Department of Mathematics and Statistics

Colloquia

Colloquia for the Department of Mathematics and Statistics are normally held in 4010 University Hall on Fridays at 4:00pm. Any departures from this are indicated below.

Light refreshments are served after the colloquia in 2040 University Hall.

Driving directions, parking information, and maps are available on the university website.

2012-2013 Colloquia

What follows is a list of speakers, talk titles and abstracts for the current academic year. Abstracts for the talks are also posted in the hallways around the departmental offices.

Spring Semester

April 26, 2013

Thomas Garrity (Williams College)

A Thermodynamic Classification of Real Numbers

Abstract: A new classification scheme for real numbers will be given, motivated by ideas from statistical mechanics in general and work of Knauf and Fiala and Kleban in particular. Critical for the classification of a real number will be the Diophantine properties of its continued fraction expansion.

Host: Alessandro Arsie

April 12, 2013

Longzhi Lin (Rugters University)

Uniformity of harmonic map heat flow at infinite time

Abstract: The harmonic map and its heat flow between Riemannian manifolds have been extensively researched, yet there are still many interesting questions to be answered. In this talk we will discuss an energy convexity along the harmonic map heat flow with small initial energy and fixed boundary data on the unit 2-disk. In particular, this gives an affirmative answer to a question raised by W. Minicozzi asking whether such harmonic map heat flow converges uniformly in time strongly in the $W^{1,2}$-topology, as time goes to infinity, to the unique limiting harmonic map. The key ingredient in the proof is a "compensated regularity" that was observed for the first time by Henry Wente in 1969.

April 5, 2013

Apoorva Khare (Stanford University)

Preserving positivity according to graphs, and absolutely monotonic functions

Abstract: We study the problem of characterizing functions, which when applied entrywise, preserve the set of positive semidefinite matrices with zeroes according to a family $\mathcal{G} = \{ G_n \}$ of graphs. The study of these and other maps preserving different notions of positivity was initiated by Schoenberg and Rudin, and has been the focus of a concerted effort throughout the past century. It now has many modern manifestations and involves an interplay between linear algebra, graph theory, convexity, spectral theory, and harmonic analysis. In addition to the inherent theoretical interest, this problem has important consequences in machine learning (via kernels), in the regularization of covariance matrices, and in the study of Toeplitz operators.

We obtain a characterization for $\mathcal{G}$ an arbitrary collection of trees; the only previous result along these lines was for all complete graphs: $\mathcal{G} = \{ K_n \}$, in terms of absolutely monotonic functions. Another result characterizes functions preserving positivity in low-rank. We provide two proofs; one via harmonic analysis and the other using elementary methods. As a consequence, we recover the result by Rudin et al. as a special case.

This is joint work with Dominique Guillot and Bala Rajaratnam (Stanford).

March 29, 2013

Mihai D. Staic (Bowling Green State University)

Homotopy Quantum Field Theory

Abstract: The first k-invariant of a topological space X was introduced by Eilenberg and MacLane in order to classify the 2-type of a space. I will discuss its construction and a recent generalization. Homotopy quantum field theories (HQFT) were introduced by Turaev. Roughly speaking an HQFT is an invariant for X-manifolds that behave functorially with respect to gluing along the boundary. I will talk about the role of the first k-invariant in the context of 2-dimensional HQFT's.

Host: A. Tikaradze

March 22, 2013

Mike Field (Rice University)

Asynchronous Networks

Abstract: The N-body problem of Newtonian mechanics provides a canonical example of a synchronous network of interacting dynamical systems. Although work by dynamicists on network dynamics typically assumes networks are synchronous, most networks arising in contemporary engineering, biology, computer systems, etc. are asynchronous and behave quite differently from classical synchronous networks.

After discussing some of the basic properties one can expect of synchronous and asynchronous networks, we describe a precise and simple definition of an asynchronous network of interacting dynamical systems (that includes synchronous networks as a very special case). We illustrate with a non-trivial example of an adaptive asynchronous network inspired by ideas from computational neuroscience and learning theory.

We conclude by indicating some mathematical challenges posed by asynchronous networks and their possible resolution.

Hosts: Alessandro Arsie and Maria Leite

March 15

Xiaodong Cao (Cornell University)

Ricci Flow and its Singularities

Abstract: In this talk, I will start with a brief introduction to the Ricci flow, then I will discuss a few analytic and geometric techniques used in the study of singularities. Finally, I will survey some recent development on the study of Ricci flow singularities.

