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Mathematics & Statistics
Colloquium Archive

# Colloquia — Spring 2016

## Friday, May 6, 2016

Title
Speaker

Time
Place
Sponsor

Steklov's problem for orthogonal polynomials
Serguei Denisov
University of Wisconsin-Madison
3:00pm-4:00pm
CMC 130
E. A. Rakhmanov

Abstract

The problem of Steklov is the problem of estimating the uniform norm of polynomials orthogonal with respect to a measure whose density is bounded away from zero. We will discuss its history, recent results, and some open problems.

## Friday, April 29, 2016

Title
Speaker

Time
Place
Sponsor

Recent developments of Hom-Type algebra structures
Abdenacer Makhlouf
Université de Haute Alsace
3:00pm-4:00pm
CMC 130
Mohamed Elhamdadi

Abstract

The main feature of Hom-algebras is that the identities are twisted by a homomorphism. The main structures are Hom-Lie algebras and Hom-associative algebra generalizing classical Lie and associative algebras. Hom-Lie algebras appeared naturally in the study of $$q$$-deformations of Witt and Virasoro algebras. They are also related to $$\sigma$$-derivations.

In the last years, many concepts and properties from classical algebraic theories have been extended to the framework of Hom-structures.

In this talk, we deal with a recent generalization involving two linear maps. Such structures appeared naturally when studying new type of categories. We mainly discuss constructions and representations of BiHom-associative algebras, BiHom-Lie algebras and BiHom-bialgebras, as well as some constructions of twisted tensor products and smash products.

## Friday, April 22, 2016

Title
Speaker

Time
Place
Sponsor

Riemann surfaces and crystallographic groups
Yukio Matsumoto
University of Tokyo
3:00pm-4:00pm
CMC 130
Masahiko Saito

Abstract

A Riemann surface is a closed surface with a complex structure, while a crystallographic group is an isometry group acting on a Euclidean space $$\mathbb{E}^n$$ whose translation subgroup forms an $$n$$-dimensional lattice. For example, a so-called wallpaper group is a two dimensional crystallographic group.

This talk will report our recent discovery that certain crystallographic groups on $$\mathbb{E}^{3g-3}$$ naturally arise from Teichmüller space of Riemann surfaces of genus $$g$$.

## Thursday, April 21, 2016

Title
Speaker

Time
Place
Sponsor

Chord index, its definition, application and generalization
Zhiyun Cheng
Beijing Normal University and George Washington University
3:30pm-4:30pm
NES 104
Mohamed Elhamdadi

Abstract

In this talk I will present some development of virtual knot theory in recent years. In particular we will focus on the parity invariant and its generalization, the chord index. Several applications of the chord index will be discussed. If time permitting, I will give some general construction of the chord index.

## Friday, April 15, 2016

Title
Speaker

Time
Place
Sponsor

Is Dispersion a Stabilizing or Destabilizing Mechanism?
Edriss Titi
Texas A&M University
4:00pm-5:00pm
CMC 130
TBA

Abstract

In this talk I will present a unified approach for the effect of fast rotation and dispersion as an averaging mechanism for, on the one hand, regularizing and stabilizing certain evolution equations, such as the Navier-Stokes and Burgers equations. On the other hand, I will also present some results in which large dispersion acts as a destabilizing mechanism for the long-time dynamics of certain dissipative evolution equations, such as the Kuramoto-Sivashinsky equation.

Title
Speaker

Time
Place
Sponsor

Teaching and Learning in Student-Centered Classrooms — What Do We Know?
Karen Keene
North Carolina State
3:00pm-4:00pm
CMC 130
Milé Krajčevski

Abstract

We shall present results from our work in inquiry-oriented differential equations. We will discuss what researchers have found about how students think about solutions to systems of differential equations. Then we will talk about some of the preliminary work from an NSF-funded project supporting faculty as they consider making instructional change to more active learning in Differential Equations, Abstract Algebra and Linear Algebra.

