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Title: TBA Speaker: Mustafa Hajij Time: 3:00pm‐4:00pm Place: CMC 204

TBA

Title: On graphs defined by systems of equations Speaker: On graphs defined by systems of equations Felix Lazebnik, University of Delaware Time: 3:00pm‐4:00pm Place: CMC 204

In this talk we consider a large class of graphs and digraphs defined by certain systems of polynomial equations over commutative rings.

Some specializations of these constructions over fields turned out to be useful in extremal graph theory, finite geometries and Ramsey theory, and they also led to interesting new algebraic questions. We discuss motivation for the constructions, survey some main old and new results, and pose several open problems.

Title: Determining forms and data assimilation Speaker: Michael S. Jolly, Indiana University Time: 3:00pm‐4:00pm Place: CMC 204

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.

Title: Compiling quantum algorithms Speaker: Michele Mosca, Institute for Quantum Computing University of Waterloo Time: 3:00pm‐4:00pm Place: CMC 204

To capture the power of quantum algorithms in practice requires translating high-level instructions into low-level machine instructions. An important part of this sequence of transformations is the synthesis and optimization of high-level operations in terms of fault-tolerant quantum gates.

A number of interesting mathematical problems emerge, including matroid partitioning, decoding Reed-Muller codes, and problems in algorithmic number theory.

I will overview the current state of global efforts to build a large-scale fault-tolerant quantum computer, and of quantum compiling, and give some examples of the mathematical tools being developed and applied for the compiling of quantum algorithms.

Title: Tiling directed graphs with tournaments Speaker: Theodore Molla, University of Illinois at Urbana-Champaign Time: 12:00pm‐1:00pm Place: CMC 109

The Hajnal-Szemeredi Theorem is a celebrated classical result in extremal graph theory. It states that for any positive integer \(r\) and any multiple \(n\) of \(r\), if \(G\) is a graph on \(n\) vertices in which every vertex has \((1-1/r)n\) neighbors, then \(G\) can be partitioned into \(n/r\) vertex-disjoint copies of the complete graph on \(r\) vertices. We will discuss analogues of this theorem for directed graphs and some related conjectures.

This is joint work with Andrzej Czygrinow, Louis DeBiasio, and Andrew Treglown.

Title: Sequential Test Planning for Polymer Composites Speaker: Yili Hong, Department of Statistics Virginia Tech Time: 3:00pm‐4:00pm Place: CMC 204

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.

Title: Linear and Orbital Stability of Solutions to the VFE and the VFE Hierarchy Speaker: Stéphane Lafortune, College of Charleston Time: 3:00pm‐4:00pm Place: CMC 204

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.

Title: Study of the chaotic vibration of the 2D non-strictly hyperbolic equation Speaker: Jing Tian Time: 2:00pm‐3:00pm Place: CMC 204

In this talk, I will mainly discuss the chaotic vibration phenomenon of the 2D non-strictly hyperbolic equation due to an energy-injection boundary condition and a distributed self-regulation boundary condition. By using the method of characteristics, we give a rigorous proof of the chaotic vibration phenomenon of the 2D non-strictly hyperbolic equation. We have also found a regime of the parameters when the chaotic vibration phenomenon occurs by applying the period-doubling bifurcation theorem. Numerical simulations will be provided.

Title: Role of the Ph.D. in Data Science Speaker: Jennifer Lewis Priestley, Director of the Center for Statistics and Analytical Services Department of Statistics and Analytical Sciences Kennesaw State University Time: 1:45pm‐2:45pm Place: ENC 1002

As the demand for deep analytical skills continues to outpace the supply of talent, universities across the country have responded by working with the private and public sectors to develop programs in data science to help close the talent gap. This talk will provide an overview of the landscape of data science programs, with particular emphasis on the nascent doctoral programs in data science.

Title: Kernel methods for Koopman mode analysis and prediction: Ergodic and skew-product systems Speaker: Dimitris Giannakis, Courant Institute of Mathematical Sciences Time: 3:00pm‐4:00pm Place: CMC 204

We discuss a framework for dimension reduction, mode decomposition, and nonparametric forecasting of data generated by ergodic and skew-product dynamical systems. This framework is based on a representation of the Koopman group of unitary operators governing dynamical evolution in a smooth orthonormal basis acquired from time-ordered data through the diffusion maps algorithm. Using this representation, we compute Koopman eigenfunctions through a regularized advection-diffusion operator, and employ these eigenfunctions in dimension reduction maps with projectible dynamics and high smoothness for the given observation modality. We also use this basis to build nonparametric forecast models for arbitrary probability densities and observables. In the first part of the talk, we focus on the ergodic case and discuss a connection between Koopman operators and diffusion operators obtained from Takens delay-coordinate mapped data that helps improve the efficiency and noise robustness of numerically computed Koopman eigenfunctions. Then, we discuss extensions of this framework to skew-product systems; our primary motivation being the identification and prediction of coherent spatiotemporal patterns in time-dependent fluid flows. We illustrate how Koopman operators for skew-product systems lead to a natural definition of global coherent patterns through their eigenfunctions, as well as model-free prediction schemes for these patterns. This work is in collaboration with Tyrus Berry (George Mason), Shuddho Das (NYU), Matina Gkioulidou (Johns Hopkins APL), John Harlim (Penn State), and Joanna Slawinska (Rutgers).

Title: Universal lower bound on energy in for spherical and Hamming codes Speaker: Peter D. Dragnev, Indiana University–Purdue University Fort Wayne Time: 2:00pm‐3:15pm Place: CMC 108

A finite subset of the unit sphere or the Hamming space is called a code. If we interpret points in a code as particles interacting with each other according to a given potential function, the sum of all possible pair-wise interactions is called potential energy. Configurations minimizing the potential energy have wide applications in physics, chemistry, biology, computer science, information transfer, etc. Under some general assumptions on the potential function we establish a universal lower bound on that energy that depends on the cardinality of the code and the underlying dimension of the vector space.

Title: Impugning Alleged Randomness Speaker: Yuri Gurevich, Microsoft Research Redmond, WA Time: 3:00pm‐4:00pm Place: CMC 204

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.

Title: On a group theoretic generalization of the Morse-Hedlund theorem Speaker: Svetlana Puzynina, Institut de Recherche en Informatique Fondamentale Time: 3:00pm‐4:00pm Place: CMC 204

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.

Title: Wavelet Modeling: All That Scaling Speaker: Branislav Vidakovic, Georgia Institute of Technology Time: 3:00pm‐4:00pm Place: CMC 204

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.

Title: Coloring and Covering — Geometric Graphs and Hypergraphs Speaker: Torsten Ueckerdt, Karlsruhe Institute of Technology Karlsruhe, Germany Time: 2:00pm‐3:00pm Place: CPR 338

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.

Title: Structure and Pseudorandomness in Large Objects Speaker: Pooya Hatami, DIMACS and Rutgers University Time: 2:00pm‐3:00pm Place: CPR 338

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.

Title: Automaton groups and square complexes Speaker: Ievgen Bondarenko, Taras Shevchenko National University of Kyiv Kyiv, Ukraine Time: 3:00pm‐4:00pm Place: CMC 204

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.

Title: On Vassiliev Invariants for Knots in the Solid Torus Speaker: Khaled T. Bataineh, Jordan University of Science and Technology Time: 3:00pm‐4:00pm Place: CMC 204

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.

Title: Zeros of optimal polynomial approximants: Jacobi matrices and Jentzsch-type theorems Speaker: Daniel Seco, Universitat de Barcelona Time: 3:00pm‐4:00pm Place: CMC 204

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.