February 15, 2012 (Shoemaker Lecture III, Friday, 4:00-5:00pm in UH-4010)

Robert Lund (Clemson University)

Periodic Time Series

Abstract: This talk overviews modeling and inference procedures for time series data with periodic means and autocovariances. Such series arise in environmetrics, meteorology, astronomy, engineering, economics, health, and ecology.

The class of periodic autoregressive moving-average (PARMA) models is introduced to describe series with second-order periodicities. PARMA models are compared and contrasted to seasonal autoregressive moving-average models.

Testing for the presence of second order periodicities, asymptotic properties of PARMA parameter estimators, and parsimonious PARMA modeling are central issues. Several environmental applications of the developed methods are given.

February 14, 2012 (Shoemaker Lecture II, Thursday, 4:00-5:00pm in UH-4010)

Robert Lund (Clemson University)

A New Way of Modeling Integer Count Time Series

Abstract: This talk proposes a new but simple method of modeling stationary time series of integer counts. Previous work has focused on thinning methods and classical time series autoregressive moving-average (ARMA) difference equations; in contrast, our methods bypass ARMA tactics altogether by using a stationary renewal process to generate a correlated sequence of Bernoulli trials.

By superpositioning independent copies of such processes, stationary series with binomial, Poisson, geometric, or any other discrete marginal distribution are easily constructed. Excursions into multivariate count series and models with periodic features are considered.

The models are naturally parsimonious, can have negative autocorrelations, and can be fitted via one-step-ahead linear prediction techniques for stationary series. As examples, count models with binomial marginal distributions are fitted to observed counts of rainy/precipitation days in consecutive weeks at Key West, Florida, and Coldfoot, Alaska.

February 13, 2012 (Shoemaker Lecture I, Wednesday, 4:00-5:00pm in UH-4010)

Robert Lund (Clemson University)

Multiple Changepoint Detection

Abstract: This talk presents a method to estimate the number of changepoint times and their locations in time-ordered data sequences. A penalized likelihood objective function is developed from minimum description length information theory principles.

Optimizing the objective function yields estimates of the changepoint numbers and location times. Our model penalty is based on the types of model parameters and where the changepoint(s) lie, but not the total number of model parameters (such as classical AIC and BIC methods). Specifically, changepoints that occur relatively closely are penalized more heavily.

Our methods allow for autocorrelation in the observations and general mean shifts at each changepoint time. A genetic algorithm, which is an intelligent random walk search, is developed to rapidly optimize the penalized likelihood. Several applications to climatic time series are given.

February 1, 2013

Ignacio Uriarte-Tuero (Michigan State University)

Two conjectures of Astala on distortion of sets under quasiconformal maps and related removability problems

Abstract: Quasiconformal maps are a certain generalization of analytic maps that have nice distortion properties. They appear in elasticity, inverse problems, geometry (e.g. Mostow's rigidity theorem)... among other places.

In his celebrated paper on area distortion under planar quasiconformal mappings (Acta 1994), Astala proved that if $E$ is a compact set of Hausdorff dimension $d$ and $f$ is $K$-quasiconformal, then $fE$ has Hausdorff dimension at most $d' = \frac{2Kd}{2+(K-1)d}$, and that this result is sharp. He conjectured (Question 4.4) that if the Hausdorff measure $\mathcal{H}^d (E)=0$, then $\mathcal{H}^{d'} (fE)=0$.

UT showed that Astala's conjecture is sharp in the class of all Hausdorff gauge functions (IMRN, 2008).

Lacey, Sawyer and UT jointly proved completely Astala's conjecture in all dimensions (Acta, 2010). The proof uses Astala's 1994 approach, geometric measure theory, and new weighted norm inequalities for Calderon-Zygmund singular integral operators which cannot be deduced from the classical Muckenhoupt $A_p$ theory.

These results are related to removability problems for various classes of quasiregular maps. I will mention sharp removability results for bounded $K$-quasiregular maps (i.e. the quasiconformal analogue of the classical Painleve problem) recently obtained jointly by Tolsa and UT.

I will further mention recent results related to another conjecture of Astala on Hausdorff dimension of quasicircles obtained jointly by Prause, Tolsa and UT.

