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Mathematics & Statistics

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Asymptotics of multiple orthogonal polynomials for cubic weight

Andrei Martínez-Finkelshtein

Baylor University/Universidad de Almería

3:00pm-4:00pm

CMC 130

Evguenii Rakhmanov

**Abstract**

We consider the type I and type II multiple orthogonal polynomials (MOPs), satisfying non-hermitian orthogonality with respect to the weight \(\exp(-z3)\) on two unbounded contours on the complex plane. Under the assumption that the orthogonality conditions are distributed with a fixed proportion \(\alpha\), we find the detailed (rescaled) asymptotics of these MOPs, and describe the phase transitions of this limit behavior as a function of \(\alpha\). This description is given in terms of the vector critical measure, the saddle point of the energy functional comprising both attracting and repelling forces. These critical measures are characterized by a cubic equation (spectral curve), and their components live on trajectories of a canonical quadratic differential on the Riemann surface of this equation. The structure of these trajectories and their deformations as function of \(\alpha\) plays the crucial role. This is a joint work with Gulherme L. Silva (University of Michigan, Ann Arbor).

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CSIDH: an efficient post-quantum commutative group action

Chloe Martindale

TU Eindhoven

The Netherlands

3:00pm-4:00pm

CMC 130

Marian Hernandez Viera

**Abstract**

CSIDH, or ‘commutative supersingular isogeny Diffie-Hellman’ is a new isogeny-based protocol of Castryck, Lange, Martindale, Panny, and Renes. The Diffie-Hellman style scheme resulting from the group action is the first practical post-quantum non-interactive key exchange. We will describe the CSIDH protocol, give an overview of the security analysis, and outline some potential applications.

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Clifford Algebras as Images of Group Algebras of Salingaros Vee Groups

Rafaɫ Abɫamowicz

Tennessee Technological University

2:00pm-3:00pm

CMC 130

Lesɫaw Skrzypek

**Abstract**

In [9–11], Salingaros defined five families \(N_{2k-1}\), \(N_{2k}\), \(\Omega_{2k-1}\), \(\Omega_{2k}\) and \(S_k\) of finite 2-groups \(G_{p,q}\) related to Clifford algebras \(C\ell_{p,q}\). For each \(k\ge 1\), the group \(N_{2k-1}\) is a central product \(\left(D_8\right)^{\circ k}\) of \(k\) copies of the dihedral group \(D_8\) and the group \(N_{2k}\) is a central product \(\left(D_8\right)^{\circ (k-1)}\circ Q_8\), where \(Q_8\) is the quaternion group. Both groups \(N_{2k-1}\) and \(N_{2k}\) are extra-special \cite{gorenstein,mckay} while \(\Omega_{2k-1}\cong N_{2k-1}\circ\left(C_2\times C_2\right)\), \(\Omega_{2k}\cong N_{2k}\circ\left(C_2\times C_2\right)\), and \(S_{k}\cong N_{2k-1}\circ C_4\cong N_{2k}\circ C_4\), where \(C_2\) and \(C_4\) are cyclic groups of order 2 and 4, respectively (see also [5]).

In [2], Abɫamowicz *et al.* proved that every Clifford algebra \(C\ell_{p,q}\) is \(\mathbb{R}\)-isomorphic to a quotient of a group algebra \(\mathbb{R}\left[G_{p,q}\right]\) modulo an ideal \(\mathcal{J}=(1+\tau)\) where \(\tau\) is a central element of order 2. Thus, Clifford algebras \(C\ell_{p,q}\) can be classified by the Salingaros classes in accordance with the well-known Periodicity of Eight [8]. One goal of this colloquium is to present this new classification.

The second goal is to present a follow-up result [1] that spinor modules of Clifford algebras in classes \(N_{2k-1}\) and \(\Omega_{2k-1}\) are uniquely determined by irreducible nonlinear characters of the corresponding vee groups. This result should be extendable to the remaining three classes.

