2015-16 Colloquia

Tuesdays at 4:10 in
Carver 202 or as announced

 

For more information, contact
David Herzog
Pablo Raúl Stinga

 
 

Spring 2016

January 19: Lee Altenberg (KLI, Austria) The Deep Connection between Mutational Robustness and Mutational Time Dynamics.

January 26: Harbir Antil (George Mason University) Optimal Control of Free Boundary Problems

January 28: Joy Zhou (MBI) Spatiotemporal dynamics of a population under environmental changes **Talk in 1213 Hoover at 4:10pm

February 2: Zdenek Dvorak (Charles University, Prague) Classes of graphs with sublinear separators

February 9: Daomin Cao (Chinese Academy of Sciences) The Steady Vortex Solutions for Euler Equations in 2 Dimensions

February 23: Camelia Pop (U. Minnesota) Transition Probabilities for Degenerate Diffusions Arising in Population Genetics

March 8: Nicolai Krylov (U. Minnesota) Recent progress in the theory of fully nonlinear elliptic second-order equations

March 22: Krishna B. Athreya (Iowa State University) Glivenko-Cantelli theorem for delayed regenerative sequences and application to functionals of Harris Markov chains

March 29: Ovidiu Savin (Columbia) Elliptic PDEs in two dimensions

April 5: Scott McKinley (Tulane) Anomalous Diffusion and Random Encounters in Biological Fluids

April 12: Raul Curto (U. Iowa) Berger measures for transformations of subnormal weighted shifts

April 19: Mary Ann Horn (NSF/Vanderbilt) Mathematical challenges arising from the questions of controllability and stabilization for complex elastic structures

April 26: Jasmine Foo (U. Minnesota) Evolutionary dynamics of cancer

 

Spring 2016 Abstracts

January 19: Lee Altenberg (KLI, Austria)

Title: The Deep Connection between Mutational Robustness and Mutational Time Dynamics
Abstract: The production of genetic variation is essential for the evolutionary process, but inescapably much of this variation is deleterious, and depresses the average fitness of a population below its maximal value. Haldane (1937) found for some simple models that, counterintuitively, this depression in fitness — the genetic load — was independent of the selection coefficients, and determined instead by the mutation rate. Departures from Haldane's principle were found in 1999 due to the evolution of mutational robustness on neutral networks of genotypes. The genetic load was found to be determined by the topology of the neutral network. No quantification of how the topology determines the genetic load has been forthcoming. Here, bounds are placed on the genetic load through the eigenvalues and eigenvectors of the mutation matrix. The treatment goes beyond neutral networks to arbitrary fitness landscapes and reversible mutation matrices. The mutational relaxation time for a perturbation of genotype frequencies has a direct relationship to the mutational robustness under the same perturbation of genotype fitnesses. By taking a general approach, the behavior of different kinds of mutation — point mutation, copy number change, epigenetic mutation, as well as non-genetic information transmission such as dispersal — can be compared all within a unified framework, and their levels of robustness characterized.

January 26: Harbir Antil (George Mason University)

Title: Optimal Control of Free Boundary Problems
Abstract: Over the last decade, numerous fields have been revolutionized by surface tension driven phenomena. For instance, electrowetting provided advances in lab-on-chips; special fluids like ferrofluids have led to the design of magnetically guided drug delivery systems. To fully take advantage of these processes, one must be able to control them. However, the PDEs that govern them are nonlinear, multiscale with typically unknown domains (free boundary problems) with a Young-Laplace equation on the free boundary to account for surface tension. We will discuss the analysis and approximation of the control of a model free boundary problem with surface tension. We will conclude with a novel approach to realize the regularity of a Stokes problem with Navier slip boundary conditions which naturally appears in Stokes free boundary problem.

January 28: Joy Zhou (MBI)

Title: Spatiotemporal dynamics of a population under environmental changes
Abstract: We live in an environment that is constantly changing. On a large time scale, climate change has a global effect on the dynamics of plant populations. On a smaller scale, there are seasonal changes of local habitats, for example, flooding and drying of wetland habitats. In this talk, I will present a spatial perspective of the effects of environmental changes. What happens when the suitable habitat of a population changes its location, or its size over time? Are there limits of the population’s ability to cope with these spatial changes? How does the life history of plant species affect their persistence in the presence of environmental change? I will present a set of mathematical models aiming at answering these questions.

