## Spring 2021 Department Seminars

### Tutorial on Network Data Analysis I

#### The meeting is available in Zoom 913 1593 2940 (passcode: 751056)

Speaker: Prof. In-Jae Kim, Minnesota State University, Mankato

Abstract: In recent years, network data have been generated in various places, and many researchers from diverse disciplines have been investigating the structures and dynamics of networks. Due to the broad spectrum of network data analysis applications ranging from social sciences to national security, many network data analysis topics are interdisciplinary. To help understand the theory behind network data analysis and promote interdisciplinary collaboration, I plan to present a series of tutorials about network data analysis. This first tutorial will discuss network centralities quantifying how central (or influential) nodes of a network are. I’d like to thank the CSET Dean’s Office for supporting the development of this tutorial series.

### Euclidean and Non-Euclidean Geometries: Using Groupwork and IBL Methods in a Textbook-Free Approach to Teaching Non-Math Majors Logic and Mathematical Proof

#### The meeting is available in Zoom 933 4157 0158 (passcode: 514677)

Virtual teatime will be during 3:40-4:00pm in the same Zoom room.

Speaker: Dr. Heidi Andersen, University of Michigan-Shanghai Jiao Tong University Joint Institute (UM-SJTU JI)

Abstract: At the University of Dallas (UD), every student not seeking to complete a math major must complete at least one math course from a limited list of options. Most students majoring outside of STEM fields take Euclidean and Non-Euclidean Geometries, the main purpose of which is to instill mathematical thinking through direct experience. Students in this course are expected to learn the basic tenets of mathematical logic and to master some basic styles of mathematical proof. I taught this course for six consecutive semesters, to groups of students with mixed mathematical abilities and levels of comfort, and did it all without a traditional textbook. In this talk, I will describe this difficult process of daily experimentation and blue-sky curriculum design and how I turned around unsuccessful classroom experiences in the first half of my time at UD into much more successful classroom experiences in the second half. I will also describe how the pandemic affected my final semester of this course, and provide suggestions for how the curricular materials and activities can be adapted for remote teaching.

## Fall 2020 Department Seminars

### Towards a Kernel-based Projection Method for the Navier-Stokes Equations

#### The meeting is available in Zoom 962 5550 0409 (passcode: 789069)

Virtual teatime will be during 3:40-4:00pm in the same Zoom room.

Speaker: Dr. Nicholas Fisher, Minnesota State University, Mankato

Abstract: We propose an analytically divergence-free quasi-interpolation scheme for the incompressible Navier-Stokes equations. First, we construct the divergence-free quasi-interpolant and derive corresponding approximation orders to both the approximated function and its derivatives. Then, we extend our techniques to the curl-free setting and establish a method for performing Leray projections via the divergence-free quasi-interpolation scheme. Finally, we show how the quasi-interpolant based Leray projection can be used to construct a low-order mesh-free projection method for the Navier-Stokes equations. This project is a work in progress.

### Symplectic Isotopy Problem in Dimension 4

#### The meeting is available in Zoom 951 5617 3163 (passcode: 095649)

Virtual teatime will be during 3:40-4:00pm in the same Zoom room.

Speaker: Dr. Jun Li, Department of Mathematics, University of Michigan

Abstract: We will discuss some recent development for the isotopy problem of 4-dimensional symplectic manifolds, and how these root in dynamics and topology in dimension 2. A brief introduction to symplectic geometry will be provided at the beginning of the talk.

### A friendly introduction to Lie groupoids

#### The meeting is available in Zoom 968 1926 3386 (passcode: 822208)

Virtual teatime will be during 3:40-4:00pm in the same Zoom room.

Speaker: Dr. Seth Wolbert, Minnesota State University, Mankato

Abstract: In the study of differential geometry, there are many desirable constructions from topology that lack a smooth counterpart. In particular, the quotient of a manifold by a smooth Lie group action is rarely again a smooth manifold and is often in fact a quite nasty topological space! One method for smoothly modelling certain types of singular spaces is via Lie groupoids. This construction fuses category theory and smooth manifolds, yielding a model which is built entirely from smooth manifolds and their maps while also serving as a model for certain types of singular spaces. This talk will focus on introducing Lie groupoids in as painless a manner as possible with plenty of examples, including Lie group quotients, flows of vector fields, and orbifolds, to help ease the listener into the concept and hopefully motivate some higher level discussion on the notion of (but certainly not the definition of!) differentiable stacks. I will assume the listener has a good idea of what a differentiable manifold is. I will not assume any knowledge of category theory, though such knowledge will definitely aid understanding of the underlying definitions being laid out.

### Long-time Asymptotic Expansion for Solutions of Non-autonomous Differential Equations

#### The meeting is available in Zoom 922 2428 4599 (passcode: 928194)

Virtual teatime will be during 3:40-4:00pm in the same Zoom room.

Speaker: Dr. Dat Cao, Minnesota State University, Mankato

Abstract: We study the long-time behavior of solutions to nonlinear ordinary differential equations (ODE) with time-decaying forcing functions. The nonlinear term can be, but not restricted to, any smooth vector field which, together with its first derivative, vanishes at the origin. The forcing function can be approximated, as time tends to infinity, by a series of functions which are combinations of exponential, power, and iterated logarithmic functions. We prove that any decaying solution admits an asymptotic expansion, as time tends to infinity, of the same type as the force's. This talk is accessible to graduate and undergraduate students with some background in ODE. This is a joint work with Luan Hoang (Texas Tech University).

## Fall 2019 Department Seminars

### Creating Online Course for Math 130

#### Tuesday, December 3rd, 4:00-4:50pm, WH 286

Refreshment will be provided at room WH 291 at 3:30-4:00pm.

