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Mathematics & Statistics

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Small sample GEE estimation of regression parameters for Longitudinal Data

Sudhir Paul

University of Windsor

Windsor, Canada

3:45pm-4:45pm

CMC 204

Gangaram Ladde

**Abstract**

Longitudinal (clustered) response data arise in many bio-statistical applications which, in general, cannot be assumed to be independent. Generalized estimating equation (GEE) is a widely used method to estimate marginal regression parameters for correlated responses. The advantage of the GEE is that the estimates of the regression parameters are asymptotically unbiased even if the correlation structure is misspecified, although their small sample properties are not known. In this paper, two bias adjusted GEE estimators of the regression parameters in longitudinal data are obtained when the number of subjects is small. One is based on a bias correction, and the other is based on a bias reduction. Simulations show that the performances of both the bias-corrected methods are similar in terms of bias, efficiency, coverage probability, average coverage length, impact of misspecification of correlation structure, and impact of cluster size on bias correction. Both these methods show superior properties over the GEE estimates for small samples. Further, analysis of data involving a small number of subjects also shows improvement in bias, MSE, standard error, and length of the confidence interval of the estimates by the two bias adjusted methods over the GEE estimates. For small to moderate sample sizes (\(N ≤ 50\)), either of the bias-corrected methods GEEBc and GEEBr can be used. However, the method GEEBc should be preferred over GEEBr, as the former is computationally easier. For large sample sizes, the GEE method can be used.

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TBA

Michael Kosorok

University of North Carolina at Chapel Hill

3:00pm-4:00pm

CMC 130

Chris Tsokos

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TBA

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Multi-crossing Number of Knots: Turning Knots into Flowers

Colin Adams

Williams College

3:00pm-4:00pm

CMC 130

Nagle Lecure Series Committee

**Abstract**

Knots have traditionally been depicted using projections with crossings where two stands cross. But what if we allow three strands to cross at a crossing? Or four strands? Can we find projections of any knot with just one of these multi-crossings? We will discuss these generalizations of traditional invariants to multi-crossing numbers, ubercrossing numbers and petal numbers and their relation to hyperbolic invariants.

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TBA

Sayan Mukherjee

Duke University

3:00pm-4:00pm

CMC 130

K. M. Ramachandran

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TBA

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Recent Advances in Delay-Differential Equations

John Mallet-Parret

Brown University

3:00pm-4:00pm

CMC 130

Yuncheng You

**Abstract**

In this talk we discuss some aspects of delay-differential equations. The historical roots of the subject date from the earliest twentieth century. At that time much of the focus was on linear equations arising in applications in science and engineering, and the methods were often formal and ad hoc. Beginning in the 1960's more attention was paid to nonlinear systems, and a firm theoretical foundation based on infinite-dimensional dynamical systems was established. What has emerged since then is a body of theory with a rich mathematical structure that draws from numerous areas, including dynamical systems, functional analysis, and topology, and which retains close ties with applications. We shall discuss various recent results and ongoing research in delay equations, and we shall also mention some open problems in the field.

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Rogue waves in integrable systems

Jianke Yang

University of Vermont

3:00pm-4:00pm

CMC 130

Wen-Xiu Ma

**Abstract**

Rogue waves are instantaneous large localized waves which arise from a constant background and then retreat back to the same background. They attracted significant interest in recent years due to their relevance to physical phenomena such as freak waves in the ocean and extreme waves in optics. In integrable systems, such waves admit exact analytical expressions and represent a special but important class of solutions. In this talk, we report our work on rogue waves in integrable systems, including the nonlinear Schröedinger equation, the Davey-Stewartson equations, the Ablowitz-Ladik equation and a nonlocal parity-time-symmetric nonlinear Schröedinger equation.

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Quandle and its applications to knots

Takefumi Nosaka

Tokyo Institute of Technology

Tokyo, Japan

3:00pm-4:00pm

CMC 130

Masahico Saito

**Abstract**

Quandle is a certain algebraic system, and has applications to knot theory, including “quandle cocycle invariant” which was defined by M. Saito, etc. I show that every cohomology (bilinear) pairing of any knot can be described by a quandle cocycle invariant, and give some relations to classical knot-invariant. In this talk, I begin by introducing quandles, and explain the basic and benefits. Furthermore, I briefly present some of my results, and applications to low-dimensional topology.