The talk will be self-contained and accessible to graduate students.

Host: Zeljko Cuckovic

January 25, 2013

Libin Rong (Oakland University)

Hepatitis C virus dynamics: modeling and implications

Abstract: Chronic hepatitis C virus (HCV) infection remains a world-wide public health problem. Therapy with interferon plus ribavirin leads to viral clearance in about 50% of treated patients. New treatment using direct-acting antiviral agents (DAAs) has the potential to cure patients unresponsive to the interferon-based therapy. In this talk, I will review mathematical models used to study HCV dynamics under interferon-based therapy and introduce new models for DAAs. Treatment implications related to modeling results will be discussed.

Host: Maria Leite

January 18, 2013

Friedrich Haslinger (University of Vienna)

Compactness of the $\overline{\partial}$-Neumann operator

Abstract: We consider the $\overline{\partial}$-Neumann operator $$N : L^2_{(0,q)}(\Omega ) \longrightarrow L^2_{(0,q)}(\Omega ),$$ where $\Omega \subset \mathbb{C}^n$ is bounded pseudoconvex domain, and $$N_\varphi : L^2_{(0,q)}(\Omega , e^{-\varphi}) \longrightarrow L^2_{(0,q)}(\Omega , e^{-\varphi}),$$ where $\Omega \subseteq \mathbb{C}^n$ is a pseudoconvex domain and $\varphi $ is a plurisubharmonic weight function.

Using a general description of precompact subsets in $L^2$-spaces we obtain a characterization of compactness of the $\overline{\partial}$-Neumann operator, which can be applied to related questions about Schršdinger operators with magnetic field and Pauli and Dirac operators and to the complex Witten Laplacian. In this connection it is important to know whether the Fock space $$\mathcal{A}^2 (\mathbb{C}^n, e^{-\varphi }) =\{ f : \mathbb{C}^n \longrightarrow \mathbb{C} {\text{ entire }} : \int_{\mathbb{C}^n} \vert f\vert^2 e^{-\varphi } d\lambda < \infty \}$$ is infinite-dimensional, which depends on the behavior at infinity of the eigenvalues of the Levi matrix of the weight function $\varphi.$

In addition we discuss obstructions to compactness of the $\overline{\partial}$-Neumann operator.

References

  1. Haslinger, Compactness for the $\overline{\partial}$-Neumann problem - a functional analysis approach, ESI -preprint 2208, arXiv:0912.4406 , Collectanea Mathematica 62 (2011), 121-129.
  1. Haslinger, Compactness of the $\overline{\partial}$-Neumann operator on weighted (0,q)-forms. ESI preprint 2291, arXiv: 1012.433 , Proceedings of the IWOTA Conference 2010, Birkhauser Verlag, 2012, Operator Theory, Advances and Applications 221, 413-420.

Host: Zeljko Cuckovic

Fall Semester

November 30, 2012

Hugh Montgomery (University of Michigan)

Primes, zeros and random matrix theory

Abstract: First we describe how the Riemann zeta function is connected to prime numbers, and how the distribution of primes is expressed in terms of zeros of the zeta function.

Then we turn to the issue of the distribution of the differences between these zeros, work of mine from 41 years ago. At that time I had a chance encounter with Freeman Dyson, who recognized that my results coincided with corresponding results in random matrix theory. Since then, random matrix theory has become a major tool for creating and testing conjectures concerning the zeta function.

Host: Paul Hewitt

November 16, 2012 (Shoemaker Lecture III)

Duong Phong (Columbia University)

Monge-Ampere Equations III

Monge-Ampere equations in complex function theory and Kahler geometry

November 15, 2012 (Shoemaker Lecture II, Thursday, 4:00-5:00pm in ST-S-0131)

Duong Phong (Columbia University)

Monge-Ampere Equations II

Basic existence and regularity results of Monge-Ampere equations

November 14, 2012 (Shoemaker Lecture I, Wednesday, 4:00-5:00pm in FH-2100)

Duong Phong (Columbia University)

Monge-Ampere Equations I

Examples, notions of generalized solutions and a priori estimates of Monge-Ampere equations

November 9, 2012 (Bowman Oddy 1045)

James Bence (Michigan State University)

Coping with variability: experiences with stock assessment and management models for the Great Lakes