February 2: Zdenek Dvorak (Charles University, Prague)

Title: Classes of graphs with sublinear separators
Abstract: We say that a subgraph-closed class C of graphs has strongly sublinear separators if there exists a function f(n)=O(n^{1-epsilon}) for some epsilon>0 such that every n-vertex graph in C has a balanced separator of order at most f(n). Examples include planar graphs and more generally all graphs avoiding a fixed minor, as well as some geometrically inspired graph classes. The property of having sublinear separators leads to natural divide-and-conquer algorithms, and thus it is of significant interest in computer science. However, the structural implications of this property are less clear, and in particular we have no good description of graph classes with strongly sublinear separators. In the talk, I will present some partial results in this direction. Joint work with Sergey Norin.

February 9: Daomin Cao (Chinese Academy of Sciences)

Title: The Steady Vortex Solutions for Euler Equations in 2 Dimensions
Abstract: In this talk, the speaker will talk about the existence of solutions with small vorticity set. He will first recall the two main methods used to obtain the steady solutions of Euler equation. Next we will explain the relation between the existence of critical points of Kirchhoff - Routh function and the existence of steady solutions of Euler equations. Lastly he will present some of the results obtained in his two recent papers with Zhongyuan Liu, Juncheng Wei and Shuangjie Peng, Shusen Yan respectively. The solutions are obtained by using Lyapunov -Schmidt reduction to semilinear elliptic equations with nonlinearities like Heviside functions.

February 23: Camelia Pop (U. Minnesota)

Title:Transition Probabilities for Degenerate Diffusions Arising in Population Genetics
Abstract: We study the transition probabilities of a class of degenerate diffusions arising as models for gene frequencies in population genetics. The processes we consider are a generalization of the classical Wright-Fisher model, and they are defined through their infinitesimal generator, which is a boundary-degenerate operator defined on compact manifolds with corners, of which simplices and convex polyhedra are particular examples. Under suitable conditions, we find that the transition probabilities have a singular structure that described the absorbing and reflecting behavior of the underlying diffusion on the boundary components of the compact manifold with corners. This is joint work with Charles Epstein.

March 8: Nicolai Krylov (U. Minnesota)

Title:Recent progress in the theory of fully nonlinear elliptic second-order equations
Abstract: We will discuss some of the history of the theory of fully nonlinear elliptic second-order equations starting around 1972, when the first result appeared about the solvability in the class of functions with bounded second-order derivatives. Then we will discuss some results about the solvability in the class of functions whose second-order derivatives are H\"older continuous. Finally, we will discuss some recent results about the solvability in the class of functions whose second-order derivatives are summable to some power larger that the space dimension.

March 22: Krishna B. Athreya (Iowa State University)

Title:Glivenko-Cantelli theorem for delayed regenerative sequences and application to functionals of Harris Markov chains. (joint work with Vivek Roy, ISU stat dept)
Abstract: Birkhoff's ergodic theorem (also Kolmogorov's strong law of large numbers) yields via Polya's theorem the Glivenko-Cantelli theorem that says: For independent and identically distributed random variables the empirical distribution function converges in sup norm to the true distribution function almost surely.In this talk we derive a version of this for a general delayed regenerative sequence of random variables. We apply this to functionals of positive recurrent Harris Markov chains.

March 29: Ovidiu Savin (Columbia University)

Title: Elliptic PDEs in two dimensions
Abstract: I will present several approaches to the regularity theory of elliptic equations in two dimensions. In particular I will focus on some old ideas of Bernstein and their application to the non-isotropic two-phase free boundary problem and to the Bellman equation in two dimensions.

April 5: Scott McKinley (Tulane University)

Title: Anomalous Diffusion and Random Encounters in Biological Fluids
Abstract: The last twenty years have seen a revolution in tracking data of biological agents across unprecedented spatial and temporal scales. An important observation from these studies is that path trajectories of living organisms can appear random, but are often poorly described by classical Brownian motion. The analysis of this data can be controversial because practitioners tend to rely on summary statistics that can be produced by multiple, distinct stochastic process models. Furthermore, these summary statistics inappropriately compress the data, destroying details of non-Brownian characteristics that contain vital clues to mechanisms of transport and interaction. In this talk, I will survey the mathematical and statistical challenges that have arisen from recent work on the movement of foreign agents, including viruses, antibodies and synthetic microparticle probes, in human mucus. My collaborators and I have demonstrated that the behavior of individual particles is well-described by the integrated Generalized Langevin Equation, a Gaussian process that features tunable autocorrelation in time. Physicists have postulated that the memory in particle paths is related to certain viscoelastic features of the fluid environment. I will detail the stochastic PDE framework necessary to probe this hypothesis and detail some successes (and failures!) in describing population scale dynamics in such a way to predict the rate at which these agents penetrate the human body's first line of defense.