Speaker: Sarah Lanand, Minnesota State University, Mankato

Abstract: Many of our students are taking more of their courses online, especially during the summer months when they are not living in Mankato. To meet the demands of students, as well as to increase summer enrollment, I requested to create an online version of Math 130 (Finite Mathematics & Introductory Calculus). In this talk, I will share how I planned, created, and implemented the materials created for the online course during the summer of 2019. I will also share some specific resources that could be helpful for designing future online courses.

### Experience in 2019 Workshop at Center for Project-Based Learning

#### Tuesday, September 24th, 4:00-4:50pm, WH 286

Refreshment will be provided at room WH 291 at 3:30-4:00pm.

Speaker: Dr. Ke Zhu, teamed with Dr. Hongxia Yin and Dr. Deepak Sanjel, Department of Mathematics and Statistics, Minnesota State University, Mankato

Abstract: In this talk, we will share our experience from the recent workshop at Center for Project-Based Learning in Worcester Polytechnic Institute (WPI). We will explain the paradigm of project-based learning, list the essential project design elements, and describe how they are implemented in WPI through examples. We will also share the resources for PBL design we learned from this workshop.

## Spring 2019 Department Seminars

### Machine Learning Techniques to Predict Churn Rate

#### Tuesday, April 23rd, 4:00-4:50pm, WH 284

Refreshment will be provided at room WH 291 at 3:30-4:00pm.

Speaker: Dr. Deepak Sanjel, Department of Mathematics and Statistics, Minnesota State University, Mankato

Abstract: Predicting customer churn rate in subscription-based companies is of great importance. Data Science can be used to dig into what led the customer to leave, and what can be done to prevent a similar customer churning in the future.
Traditional Regression based statistical models have been used in literature. We will discuss how Machine Learning techniques such as Decision Trees (CART), Bagging, Boosting, Random Forest, Support Vector Machines (SVM), and Naïve Bayes Methods are used to improve predictions.
Case studies with customer churn and employee retention data will be discussed, and comparisons of the accuracy between different types of models will be made.

### Prescribing the Q'-curvature on CR three manifold

#### Monday, March 25th, 2:00-2:50pm, WH 284

Speaker: Dr. Ali Maalaoui, Department of Mathematics and Natural Sciences, American University of Ras al Khaimah

Abstract: In CR-geometry as in conformal geometry, there are different kinds of curvatures that appear under conformal change of the contact structure. In this talk I will be discussing the problem of prescribing the Q'-curvature on three dimensional CR-manifolds. This problem is tied to a Moser-Trudinger type inequality in the CR-setting and we will show that under certain assumptions we can prescribe a projected part of the Q' curvature.

### Quadrature Domains and Equilibrium on the Sphere

#### Tuesday, March 19th, 2:00-2:50pm, WH 284

Speaker: Dr. Alan Legg, Department of Mathematical Sciences, Purdue University Fort Wayne

Abstract: Imagine the unit sphere in Euclidean space as a conductor, and place a unit positive charge on the sphere which is free to redistribute into the configuration of minimal logarithmic energy. Then place finitely many positive point charges onto the sphere to constitute an external field. How will the initial unit charge redistribute in the presence of the new point charges? By using tools of potential theory and complex analysis, we will show that the answer involves special domains in the plane called quadrature domains.

### Jacobian and Hessian determinants in the sense of distributions and some stability results

#### Tuesday, March 18th, 2:00-2:50pm, WH 284

Speaker: Dr. Eric Baer, Department of Mathematical Sciences, Carnegie Mellon University

Abstract: The determinant of the Jacobian matrix of a mapping $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ arises in many contexts in analysis, perhaps most notably as the change of variables factor for integration on $\mathbb{R}^n$. In solid and fluid mechanics, this quantity is closely related to notions of incompressibility and interpenetration of matter. Its interpretation in the sense of distributions is closely connected to delicate regularity issues in real and complex analysis, and at the level of mechanics corresponds to understanding formation of holes/cavities in the deformed material.

When the mapping $f$ is itself the gradient of a function $g:\mathbb{R}^n\rightarrow\mathbb{R}$, the Hessian matrix of $g$ enters the picture. We will give an overview of these ideas, discuss some connections with topics in real and complex analysis, and describe joint work with D. Jerison, where we established optimal continuity results for the action of the Hessian determinant as a distribution on fractional-order spaces, inspired by work of Brezis and Nguyen. The relevant fractional order spaces are Besov spaces, which arise as traces of Sobolev functions, functions having generalized derivatives which play a key role in modern analysis and partial differential equations.

### Long-time asymptotic expansions for solutions of Navier-Stokes equations

#### Monday, March 11th, 4:00-4:50pm, WH 286A

Speaker: Dr. Dat Cao, Department of Mathematics and Statistics, Texas Tech University

Abstract: We discuss the long-time behavior of solutions to the three-dimensional incompressible Navier-Stokes equations with periodic boundary conditions. We investigate the asymptotic expansions of Foias-Saut type for all Leray-Hopf weak solutions. It is shown that if the force has an asymptotic expansion, as time goes to infinity, with respect to certain families of decaying functions in Sobolev-Gevrey space, then any weak solution admits an asymptotic expansion of the same type. In particular, we establish the expansions in terms of power decaying functions and logarithmic and iterated logarithmic decaying ones. This is a joint work with Luan Hoang.

### A non-singular volume-preserving dynamical system on R3, with all trajectories bounded

#### Thursday, February 28th, 4:00-4:50pm, WH 286A.

Refreshment will be provided at room WH 291 at 3:30-4:00pm.