Joint Colloquium with the Department of Environmental Sciences

November 2, 2012

Faruk Abi-Khuzam (American University of Beirut and University of Toledo)

The Geometry of Zero-Free Regions and Growth of Entire Functions

Abstract: Let $f : \mathbb{C} \rightarrow \mathbb{C}$ be a transcendental entire function and $s_n$ its sequence of sections. We present a study of the interplay between the geometry of zero-free regions of $f$ and its sections, and the growth of $f$ . If the Maclaurin coefficients of $f$ are positive, $M(r;f)$ is its maximum modulus, and $f(re^{i\theta})=0, r > 0$, we demonstrate the "uncertainty" inequality $$ \sin^2(\theta/2)\cdot\frac{d^2 \log M(r;f)}{d(\log r)^2} \ge \frac{1}{4} $$ and show how it leads, through the introduction of an index of growth $\beta \in [0,\infty)$, generally larger than the order of $f$ , to a unified approach to known results about zero-free regions, and supplies new information in the outstanding cases where $f$ is of order $0$ or $\infty$. Of special interest is the inequality $\beta\le\frac{1}{2}$, in which case almost all zeros of $f$ lie on one ray.

Host: Zeljko Cuckovic

October 26, 2012 (FH-2100)

Lawrence Anderson (University of Toledo - Dept of Astronomy and Physics)

Radiation Hydrodynamics

Abstract: Radiation hydrodynamics refers to the study of fluid behavior (in particular gasses and plasmas) in the presence of a photon radiation field capable of applying pressure forces on the fluid flow. In astronomy, radiation pressure driven flows occur on the surfaces and in the surrounding envelopes of very luminous stars.

At this time, most computational modeling is limited to calculations assuming mean averages over the radiation spectrum, simply because the three-dimensional treatment of radiation absorption, scattering, emission, and propagation is so numerically intensive. However, even small accelerations of the fluid atoms Doppler shift spectral features so those atoms become exposed to different photons coming from far away. This new exposure further accelerates the atoms.

I will discuss a new code designed to study the turbulence driven by these Doppler shifts, and the spectral signatures that result.

After the talk there will be an informal dinner.

Host: Alessandro Arsie

October 12, 2012

Benito Chen-Charpentier (University of Texas at Arlington)

Parameter Estimation Using Polynomial Chaos

Abstract: When modeling biological processes, there are always errors, uncertainties and variations present. It is important to quantify these uncertainties. In this paper, we consider that the coefficients in the mathematical model of are random variables, whose distribution and moments are unknown a-priori, and need be determined by comparison with experimental data.

A stochastic spectral representation of the parameters and the unknown solution stochastic process is used, together with the polynomial chaos method. The polynomial chaos representation generates a system of equations of the same type as the original model. The inverse problem of finding the coefficients is reduced to establishing the coefficients of the chaos expansions and this is done using maximum likelihood estimation.

In particular, in modeling biofilms, there are variations in the structure and in the bacterial behavior, measurement errors, and uncertainties in the processes.The biofilm growth model is given by a parabolic partial differential equation, so the polynomial chaos formulation generates a system of parabolic partial differential equations. Examples are presented.

This is a joint work with Dan Stanescu at University of Wyoming.

Host: Maria Leite

September 14, 2012 (1:00-4:00pm in GH-5300)

Joint Colloquium with the Department of Medicine

This will include a review of the baseline CORAL medications analysis along with student presentations.

September 7, 2012

Ken Meyer (University of Cincinnati)

Stability of a Hamiltonian System in a Limiting Case

Abstract: I will give a fairly simple geometric proof that an equilibrium point of a Hamiltonian system of two degrees of freedom is Liapunov stable in a degenerate case. That is the $(1:-1)$ resonance case where the linearized system has double pure imaginary eigenvalues $\pm i \omega$, $\omega \ne 0$ and the Hamiltonian is indefinite. The linear system is weakly unstable, but if a particular coefficient in the normalized Hamiltonian is of the correct sign then Moser's invariant curve theorem can be applied to show that the equilibrium point is encased in invariant tori and thus it is stable.

This result implies the stability of the Lagrange equilateral triangle libration point, $L_4$, in the planar circular restricted three-body problem when the mass ratio parameter is equal to $\mu_R$, the critical value of Routh.

Host: Alessandro Arsie

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