## Colloquia

Colloquia for the Department of Mathematics and Statistics are normally held in **University Hall 4010 on Fridays at 4:00 p.m.** Any departures from this are indicated below. Light refreshments are served before
the colloquia in 2040 University Hall. Driving directions, parking information and maps are available on the university website.

## 2013-2014 Colloquia

Below 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 25, 2014**

Bhupendra Paudyal (University of Toledo)

Abstract: I begin my talk with the discussion of the eigenfunctions of composition and weighted composition operators acting the space of the holomorphic functions on the unit disk. Then, I present the sufficient conditions that ensure all these eigenfunctions belong to Bloch-type spaces.

In the second part of talk, I will provide the complete description of lattice of closed invariant subspaces of shift plus Volterra operator acting on the Hardy space.

This talk is a public presentation of the Ph.D. thesis of doctoral candidate Bhupendra Paudyal.

**April 18, 2014**

Giriraj Bhattarai (University of Toledo) *Inverse Problem of Lagrangian Mechanics and normalizing a metric on a Lie group.*

Abstract: For the first part of the talk, I will discuss the inverse problem of Lagragina Mechanics. The problem consists of finding necessary and sufficient conditions for a system of second order ordinary differential equations to be the Euler-Lagrange equations of a regular Lagrangian function and in case they are, to describe all possible such Lagrangians. Jesse Douglas gave an exhaustive discussion of the problem in two degrees of freedom. I am working on the problem with three degree of freedom, especially in connection case (when right hand sides of the given system is homogeneous quadratic in velocity and given data are components of the connection). However, it is remarkable that even after so much work, not much is known about the solution of the problem for three-dimension. It turns out that the PDE system arising from the IP has all sorts of unexpected integrability conditions that makes a complete analysis difficult if not impossible.

For the second part of the talk, I will discuss how to normalize a metric on a lie group using the automorphism group of the Lie group.

This talk is a public presentation of the Ph.D. thesis of doctoral candidate Giriraj Bhattarai.

**April 11, 2014**

Souhong Wang (University of Toledo) *A semiparametric hypothesis testing method on the differences of several population
means*

Abstract: One-way analysis of variance (ANOVA) is the most commonly used method to compare three or more population means with the assumption that the populations are independent and normally distributed with equal variance. Nonparametric method is used as an alternative when the normal and equal variance assumptions are violated.

In the talk, I will discuss using a semiparametric method to test the null hypothesis that the population means are equal, by assuming a multiple sample density ratio model for the population densities. The semiparametric method, via simulation study, is slightly superior to the classical one-way ANOVA and nonparametric method when the data are normal and is significantly better than them when the data are not normal. Some simulation results and a real data example will be presented.

This talk is a public presentation of the Ph.D. thesis of doctoral candidate Souhong Wang.

**April 4, 2014**

Rafael de la Llave (Georgia Tech) *Manifolds on the verge of a hyperbolicity break down*

Abstract: As V. Arnold said, "Mathematics is an experimental science, only that the experiments are not expensive".

We will describe an experimental approach (involving rigorous results and computations in a way that they feed on each other) to explore the boundary of validity of several theorems in dynamical systems and analysis, notably the KAM theorem and the theory of normally hyperbolic manifolds. We will present motivations for the study and interpret the consequences.

This is joint work with many people. Notably, T. Blass, R. Calleja, A. Haro.

**March 28, 2014**

Nikola Petrov (University of Oklahoma) *Moving boundaries and dynamical systems*

Abstract: We consider the electromagnetic field in a one-dimensional spatial domain one of whose boundaries is stationary, while the other one is moving (quasi)periodically. We show how the boundary value problems for the field can be expressed in terms of the iterates of a map of the circle (or a map of the torus in the quasiperiodic case). After a brief review of some facts from theory of circle maps, we show how one can draw conclusions about the asymptotic behavior of the electromagnetic field from these facts. Finally, we discuss some aspects of the problem of a quantum field in a domain with moving boundaries.