Speaker: Dr. Jeff Ford, Math, Comp Sci & Statistics, Gustavus Adolphus College

Abstract: Creating volume-preserving dynamical systems with bounded trajectories presents some interesting problems. Usually when trajectories are bounded, there is some sort of attractor in the system, but if the system preserves volume, that attractor needs to have a pre-image of measure zero. We present here a smooth construction of a non-singular, volume-preserving dynamical systems on R^3, with each trajectory contained in a bounded set. We start by nesting an infinite sequence of tori, whose union is all of R^3, so that any trajectory originating in a particular torus stays in that torus. With some careful modifications, and a nice diffeomorphism of each torus, we can achieve volume-preservation, without introducing singular points and getting any unbounded trajectories.

## Fall 2018 Department Seminars

### An Efficient Numerical Method for an S-I-R Model with Directed Diffusion

#### Monday, November 19th, 3:00-4:00pm, WH 288.

Refreshment will be provided at room WH 291 at 2:30-3:00pm.

Speaker: Dr. Ruijun Zhao, Department of Mathematics and Statistics, MSU

Abstract: Population spatial movement is often modeled by random diffusion (to capture the intra-species movement) and directed diffusion (to capture the inter-species movement). In this talk, we will first introduce an S-I-R model with directed spatial movement, particularly total population moving away from crowd and susceptibles moving away from the infected. The model is a degenerated system of second-order partial differential equations, which poses challenges in designing efficient numerical methods to solve it. In the second part of this talk, we propose an efficient numerical scheme, the implicit integration factor (IIF) WENO scheme. We will discuss the order of convergence and its performance in solving this system.

### Mathematics for Sustainability

#### Friday, November 2nd, 3:00-3:50pm, WH 288.

Refreshment will be provided at room WH 291 at 2:30-3:00pm.

Speaker: Dr. Jacob Duncan (Winona State University, Mathematics & Statistics)

Abstract: Some of the biggest challenges facing humanity today stem from issues surrounding environmental degradation and social injustice, and the need to address the ramifications of these issues from a STEM perspective is greater than ever. This talk centers around the course Mathematics for Sustainability I recently developed at St. Mary’s College, Notre Dame and currently teach at Winona State University. The course develops and applies mathematical concepts and tools to quantitatively explore real-world, topical problems pertaining to environmental and social sustainability. Topics are motivated by exciting hands-on experiences – experiments, demonstrations, outdoor data collection excursions – from relevant STEM fields.

### Open Textbooks: What are they and what are other options available?

#### Monday, October 15th, 4:00-5:00pm, WH 288.

Refreshment will be provided at room WH 291 at 3:30-4:00pm.

Speaker: Dr. Namyong Lee, Department of Mathematics and Statistics, MSU

Abstract: In this talk, we present the on-going efforts by the MnSCU OER and the MSU, Mankato on open educational resources, especially the open textbooks. After a short overview of open textbooks in both legal and educational perspective, we will discuss on available options for our common service courses, including the calculus sequence. The second part is intended as an open faculty forum rather than an information session.

## 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).

## Fall 2017 Department Seminars

### Universal Turing Machine

#### Tuesday, September 26th, 4:00-5:00pm, WH 288A.

Refreshment will be provided at room WH 291 at 3:30-4:00pm.

Speaker: Dr. Kyung Il Lee, Department of Mathematics and Statistics, MSU

Abstract: Throughout the 19th Century, the diverse fields of mathematics was getting more and more abstract. Consequently, at the beginning of the 20th century, mathematics itself was challenged by the discovery of a couple of paradoxes such as Russell’s paradox. In this talk, we discuss what was identified as a challenge, namely Hilbert’s Program, and describe Turing’s solution to Entscheidungsproblem (decision problem), which is destructive to one aspect of Hilbert’s Program. Next, the idea of the modern stored-program computers will be discussed, which dates back to the notion of Turing’s universal machine conceived when Turing was answering the decision problem by a diagonalization work against the most convincing mathematical model of computation, namely Turing machine model.

### Computing Graph and Group Automorphisms with Mathematica

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

Refreshment will be provided at room WH 291 at 3:30-4:00pm.

Speaker: Prof. Dan Singer, Department of Mathematics and Statistics, MSU

Abstract:Mathematica is a powerful programming tool for implementing mathematical algorithms, manipulating large data sets, testing conjectures, and making abstract mathematical concepts concrete. I will demonstrate its use by computing all digraph automorphisms of a directed graph D. I will then find all group automorphisms of a given group G by finding all automorphisms of the associated Cayley digraph D = T(G; S) with respect to a generating set S. While no new mathematical ground is being broken here, my purpose is to demonstrate how to design and implement mathematical algorithms in software and to encourage mathematics students to consider learning Mathematica or the equivalent to do the same.

### An Efficient Method for Band-limited Extrapolation by Regularization

#### Tuesday, October 24th, 4:00-5:00pm, WH 288A.

Refreshment will be provided at room WH 291 at 3:30-4:00pm.

Speaker: Dr. Weidong Chen, Department of Mathematics and Statistics, MSU

Abstract: In this presentation, I will discuss the problem that I will attack: An Efficient algorithm for solving a band-limited extrapolation. The model is the following: a function f(t) in L^1(R) is band-limited if its Fourier transform F(w) is supported in a finite interval. Then f(t) can be recoverd from F(w) by the inverse Fourier transform by integration on that interval. The extrapolation problem is: Given f(t) on [-T,T], find f(t) outside that interval, where T>0 is a constant.

