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  Colloquium Archive

Colloquia — Spring/Summer 2022

Friday, May 20, 2022

Title
Speaker

Time
Place
Sponsor

System Identification Techniques
Ben Russo
Oak Ridge National Laboratory
3:00pm–4:00pm
CMC 130
J. Rosenfeld










Abstract

A dynamical system is given as \(\dot x=f(x)\), where \(x:[0,T]\to\mathbb{R}^n\) is the system state and \(f:\mathbb{R}^n\to\mathbb{R}^n\) is some function. Dynamical systems are prevalent in the sciences, such as engineering, biology, neuroscience, physics, and mathematics. However, in many cases even physically motivated dynamical systems can have unknown parameters (i.e., a gray box), such as mass and length of mechanical components, or the dynamics may be completely unknown (i.e., a black box). In such cases, system identification methods are leveraged to gain estimates on the dynamics of the system based on data generated by the system itself. In this talk, we'll go over some current techniques in non-linear system identification which use some tools from functional analysis.

Monday, May 16, 2022

Title
Speaker

Time
Place
Sponsor

Membership problems for \(2\times 2\) matrices over rational numbers
Volker Diekert
Universität Stutgart
2:00pm–3:00pm
CMC 109
N. Jonoska

Abstract

This work relates numerical problems on matrices over the rationals to symbolic algorithms on words and finite automata. Using exact algebraic algorithms and symbolic computation, we prove various new decidability results for \(2\times 2\) matrices over \(Q\). For that, we introduce the concept of flat rational sets: if \(M\) is a monoid and \(N\) is a submonoid, then flat rational sets of \(M\) over \(N\) are finite unions of the form \(L_0g_1L_1\cdots g_t L_t\) where all \(L_i\)'s are rational subsets of \(N\) and \(g_i\in M\). We give quite general sufficient conditions under which flat rational sets form an effective relative Boolean algebra. As a corollary, we obtain that the emptiness problem for Boolean combinations of flat rational subsets of \(GL(2,Q)\) over \(GL(2,Z)\) is decidable (in singly exponential time). It is possible that such a strong decidability result cannot be pushed any further inside \(GL(2,Q)\).

Our results improve all known decidability results for \(2\times 2\) matrices over \(Q\), and it also supports them with concrete complexity bounds for the first time.

A conference abstract appeared in the proceedings of ISSAC 2020.

Wednesday, May 4, 2022

Title
Speaker


Time
Place
Sponsor

Polynomial identities, group rings and enveloping algebras
Hamid Usefi
Memorial University of Newfoundland
St. John’s, Canada
11:00am–12:00pm
CMC 130
M. Elhamdadi

Abstract

I will talk about the development of the theory of polynomial identities initiated by important questions such as Burnside's asking if every finitely generated torsion group is finite. The field was enriched by contributions of many great mathematicians. Most notably Lie rings methods were developed and used by Zelmanov in the 1990s to give a positive solution to the restricted Burnside problem which awarded him the Fields medal. It has been of great interest to expand the theory to other varieties of algebraic structures. In particular, I will review when a group algebra or enveloping algebra satisfy a polynomial identity.

Friday, April 29, 2022

Title
Speaker




Time
Place
Sponsor

Zagier’s formula for multiple zeta values and its odd variant revisited
Cezar Lupu
Beijing Institute of Mathematical Sciences and Applications (BIMSA)
& Yau Mathematical Sciences Center (YMSC)
Tsinghua University
Beijing, China
3:00pm–4:00pm
Zoom Meeting
R. Teodorescu

Abstract

In this talk, we revisit the famous Zagier formula for multiple zeta values (MZV's) and its odd variant for multiple \(t\)-values which is due to Murakami. Zagier's formula involves a specific family of MZV's which we call nowadays the Hoffman family, $$ H(a,b)=\zeta(\underbrace{2, 2, \dotsc, 2}_{\text{$a$}}, 3, \underbrace{2, 2, \dotsc, 2}_{\text{$b$}}), $$ which can be expressed as a \(\mathbb{Q}\)-linear combination of products \(\pi^{2m}\zeta(2n+1)\) with \(m+n=a+b+1\). This formula for \(H(a,b)\) played a crucial role in the proof of Hoffman's conjecture by F. Brown, and it asserts that all multiple zeta values of a given weight are \(\mathbb{Q}\)-linear combinations of MZV's of the same weight involving \(2\)'s and \(3\)'s.