## Friday, April 8, 2016

Title
Speaker

Time
Place
Sponsor

The first exit problem of dynamical systems perturbed by Brownian motion or a stable process
Michael A. Högele
Universidad de Los Andes
Bogotá, Colombia
3:00pm-4:00pm
CMC 130
Yuncheng You

Abstract

This colloquium talk is dedicated to the presentation of a classical problem in probability theory, which has gained new interest due to recent studies in paleoclimatology. We start with an introduction to Brownian motion and L evy processes, in particular stable processes. In the sequel we explain the first exit problem of a gradient dynamical system perturbed by a Brownian motion or a stable process from the domain of attraction of a stable state when the intensity of the stochastic process tends to zero. After this we generalize this reasoning to a generic class (Morse-Smale) of hyperbolic non-gradient system with only a global attractor. In the end we illustrate this result for the celebrated van-der-Pol oscillator perturbed by a stable process.

## Friday, April 1, 2016

Title
Speaker

Time
Place
Sponsor

Linear Codes over Galois Ring $$\mathrm{GR}(p^2,r)$$
Keqin Feng
Tsinghua University
Beijing, China
3:00pm-4:00pm
CMC 130
Xiang-dong Hou

Abstract

Linear Codes over finite rings become one of hot topics in coding theory after Hommons et al. discovered in 1994 that several remarkable nonlinear binary codes with linear like properties are the images of Gray map of linear codes over $$\mathbb{Z}_4$$. Many codes over finite rings have been constructed and classified since then, but in general, to determine their minimum distance is a difficult problem. In this talk we consider two series of linear codes $$C(G)$$ and $$C'(G)$$ over Galois ring $$R=\mathrm{GR}(p^2,r)$$ where $$G$$ is a subgroup of the unit group of extension ring of $$R$$. We determine their minimum distance and weight distributions by using the Gauss sums on Galois ring. Then we use the homogeneous weight and generalized Gray map, we get a series of remarkable $$p$$-ary nonlinear codes with good Hamming minimum distance.

Title
Speaker

Time
Place
Sponsor

Health Information Sharing with Differential Privacy
Li Xiong
Emory University
1:00pm-2:00pm
CMC 130
Kaiqi Xiong

Abstract

Enabling large-scale analysis of clinical and public health data while protecting privacy of human subjects has been a key challenge in biomedical research. The current de-identification approach is subject to various re-identification and disclosure risks and does not provide sufficient privacy protection for patients. In this talk, I will give an overview of our work on sharing statistical or synthetic health data that enables medical research while protecting individual privacy with the state-of-art differential privacy framework. I will present our technical solutions for handing different types of data including relational, sequential, and time series data. I will also present case studies using real public health datasets and demonstrate the feasibility as well as challenges of applying the differential privacy framework on medical data.

## Wednesday, March 30, 2016

Title
Speaker

Time
Place
Sponsor

Becoming a Data Scientist
Michael Rappa
North Carolina State University
3:30pm-4:30pm
CHE 103
K. M. Ramachandran

Abstract

What is a data scientist and how does a student with the ambition become one? This talk will examine the rise of the data scientist as a major new professional occupation over the past decade. It will explore the necessary skills needed to practice data science in industry and government, and how these skills can be acquired—in formal academic programs and informal individualized educational pathways. Data on market demand and average compensation for data scientists and will be provided based on the experience of over 500 graduates of the Institute for Advanced Analytics since 2007.

## Friday, March 25, 2016

Title
Speaker

Time
Place
Sponsor

Polytopes of Stochastic Tensors
Fuzhen Zhang
Nova Southeastern University
Fort Lauderdale
3:00pm-4:00pm
CMC 130
Wen-Xiu Ma

Abstract

A square matrix is doubly stochastic if its entries are all nonnegative and each row and column sum is 1. A celebrated result known as Birkhoff's theorem about doubly stochastic matrices states that an $$n\times n$$ matrix is doubly stochastic if and only if it is a convex combination of some $$n\times n$$ permutation matrices (a.k.a. Birkhoff polytope).

