain' Ramanujan's partition congruences in an unexpected way.
April 25, Tuesday, Prof. Marton Balazs,
Title: The four outfits and the fluctuations of the simple exclusion
process
Abstract:
The subject of the talk will be the totally asymmetric simple exclusion
process, which is the simplest stochastic interacting particle system. Many
aspects of this process are well-known, which makes it possible to give four
different representations of it. I will use two of them to explain results
about the initial state's normal fluctuations being transported along some kind
of characteristic lines.
It is natural that these fluctuations disappear along the characteristic lines
themselves. This is the point where the much more exotic dynamical fluctuations
kick in. To investigate these, two more representations of the model will be
used. Different pieces of the puzzle come from those four different outfits,
and finally form a probabilistic proof of the fact that the dynamical fluctuations
scale with the 1/3-rd power of time.
The talk will aim at a general mathematics audience.
April 27, Thursday, Dr. Jayadev Athreya, University of
Chicago
Title: Billiards in rational-angled polygons
Abstract: The study of billiard flows in euclidean rational angled polygons
is closely related to the dynamics of group actions on certain moduli spaces of
geometric structures on surfaces. We will explain this connection, and discuss
the results that can be obtained using it.
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