A regularized spectral estimation formula and a regularized iterative algorithm for band-limited extrapolation are presented. The ill-posedness is taken into account. First the Fredholm equation is regularized. Then it is transformed to a differential equation in the case where the time interval is R. A fast algorithm to solve the differential equation is given by the finite difference, and a regularized spectral estimation formula is obtained. Then a regularized iterative extrapolation algorithm is introduced and compared with the Papoulis and Gerchberg algorithm.

#### Thursday, November 2nd, 5:00-6:00pm, WH 285.

Speaker: Prof. Dan Singer, Department of Mathematics and Statistics, MSU

Abstract: Let a, b, and c be integers. We will demonstrate how to find all integer pairs (x,y) that satisfy the quadratic Diophantine equation ax^2+by=c. We will also provide criteria for deciding whether or not any solution to this equation can be found. We will introduce elementary concepts from number theory as needed: Euclid's algorithm, properties of prime numbers, the Chinese Remainder Theorem, the theorems of Fermat and Wilson, Euler's criterion for quadratic residues, and the Gauss Reciprocity Theorem.

### High Fidelity Simulations of Flow Over a Biofilms based on the Cahn-Hilliard and Navier-Stokes Equations

#### Tuesday, November 21st, 4:00-5:00pm, WH 288A.

Refreshment will be provided at room WH 291 at 3:30-4:00pm.

Speaker: Nathan McClanahan from South Dakota State University (Presenter), joint work with Nicholas Stegmeier, Jeffrey Doom, and Jung-Han Kimn

Abstract: Biofilms are attached microbial communities made of many different components. Biofilms are found throughout nature as well as industrial and medical settings. Understanding how biofilms spread is important in prevention and treatment of diseases and contamination. To model a biofilm we used an energy based model, starting with the Cahn-Hilliard equation using the Flory-Huggins equation. We will give a brief description of the background of these equations. We will discuss the implementation of efficient parallel simulation procedures based on parallel numerical algorithms and toolkits including PETSc (Portable Extensible Toolkit for Scientific Computing) which is developed at Argonne National Laboratory. We will also discuss the ongoing collaborative work to combine the Cahn-Hilliard and Navier-Stokes equations into a single system. This system will use the Navier-Stokes equation to handle the flow outside of the biofilm and the Cahn-Hilliard equation to model the interface between the fluid and biofilm. Results consistent with observations in nature will be discussed as well as future work and applications of the combined model.

## Spring 2017 Department Seminars

### Parametric analysis of “Hemophilia A” and Analytical Modeling of Democracy

#### Friday, April 14th, 11:00-11:50am, AH 310.

Speaker: A K M Raquibul Bashar, University of South Florida

Abstract: This study is to show parametric analysis of a rare disease ‘Hemophilia A’ using the data from Centers for Disease Control and Prevention (CDC). This parametric analysis will enable us to answer some of the basic critical questions such as: What is the distribution of severity levels of patients? Is there any dependency between and among severity levels, inhibitor history, and races? Those were answered by some classical parametric analysis of categorical variables.

Also, this presentation will show a statistical model constructed by using data from Economics Intelligence Unit’s (EIU) that is being collected from 167 countries around the world. This analytical model results in predicting the democracy score which can be used to rank the countries around the globe to categories one the classifications defined by EIU as ‘full democracy’, ‘flawed democracy’, ‘hybrid democracy’, and ‘authoritarian regimes’. The EIU performs descriptive analysis to classify the countries around the globe. We developed a statistical model using the EIU data to estimate the democracy score and proceed to classify the countries. The proposed statistical model is of high quality that will reflect on the accuracy of our classification.

### Classification of protein binding ligands using their structural dispersion

#### Monday, April 10th, 9:00-9:50am, Nelson Hall 003.

Speaker: Dr. Galkande Premarathna, Texas Tech University

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. The main purpose of this research is to gain a better understanding of how structure relates to binding activity and to classify proteins according to function via structural information. We approach the problem from the dataset compiled by Kahraman et al (2007) and extended Kahraman dataset. There we calculated the covariance matrices of site’s coordinates which use the distance of each atom to the center of mass and calculate the distance from an atom to the 1st, 2nd and 3rd principal axis. Then, we performed classification on these matrices using a variety of techniques, including nearest neighbor. Finally, we compared the performance of this model based technique with alignment based techniques.

### Model Average Versus Model Selection: A Bayes Perspective

#### Friday, April 7th, 11:00-11:50am, AH 310.

Speaker: Dr. Tri Le (speaker) and Bertrand Clarke, University of Nebraska-Lincoln

Abstract: We compare the performance of five model average predictors -- stacking, Bayes model averaging, bagging, random forests, and boosting -- to the components used to form them. In all five cases we provide conditions under which the model average predictor performs as well or better than any of its components. This is well known empirically, especially for complex problems, although few theoretical results seem to be available. Moreover, all five of the model averages can be regarded as Bayesian. Stacking is the Bayes optimal action in an asymptotic sense under several loss functions. Bayes model averaging is known to be the Bayes action under squared error. We show that bagging can be regarded as a special case of Bayes model averaging in an asymptotic sense. Random forests are a special case of bagging and hence likewise Bayes. Boosted regression is a limit of Bayes optimal boosting classifiers. We have limited our attention to the regression context since that is where model averaging techniques differ most often from current practice.