We study the counterpart of the Birkhoff's theorem for higher dimensions. An $$n\times n\times n$$ stochastic tensor is a nonnegative array (hypermatrix) in which every sum over one index is 1. We study the polytope ($$O$$) of all these tensors, the convex set ($$L$$) of all tensors with some positive diagonals, and the polytope ($$T$$) generated by the permutation tensors. We show that $$L$$ is almost the same as $$O$$ except for some boundary points. We also present an upper bound for the number of vertices of $$O$$.

## Friday, March 11, 2016

Title
Speaker

Time
Place
Sponsor

New theorems at the interface of representation theory and number theory
Ken Ono
Emory University
3:00pm-4:00pm
CMC 130
Nagle Lecture Committee

Abstract

The speaker will discuss recent work on Moonshine and the Rogers-Ramanujan identities. The Rogers-Ramanujan identities are two peculiar identities which express two infinite product modular forms as number theoretic $$q$$-series. These identities give rise to the Rogers-Ramanujan continued fraction, whose values at CM points are algebraic integral units. In recent work with Griffin and Warnaar, the speaker has obtained a comprehensive framework of identities for infinite product modular forms in terms of Hall-Littlewood $$q$$-series. This work characterizes those integral units that arise from this theory. In a related direction, the speaker revisits the classical Moonshine Theorem which asserts that the coefficients of the modular $$j$$-functions are dimensions of virtual characters for the Monster, the largest of the simple sporadic groups. There are 194 irreducible representations of the Monster, and it has been a longstanding open problem to determine the distribution of these representations in Moonshine. In joint work with Griffin and Duncan, the speaker has obtained exact formulas for these distributions.

Title
Speaker

Time
Place
Sponsor

Rank one perturbations and Anderson models
Constanze Liaw
Baylor University
2:00pm-3:00pm
CMC 130
Alan Sola

Abstract

We consider two perturbation problems of very different type: (1) For rank one perturbations, we simply add to a fixed operator a vector projection onto a one dimensional subspace. Though falling into a restricted category of classical perturbation theory, important questions about the spectrum of the perturbed operator remain open. (2) Presenting a generalization of the discrete random Schröedinger operator, Anderson models contain a probabilistic component which causes the perturbation to be a non-compact operator (almost surely). As a result, none of classical perturbation theory applies here.

We present several links between these two perturbation problems. Some connections are identified as viable candidates to solve a long-standing conjecture (Anderson Localization Conjecture) about the spectral properties of Anderson models.

## Friday, February 26, 2016

Title

Speaker

Time
Place
Sponsor

Cavitation of spherical bubbles with surface tension and viscosity and connection with FRW cosmological equations
Stefan C. Mancas
Embry-Riddle Aeronautical University
3:00pm-4:00pm
CMC 130
Razvan Teodorescu

Abstract

In this talk an analysis of the Rayleigh-Plesset (RP) equation for a three dimensional vacuous bubble in water is presented. When the e ects of surface tension are neglected we find the radius and time of the evolution of the bubble as parametric closed-form solutions in terms of hypergeometric functions. By including capillarity we show the connection between RP equation and Abel's equation, and we present parametric rational Weierstrass periodic solutions for nonzero surface tension. When viscosity is present we present only numerical solutions. We also show the connection between the RP equation and Einstein's field equations for spatially curved FRW cosmology.

Title
Speaker

Time
Place
Sponsor

On the Emergence of Navier-Stokes-alpha Model for Channel Flow
Jing Tian
Texas A&M University
2:00pm-3:00pm
CMC 130
Yuncheng You

Abstract

Navier-Stokes equations (NSE) have been widely used to describe the motion of viscous incompressible fluid flow. As an averaged version of the NSE, Navier-Stokes-alpha model (NS-alpha) has good mathematical properties as well as nice experimental matches. Therefore, we take the NS-alpha as a favorable approximation for dynamics of appropriately averaged turbulent fluid flow. It is of intrinsic interest to find the transition mechanism between the NSE and the NS-alpha.