The last goal is to present an idea of extending this work to finite 3-groups whose group algebras are \(\mathbb{Z}_3\)-graded. This is so that one could define new \(\mathbb{Z}_3\)-graded Clifford-like algebras as the quotients of the these group algebras. The first step in this direction has already been done [4].

**References**

- R. Abɫamowicz: Spinor Modules of Clifford Algebras in Classes \(N_{2k-1}\) and \(\Omega_{2k-1}\) are Determined by Irreducible Nonlinear Characters of Corresponding Salingaros Vee Groups, Adv. Applied Clifford Algebras (2018) 28:51.
- R. Abɫamowicz, V. S. M. Varahagiri, A. M. Walley: A Classification of Clifford Algebras as Images of Group Algebras of Salingaros Vee Groups, Adv. Applied Clifford Algebras (2018) 28:38.
- R. Abɫamowicz and K. D. G. Maduranga: Representations and Characters of Salingaros' Vee Groups of Low Order, Bull. Soc. Sci. Lettres Łódź Sér. Rech. Déform. Vol. 66, no. 1 (2016) 43–75.
- D. Blythe: On \(\mathbb{Z}/3 \mathbb{Z}\) Graded Group Algebras of Extra-Special Groups of order 27, Master Thesis, Department of Mathematics, TTU, Cookeville, TN, 2018.
- Z. Brown: Group Extensions, Semidirect Products, and Central Products Applied to Salingaros Vee Groups Seen As 2-Groups, Master Thesis, Department of Mathematics, TTU, Cookeville, TN, 2015.
- D. Gorenstein, Finite Groups, Chelsea Publishing Company, New York, 1980.
- C. R. Leedham-Green and S. McKay: The Structure of Groups of Prime Power Order, Oxford University Press, Oxford, 2002.
- P. Lounesto: Clifford Algebras and Spinors, Cambridge University Press, Cambridge, 2001.
- N. Salingaros: Realization, extension, and classification of certain physically important groups and algebras, J. Math. Phys. 22 (1981), 226–232
- N. Salingaros: On the classification of Clifford algebras and their relation to spinors in \(n\) dimensions, J. Math. Phys. 23 (1) (1982), 1–7.
- N. Salingaros: The relationship between finite groups and Clifford algebras, J. Math. Phys. 25 (4) (1984), 738–742.

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On some extremal problems in complex analysis

Alex Stokolos

Georgia Southern University

3:00pm-4:00pm

CMC 130

Dima Khavinson

**Abstract**

Many celebrated results in Complex Analysis state solutions to some extremal problems. For instance, let me mention Koebe one quarter theorem and Bieberbach conjecture (now De Brange Theorem). In the talk I will discuss the polynomial version of these theorems and related questions. The lecture will be accessible to everyone who has taken a standard complex analysis course.

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Recent advances in conformal random aggregation

Alan Sola

Stockholm University

4:00pm-5:00pm

CMC 130

**Abstract**

Conformal mapping models of random growth bring the powerful machinery of complex analysis to bear on two-dimensional variants of aggregation phenomena, where particles form complicated clusters by undergoing random motion in the plane. The most physically relevant instances of such models remain inaccessible to rigorous analysis, but the last years have seen some tentative progress on describing the local and global geometries of growing aggregates. In my talk, I will survey some of these recent developments.

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Mathematics of novel imaging techniques

Peter Kuchment

Texas A&M University

3:00pm-4:00pm

CMC 130

Nagle Lecture Series Committee

**Abstract**

The talk will provide a excursion into the mathematics of hybrid imaging, reconstructions with internal information and, time permitting, Compton camera imaging.

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Introduction to the edit distance in graphs

Ryan Martin

Iowa State University

3:00pm-4:00pm

CMC 130

Theo Molla

**Abstract**

The edit distance is a very simple and natural metric on the space of graphs. In the edit distance problem, we fix a hereditary property of graphs and compute the asymptotically largest edit distance of a graph from the property. This quantity is very difficult to compute directly but in many cases, it can be derived as the maximum of what is known as the edit distance function.

Szemerédi's regularity lemma, strongly-regular graphs, constructions related to the Zarankiewicz problem — all these play a role in the computing of edit distance functions.