April 12: Raul Curto (University of Iowa)

Title: Berger measures for transformations of subnormal weighted shifts.
Abstract: We study unilateral weighted shifts on the Hilbert space l2. A subnormal unilateral weighted shift may be transformed to another shift in various ways, such as taking the p-th power of each weight or forming the Aluthge transform. We determine in a number of cases whether the resulting shift is subnormal, and, if it is, we find a concrete representation of the associated Berger measure. We do this directly for finitely atomic measures, and using both Laplace transform and Fourier transform methods for more complicated measures. Alternatively, the problem may be viewed in purely measure-theoretic terms as the attempt to solve moment matching equations such as (ʃ tn dμ(t))2 = ʃ tn dν (t) (n=0, 1, …), for one measure given the other. The talk is based on joint work with George R. Exner (Bucknell Univ.).

April 19: Mary Ann Horn (Vanderbilt/NSF)

Title: Mathematical challenges arising from the questions of controllability and stabilization for complex elastic structures.
Abstract: In the study of control and stabilization of dynamic elastic systems, a signicant challenge is the ability to rigorously address whether linked dynamic structures can be controlled using boundary feedback alone. When a structure is composed of a number of interconnected elastic elements or is modelled by a system of coupled partial differential equations, the behavior becomes much harder to both predict and to control. Structures composed of multiple layers or components of dierent dimensions pose serious challenges because the energy transferred through the interfaces between components can lead to uncontrollable behavior. This talk focuses on issues that arise when attempting to understand the control and stability of such complex systems.

April 26: Jasmine Foo (U. Minnesota)

Title: Evolutionary dynamics of cancer
Abstract: In this talk I will give an overview of some questions in cancer biology that can be studied using mathematical modeling from an evolutionary perspective. I will provide some examples from recent work using probabilistic tools such as branching processes and interacting particle systems. For example, how and when does drug resistance arise during treatment, and can we design novel drug combination strategies to prevent it? How does cancer arise from healthy tissue, and can we predict the occurrence of secondary tumors after surgery?

 

Fall 2015 Abstracts

December 8: Matthew Moore (Vanderbilt University)

Title: Decidability in general algebra
Abstract: Tarski's problem is the following: given a finite algebra, decide if it is finitely axiomatizable. In the 90s this problem was shown to be undecidable (that is, no algorithm solving it can exist). In the first half of the talk we demonstrate another undecidable algebraic property and use it to give an independent proof that Tarski's problem is undecidable. In the second half of the talk we discuss the (un)decidability of various other algebraic properties, both known and conjectured.

December 1: Farhad Jafari (University of Wyoming)

Title: Sums of squares, positive definiteness and moment problems in several variables: a case of algebra extension
Abstract: Many problems of mathematical physics (and, in fact, analysis) are problems about sums of positive operators. In this talk, we trace this problem to its early days and show how attention to the algebra, and its extensions, plays a fundamental role in this problem. Several representation theorems on the moment completion problem will be given, and some open questions will be posed.

November 17: David Nualart (University of Kansas)

Title: The Malliavin calculus and its applications
Abstract: The purpose of this talk is to present an elementary introduction to the stochastic calculus of variations or Malliavin calculus. This is a differential calculus on a Gaussian space introduced by Paul Malliavin in the 70s to provide a probabilistic proof of Hormander's hypoellipticity theorem. We will also discuss a recent application of Malliavin calculus, combined with Stein's method, to normal approximations.

November 3: Shuwang Lu (Illinois Institute of Technology)

Title: The Hele-Shaw problem and its computational challenges
Abstract: In the first part of this talk, I will give a brief introduction on a classical moving boundary problem---the Hele-Shaw problem. From a computation point of view, it is very expensive to perform long-time simulations for expanding fingering patterns because of (1) complex interface formed by successive finger tip-splitting (Saffman-Taylor instability); (2) the intrinsic slow growth for a fixed injection rate. Here I present a time and space rescaling scheme, which can significantly reduces the computation time, and enables one to accurately compute the very long-time dynamics of moving interfaces. In the second part of the talk, I will discuss shrinking fingering patterns produced by lifting the top plate of a Hele-Shaw cell. This new setup gives a transient shrinking instability and featuring different dynamic patterns.

