Manoj B Karki (University of Toledo)

Invariant metrics on four-dimensional Lie groups

Abstract: An invariant metric on a Lie group is a Riemannian metric that is invariant under either left or right translations. The curvature properties of an invariant metric are simpler than an arbitrary metric although still very complicated. In order to simplify the metric as much as possible we use automorphisms of the associated Lie algebra. Some formulas for curvature are given and then we list the Ricci tensor in reduced form for each of the four dimensional Lie algebras, decomposable as well as indecomposable. We will apply the results to find all the Lie-Einstein metrics in dimension four and also explain the significance of the term "Einstein".

Manoj B Karki (University of Toledo)

Invariant metrics on four-dimensional Lie groups

Abstract: An invariant metric on a Lie group is a Riemannian metric that is invariant under either left or right translations. The curvature properties of an invariant metric are simpler than an arbitrary metric although still very complicated. In order to simplify the metric as much as possible we use automorphisms of the associated Lie algebra. Some formulas for curvature are given and then we list the Ricci tensor in reduced form for each of the four dimensional Lie algebras, decomposable as well as indecomposable. We will apply the results to find all the Lie-Einstein metrics in dimension four and also explain the significance of the term "Einstein".

Yunus E. Zeytuncu (University of Michigan-Dearborn)

What is a Math Circle?

Abstract: This talk is an introduction on how to start a math circle. I will share my recent experience in initiating the Maize and Blue Math Circle at University of Michigan-Dearborn. I will also demonstrate a sample math circle activity where attendees will be expected to pretend like middle school students.

Anthony Bloch (University of Michigan)

Geometry of Integrable Systems

Abstract: In this talk I will discuss the geometry of integrable systems of Toda type and compare to integrable systems of rigid body. I will discuss in particular the finite Toda lattice and its relationship to gradient flows and the geometry of convex polytopes as well as the generalized Flaschka map.

Stephen Bell (Purdue University)

Complexity in complex analysis and Khavinson-Shapiro conjectures

Abstract: How much computational effort does it take to find classical objects of complex analysis like the Poisson kernel? I will explain my quest to get my hands on these objects that involves new ways of looking at the Riemann Mapping Theorem. Solving the Dirichlet problem can be as easy as the method of partial fractions from freshman calculus!

Stephen Bell (Purdue University)

Bergman coordinates, quadrature domains, and Riemann Mapping Theorems

The unit disc in the plane is called a quadrature domain because the average of an analytic function over the disc with respect to area measure yields the value of the function at the origin. That is and is not as special as it sounds.

Stephen Bell (Purdue University)

A surefire way to find new results about old things

Abstract: In this first talk, I will tell the story of how my PhD thesis advisor, Norberto Kerzman, and his mentor, Eli Stein, discovered a new property of the centuries old Cauchy integral and how it has influenced the way I think about complex analysis. I have tried to use the Kerzman-Stein modus operandi in my own research, and once in a while, it has led me to find shiny new things in moldy corners of the basement of complex analysis.

Gerard Thompson (University of Toledo)

Classifying of groups of small order

Abstract: This elementary talk will focus on the definition of a group in abstract algebra and particularly the classification of groups of small order.

Henry Wente (University of Toledo)

Geometric Proof of a uniqueness theorem for Liouville differential Equation due to W. Chen and C. Li

Abstract: We present a new argument to reprove the uniqueness result for solutions on $\mathbb{R}^2$ to the Liouville equation $\Delta \omega+e^{2\omega}=0$.

Our proof is based on the construction of constant mean immersions along with a regularity result for the H-surface equation (Joint work with Mao-Pei Tsui).

Anna Mazzucato (Penn State University)

Optimal mixing by incompressible flows

Abstract: I will discuss mixing of passive scalars by incompressible flows and measures of optimal mixing. In particular, I will present recent results concerning examples of flows that achieve the optimal theoretical rate in the case of flows with a prescribed bound on certain Sobolev norms. These examples are related to loss of regularity for solutions to linear transport equations.

This is joint work with Giovanni Alberti (Pisa) and Gianluca Crippa (Basel).

Evgeny A. Poletsky (Syracuse University)

Holomorphic envelopes, homotopical Oka principle and extensions of complex manifolds

Abstract: Given a complex manifold $X$ its holomorphic envelope $\widetilde{X}$ is the largest complex manifold where all holomorphic functions extend. It was proved that envelopes exist for Riemann domains. However, the proof is very non-constructive.