The talk does not require background of dynamical system or physics.

**March 21, 2014**

Michael Levine (Purdue University) *Nonparametric regression with rescaled time series errors*

Abstract: We consider a heteroscedastic (nonparametric regression model with an) autoregressive error process of finite known order $p$. The heteroscedasticity is incorporated using a scaling function (defined at uniformly spaced design points on an interval [0,1]). We provide an innovative nonparametric estimator of the variance function and establish its consistency and asymptotic normality. We also propose a semiparametric estimator for the vector of autoregressive error process coefficients that is consistent at the square root of the sample size rate and asymptotically normal for a sample size $T$. Explicit asymptotic variance covariance matrix is obtained as well. Finally, the finite sample performance of the proposed method is tested in simulations.

**February 28, 2014**

Seungmoon Hong (University of Toledo) *Two approaches to link invariants*

Abstract: I will introduce two approaches to link invariants. One is obtained from (generalized) Yang-Baxter operator by considering representation of braid groups and then taking trace of resulting images. The other is obtained directly from ribbon categories by decorating given link with an object in the category and regarding the link as a morphism in Hom-space. We will see how these two approaches are related in the case that one obtains a (generalized) Yang-Baxter operator from a ribbon category. No prerequisite is required to follow the discussion.

**February 21, 2014**

Sonmez Sahutoglu (University of Toledo) *Some operator theoretic problems on pseudoconvex domains*

Abstract: I will discuss several recent results about compactness of Hankel operators as well as Axler-Zheng theorem on Bergman spaces over pseudoconvex domains in $\mathbb{C}^n$.

The results presented in this talk are joint work with Mehmet Celik and Zeljko Cuckovic.

**February 14, 2014**

Trieu Le (University of Toledo) *Finite Rank Moment Matrices and Toeplitz Operators*

Abstract. Let $\mu$ be a complex Borel measure on the plane such that all entries in the moment matrix $M[\mu]=\big(\int_{\mathbb{C}}z^k\overline{z}^j\,d\mu\big)_{k,j\geq 0}$ exist. It is well known that under certain conditions, $M[\mu]$ is the zero matrix if and only if $\mu$ is the zero measure. We shall discuss the problem of classifying the measures $\mu$ for which $M[\mu]$ has finite rank. This problem is equivalent to the finite rank problem in the theory of Toeplitz operators, which was open for quite some time. The case where $\mu$ has a bounded support was solved by Luecking in 2008 while progresses on the unbounded case have just been made very recently.

**Fall Semester**

**November 15, 2013**

Mitya Boyarchenko (University of Michigan) *Complex Analysis and Financial Mathematics*

Abstract: The talk will be about the part of financial mathematics that deals with stochastic models of asset prices and with computing the values of various derivative securities in these models. Examples of such securities include stock/index options (European, American, barrier, etc.), interest rate derivatives, credit derivatives, commodity derivatives, and many others.

In a large number of these models, the prices of securities one wishes to calculate are not known in closed form, but instead can be expressed as inverse Fourier-Laplace transforms of analytic functions of one or several variables. The algorithms that are commonly used to numerically perform the Fourier-Laplace inversion are often slow and/or unreliable. We will present a new method, developed recently by S. Boyarchenko and S. Levendorskii, which is based on a conformal deformation of the contours of integration in the standard Fourier-Laplace inversion formulas. In most cases this method greatly improves the convergence properties of the integrals and allows for efficient error control.

No previous knowledge of financial mathematics will be assumed. Some prior experience with probability theory will be helpful, but not strictly necessary either. The main tool that will be used is the theory of analytic functions of one complex variable.

**November 8, 2013**

Ryad Ghanam (University of Pittsburgh at Greensburg) *Representations of Low Dimensional Lie Algebras and Applications*

Abstract: In this talk I will report on the progress of the problem of finding linear representation for low-dimensional Lie algebras. For each Lie algebra of dimension less than or equal to six, we will give a matrix Lie group whose Lie algebra is the given algebra in the list. I will also report on the progress of the finding representations for the seven-dimensional nilpotent Lie algebras. As an application, I will show how Lie algebra plays a major role in understanding and solving partial differential equations.