### Multiple Imputation for Missing Data and Disclosure Limitation

#### Tuesday, April 11th, 11:00-11:50am, WH 284.

Speaker: Dr. Christine Kohnen, Duke University

Abstract: Multiple Imputation (Rubin, 1987) and its underlying principles can be used in applications ranging from missing data to disclosure limitation. Regardless of the scenario, valid inferences can be obtained using either the standard combining rules and variance estimates of multiple imputation or variants of the rules derived specifically for disclosure limitation. The focus of this seminar will be on two applications of multiple imputation. The first uses multiple imputation to help determine an overall household income distribution of a set of students, where roughly 30% of the data are unknown. The second application is based on multiple imputation for disclosure limitation, such that sensitive and non-sensitive data can be released through the creation of partially synthetic data. The seminar will conclude with a discussion on how these two methods can be combined for use in other applications.

### Mathematical Methods and Modeling in the National Security Sciences

#### Monday, March 27th, 4:00-5:00pm, WH 284A.

Refreshment will be provided at room WH 291 at 3:30-4:00pm.

Speaker: Dr. Aaron Luttman, Manager, Diagnostic Research and Material Studies Nevada National Security Site

Abstract: While most people are familiar with many of the military aspects of national security, the scientific enterprise in support of national security is less well known. The National Nuclear Security Administration (NNSA) is a semi-autonomous agency within the U.S. Department of Energy that oversees the nation’s nuclear security science, from nuclear non- and counter-proliferation technologies to nuclear emergency response (like the Fukushima disaster in Japan) to the science of maintaining the U.S. nuclear weapons stockpile. The NNSA supports a scientific enterprise of more than 50,000 scientists, technicians, and engineers, and, in this presentation, we will introduce some of the latest scientific developments that are underway in support of U.S. nuclear security, including current mathematical research associated with the chemistry and physics of dynamic material studies, which involves explosively-driven experimentation in material science. In addition to some actual mathematical case studies at the cutting edge of nuclear security science, we will discuss some of the national policies that drive the science as well as how new graduates in science, technology, engineering, and mathematics can get involved in this research through internships and support for graduate studies.

### Mathematica Demonstration

#### Thursday, April 13th, 11:00-11:50pm, including Q & A, WH 284.

Speaker: Matt Woodbury from Wolfram Research in Education and Research

Abstract: I will begin with a technical overview of Mathematica, as well as briefly touching on the creation of Wolfram|Alpha. Next, we discuss emerging trends in technology and what is currently available (or being developed) to support those trends. Then, to give you a sense of what's possible, I'll discuss how other organizations use these tools for teaching and research.

## Fall 2016 Department Seminars

### Parallel algorithms based on domain decomposition methods and their implementations

#### Tuesday, November 22nd, 4:00-5:00pm, WH 284A.

Refreshment will be provided at room WH 291 at 3:30-4:00pm.

Speaker: Professor Jung Han Kimn, Department of Mathematics & Statistics, South Dakota State University

Abstract: Parallel algorithms are essential to improve the performance of numerical procedures for large size simulation of many application problems. In this talk, we present how efficient parallel algorithms can be designed and implemented using the mathematical idea based on domain decomposition methods. Domain decomposition methods comprise an important class of parallel algorithms that is naturally parallel and flexible in their application to a wide range of scientific and engineering problems. The major ingredients of this talk are 1) a brief introduction to domain decomposition methods, 2) the time decomposition procedures for application problems from computational physics, 3) the parallel implementation results of a CFD (Computational Fluid Dynamics) simulation, and 4) the parallel implementation using PETSc (Portable, Extensible Toolkit for Scientific Computation) of the Argonne National Laboratory.

### Hard Lefschetz Property of Symplectic Structures on Compact Kaehler Manifolds

#### Monday, November 7th, 4:00-5:00pm, WH 284A.

Refreshment will be provided at room WH 291 at 3:30-4:00pm.

Speaker: Heidi Andersen, Ph.D. candidate, Department of Mathematics, University of Minnesota

Abstract: In 1996, B. Khesin and D. McDu questioned whether or not there exists a path of symplectic forms such that the dimension of the space of cohomology classes of symplectic harmonic k-forms varies along the path. Recently, Y. Cho provided an answer by constructing a compact, simply-connected Kaehler manifold of complex dimension 3, which possesses a symplectic form that does not satisfy the hard Lefschetz property yet is symplectically deformation equivalent to the Kaehler form; the first example of its kind. We generalize the construction to complex dimension 4, and consider further applications.

### Testing for Vector White Noise Using Maximum Cross Correlations

#### Thursday, October 13th, 4:00-5:00pm, WH 284A.

Refreshment will be provided at room WH 291 at 3:30-4:00pm.

Speaker: Professor Wen Zhou, Department of Statistics, Colorado State University

Abstract: We propose a new omnibus test for vector white noise using the maximum absolute auto-/cross-correlations of the component series. Based on the newly established approximation by the L_infty-norm of a normal random vector, the critical value of the test can be evaluated by bootstrapping from a multivariate normal distribution. In contrast to the conventional white noise test, the new method is proved to be valid for testing the departure from non-IID white noise. We illustrate the accuracy and the power of the proposed test by simulation, which also shows that the new test outperforms the three multivariate versions of the Box-Pierce portmanteau test especially when the dimension of time series is high. The numerical results also indicate that the performance of the new test can be further enhanced when it is applied to the pre-transformed data obtained via the time series principal component analysis proposed by Chang, Guo and Yao (2015).

### Merging mixture components for clustering

#### Thursday, October 6th, 4:00-5:00pm, WH 284A.

Refreshment will be provided at room WH 291 at 3:00-4:00pm.

Speaker: Professor Melnykov, Department of Information Systems, Statistics and Management Science, University of Alabama

Abstract: Finite mixture models are well-known for their flexibility in modeling heterogeneity in data. Model-based clustering is an important application of mixture models that assumes that each mixture component distribution can adequately model a particular group of data. Unfortunately, when more than one component is needed for each group, the appealing one-to-one correspondence between mixture components and groups of data is ruined and model-based clustering loses its attractive interpretation. Several remedies have been considered in literature. We discuss the most promising recent results obtained in this area and propose a new algorithm that finds partitionings through merging mixture components relying on their pairwise overlap. Extensions of the developed technique are considered in the context of clustering large datasets.