In this talk, we focus on a restricted class of the fluid flow with the hypothesis that the turbulence described by the NS-alpha equations is partly due to the roughness of the walls. We present the transition from the Navier-Stokes equations to the Navier-Stokes-alpha model by introducing a Reynolds type averaging. This is a joint work with C. Foias and B. Zhang.

## Friday, February 19, 2016

Title
Speaker

Time
Place
Sponsor

Visualizing the Fourth Dimension
Thomas Banchoff
Brown University
3:00pm-4:00pm
CMC 130
Milé Krajčevski

Abstract

Students and teachers of geometry have always relied on diagrams and models for visualizing relationships in the plane and in ordinary three-dimensional space, and for almost two hundred years mathematicians have struggled to find ways to deal with geometric phenomena in a fourth spatial dimension. All that changes with computer graphics.

We can now see and manipulate objects in new ways and share insights with our students and colleagues, in mathematics as well as in physics, philosophy, literature and modern art. Salvador Dali had a particular fascination with science and mathematics and this presentation will display a number of his favorite four-dimensional images.

## Friday, February 12, 2016

Title
Speaker

Time
Place
Sponsor

Tiling and spectral are equivalent in $$\mathbb{Z}_p^2$$
Azita Mayeli
QCC and the Graduate Center, CUNY
3:00pm-4:00pm
CMC 130
Arthur Danielyan

Abstract

The equivalence relation between tiling and spectral property of a set has its root in the Fuglede Conjecture a.k.a. Spectral Set Conjecture in $$\Bbb R^d$$, $$d\geq 1$$. In 1974, Fuglede stated that a bounded Lebesgue measurable set $$\Omega\subset\Bbb R^d$$, with positive and finite measure, tiles $$\Bbb R^d$$ by its translations if and only if the Hilbert space $$L^2(\Omega)$$ possesses an orthogonal basis of exponentials. A variety of results were proved for establishing connection between tiling and spectral property for some special cases of $$\Omega$$. However, the conjecture is false in general for dimensions $$3$$ and higher.

In this talk, we will define the tiling and spectral sets $$E\subseteq\Bbb Z_p\times\Bbb Z_p$$, $$p$$ prime, and show that these two properties are equivalent for $$E$$. In other words, we prove that the Fuglede Conjecture holds for $$\Bbb Z_p\times \Bbb Z_p$$.

## Friday, February 5, 2016

Title
Speaker

Time
Place
Sponsor

The phenomena of heavy tails in physical models including random matrices
Paul Jung
University of Alabama at Birmingham
3:00pm-4:00pm
CMC 130
Seung-Yeop Lee

Abstract

We will discuss a toy model of heavy tails and show how this does not follow central limit behavior. We will then see how this relates to models in physics including random matrices. In the random matrix setting, we equate limiting spectral distributions (LSD) to spectral measures of rooted graphs. The LSD result also includes matrices with i.i.d. entries (up to self-adjointness) having infinite second moments, but following central limit behavior. In this case, the graph is the natural numbers rooted at one, so the LSD is well-known to be the semi-circle law.

## Friday, January 29, 2016

Title
Speaker

Time
Place
Sponsor

Graph Polynomials motivated by Gene Assembly
Hendrik Jan Hoogeboom
University of Leiden
the Netherlands
3:00pm-4:00pm
CMC 130
Nataša Jonoska

Abstract

The interlace polynomial was discovered by Arratia, Bollobas, and Sorkin by studying DNA sequencing methods. Its definition can be traced from 4-regular graphs (the Martin polynomial), to circle graphs and finally to arbitrary graphs.