In this talk, we give an overview of some of the major results in the literature and connections to other problems in extremal graph theory.

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Manifolds

Erik Pedersen

Binghamton University

3:00pm-4:00pm

CMC 130

Mohamed Elhamdadi

**Abstract**

In the talk I will discuss the way we have come to understand manifolds over the last 60 years.

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Dynamic Programming Principles for Nonlinear Elliptic Equations

Juan Manfredi

University of Pittsburgh

3:00pm-4:00pm

CMC 130

Thomas Bieske

**Abstract**

Under appropriate hypothesis for a real valued function \(u\), the following conditions are equivalent

- \(u\) satisfies a mean value property; for example \(u(x)=\frac{1}{|B|}\int_{B}\,u(y)\,dy\).
- \(u\) satisfies a Dynamic Programming Principle associated to a game or control problem; for example \(u(x)=\mathbb{E}^x\left[u\left(B_\tau\right)\right]\).
- \(u\) solves a partial differential equation; for example \(\Delta u=0\).

These conditions can also be stated at the discrete parameter level. We approximate $$ u=\lim_{\epsilon\to 0}u_\epsilon $$ and consider discrete versions of two of the conditions above:

- \(u_\epsilon\) satisfies a mean value property; for example \(u_\epsilon(x)=\frac{1}{|B_\epsilon(x)|}\int_{B_\epsilon(x)}\,u_\epsilon(y)\,dy\), for fixed \(\epsilon>0\) and all balls of radius \(\epsilon\).
- \(u_\epsilon\) satisfies a Dynamic Programming Principle associated to a game or control problem; for example \(u_\epsilon(x)=\mathbb{E}^nx\left[u_\epsilon\left(X^\epsilon_\tau\right)\right]\), where \(X^\epsilon\) is random walk of step-size \(\epsilon>0\).

We can further discretize the domain of \(u\) to get approximations \(u^h_\epsilon\) defined in lattices \(h\mathbb{Z}^n\) that satisfy a discrete mean value property often called a scheme in Numerical Analysis.

We will present non-linear versions of the conditions above and will provide a unified strategy to show that these discretizations obtained via dynamic programming principles, stochastic games, mean value properties, and schemes converge to the solution of the corresponding Dirichlet problem for many classes of non-linear elliptic equations, including the \(p\)-Laplace operator and its variants, and the Maximal Pucci operator.

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Modules over infinite dimensional algebras

Sergio Lopez-Permouth

Ohio University

2:00pm-3:00pm

CMC 130

Xiang-dong Hou

**Abstract**

Given an algebra \(A\) over a field \(F\), a basis \(B\) for \(A\) is said to be amenable if one can naturally extend the \(A\)-module structure on the \(F\)-vector space \(F^{(B)}\) to the vector space \(F^B\). A basis \(B\) is congenial to another one \(C\) if infinite linear combinations of elements of \(B\) translate in a natural way to infinite linear combinations of elements of \(C\). While congeniality is not symmetric in general, when two bases \(B\) and \(C\) are mutually congenial then \(B\) is amenable if and only if \(C\) is amenable and, in that case, the module structures obtained on \(F^B\) and \(F^C\) are isomorphic. We will present these definitions including a recent interpretation of these notions in topological terms that is part of the doctoral dissertation of my student Benjamin Q. Stanley.

An interesting feature of congeniality is that (not necessarily mutual) congeniality between amenable bases yields an epimorphism of the modules they induce; the epimorphism is one-to-one only if the congeniality is mutual.

An amenable basis \(B\) is simple if it is not properly congenial to any other amenable basis. Projective amenable bases are defined similarly in a dual fashion. We will discuss what is known and not known about simple and projective bases.

The results in this presentation are due to collaborations with several authors including L. Al-Essa, P. Aydogdu, R. Muhammad, N. Muthana, B. Stanley, and J. Díaz Boils.

