Spring 2010 Department Colloquia
Tuesdays at 4:10 p.m. in 268 Carver Hall unless otherwise noted
For more information contact Anastasios Matzavinos or Alexander Roitershtein
February 9 - Ulrica Wilson (Morehouse College) on Division algebras: a case for pure mathematics. Host: Leslie Hogben
March 2 in 305 Carver- Miller Lecture: Distinguished University Professor Avner Friedman (Ohio State University) on What is mathematical biology and how useful is it? Host: A. Matzavinos. Co-sponsored by EEOB, BBMB, the Department of Statistics and the Department of Mathematics.
March 30 - Xiaoming Wang, Florida State University Host: Paul Sacks
March 31 - (Wednesday) Reserved by Yiu Poon.
April 15 (Thursday) - Yiannis Moschovakis (UCLA) The axiomatic derivation of absolute lower bounds. Hosts: A. Matzavinos (Math) & Lu Ruan (CS). This is a Distinguished Lecture co-hosted with the Computer Science Department. Howe Hall, Alliant Energy-Lee Liu Auditorium.
April 19 (Monday) - Chihoon Lee (Colorado State) Host: Ananda Weerasinghe
All 2009-2010 Colloquia Programs
Abstracts
Avner Friedman (Ohio State University) on What is mathematical biology and how useful is it?
I shall define what is meant by “mathematical biology,” and then proceed to illustrate the degree of its usefulness by examples taken from projects developed at the Mathematical Biosciences Institute: chronic wound healing; modeling of the immune rheostat of macrophages in the lung in response to infection; neointimal hyperplasia occurring in dialysis, tuberculosis as a disease with prognosis which depends on the age of the patient, and viral treatment of glioblastoma. All these examples are modeled by systems of differential equations, and the challenges are:
1) Researching the biological literature in order to set up a mathematical model;
2) Determining the rate parameters;
3) Simulating the model.
The final test is to show good fit with experimental results, after which the model can be used to suggest new biologically testable hypotheses.
Ulrica Wilson (Morehouse College) on Division algebras: a case for pure mathematics
What are all of the different types of division algebras? This question is far from being answered, but there is much that can be said. One strategy is to identify all the possible constructions of division algebras over a particular field. For example, thanks to Frobenius, we know that there are exactly two R-division algebras, R itself, and Hamilton’s quaternions. This kind of classification is optimal because we have an explicit list of R-division algebras(up to isomorphism). Classifying division algebras over other fields has proven to be much more difficult. In this talk we discuss the problem of classifying finite-dimensional division algebras and illustrate the importance of studying such abstract objects.
Nick Tanushev (University of Texas - Austin) Computations of high frequency waves
We will begin with some motivation behind asymptotic solution methods and the necessity of high frequency wave solutions in science and industry.
More specifically, we will discuss methods based on Gaussian beams.
Gaussian beams are asymptotic high frequency wave solutions of hyperbolic partial differential equations that are concentrated on a single curve through space-time. The development of Gaussian beams as a computational tool has gained high popularity in the last few years, in large part due to the fact that Gaussian beam methods provide a globally valid solution, even at caustic regions. We will discuss the basic construction of Gaussian beams, the analytical results that are needed to show their validity, some challenges that arise in using Gaussian beams as a numerical tool and finally some recent results and open questions.
