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Mathematics & Statistics

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TBA

Felix Lazebnik

University of Delaware

3:00pm-4:00pm

CMC 204

Xiang-dong Hou

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TBA

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Chad Giusti

University of Pennsylvania

3:00pm-4:00pm

CMC 204

Nataša Jonoska

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TBA

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Determining forms and data assimilation

Michael S. Jolly

Indiana University

3:00pm-4:00pm

CMC 204

Yuncheng You

**Abstract**

A determining form for a dissipative partial differential equation is an ordinary differential equation in a certain trajectory space where the solutions on the global attractor of the PDE are readily recognized. It is an ODE in the true sense of defining a vector field which is (globally) Lipschitz. We discuss two types of determining forms: one where solutions on the global attractor of the PDE are traveling waves, and another where they are steady states. Each determining form is related to a certain approach to data assimilation, i.e., the injection of a coarse-grain time series into the model in order to recover the matching full solution. Applications have been made to the 2D incompressible Navier-Stokes, damped-driven nonlinear Schrödinger, damped-driven Korteveg-de Vries and surface quasigeostrophic equations.

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Sequential Test Planning for Polymer Composites

Yili Hong

Department of Statistics

Virginia Tech

3:00pm-4:00pm

CMC 204

Lu Lu

**Abstract**

Polymer composite materials are widely used in areas such as aerospace and alternative energy industries, due to their lightweight and comparable levels of strength and endurance. To ensure that the material can last long enough in the field, accelerated cyclic fatigue tests are commonly used to collect data and then make predictions for the field performance. Thus, a good testing strategy is desirable for evaluating the property of polymer composites. While there has been a lot of development in optimum test planning, most of the methods assume that the true parameter values are known (i.e., the true values are used as the planning values). However, in reality, the true model parameters may depart from the planning values. In this paper, we propose a sequential strategy for test planning, and use a Bayesian framework for the sequential model updating. We also use extensive simulations to evaluate the properties of the proposed sequential test planning strategy. Finally, we compare the proposed method to traditional optimum designs. Our results show that the proposed strategy is more robust and efficient, as compared to the optimum designs, when true values of parameters are unknown.

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Linear and Orbital Stability of Solutions to the VFE and the VFE Hierarchy

Stéphane Lafortune

College of Charleston

3:00pm-4:00pm

CMC 204

Wen-Xiu Ma

**Abstract**

By the term vortex filament, we mean a mass of whirling fluid or air (e.g. a whirlpool or whirlwind) concentrated along a slender tube. The most spectacular and well-known example of a vortex filament is a tornado. A waterspout and dust devil are other examples. In more technical applications, vortex filaments are seen and used in contexts such as superfluids and superconductivity. One system of equations used to describe the dynamics of vortex filaments is the Vortex Filament Equation (VFE). The VFE is a system giving the time evolution of the curve around which the vorticity is concentrated. In this talk, we develop a framework for studying the linear and orbital stability of VFE solutions, based on the correspondence between the VFE and the NLS provided by the Hasimoto map. This framework is applied to VFE solutions that take the form of soliton solutions or closed vortices. If time permits, we will also tackle the case of solutions to other members of the VFE hierarchy of integrable equations.

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Dimitris Giannakis

Courant Institute of Mathematical Sciences

3:00pm-4:00pm

CMC 204

Jing Tian

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TBA

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Yuri Gurevich

Microsoft Research

Redmond, WA

3:00pm-4:00pm

CMC 204

Greg McColm

**Abstract**

According to a 1985 issue of New York Times, “The New Jersey Supreme Court today caught up with the Essex County Clerk and a Democrat who has conducted drawings for decades that have given Democrats the top ballot line in the county 40 times out of 41 times.” But the clerk wasn't found guilty. Here's another case of that sort, from a different part of the world. In the 1980s the Israeli tax authorities encouraged the public to request invoices from plumbers, painters, etc. and send the invoices in; big prices were ruffled off. Suprisingly, a big price went to none other than the Director of Customs and VAT. The operation collapsed but the director wasn't punished.

You may be convinced that such lotteries are rigged, but how would you argue that in the court of law? Yes, the probability of the suspicious outcome is negligible. However the probability of any particular outcome is negligible. What can you say? We attempt to furnish you with an argument.

Only most rudimentary probability theory will be presumed.

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On a group theoretic generalization of the Morse-Hedlund theorem

Svetlana Puzynina

Institut de Recherche en Informatique Fondamentale

Paris, France

3:00pm-4:00pm

CMC 204

Dima Savchuk

**Abstract**

In their 1938 seminal paper on symbolic dynamics, Morse and Hedlund proved that every aperiodic infinite word \(x\) contains at least \(n+1\) distinct factors (i.e., blocks of consecutive symbols) of each length \(n\). They further showed that an infinite word \(x\) has exactly \(n+1\) distinct factors of each length \(n\) if and only if \(x\) is binary, aperiodic and balanced, i.e., \(x\) is a Sturmian word. In this talk I will present a concept of words complexity via group actions and discuss generalizations of the Morse-Hedlund theorem.

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Wavelet Modeling: All That Scaling

Branislav Vidakovic

Georgia Institute of Technology

3:00pm-4:00pm

CMC 204

K. M. Ramachandran

**Abstract**

In this overview talk we focus on the wavelet-based estimation of scaling indices of self-similar time series and images. This estimation is conducted in multiscale domains. We consider a range of wavelet and wavelet-like decompositions: orthogonal, nondecimated, wavelet packets, complex-number decompositions, autocorrelation shells of wavelets, and spherical wavelets. They all result in a hierarchy of imbedded multiresolution subspaces that could produce a valid multiscale spectra. Like in the Fourier transforms where the linear decay of the log-power spectra over the frequencies characterizes the regularity/smoothness of a time series/image, the decay of the log-average squared wavelet coefficients leads to an alternative and arguably more local and stable measure of signal/image regularity. We provide examples from medicine, finance, and geosciences in which the scaling indices turn out to be useful in tasks of statistical learning. In the talk we also overview some traditional results, some results from the past research of the speaker and his collaborators, as well as some interesting results from the ongoing research. We will point out at several interesting avenues for possible future research.