### Symplectic toric manifolds and applications

#### Monday, September 26th, 4:00-5:00pm, WH 284A.

Refreshment will be provided at room WH 218 at 3:30-4:00pm.

Speaker: Jun Li, Ph.D. candidate, Department of Mathematics, University of Minnesota

Abstract: We will focus on the theory of toric geometry from the symplectic side. Toric varieties form an important and rich class of examples, which often provide a testing ground for theorems. In this talk, a fundamental classifying theorem (by Atiyah-Guillemin-Sternberg;Delzant) of symplectic toric manifolds will be reviewed, while new applications to solve problems in symplectic topology will be sketched. Throughout the talk, we will provide many examples and pictures.

## Spring 2016 Department Seminars

### Intrinsic Volumes and Approximations

#### Thursday, April 28th, 4:00-5:00pm, WH 286A.

Refreshment will be provided at room WH 291 at 3:30-4:00pm.

Speaker: Professor Brandon Rowekamp, Department of Mathematics and Statistics, MSU

Abstract: When approximating shapes it is possible that some measurements may be preserved by the approximation while others are not. For example, an approximation of a circle might have an area close to the correct value, but have the length of the perimeter very different from the correct value. To guarantee that an approximation preserves all important quantities, it is only necessary to calculate a small number of "essential" measurements. These are the "intrinsic volumes" of the shape. While the study of intrinsic volumes has many complexities, the volumes themselves can be found in simple cases by formulas for the volumes of tubes. In this talk I will demonstrate the calculations from the simpler cases, describe how the process could be generalized to more complicated cases, and discuss the fundamental results (such as the Hadwiger Characterization Theorem) which make intrinsic volumes extremely important in geometric approximation.

### Dynamics, Floer Homology, and Knot Invariants

#### Thursday, April 14th, 4:15-5:15pm, AH 316.

Refreshment will be provided at room WH 291 at 3:30-4:15pm.

Speaker: Dr. Ke Zhu, Department of Mathematics and Statistics, MSU

Abstract: Classic topology has its root in dynamic systems. Concepts like rotation number already appeared in Poincaré’s early work in celestial mechanics. In 80’s, Floer developed a homology suited for studying Hamiltonian dynamics. Many conjectures about number of periodic orbits were solved by Floer homology and its variants. In this talk, I will explain my work on thick-thin decomposition of Floer homology, which built the chain level isomorphism to quantum cohomology in symplectic manifolds (joint work with Yong-Geun Oh), and also give the application to knot invariants.

### On the Homological Arnold Chord Conjecture

#### Tuesday, April 12th, 4:15-5:15pm, TR-E 224.

Refreshment will be provided at room WH 291 at 3:30-4:15pm.

Speaker: Dr. Roman Golovko, Alfred Renyi Institute of Mathematics

Abstract: In 1986, V.I. Arnold formulated a conjecture which up to this day is remarkably little understood. It concerns the dynamics of a Reeb flow in relation to a Legendrian submanifold. More precisely, it says that for a horizontally displaceable Legendrian submanifold L of the contactization of a Liouville manifold, the number of Reeb chords on L is bounded from below by half of the Betti numbers of L. We will discuss several recent contributions to this conjecture and the way to prove its generalization when L admits an exact Lagrangian filling. This is joint work with G. Dimitroglou Rizell.

### Linking Numbers, Configuration Spaces, and Homotopical Invariants of Links and Knots

#### Friday, April 8th, 4:15-5:15pm, AH 304.

Refreshment will be provided at room WH 291 at 3:30-4:15pm.

Speaker: Dr. Robin Koytcheff, University of Massachusetts-Amherst

Abstract: Knots and links have been studied by mathematicians at least since the time of Gauss, who defined the linking number of two closed curves. I will describe this invariant of links in elementary terms, by counting crossings. I will then reformulate it as the degree of a map of surfaces and ultimately in terms of homotopy classes of maps of configuration spaces. This latter description generalizes to an invariant of n-strand links which ignores knotting but conjecturally detects all linking phenomena. This is the subject of recent joint work with Fred Cohen, Rafal Komendarczyk, and Clay Shonkwiler, where we proved a certain analogue of this conjecture. Finally, I will briefly discuss related joint work with Budney, Conant, and Sinha on knot invariants. The latter work develops a relationship between the Taylor tower for the space of knots and finite-type knot invariants.

### Application of Survival Analysis in Actuarial and Medical Research

#### Thursday, April 7th, 4:00-5:00pm, WH 304.

Refreshment will be provided at room WH 291 at 3:30-4:00pm.

Speaker: Professor Deepak Sanjel and Prashant K. C., Department of Mathematics and Statistics, MSU

Abstract: Survival Analysis is heavily used in clinical trials to study the rate of death (Hazard rate) or the Survival probability in patients after they have received treatments or a organ transplant. Survival models can also be very useful in analyzing insurance claim data to calculate the risk of claim and to determine the insurance premium, such as life or auto insurance. We are interested in the relationship between lifespan (response variable) and some predictive covariates. Applications of some of the commonly used survival methods such as Kaplan-Meier estimator, Nelson Alan Actuarial estimator and the CoxPH model will be discussed in connection with actuarial and medical application. These result will be compared with Bayesian approach using the Markov Chain Monte Carlo (MCMC) method. In this talk we will use the survival models in two case studies: * Actuarial Applications: Auto insurance claim data. * Medical Applications: Estimate and compare the survival rate of primary biliary cirrhosis (PBC) patient data from the Mayo clinic.