**October 18, 2013**

Yunus Zeytuncu (University of Michigan-Dearborn) *Analytic properties of canonical operators and geometry of domains in $\mathbb{C}^n$*

Abstract: Analytic properties of canonical operators, such as the $\overline{\partial}$-Neumann operator, the Bergman projection operator and the Szego projection operator, depend on the geometric structure of the domain in $\mathbb{C}^n$. In particular, $L^p$ and Sobolev regularity of these operators are determined by the smoothness of the boundary and the order of contact of analytic curves to the boundary.

We will present a sample of old and new results establishing these relations. We will also mention a few open problems.

**October 11, 2013**

Mark Hoefer (North Carolina State University) *Dispersive Hydrodynamics*

Abstract: The dynamics of media modeled by partial differential equations of hydrodynamic type modified by weak dispersion encompass a number of intriguing nonlinear phenomena occurring in a wide range of physical systems. The dispersionless hydrodynamic "core" of these models can yield singularities, e.g., gradient catastrophe, which require regularization according to the underlying dispersive processes in the medium. Such regularization provides a description of dispersive shock waves (DSWs). Owing to their expanding, oscillatory structure and the underlying dissipationless medium in which they propagate, DSWs are strikingly distinct from classical, dissipatively regularized shock waves. This talk will review recent results on the mathematical description of DSWs and applications.

Host: Alessandro Arsie

**September 27, 2013**

Zbigniew Blocki (Jagiellonian University) *Hörmander's Estimate, Some Generalizations and New Applications*

Abstract: Lars Hörmander proved his estimate for the dbar-equation in 1965 and it is one the most important results in Several Complex Variables (SCV). New applications have emerged recently, also outside of SCV. We will present three of them: Ohsawa-Takegoshi extension theorem with optimal constant, one-dimensional Suita Conjecture and Nazarov's approach to the Bourgain-Milman inequality from convex analysis.

Host: Zeljko Cuckovic

**September 13, 2013 (Shoemaker Lecture III)**

Simon Brendle (Stanford University) *Partial Differential Equations in Geometry - Hamilton's Ricci Flow and the Sphere
Theorem*

Abstract: A central theme in geometry is the study of manifolds and their curvature. In this lecture series, we will discuss how techniques involving partial differential equations have shed light on several longstanding problems in global differential geometry. In particular, we will discuss the Ricci flow approach to the Sphere Theorem, as well as applications to minimal surfaces.

Host: Mao-Pei Tsui

**September 12, 2013 (Shoemaker Lecture II, Thursday, 4:00-5:00pm in BO 1049)**

Simon Brendle (Stanford University) *Partial Differential Equations in Geometry - The Yamabe Problem in Conformal Geometry*

Abstract: A central theme in geometry is the study of manifolds and their curvature. In this lecture series, we will discuss how techniques involving partial differential equations have shed light on several longstanding problems in global differential geometry. In particular, we will discuss the Ricci flow approach to the Sphere Theorem, as well as applications to minimal surfaces.

Host: Mao-Pei Tsui

**September 11, 2013 (Shoemaker Lecture I, Wednesday, 4:00-5:00pm in ST-S 0129)**

Simon Brendle (Stanford University) *Partial Differential Equations in Geometry - Minimal Surfaces in the Three-Sphere
and Lawson's Conjecture*

Abstract: A central theme in geometry is the study of manifolds and their curvature. In this lecture series, we will discuss how techniques involving partial differential equations have shed light on several longstanding problems in global differential geometry. In particular, we will discuss the Ricci flow approach to the Sphere Theorem, as well as applications to minimal surfaces.

A reception will be held following the talk in Libbey Hall at 5:30 pm

Host: Mao-Pei Tsui