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Coloring and Covering — Geometric Graphs and Hypergraphs

Torsten Ueckerdt

Karlsruhe Institute of Technology

Karlsruhe, Germany

2:00pm-3:00pm

CPR 338

Brian Curtin

**Abstract**

Geometrically defined graphs and hypergraphs are a classical topic in discrete mathematics. In fact, the Four-Color-Problem for planar graphs is generally recognized as the driving force that led to the development of modern graph theory. Nowadays, some of the most intriguing areas of combinatorics concern graphs, hypergraphs and partially ordered sets that arise from geometric settings, the majority of which seeks to color or cover the elements at hand. The interest in combinatorial geometry stems not only from its beauty and complexity, but also from the fact that geometric arrangements play a central role in many sciences, such as physics, biology and computer science, as well as in many applications, such as geographical maps, sensor networks, chip designs, or resource allocations.

In this talk, we present a variety of geometric coloring and covering problems, including as diverse concepts as cover decomposability problems, online coloring problems, questions of representability, and arboricities. We will see several results bridging geometry and combinatorics and how both fields can drive one another towards more advancement. Specifically, we start with finite point sets \(X\) in \(\mathbb{R}^2\) and the graphs with vertex set \(X\) whose edges correspond to those pairs \(u,v\) of points such that there is an axis-aligned equilateral triangle \(T\) with \(T\cap X=\{u,v\}\). Switching back and forth between geometry and combinatorics several times, we shall close with recent investigations on decomposing the edge set of a planar graph into forests of given size, maximum degree or diameter.

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Structure and Pseudorandomness in Large Objects

Pooya Hatami

DIMACS and Rutgers University

2:00pm-3:00pm

CPR 338

Brendan Nagle

**Abstract**

The dichotomy between pseudorandomness and structure has proven to be a useful point of view in the study of large complex objects (e.g., a large graph, a function on a space, a set of integers, etc). In this talk I will mostly focus on higher-order Fourier analysis over finite fields which is a powerful theory that uses this phenomenon.

I will describe several interesting questions about the structure and distribution of low-degree polynomials that arise naturally in this context. I will also discuss some applications of our answers to these questions, for example, in the analysis of several simple heuristic algorithms for algebraic tasks such as testing whether a given polynomial admits a prescribed decomposition.

If time allows, I will briefly talk about a few other problems in pseudorandomness.

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Automaton groups and square complexes

Ievgen Bondarenko

Taras Shevchenko National University of Kyiv

Kyiv, Ukraine

3:00pm-4:00pm

CMC 204

Dmytro Savchuk

**Abstract**

Any automaton-transducer gives rise to a square complex: one can take a unit square with labeled and oriented edges for each arrow in automaton and glue these squares to get a complex. Transitions in automaton correspond to relations in the fundamental group of the associated square complex. In this talk, based on a joint work with Bohdan Kivva, I will discuss the connection between groups generated by automata, tiling properties of associated collection of squares, and residual properties of the fundamental groups of these square complexes. In particular, I will show how to construct square complexes with non-residually finite \(\mathrm{CAT}(0)\) fundamental group from any bireversible automaton with infinite automaton group.

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On Vassiliev Invariants for Knots in the Solid Torus

Khaled T. Bataineh

Jordan University of Science and Technology

3:00pm-4:00pm

CMC 204

Mustafa Hajij

**Abstract**

Vassiliev invariants (or finite type invariants), discovered around 1989, provided a new way of looking at knots. A Vassiliev invariant of order \(m\) is a knot invariant that can be extended (in a precise manner) to an invariant of certain singular knots that vanishes on singular knots with \(m+1\) singularities and does not vanish on some singular knot with '\(m\)' singularities.

Michael Polyak and Oleg Viro gave a description of the first nontrivial invariants of orders 2 and 3 for knots in the Euclidean space by means of Gauss diagrams. We give our description of the infinite families of Vassiliev invariants of orders 1 and 2 for knots in the solid torus with zero winding number.

For the order 1 invariants we give two ways of describing these invariants. One of them uses decorated Gauss diagrams, and the other uses techniques of lifting the solid torus into its universal cover and applying linking numbers.

For the order 2 invariants we introduce a natural filtration in the space of knots and singular knots in the solid torus, and start the study of the Vassiliev invariants of order 2 with respect to this filtration. The main result states that any such invariant within the second term of this filtration in the space of knots with zero winding number is a linear combination of seven explicitly described decorated Gauss diagram invariants. This introduces a basis (and a universal invariant) for the Vassiliev invariants of order 2 in the second term. Then we formalize the problem of exploring the set of all invariants of order 2 for knots with zero winding number.

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Zeros of optimal polynomial approximants: Jacobi matrices and Jentzsch-type theorems

Daniel Seco

Universitat de Barcelona

3:00pm-4:00pm

CMC 204

Catherine Bénéteau

**Abstract**

I present a recent work with Bénéteau, Khavinson, Liaw and Simanek where we study the structure of the zeros of polynomials appearing in the study of cyclicity in Hilbert spaces of analytic functions. We find the minimum possible modulus of occurring zeros via a nonlinear extremal problem associated with norms of Jacobi matrices. We examine global properties of these zeros and prove Jentzsch-type theorems describing where they accumulate.