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Spanning cycles and 2-factors in hypergraphs

Andrzej Czygrinow

Arizona State University

3:00pm-4:00pm

CMC 130

Theo Molla

**Abstract**

Dirac's theorem states that every graph on n vertices with minimum degree at least \(n/2\) has a spanning cycle. There are many related results on 2-factors in graphs which generalize this fundamental fact. In this talk we will discuss possible analogs and generalizations of these results to hypergraphs. It turns out that even in the case of 3-uniform hypergraphs there are a few different ways of approaching this problem, as cycles and degrees can be defined in multiple ways.

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Computer Networking and Security Studies for Cyber-Physical Infrastructures

Kaiqi Xiong

3:00pm-4:00pm

CMC 130

**Abstract**

As the International Data Corporation (IDC) predicted, IT spending will reach $5 billion by 2020 — $1.7 billion more than today — driven by the 3rd cyber-physical infrastructures consisting of cloud, mobile, social media, and big data technologies. Computer networking and security plays a very important role in cyber-physical infrastructures. Furthermore, the United Nations predicted in 2016 that an estimated 60% of the world’s population would live in urban areas and one in every three people would settle down in a city by 2030. With a large portion of the population moving into urban areas, people will face various unprecedented challenges interacting with cyber-physical infrastructures. In this talk, I will give an overview of the research in my group and discuss several key computer networking and security challenges we have studied and will continue to address. Then, I will outline a few research topics with potential collaboration in the cyber-physical infrastructures.

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What Are Zeta Functions?

Daqing Wan

University of California, Irvine

3:00pm-4:00pm

CMC 130

Xiang-dong Hou

**Abstract**

Zeta functions are central topics in number theory and arithmetic algebraic geometry. They can be viewed as generating functions for counting “points” (or “solutions”) of polynomial equations, thus contain deep arithmetic and geometric information of the equations. We will explain various zeta functions from this point-of-view, including the Riemann zeta function, the Hasse-Weil zeta function, and zeta functions over finite fields. This introductory talk is accessible to advanced undergraduate students and beginning graduate students.

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Topological structure in neural systems

Chad Giusti

University of Delaware

3:00pm-4:00pm

CMC 130

Nataša Jonoska

**Abstract**

Twenty-first century neuroscience has been marked by an extraordinary leap in the quantity of data produced by experiments: it is now possible in animal models to record from thousands of individual neural units for days at a stretch, and this growth appears likely to accelerate. However, like much data about biological systems, this data is still expensive to obtain, noisy, and a dramatic sub-sampling of the system being studied. Such confounds make many standard mathematical techniques difficult or impossible to apply. However, one promising recent approach to understanding such complex systems is through techniques adapted from algebraic and combinatorial topology. This language maps surprisingly well onto various qualitative characterizations of neural systems that have been discovered by experimental and theoretical neuroscientists, and provides quantitative tools for understanding and comparing the resulting models. In this talk, we will survey this connection, then focus on a couple of specific examples involving how neural systems represent information.

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Dispersive equations with potential

Marius Beceanu

SUNY at Albany

3:00pm-4:00pm

CMC 130

Razvan Teodorescu

**Abstract**

I'll present some applications of Wiener's Theorem to the study of dispersive equations with scalar potential.

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Singularities of Rational Inner Functions

Alan Sola

Stockholm University

3:00pm-4:00pm

CMC 130

Dima Khavinson

**Abstract**

This talk will be devoted to rational inner functions in two complex variables and their singularities. I will explain why this seemingly special class of functions is actually quite important, and I will provide a brief survey of the history of these functions and their one variable relatives.

In the second half of the talk, I will discuss ongoing work with Kelly Bickel and James Pascoe concerning precise descriptions of the nature of the singularities of rational inner function and a method for visualizing these singularities.