## Fall 2015 Department Seminars

### Exotic 4-Manifolds with Small Euler Characteristics

#### Wednesday, December 2nd, 4:00pm, WH288A.

Refreshment will be provided at room WH291 at 3:30-4:00pm.

Speaker: Professor Anar Akhmedov, Department of Mathematics, University of Minnesota

Abstract: It is known that many simply connected, smooth topological 4-manifolds admit infinitely many exotic smooth structures. However, the smaller the Euler characteristic, the harder it is to construct exotic smooth structure. In this talk, we will construct exotic smooth structures on various small 4-manifolds. If time permits, we will also discuss interesting applications to the geography of symplectic 4-manifolds.

### Statistical Applications in the Biomedical and Medical Studies of Organ Transplant

#### Tuesday, November 17th, 4:00pm, WH289.

Refreshment will be provided at room WH291 at 3:30-4:00m.

Speaker: Professor Chunfa Jie, Des Moines University

Abstract: Statistical analysis is a key component in the biomarker development and outcome research, which hold the promise to improve outcome and reduce need for invasive diagnostic in organ transplantation. Such statistical applications to biomedical and medical research in a transplant environment will be discussed with three studies. 1) In a prospective study, we found several proteins were associated with the presence/absence of pre-LT (liver transplant) Acute Kidney Injury and post-LT renal recovery. We also built a logistic regression model to correlate the occurrence probability of post-LT renal recovery with both pre-LT protein biomarkers and clinical variables, which may be used to predict the pre-LT kidney injury recovery after LT. 2) CMV virus is a member of the Herpes virus family that has a common infection in adults (50-90%). Using a genetically engineered CMV strain, we performed an RNA-seq analysis to investigate the host cellular gene expressions affected by the shutdown of the viral IE1/IE2 genes. We also did a Chip-seq pilot-study to analyze the acetylated histone H3 binding to viral genome as the virus undergoes epigenetic reprogramming upon acute infection. 3) Recent solid phase antibody DSA (Donor Specific Antibody) assays have found associations between DSA in deceased donor liver transplant (DDLT). Time-to-event analyses were taken to compare the association of DSA and patient/graft survival as well as other complications between LDLT (living donor liver transplant) vs. DDLT recipients, and determine if DSA is an independent predictor of outcomes in a multivariate context.

### Introduction to Category Theory

#### Tuesday, October 27th, 4:00pm, WH284A.

Refreshment will be provided at room WH291 at 3:30-4:00pm.

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

Abstract: Category theory could be defined in two ways, either (i) "framework of mathematical structures " or (ii) "abstract nonsense". The goal of the talk is to show that (i) is a better definition than (ii). The growing demand for category theory may be due to the following reasons. Firstly, it seems that mathematics needs a larger play ground than set theory as we may see from the formulation of moduli problems. In this direction, Stacks (2-categories) and infinite categories are introduced in mathematics. Secondly, a general homology theory requires reformulation of categories and consequently the concept of derived categories. Thirdly, mathematicians may need to know how to handle with limits of sheaves in certain categories (e.g., abelian categories) in a systematic way and one way to do is to use "abstract nonsense". Starting with similarity between a (classical) category and a topological space we generalize category to higher ones, and review some developments of new homology theory in a new framework.

### Stenger's Sinc Matrix Conjecture

#### Friday, October 16th, 3:30pm, WH288.

Refreshment will be provided at room WH291 at 3:00-3:30pm.

Speaker: Professor Lixing Han, Department of Mathematics, University of Michigan-Flint

Abstract: For the past few decades, Sinc approximation methods have been successfully used in handling a wide variety of computational problems. In his paper [Journal of Computational and Applied Mathematics, 86 (1997), pp. 297-310], Frank Stenger pointed out that the validity of relevant Sinc methods hinges on the assumption that all eigenvalues of the Sinc matrix I^{(-1)} are located in the open right half-plane. In light of favorable numerical evidence for each I^{(-1)} of order up to n=513, Stenger conjectured that this is the case for all I^{(-1)}, regardless how large n is. In 2003, Iyad Abu-Jeib and Thomas Shores established a partial answer to this problem. In 2013, Jianhong Xu and I gave a complete proof of the conjecture. In this talk, I will present the conjecture and give an outline of the proof. This talk is based on joint work with Jianhong Xu.

### From Projective Geometry to a Number Field

#### Friday, October 2nd, 1:00pm, WH285.

Refreshment will be provided at room WH291 at 2:00-3:00pm.

Speaker: Professor Jim Fowler, Department of Mathematics, Ohio State University

Abstract: Given a field (a number system), we can build a "geometry"---something with points and lines. For instance, starting with the field of real numbers, we consider ordered pairs of real numbers, and we get the Cartesian plane. What if we instead started with the geometry? Could we, from the points and lines, recover a field? Sometimes, yes! The geometries we start with will be "projective geometries" where any two lines meet, maybe "at infinity." How do we then get a field? Commutativity of multiplication---among the other axioms for a field---are encoded as gloriously complicated diagrams of points and lines. Desargues' theorem and Pappus' theorem show up to save the day.

### The Poincare-Birkhoff Theorem and the Billiard Problem

#### Thursday, October 1st, 4:00pm, WH284A.

Refreshment will be provided at room WH291 at 3:30-4:00pm.