Our interest in these polynomials came from the study of ciliates, an ancient group of unicellular organisms. They have the remarkable property that their DNA is stored in two vastly different types of nuclei. The two representations of the versions of the gene can be elegantly modelled using a 4-regular graph.

We give an overview of the polynomials involved, their basic properties, and their relation to the Tutte polynomial. Joint work with Robert Brijder, Hasselt Belgium.

Title
Speaker

Time
Place
Sponsor

Local Gaussian process approximation for large computer experiments
Robert B. Gramacy
Booth School of Business
University of Chicago
2:00pm-3:00pm
CMC 130
Lesɫaw Skrzypek

Abstract

We provide a new approach to approximate emulation of large computer experiments. By focusing expressly on desirable properties of the predictive equations, we derive a family of local sequential design schemes that dynamically define the support of a Gaussian process predictor based on a local subset of the data. We further derive expressions for fast sequential updating of all needed quantities as the local designs are built-up iteratively. Then we show how independent application of our local design strategy across the elements of a vast predictive grid facilitates a trivially parallel implementation. The end result is a global predictor able to take advantage of modern multicore architectures, GPUs, and cluster computing, while at the same time allowing for a non stationary modeling feature as a bonus. We demonstrate our method on examples utilizing designs sized in the tens of thousands to over a million data points. Comparisons are made to the method of compactly supported covariances, and we present applications to computer model calibration of a radiative shock and the calculation of satellite drag.

## Friday, January 22, 2016

Title
Speaker

Time
Place
Sponsor

Minimal Energy and Maximal Polarization
Edward B. Saff
Vanderbilt University
3:00pm-4:00pm
CMC 130
Vilmos Totik

Abstract

The work to be discussed has its origins in research conducted at USF some twenty years ago. It concerns minimal energy configurations as well as maximal polarization (Chebyshev) configurations on manifolds, which are problems that are asymptotically related to best-packing and best-covering.

In particular, we discuss how to generate $$N$$ points on a $$d$$-dimensional manifold that have the desirable qualities of well-separation and optimal order covering radius, while asymptotically having a given distribution. Even for certain small numbers of points like $$N=5$$, optimal arrangements with regard to energy and polarization can be a challenging problem.

## Friday, January 15, 2016

Title
Speaker

Time
Place
Sponsor

On the Complexity of Conjugacy Problem in certain Metabelian Groups
Delaram Kahrobaei
CUNY Graduate Center
3:00pm-4:00pm
CMC 130
Dima Savchuk

Abstract

We analyze the computational complexity of the conjugacy search problem in a certain family of metabelian groups. We prove that in general the time complexity of the conjugacy search problem for these groups is at most exponential. For a subfamily of groups we prove that the conjugacy search problem is polynomial. We also show that for some of these groups the conjugacy search problem reduces to the discrete logarithm We also provide experimental evidence which illustrates our results probabilistically. This is a joint work with Conchita Martinez and Jonathan Gryak.

Polycyclic and Metabelian groups have been proposed as platform for Cryptography by Eick and Kahrobaei some years ago. The results I am presenting will have potential applications in Cryptography. The interesting question would be whether such cryptosystems are resistant against quantum algorithms.

Title
Speaker

Time
Place
Sponsor

Random Topology and Random Knots
Moshe Cohen
Technion — Israel Institute of Technology
2:00pm-3:00pm
CMC 130
Mohamed Elhamdadi

Abstract

Combinatorialists use the probabilistic method to construct impossibly large graphs and study their properties. For example, how does the parameter $$p$$ of an unfair coin affect simple topological questions like the number of components?

I will present some topological applications of the probabilistic method: random walks on the Cayley graph of a group, used for example by Nathan Dunfield and William Thurston to construct random 3-manifolds; random simplicial complexes that can be used to model large data sets; and random physical walks in three-space and the knotting phenomena that occur, with applications ranging from DNA and proteins in molecular biology to polymers in chemistry.