October 27: Petr Vojtechovsky (University of Denver)

Title: Automated deduction in research mathematics
Abstract: Thanks to advances in hardware and, more importantly, software, automated deduction is becoming useful in several fields of mathematics, including non-associative algebra, my expertise. In this introductory talk I will demonstrate tools and methods of automated deduction, point out some recent improvements, and show you several projects (as an illustration) where automated deduction was essential. I will reserve a few minutes at the end for philosophical discussions.

 

October 20: Farzad Sabzikar (ISU Dept. of Statistics)

Title: Wiener integrals with respect to tempered fractional Brownian motion
Abstract: Tempered fractional Brownian motion is obtained when the power law kernel in the moving average representation of a fractional Brownian motion is multiplied by an exponential tempering factor. In this talk, we develop the theory Wiener integrals for tempered fractional Brownian motion. Along the way, we develop some basic results on tempered fractional calculus.

 

October 13: Rodica Curtu (University of Iowa)

Title: Neuronal dynamics of stream integration and stream segregation: a modeling approach
Abstract: Several acoustic features have been shown to influence perceptual organization of sound - in particular, the frequency of tones. Subjects listening to sequences of pure tones that alternate in frequency (presented in patterns of repeating doublets or repeating triplets) report alternations in perception between one auditory object (''stream integration'') and two separate auditory objects (''stream segregation''). In this talk we introduce a firing rate model for neuronal responses in primary auditory cortex during streaming of triplets, and compare its dynamics to data recorded from nonhuman primates. Then we show how the model can account for stream segregation and stream integration.

 

October 6: Hailiang Liu (ISU)

Title: Gaussian beam methods for the Helmholtz equation
Abstract: The Helmholtz equation is widely used to model wave propagation problems in application areas like electro-magnetics, geophysics and acoustics. Numerical simulation of Helmholtz becomes expensive when the frequency of the waves is high. In this talk we present the recent construction of the Gaussian beam approximations to solutions of the high frequency Helmholtz equation with a localized source, under the assumption of non-trapping rays we show error estimates between the exact outgoing solution and Gaussian beams in terms of the wave number k, both for single beams and superposition of beams. The main result is that the relative local L2 error in the beam approximations decay as k^{−N/2} independent of dimension and presence of caustics, for N-th order beams. This is a joint work with J. Ralston, O. Runborg and N. Tanushev. .

 

September 29: Steve Butler (ISU)

Title: Aspects of the normalized Laplacian matrix
Abstract: The eigenvalues of the normalized Laplacian matrix give information about the graph the matrix is associated with, including data on expansion and mixing. But the spectrum has some various quirks, for example they cannot detect the number of edges. We will give an introduction to the matrix and establish several properties including the construction of cospectral graphs.

 

September 15: David Offne (Westminster College PA)

Title: Polychromatic Colorings of the Hypercube
Abstract: Given a graph G which is a subgraph of the n-dimensional hypercube Qn, an edge coloring of Qn with r ≥ 2 colors such that every copy of G contains every color is called G-polychromatic. Denote by p(G) the maximum number of colors with which it is possible to G-polychromatically color the edges of any hypercube. Originally introduced by Alon, Krech and Szab´o in 2007 as a way to prove bounds for Tura´n type problems on the hypercube, polychromatic colorings have proven to be worthy of study in their own right. This talk will survey what is currently known about polychromatic colorings and introduce some open questions. In particular, we will discuss the best known constructions that give good lower bounds on p(G) for many graphs G, and a lemma that follows from Ramsey’s Theorem that gives good upper bounds. Exact values for p(Qd) are known for all d, but there are many graphs G for which p(G) cannot be determined using current techniques. In addition, there are many related open problems. For example, it is not known whether for all r there is a graph G such that p(G) = r. In addition there are some natural generalizations and variations of the problem that are only partially understood, and a number of questions about the relationship of polychromatic numbers to Tura´n type problems on the hypercube.

 

September 8: Speaker: Gerard Awanou (UIC)

Title: DISCRETE ALEKSANDROV SOLUTIONS OF THE MONGE-AMPERE EQUATION 

Abstract: The Monge-Amp`ere equation is a nonlinear partial differential equation which appears in a wide range of applications, e.g. geometric optics and material sciences. We present convergence results for finite difference discretizations with the weak solution in the sense of Aleksandrov. The numerical solution is computed as the minimizer of a convex functional of the gradient and under convexity and nonlinear constraints. For monotone schemes we obtain uniform convergence on compact subsets and for the standard finite difference discretization convergence of the discretization for approximate problems. The main tool used is approximation by smooth functions. Part of this talk is based on joint work with R. Awi and L. Matamba Messi.