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Mathematics & Statistics

Colloquia — Spring 2019

Friday, May 3, 2019

Title
Speaker

Time
Place
Sponsor

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).

Friday, April 26, 2019

Title
Speaker


Time
Place
Sponsor

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.

Title
Speaker

Time
Place
Sponsor

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

  1. 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.
  2. 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.
  3. 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.
  4. 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.
  5. 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.
  6. D. Gorenstein, Finite Groups, Chelsea Publishing Company, New York, 1980.
  7. C. R. Leedham-Green and S. McKay: The Structure of Groups of Prime Power Order, Oxford University Press, Oxford, 2002.
  8. P. Lounesto: Clifford Algebras and Spinors, Cambridge University Press, Cambridge, 2001.
  9. N. Salingaros: Realization, extension, and classification of certain physically important groups and algebras, J. Math. Phys. 22 (1981), 226–232
  10. N. Salingaros: On the classification of Clifford algebras and their relation to spinors in \(n\) dimensions, J. Math. Phys. 23 (1) (1982), 1–7.
  11. N. Salingaros: The relationship between finite groups and Clifford algebras, J. Math. Phys. 25 (4) (1984), 738–742.

Friday, April 19, 2019

Title
Speaker

Time
Place
Sponsor

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.

Friday, April 5, 2019

Title
Speaker

Time
Place

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.

Title
Speaker

Time
Place
Sponsor

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.

Friday, March 29, 2019

Title
Speaker

Time
Place
Sponsor

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.

Friday, March 22, 2019

Title
Speaker

Time
Place
Sponsor

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.

Friday, March 8, 2019

Title
Speaker

Time
Place
Sponsor

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.

Title
Speaker

Time
Place
Sponsor

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.

Wednesday, March 6, 2019

Title
Speaker


Time
Place
Sponsor

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.

Friday, February 15, 2019

Title
Speaker

Time
Place
Sponsor

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

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

Wednesday, February 13, 2019

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

Friday, February 1, 2019

Title
Speaker

Time
Place
Sponsor

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.

Thursday, January 24, 2019

Title
Speaker

Time
Place
Sponsor

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.

Wednesday, January 23, 2019

Title
Speaker

Time
Place
Sponsor

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.

Wednesday, January 16, 2019

Title
Speaker

Time
Place
Sponsor

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.

Monday, January 14, 2019

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

Friday, January 11, 2019

Title
Speaker

Time
Place
Sponsor

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.

Thursday, January 10, 2019

Title
Speaker

Time
Place
Sponsor

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.

Wednesday, January 9, 2019

Title
Speaker

Time
Place
Sponsor

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.

Tuesday, January 8, 2019

Title
Speaker

Time
Place
Sponsor

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.

Monday, January 7, 2019

Title
Speaker

Time
Place
Sponsor

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.