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Mathematics & Statistics

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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.

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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.

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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 \(g\).

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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.

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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.

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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.

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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.

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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.

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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.

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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 acquiredâ€”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.

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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 \(O\).

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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.

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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.

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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 eects of surface tension are neglected wefind 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.

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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.

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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.

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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 \(\Bbb Z_p\times \Bbb Z_p\).

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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.

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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.

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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.

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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.

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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.

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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.