Marshall Slemrod (University of Wisconsin-Madison) Compensated compactness and isometric embedding
A fundamental problem in differential geometry is to characterize intrinsic metrics on a two-dimensional Riemannian manifold M2 which can be realized as Isometric immersions into R3. This problem can be formulated as initial and/or boundary value problems for a system of nonlinear partial differential equations of mixed elliptichyperbolic type whose mathematical theory is largely incomplete. In this paper, we develop a general approach, which combines a fluid dynamic formulation of balance laws for the Gauss-Codazzi system with a compensated compactness framework, to deal with the initial and/or boundary value problems for isometric immersions in R3. The compensated compactness framework formed here is a natural formulation to ensure the weak continuity of the Gauss-Codazzi system for approximate solutions, which yields the isometric realization of two-dimensional surfaces in R3. As a first application of this approach, we study the isometric immersion problem for two-dimensional Riemannian manifolds with strictly negative Gauss curvature. We prove that there exists a C1,1 isometric immersion of the two-dimensional manifold in R3 satisfying our prescribed initial conditions. To achieve this, we introduce a vanishing viscosity method depending on the features of initial value problems for isometric immersions and present a technique to make the apriori estimates including the L^\infty control and H^{-1} compactness for the viscous approximate solutions. This yields the weak convergence of the vanishing viscosity approximate solutions and the weak continuity of the Gauss-Codazzi system for the approximate solutions, hence the existence of an isometric immersion of the manifold into R3 satisfying our initial conditions.
Tong Li (U Iowa) Stability of traveling waves arising from chemotaxis
Traveling wave (band) behavior driven by chemotaxis was observed experimentally by Adler and was modeled by Keller and Segel. For a quasilinear
hyperbolic-parabolic system that arises as a non-diffusive limit of the Keller-
Segel model with nonlinear kinetics, we establish the existence and nonlinear
stability of traveling wave solutions with large amplitudes. The numerical
simulations are performed to show the stability of the traveling waves under
various perturbations.
This is a joint work with Zhi-an Wang at Vanderbilt University.
Lihe Wang (U Iowa) Estimates for invariant classes of functions
We will talk about the classical DeGiorgi, Nash, Krylov and Safonov, as well as Inverse H\"older theory from the point of view of invariant classes of functions.
Codina Cotar (Technical University of Munich) Random interface models
We will first review some of the main results in gradient interface model for Hamiltonians with strictly convex potentials, results as regards surface tension, gradient Gibbs measures, scaling limits and decay of covariances. Then we will present some new results in the area, obtained for gradient interactions with non-convex potentials and for interactions with some randomness in the Hamiltonian.(Based on joint works with J-D.Deuschel and C. Kuelske).
Jennifer Paulhus (Kansas State University) Using algebra to study curves.
Finding solutions to polynomial equations is one of the oldest questions in mathematics. Arithmetic geometry is the study of number theoretic questions using ideas from modern algebra and algebraic geometry. In particular we study solutions to systems of polynomial equations over rings such as the integers or number fields (finite extensions of Q). This approach has had profound success in such varied ways as proving long open conjectures like Fermat's Last Theorem and finding applications in cryptography.
Perhaps the best known examples of arithmetic geometry come from the study of elliptic curves. The points on these curves have a beautiful, natural group structure which allows us to use results from algebra to study these groups, and in turn, the points on these curves. We will talk about these ideas as well as how to deal with more general curves which do not have this natural group structure.
Ramiro Lafuente (UMSA) Strongly clean matrix rings over C(X)
We study some special cases of rings of nxn-matrices over C(X) to determine when they are clean and when they are strongly clean. An element of a ring is clean if it can be expressed as the sum of an idempotent and a unit. The element is strongly clean if it can be expressed as the sum of an idempotent and a unit which commute. A ring is called (strongly) clean if all of its elements are (strongly) clean.
David Bortz (University of Colorado) Fragmentation and aggregation of bacterial emboli
Klebsiella pneumoniae is one of the most common causes of intravascular catheter infections, potentially leading to life-threatening bacteremia. These bloodstream infections dramatically increase the mortality of illnesses and often serve as an engine for sepsis. Our current model for the dynamics of the size-structured population of aggregates in a hydrodynamic system is based on the Smoluchowski coagulation equations.
In this talk, I will discuss the progress of several investigations into properties of our model equations. In particular, I will focus
on: a) accurate characterization of the fractal properties for the aggregates, b) a differential geometry approach to fragmentation modeling, and (time permitting) c) self-similar solutions to the equations.
Chiu-Yen Kao (Ohio State University). A spectral method with window technique for the initial value problems of the Kadomtsev-Petviashvili equation.