Speaker: Dr. Ke Zhu, Department of Mathematics and Statistics, MSU

Abstract: The Poincare-Birkhoff theorem concerns the number of fixed points of area preserving self-maps of an annulus. It has many applications to Hamiltonian dynamical systems, including the periodic trajectories on the billiard table. The theorem is also a starting point of modern symplectic topology, leading to later development like the Arnold conjecture and Floer homology. In this talk, I will draw many pictures and explain the connections.

## Spring 2015 Department Seminars

#### Tuesday, April 28th, 3:30-4:30pm, WH286A

Refreshment will be provided at room WH291 at 3:00-3:30pm.

Speaker: Professor In-Jae Kim, Department of Mathematics and Statistics, MSU

Abstract: In this talk we talk about web information retrieval. In particular we will show how the PageRank, developed by Sergey Brin and Larry Page (cofounders of Google), uses hyperlink analysis to rank webpages in WWW.

### True TRI-gonometry

#### Thursday, April 23rd, 2:00-2:50pm, WH289

Refreshment will be provided at the same room

Speaker: Professor Paul Nelson, Department of Mathematics and Statistics, MSU

Abstract: : It is well known that sin(x) and cos(x) can be defined by alternating series. From these definitions, all of the usual identities can be established. However, this theory involves only TWO functions--it should be referred to as BI-gonometry. We will investigate what happens if we start with THREE power series instead of two. This material will be presented so that undergraduates--and even calculus students--should find it interesting and understandable.

### Big Data Analysis Through the TDA Looking Glass

#### Friday, April 3rd, 3:15 - 4:00pm, WH286A

Refreshment will be provided at 3:00pm at the same room.

Speaker: Professor Namyong Lee, Department of Mathematics and Statistics, MSU

Abstract: "Big Data" is everywhere and is rapidly creeping into every part of our life. As "Big Data" is often complex and has too many features, it has many challenges in analysis. We introduce basic idea of TDA (Topological Data Analysis) and its advantage in visual analysis compare other analysis. A few concrete examples will be given.

### Modeling and Predicting Extreme Events

#### Monday, February 16th, 3:30-4:30pm, WH286A.

Refreshment will be provided at room WH291 at 3:00-3:30pm.

Speaker: Professor Deepak Sanjel, Department of Mathematics and Statistics, MSU

Abstract: Accurately modeling and predicting extreme events such as flood, fire, hurricane etc. have huge implications in many disciplines. For instance engineering, finance, earth sciences, traffic prediction etc. However, usual statistical techniques fail when dealing with extreme events. Several methods of analyzing and modeling extreme events are proposed in the literature. However, most of the proposed methods are based on the assumption of extreme value limit distributions or some related family of distributions. We purpose Bayesian approach using Markov chain Monte Carlo method to analyze extreme events and used time series statistical models for prediction. The methods are illustrated using two data sets, a time series of maximum temperatures recorded in Brisbane Australia and Minnesota River flood data (gage height in Minnesota River at Mankato from 1903 to 2013).

## Fall 2014 Department Seminars

### Three Theorems on the Normality of a Family of Analytic Functions

#### Monday, December 1st, 4:00-5:00pm, WH 286

Speaker: Russell Jahn, Department of Mathematics & Statistics, MSU

Abstract:In certain areas of mathematics, such as the iteration of rational maps of complex variables, it is beneficial to know conditions which guarantee the normality of a family of analytic functions. Here, we state three such theorems and give proofs of two of them and a proof sketch of the third.

### Smoothing Newton Method for Data Classification

#### Tuesday, October 28th, 3:30-4:30pm, WH 284

Refreshment will be provided at room WH 291 at 3:00-3:30pm.

Speaker: Professor Hongxia Yin, Department of Mathematics & Statistics, MSU

Abstract:In this research on data classification, we give a new merit function for handling the high-dimension variables in data mining problems. A smoothing Newton method is given for solving the dual of soft margin data classification problem. It is proved that the algorithm is globally convergent and locally super-linear convergent. Preliminary numerical tests show that the algorithm is promising.

### Residual spectrum of reductive groups under isogeny

#### Tuesday, October 14th, 3:30-4:30pm, WH 284

Refreshment will be provided at room WH 291 at 3:00-3:30pm.

Speaker: Dr. Wook Kim, Former Research Fellow, Korea Institute for Advanced Study

Abstract: Given an algebraic group G, we may consider the complex vector space of certain "periodic" functions from G to complex numbers (automorphic forms). There is a certain subspace which is called the residual spectrum that is parameterized by representations of Levi subgroups of G. An isogeny from one algebraic group to another is a surjective morphism with finite kernel. One may be interested in how residual spectrum behaves under isogeny. The purpose of this talk is to find a simple relationship between residual spectrum of groups in a given isogeny class, and apply it to several important cases.

### Isometric embeddings of Riemannian manifolds via heat kernel

#### Tuesday, October 7th, 3:30-4:30pm, WH 284

Refreshment will be provided at room WH 291 at 3:00-3:30pm.

Speaker: Dr. Ke Zhu, Dept. of Mathematics and Statistics, MSU

Abstract:: The Nash embedding theorem states that every Riemannian manifold can be isometrically embedded into some Euclidean space. Isometric means preserving the length of every path. For instance, bending without stretching or tearing a page of paper gives an isometric embedding of the page into Euclidean space, because curves drawn on the page retain the same arc length when the page is bent. Nash's proof involves sophisticated perturbations of the initial embedding, so not much is known about the geometry of the resulted embedding. In this talk, using the eigenfunctions of the Laplacian operator, we construct canonical isometric embeddings of compact Riemannian manifolds into Euclidean spaces, and discuss the geometry of embedded images. They turn out to have large mean curvature (intuitively, very bumpy), but the extent of oscillation is about the same at every point.