Recently, B. Jöricke suggested a direct construction of such envelopes using holomorphic homotopy of analytic disks. In general, holomorphic homotopy theory studies continuous deformations of holomorphic mappings and the major question is when one holomorphic mapping can be continuously deformed into another holomorphic mapping via holomorphic mappings. We call such mappings $h$-homotopic.

The serious studies of such questions was initiated by M. Gromov who was interested in the homotopical Oka principle: when homotopic holomorphic mappings are $h$-homotopic? Later this theory was greatly advanced by F. Forstnerič, his collaborators and students. The major advances of this theory were done on elliptic manifolds that are highly non-hyperbolic. However, Jöricke's approach works also on hyperbolic manifolds.

In the talk we will introduce all necessary notions and results and discuss the latest advancements in the holomorphic homotopy theory of analytic disks on general manifolds. They will include Jöricke's construction, a briefly presentation of Gromov's theory and then discuss other ways of extension of complex manifolds including the results of Larusson and the speaker. Finally, we will show how an $h$-analog for the fundamental group can be introduced.

The intention is to make the talk understandable for anybody familiar with complex analysis of one variable.

Host: Zeljko Cuckovic

Nate Iverson (University of Toledo)

Applications of the Perron-Frobenius theorem

Abstract: In 1907 Oskar Perron proved his theorem for positive values matrices and it was extended by Frobenius in 1912 to nonnegative matrices. We will state both the theorem and it's extension while giving easy graph theoretical conditions on which it applies. We will also briefly describe some applications including: Eigenvector centrality, PageRank, Markov Chains, Leslie matrices, The Leontief input-output model.

Rong Liu (University of Toledo)

Credit rating via Generalized Additive Partially Linear Model

Abstract: One central field of modern financial risk management is corporate credit rating in which default prediction plays a vital role. Generalized Additive Partially Linear Model (GAPLM), which is a multivariate semiparametric regression tool for non-Gaussian responses including binary and count data. We use GAPLM to make default prediction and propose spline-backfitted kernel (SBK) estimator with simultaneous confidence bands for the component functions and BIC constructed for components testing and selection. The SBK technique is both computationally expedient and theoretically reliable, thus usable for analyzing high-dimensional time series. Simulation evidence strongly corroborates with the asymptotic theory. The method is applied to estimate insolvent probability and obtain higher accuracy ratio than previous study.

Joan Remski (University of Michigan - Dearborn)

Balancing Computational Costs in Moving Mesh Methods

Abstract: Many problems in engineering and the sciences have solutions that vary greatly in only small portions of their domains. Using standard numerical solution techniques for partial differential equations (PDEs), the solution is approximated at grid points that are distributed evenly throughout the domain, with little regard to the physical properties of the solution. In this talk, we introduce adaptive techniques called moving mesh methods that allow the mesh points to tend toward the regions where the solution varies and away from the regions where the solution is essentially constant. The mesh points evolve alongside the physical solution and the motion of the mesh is controlled by another PDE with an associated monitor function. One issue with this coupled system of PDEs is that steep gradients or large function values in the physical problem will cause similar characteristics in the mesh solution. We show that for certain choices of the monitor function, we can balance this undesirable behavior between the mesh equation and the physical equation, giving a system that is overall more efficient to compute. Applications presented include a phase transition model and a reaction problem that blows up in finite time.

Mark Kalothi (CAS Associate Actuary, Progressive Insurance)

An Insider's Perspective on Actuarial Science

Abstract: A brief overview of the actuarial profession/differences between CAS and SOA. Tips on how to prepare for actuarial exams. Resume writing/what employers are looking for in your resume. If time allows I will spend a few minutes discussing examples of projects done by entry level actuarial analysts. I will take questions from the students

Host: Nate Iverson

Ovidiu Calin (Eastern Michigan University)

Transience of diffusion on the Heisenberg group

Abstract: In the absence of a general theory of diffusion on non-integrable distributions, an important role is played by the investigation on some particular examples. This talk deals with a couple of these examples. The first one is the Heisenberg diffusion, which is a degenerate diffusion with non-holonomic constraints living on the horizontal distribution of the Heisenberg group. The second example is the Grushin diffusion, also a degenerate diffusion, which moves in the plane along the Grushin distribution. A special emphasize will be put on the transience property of the Heisenberg and Grushin diffusion as well as on a stochastic variant of the Chow-Rashewski connectivity theorem.

Host: Alessandro Arsie