The Kadomtsev-Petviashvili (KP) equation is a two-dimensional dispersive wave equation which was proposed to study the stability of one soliton solution of the KdV equation under the influence of weak transversal perturbations. It is well know that some closed-form solutions can be obtained by function which have a Wronskian determinant form. It is of interest to study KP with an arbitrary initial condition and see whether the solution converges to any closed-form solution asymptotically. To reveal the answer to this question both numerically and theoretically, we consider different types of initial conditions, including one-line soliton, V-shape wave and cross-shape wave, and investigate the behavior of solutions asymptotically. We provides a detail description of classification on the results.
The challenge of numerical approach comes from the unbounded domain and unvanished solutions in the infinity. In order to do numerical computation on the finite domain, boundary conditions need to be imposed carefully.
Due to the non-periodic boundary conditions, the standard spectral method with Fourier methods involving trigonometric polynomials cannot be used. We proposed a new spectral method with a window technique which will make the boundary condition periodic and allow the usage of the classical approach. We demonstrate the robustness and efficiency of our methods through numerous simulations.
Vanja Dukic (University of Chicago) Tracking Flu Epidemics Using Google Trends and Particle Learning Algorithms
In this paper we introduce a state-space tracking approach, based on particle learning (PL) for classic compartmental epidemics models (such as, for example, the susceptible-exposed-infected-recovered (SEIR)). The proposed approach is particularly well-suited to on-line learning and surveillance of infectious diseases as it is capable of assessing the odds of an epidemic at each time point, while simultaneously accounting for uncertainty in disease parameters and producing real-time predictive distributions. As compared to the now widely used MCMC-based methods, the PL method, which is based on a clever use of an essential state vector, is easier to implement, computationally faster, as well as more readily generalizable to problems with complex dynamics. In particular, we show how the PL approach, in combination with Bayes Factors, can be used as an on-line diagnostic and surveillance tool for tracking influenza using the Google Flu Trends data. We take a closer look at the spread of flu in the US during 2003-2009, and in New Zealand during 2006-2009, with a special emphasis on the recent epidemic season.
H. Tracy Hall (Brigham Young University) What is Quantum computation? Host: Leslie Hogben
Computer scientists assume that all computers work essentially the same way; cryptographers assume that factoring primes is difficult; and physicists assume that the universe obeys the rules of quantum mechanics. More than a decade ago, Peter Shor discovered an algorithm for a "quantum computer" that shows that at least one assumption is wrong. We still don't know whether such a machine will ever be built, but there is a rigorously defined model of how it should behave, and that model has opened a field of research. I will try to give a mathematician's introduction to quantum computation, particularly from the viewpoint of linear algebra.
Jim Ralston (UCLA) Gaussian beams
Computation of high frequency solutions to wave equations is important in many applications, and resolving wave oscillations is notoriously difficult. At present there is considerable interest in using super-positions of Gaussian beams for these computations in the presence of caustics. Gaussian beams originated in work on resonances in lasers by V.M. Babich and M.M. Popov in the late '60s, but they are just beginning to be widely used.
In this talk Professor Ralston will address important issues related to Gaussian beams, such as the possibilities and limitations of beams as well as some applications.
James Ralston is a mathematics professor at he University of California, Los Angeles (UCLA). He is known for his theoretical contributions to partial differential equations as well as his applied contributions to diverse areas including inverse problems, and Gaussian beams. He received his PhD in mathematics at Stanford in 1968, began his scientific career at the Courant Institute of Mathematical Sciences for three years and has been at UCLA ever since.
Yekaterina Epshteyn (Carnegie Mellon University) Chemotaxis and numerical methods for chemotaxis models
In this work, first we will discuss several chemotaxis models
including the classical Keller-Segel model.
Chemotaxis is the phenomenon in which cells, for example bacteria, and
other single-cell or multicellular organisms direct their movements
according to certain chemicals in their environment. The mathematical
models of chemotaxis are usually described by highly nonlinear time
dependent systems of PDEs. Therefore, accurate and efficient
numerical methods are very important for the validation and analysis
of these systems.
