# Seminars

Page address: http://cset.mnsu.edu/mathstat/seminars/## Spring 2018 Department Seminars

### Steklov representations of solutions of Laplacian boundary value problems

#### Monday, April 16th, 4:00-5:00pm, WH 288A.

**Speaker:** Dr. Manki Cho, School of Mathematical Sciences, Rochester Institute of Technology

**Abstract:** Laplace's equation as an example of elliptic partial differential equations arises in the modeling of electromagnetism, astronomy, and fluid dynamics. The solutions of Laplace's equations are the harmonic functions which can describe the behavior of electric and fluid potentials as well as the heat conduction. One goal is to understand how the boundary condition affects on the solution on the domain. A classical result is the Dirichlet to Neumann solution map. This talk describes the trace space of harmonic functions subject to non homogeneous boundary conditions using Steklov eigenfunctions which provides the Neumann to Dirichlet map. In particular, we highlight a novel application of Steklov expansion method for Laplacian boundary value problems in terms of orthonormal bases of the harmonic spaces

### Isoperimetric sets inside almost-convex cones

#### Thursday, April 10th, 4:00-5:00pm, WH 284.

**Speaker:** Dr. Eric Baer, Department of Mathematics, University of Wisconsin-Madison

**Abstract:** The isoperimetric inequality is a classical result showing that for sets of
fixed volume, perimeter (or surface area) is minimized when the set is a
ball. In dimensions three and higher, a natural setting for results of this
type is E. De Giorgi's framework of sets of finite perimeter (a setting where
one has advantageous compactness properties).

We discuss a recent result showing that a characterization of isoperimetric sets (that is, sets minimizing a relative perimeter functional with respect to a fixed volume constraint) inside convex cones as sections of balls centered at the origin (originally due to P.L. Lions and F. Pacella) remains valid for a class of "almost-convex" cones. Key tools include compactness arguments and recent progress in stability results for geometric variational problems, combined with the use of sharp characterizations of lower bounds for the first nonzero Neumann eigenvalue associated to (geodesically) convex domains in the hemisphere. The work we describe is joint with A. Figalli. The talk will conclude with a brief description of my ongoing and future work.

### Abstract Harmonic Analysis and L-functions

#### Thursday, April 9th, 4:00-5:00pm, WH 288A.

**Speaker:** Dr. Wook Kim, Department of Mathematics & Statistics, MSU

**Abstract:** L-functions are complex analytic functions with certain properties such as Euler product and functional equations. We shall review these properties by using the Riemann zeta function, a prototype of L-function. In Langlands-Shahidi method, an L-function appears as the zeroth Fourier coefficient of an Eisenstein series attached to an automorphic representation. Certain properties of L-functions are obtained from those of the Eisenstein series. Since an automorphic function can be though of as a periodic function, this theory is considered as an abstract harmonic analysis. We shall discuss how an L-function can be obtained in this abstract harmonic analysis setting and a few results of my research on L-functions. Finally, we consider a possible generalization or extention of this theory which is my future research topic.

### The Chern-Yamabe problem

#### Thursday, April 5th, 10:00-10:50pm, TRC 122.

**Speaker:** Dr. Mehdi Lejmi, Department of Mathematics, Bronx Community College in New York

**Abstract:** On a complex manifold, the holomorphic structure of the Hermitian tangent bundle viewed as a holomorphic vector bundle corresponds to the (0,1)-part of the Chern connection. In this talk, we study the scalar curvature of the Chern connection and its difference with Riemannian scalar curvature. We also discuss an analogue of the Yamabe problem in the almost-Hermitian setting. This is joint works with M. Upmeier and A. Maalaoui.

### Positivity of truncated Toeplitz operators via Berezin transform on certain model subspaces of the Hardy space

#### Thursday, March 29th, 10:00-10:50am, TRC 122.

**Speaker:** Mr. Krishna Subedi, Department of Mathematics, University of Toledo

**Abstract:** We study the relationship between the positivity of truncated Toeplitz operators and the Berezin transform of their symbols. In particular, I will show positivity of the Berezin transform of the real-valued $L^infty(mathcal{T}$ implies the positivity of the Truncated Toeplitz Operators corresponding to the inner function u=z^2, however, the statement is not true in general if the corresponding inner functions are u=z^n for n>2.

### Bi-Lagrangian structures and Teichmüller theory

#### Tuesday, March 27th, 2:00-2:50pm, TRC 122.

**Speaker:** Dr. Brice Loustau, Department of Mathematics, Rutgers University-Newark

**Abstract:** A Bi-Lagrangian structure in a smooth manifold consists of a symplectic form and a pair of transverse Lagrangian foliations. Equivalently, it can be defined as a para-Kähler structure, that is the para-complex equivalent of a Kähler structure. Bi-Lagrangian manifolds have interesting features that I will discuss in both the real and complex settings. I will proceed to show that the complexification of a real-analytic Kähler manifold has a natural complex bi-Lagrangian structure, and showcase its properties. I will then specialize this discussion to moduli spaces of geometric structures on surfaces, which have a rich symplectic geometry. I will show that some of the recognized geometric features of these moduli spaces are formal consequences of the general theory, while revealing other new geometric features; as well as deriving several well-known results of Teichmüller theory by pure differential geometric machinery. Time permits, I will also mention the construction of an almost hyper-Hermitian structure in the complexification of any real-analytic Kähler manifold, and compare it to the Feix-Kaledin hyper-Kähler structure. This is joint work with Andy Sanders.

### Classification of protein-ligand binding using their structural dispersion

#### Tuesday, February 27th, 4:30-5:30pm, WH 284.

Refreshment will be provided at room WH 291 at 4:00-4:30pm.

**Speaker:** Dr. Galkande (Iresha) Premarathna, Department of Mathematics and Statistics, MSU

**Abstract:** It is known that a protein’s biological function is in some way related to its physical structure. Many researchers have studied this relationship both for the entire backbone structures of proteins as well as their binding sites, which are where binding activity occurs. However, despite this research, it remains an open challenge to predict a protein’s function from its structure. There are many useful applications from protein function predictions, such as effective drug discovery with fewer side effects, development of structure-based drug designs, disease diagnosis, and many more.

This presentation will discuss how this ligand-binding protein prediction problem is approached by taking a higher level object oriented approach that summarizes the description of the binding site, so that it reduces the amount of information lost compared to most of the other approaches. Thereby, a model-based method is considered, where the nonparametric model is implemented by using the features of the binding sites for a given ligand group for understanding and classification purposes. Then the results obtained using the model-based approach are compared to the alignment-based method used by Ellingson and Zhang (2012) and Hoffmann et al. (2010).