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Quandle theory, a language for symmetries

Ayumu Inoue

Tsuda University

Japan

2:00pm-3:00pm

CMC 108

Masahiko Saito

**Abstract**

A quandle is an algebraic system, which is a non-empty set equipped with a binary operation satisfying several axioms. It is well-known that this algebra has good chemistry with knot theory, because the axioms are closely related to the Reidemeister moves which are fundamental local moves of knot diagrams.

On the other hand, it seems to be lesser-known that this algebra also has high compatibility with symmetries. While a group describes whole symmetries of an object, a quandle does some limited symmetries of those. In this talk, the speaker introduces quandle theory from the aspect of a language for symmetries.

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Who asked *us*? How the theory of computing answers questions that weren’t about computing

Jack Lutz

Iowa State University

3:00pm-4:00pm

CMC 130

Nataša Jonoska

**Abstract**

It is rare for the theory of computing to be used to answer open mathematical questions whose statements do not involve computation or related aspects of logic. This talk discusses recent developments that do exactly this. After a brief review of algorithmic information and dimension, we describe the *point-to-set principle* (with N. Lutz) and its application to two new results in geometric measure theory. These are

- N. Lutz and D. Stull's strengthened lower bounds on the Hausdorff dimensions of generalized Furstenberg sets and
- N. Lutz's extension of the fractal intersection formulas for Hausdorff and packing dimensions in Euclidean spaces from Borel sets to arbitrary sets

This talk will be held in conjunction with the Discrete Mathematics seminar.

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Ideals Lattices and Applications

Ha Tran

Concordia University of Edmonton

3:00pm-4:00pm

CMC 130

Nataša Jonoska

**Abstract**

Ideal lattices form a powerful tool not only for computational number theory but also for cryptography and coding theory, thanks to their underlying structures that enable a variety of useful constructions. In this talk, we will first discuss ideal lattices and their application in computational number theory such as computing important invariants of a number field (the class number, the class group and the unit group). Then we will discuss an application of ideal lattices in cryptography: constructing cryptosystems that are conjectured to be secure under attacks by quantum computers. An application of ideal lattices in coding theory — minimizing the value of the inverse norm sums — will be presented after that. Finally, we will discuss some open problems relating to this topic.

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Means of positive matrices

Raluca Dumitru

University of North Florida

2:30pm-3:30pm

ISA 3050

Boris Shekhtman

**Abstract**

Matrix means have received a lot of attention in the past years. A notion easy to understand for numbers, it created a lot of interesting problems when extended to positive matrices, due in part to matrix multiplication not being commutative, and secondly because the order relation on positive matrices presents some challenges. In this talk, we will give an introduction to the theory of matrix means and present some of our results related to their geometry, inequalities between matrix means, and characterizations of matrix monotone functions.

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On Schur's Exponent Conjecture

Viji Thomas

Indian Institute of Science Education and Research Thiruvananthapuram

2:00pm-3:00pm

CMC 108

Xiang-dong Hou

**Abstract**

Schur's exponent conjecture states that the exponent of the second homology group \(H_2(G,Z)\) with coefficients in the integers divides the exponent of the group \(G\). We will first explain the progress made so far on the conjecture. In the second part, we will mention our contribution to the conjecture. Finally we will describe the approach taken by us to attack the conjecture. This is joint work with my Ph.D. students Ammu E. Antony and Komma Patali. We will try to make the talk accessible for graduate students.

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Similarity of matrices from an arithmetic point-of-view

Tommy Hofmann

Technische Universität Kaiserslautern

2:00pm-3:00pm

CMC 108

Dima Savchuk

**Abstract**

Similarity of matrices is a fundamental notion in linear algebra and an indispensable tool when investigating linear operators. In this talk we will consider similarity of matrices in connection with arithmetic questions, like the integrality of the matrix entries. By employing tools from number theory and representation theory, we will explain how these problems can be solved both in theory and practice.

This talk will be held in conjunction with the Discrete Mathematics seminar.