Furthermore, it is known that the solutions of chemotaxis models may
blow up or may exhibit very singular spiky behavior. Capturing such
solutions numerically is a challenging problem.
In our work we propose a family of new high-order interior penalty
discontinuous Galerkin methods for the Keller-Segel chemotaxis model
with parabolic-parabolic coupling. As it can be shown the convective
part of this model is of a mixed hyperbolic-elliptic type, which may
cause severe instabilities when the studied system is solved by
straightforward numerical methods. Therefore, the first step in the
derivation of the proposed methods is made by introducing the new
variable for the gradient of the chemoattractant concentration and by
reformulating the original Keller-Segel model in the form of a
convection-diffusion-reaction system. We then design interior penalty
discontinuous Galerkin methods for the rewritten Keller-Segel system.
Our methods employ the central-upwind numerical fluxes, originally
developed in the context of finite-volume methods for hyperbolic
systems of conservation laws.
We prove error estimates for the proposed high-order discontinuous
Galerkin schemes. Our proof is valid for pre-blow-up times since we
assume boundedness of the exact solution.
Some numerical experiments to demonstrate the stability and high
accuracy of the proposed methods and comparison with other methods
will be presented.
Arash Rafiey (Simon Fraser University, Canada) Proper interval digraphs, min-max orderings, and their connection with minimum cost homomorphism problem
Digraphs with Min-Max orderings are digraph analogues of proper interval graphs and bigraphs. They can be equivalently described by a geometric representation with two inclusion-free families of intervals, and we call them monotone proper interval digraphs. We give a forbidden structure characterization of monotone proper interval digraphs which implies a polynomial time recognition algorithm.
We show the connection of these digraphs with an optimization problem, so called minimum cost homomorphism problem. The minimum cost homomorphism problem is motivated by a real-world problem in defense logistics.
Jerrold R. Griggs (University of South Carolina), Venn diagrams, necklaces, and chain decompositions of posets.
Symmetric Venn diagrams for n sets have been considered for years by several researchers, including Henderson, Grünbaum, Ruskey, Edwards, Hamburger, and Wagon. The existence of such diagrams for n sets is possible only for primes n, and they were initially constructed for primes n ≤ 7. A breakthrough was made when Hamburger devised a construction for n = 11 in 1999. For his inspiration he credited the Greene-Kleitman bracketing construction of a symmetric chain decomposition (SCD) of the Boolean lattice Bn of all subsets of [n] := {1, . . . , n}, ordered by inclusion. In my study, it became apparent that one might be able to apply the bracketing construction to produce Venn diagrams for all primes n, a project which was successfully completed (G.-Killian-Savage 2004): The key ingredient of the proof is the construction of a SCD of the “Necklace Poset” Nn, n prime, in which each element consists of a subset of [n] and its cyclic rotations. That is, Nn is the quotient poset Bn/Zn, consisting of orbits of the Boolean lattice Bn under the action of the cyclic group Zn. Researchers were convinced that the Necklace Poset should actually have a SCD for all n, prime or composite, but the previous method worked only for primes n. Finally, my student Kelly Kross Jordan (Ph.D., 2008) devised an insightful new method to prove that Nn does indeed have a SCD for general n. Tantalizing challenges remain open both on SCD’s for general quotient posets Bn/G and on constructing “simple” symmetric Venn diagrams.
Chun Liu (IMA, University of Minnesota), Energetic variational approaches in calcium and sodium channels
Ion channels are key components in a wide variety of biological
processes, such as nerve impulse, cardiac and muscle contraction,
regulating the secretion of hormones into the bloodstream. Ion channels are a frequent target in the search of new drugs.
Ion channels, like enzymes, have their specific properties: potassium, sodium, calcium, and chloride channels allow only that type of ions to move through the pores. This selectivity is the key to all those biological process mentioned above.