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Computational approaches to nonlinear inverse scattering problems

Howard Levinson

University of Michigan

3:00pm-4:00pm

CMC 130

Catherine Bénéteau

**Abstract**

Data obtained from scattering experiments contains valuable information regarding the internal structure of opaque objects. Reconstructing the structure from these potentially large data sets is a difficult computational task. In this talk, I describe two new algorithms for solving this nonlinear inverse scattering problem. The first algorithm directly addresses the issue and is well-suited for large data sets, which are increasingly common with modern experimental techniques. The second algorithm concerns the compressed sensing problem—where sparsity of the target can be used to reduce the necessary number of measurements. In the final part of the talk, I introduce an experimental approach in fluorescence microscopy that can achieve subwavelength resolution from limited data measurements.

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Particle models for boundary coarsening in networks

Joseph Klobusicky

Rensselaer Polytechnic Institute

2:00pm-3:00pm

CMC 109

Sponsor Name

**Abstract**

We construct a family of stochastic particle systems which models the coarsening of two-dimensional networks through mean curvature. The limiting kinetic equations of these models, describing distributions of grain areas and topologies, are shown to be well-posed. Evidence for the exponential convergence of the empirical densities of the particle system to solutions of the kinetic equations is provided through several minimal models. The framework for the particle system is general enough to allow for various assumptions proposed in the 1980’s and 1990’s concerning facet exchange and first order neighbor correlations. Particle system models for several different assumptions are compared against direct simulations.

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Optimal Transport and Topology in Data Science

Thomas Needham

Ohio State University

11:00am-12:00pm

CMC 109

K. Ramachandran

**Abstract**

The optimal transport problem seeks the cost-minimizing plan for moving materials to building sites. It was first formulated precisely by Monge in the 1700s and has since developed into its own sophisticated subfield of pure mathematics. Recent advances in theory and algorithm design have transformed optimal transport into a viable tool for analyzing large datasets. In this talk, I will describe a way to compare general abstract metric spaces using ideas from optimal transport and demonstrate an application to feature matching of anatomical surfaces. Along the way, I will formulate several natural inverse problems in geometry and graph theory whose solutions are obtained via tools from the rapidly-developing field of topological data analysis.

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Interfaces between Reproducing Kernel Herbert Spaces, Systems Theory, and Optimal Control

Joel Rosenfeld

Vanderbilt University

2:00pm-3:00pm

CMC 109

Razvan Teodorescu

**Abstract**

In this talk we will examine connections between data science, systems theory and optimal control. The talk will discuss various aspects of reproducing kernel Hilbert spaces, including densely defined multiplication operators and an approximation framework for the online estimate of an approximate optimal controller. Finally, the talk will conclude by demonstrating a connection between densely defined operators over reproducing kernel Hilbert spaces and optimal control theory through a new kernel function inspired by occupation measures.

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Using functional data analysis to exploit high-resolution “Omics” data

Marzia Cremona

Pennsylvania State University

3:30pm-4:30pm

CMC 120

Catherine Bénéteau

**Abstract**

Recent progress in sequencing technology has revolutionized the study of genomic and epigenomic processes, by allowing fast, accurate and cheap whole-genome DNA sequencing, as well as other high-throughput measurements. Functional data analysis (FDA) can be broadly and effectively employed to exploit the massive, high-dimensional and complex “Omics” data generated by these technologies. This approach involves considering “Omics” data at high resolution, representing them as “curves” of measurements over the DNA sequence.

I will demonstrate the effectiveness of FDA in this setting with two applications.

In the first one, I will present a novel method, called probabilistic K-mean with local alignment, to locally cluster misaligned curves and to address the problem of discovering functional motifs, i.e., typical “shapes” that may recur several times along and across a set of curves, capturing important local characteristics of these curves. I will demonstrate the performance of the method on simulated data, and I will apply it to discover functional motifs in “Omics” signals related to mutagenesis and genome dynamics.

In the second one, I will show how a recently developed functional hypothesis test, IWTomics, and multiple functional logistic regression can be employed to characterize the genomic landscape surrounding transposable elements, and to detect local changes in the speed of DNA polymerization due to the presence of non-canonical 3D structures.