Selectivities in both calcium and sodium channels can be described by the reduced models, taking into consideration of dielectric coefficient and ion particle sizes, as well as their very different primary structure and properties. The side-chains are represented only as charged spheres (calcium channel EEEA/EEEE; sodium channel DEKA). These self-organized systems will be modeled and analyzed with energetic variational approaches (EnVarA) that were motivated by classical works of Rayleigh and Onsager, and had been employed successfully in various complex fluids recently. The resulting/derived multiphysics-multiscale systems automatically satisfy the Second Laws of Thermodynamics and the basic physics that are involved in the system, such as the microscopic diffusion, the electrostatics and the macroscopic conservation of momentum, as well as the physical boundary conditions.
Our current numerical simulation of the system, appling to EEEE/DEKA channels, have produced the binding like real calcium/sodium channels.
Time dependent simulation gives the currents resemble the /m3 / or /n4 / variables in the classical Hodgkin-Huxley description of voltage activated sodium or potassium channels.
In this talk, I will discuss the some of the related biological, physics, chemistry and mathematical issues.
2009-2010 Colloquia Programs
September 8 - Chun Liu (IMA, University of Minnesota), Energetic variational approaches in calcium and sodium channels . Host: Hailiang Liu
Tuesday, September 15 at 4:10 p.m. Leslie Hogben (ISU) Sign patterns that require eventual positivity of require eventual nonnegativity
September 29 - Carolina Distinguished Professor and Chair Jerrold R. Griggs (University of South Carolina), Venn diagrams, necklaces, and chain decompositions of posets. Hosts: Maria Axenovich and Ryan Martin
Thursday, October 1- Arash Rafiey (Simon Fraser University, Canada) Proper interval digraphs, min-max orderings, and their connection with minimum cost homomorphism problem 4:10 p.m. in Carver 204 (to be confirmed)
October 6 - Dr. Yekaterina Epshteyn (Carnegie Mellon University). Chemotaxis and numerical methods for chemotaxis models. Host: Jue Yan
October 9 (FRIDAY in Carver 202) - Jim Ralston (UCLA) Gaussian beams Host: Hailiang Liu
October 13 - H. Tracy Hall (Brigham Young University) What is Quantum computation? Host: Leslie Hogben
October 19 (MONDAY) joint with Statistics in 3105 Snedecor at 4:10 p.m. Vanja Dukic (University of Chicago) Tracking Flu Epidemics Using Google Trends and Particle Learning Algorithms Host: Alex Roiterhstein
October 20 - Chiu-Yen Kao (Ohio State University). A spectral method with window technique for the initial value problems of the Kadomtsev-Petviashvili equation. Host: Tasos Matzavinos
October 21 (WEDNESDAY in 290) - David Bortz (University of Colorado) Fragmentation and aggregation of bacterial emboli Host: Alex Roitershtein
October 22 (THURSDAY in 204) - Ramiro Lafuente (UMSA) Strongly clean matrix rings over C(X) Host: Wolfgang Kliemann
October 27 in Carver 290 - Jennifer Paulhus (Kansas State University) Using algebra to study curves. Host: Ling Long
November 4 (Wednesday at 3:10 in 290) Codina Cotar (Technical University of Munich) Random interface models. Host: Alex Roitershtein
November 5 (Thursday) - Dale Olesky (University of Victoria) Mv-matrices: A generalization of M-matrices based on eventually nonnegative matrices Host: Leslie Hogben
November 20 (Friday at 3:30 p.m. in 268 Carver)Tong Li (U Iowa) Stability of traveling waves arising from chemotaxis. Host: Hailiang Liu
November 20 (Friday at 4:15 p.m. in 268 Carver) Lihe Wang (U Iowa) Estimates for invariant classes of functions. Host: Hailiang Liu
December 1 - Marshall Slemrod (University of Wisconsin-Madison) Compensated compactness and isometric embedding. Host: Hailiang Liu
January 26 - Nick Tanushev (University of Texas - Austin) Computations of high frequency waves xx Host: Hailiang Liu