Similarly, in the case of multiple \(t\)-values (the odd variant of multiple zeta values), very recently, Murakami proved a version of Brown's theorem (Hoffman's conjecture) which states that every multiple zeta value is a \(\mathbb{Q}\)-linear combination of elements \(\{t(k_{1},\dotsc, k_{r}): k_{1}, \dotsc, k_{r}\in \{2, 3\}\}\). Again, the proof relies on a Zagier-type evaluation for the Hoffman's family of multiple \(t\)-values, $$ T(a,b)=t(\underbrace{2,2,\dotsc,2}_{\text{$a$}}, 3, \underbrace{2,2,\dotsc,2}_{\text{$b$}}). $$ We show the parallel of the two formulas for \(H(a,b)\) and \(T(a,b)\) and derive elementary proofs by relating both of them to a surprising cotangent integral. Also, if time will allow, we give a brief account on how these integrals can provide us with some arithmetic information about \(\frac{\zeta(2k+1)}{\pi^{2k+1}}\). This is a joint work with Li Lai and Derek Orr.

Thursday, April 28, 2022

Title
Speaker


Time
Place
Sponsor

Ternary Algebras and Applications
Viktor Abramov
Institute of Mathematics and Statistics
University of Tartu
11:00am–12:00pm
NES 102
M. Elhamdadi

Abstract

We propose a unital associative algebra, which is motivated by a generalization of the Pauli exclusion principle proposed within the framework of the quark model. The generators of this algebra satisfy the following relations: The sum of squares of all generators is equal to zero (binary relation) and the sum of cyclic permutations of factors in any triple product of generators is equal to zero (ternary relations). We study the structure of this algebra in the case of three generators and calculate the dimensions of spaces spanned by homogeneous monomials. It is shown how the algebra we propose is related to irreducible representations of the rotation group. Particularly we show that the 10-dimensional space spanned by triple monomials is the space of a double irreducible unitary representation of the rotation group. We use ternary \(q\)-commutators, where \(q\) is a primitive 3rd order root of unity, to split the 10-dimensional space spanned by triple monomials into a direct sum of two 5-dimensional subspaces. We endow these subspaces with a Hermitian scalar product by means of an orthonormal basis of triple monomials. In each subspace there is an irreducible unitary representation \(SO(3)\to SU(5)\). We calculate the matrix of this representation and the structure of the matrix indicates a possible connection between our algebra and the Georgi-Glashow model for elementary particles.

Friday, April 22, 2022

Title
Speaker

Time
Place
Sponsor

Broken Rays, Cones, and Stars in Tomography
Gaik Ambartsoumian
University of Texas at Arlington
3:00pm–4:00pm
CMC 130
D. Savchuk

Abstract

Mathematical models of various imaging modalities are based on integral transforms mapping a function (representing the image) to its integrals along specific families of curves or surfaces. Those integrals are generated by external measurements of physical signals, which are sent into the imaging object, get modified as they pass through its medium and are captured by sensors after exiting the object. The mathematical task of image reconstruction is then equivalent to recovering the image function from the appropriate family of its integrals, i.e. inverting the corresponding integral transform (often called a generalized Radon transform). A classic example is computerized tomography (CT), where the measurements of reduced intensity of X-rays that have passed though the body correspond to the X-ray transform of the attenuation coefficient of the medium. Image reconstruction in CT is achieved through inversion of the X-ray transform.

In this talk, we will discuss several novel imaging techniques using scattered particles, which lead to the study of generalized Radon transforms integrating along trajectories and surfaces containing a “vertex”. The relevant applications include single-scattering X-ray tomography, single-scattering optical tomography, and Compton camera imaging. We will present recent results about injectivity, inversion, stability and other properties of the broken ray transform, conical Radon transform and the star transform.

Monday, April 18, 2022

Title
Speaker

Time
Place
Sponsor

Cyclotomic points in backward orbits and unlikely intersections
Andrea Ferraguti
Scuola Normale Superiore di Pisa
4:00pm–5:00pm
NES 102
J. Biasse

Abstract

Let \(K\) be a number field and \(f\in K(x)\) a rational function. A celebrated theorem of Northcott implies that the set of preperiodic points of \(f\) defined over \(K\) is finite. In 2007 Dvornicich and Zannier proved, via an ingegnous application of the torsion coset theorem, that the same holds true for preperiodic cyclotomic points lying in the cyclotomic closure of \(K\), unless the map \(f\) is special. On the other hand a recent conjecture of Andrews and Petsche asserts that the backward orbit of a point \(a\) in \(K\) via \(f\) consists entirely of abelian points only if the pair \((f,a)\) is special. In this talk, we will explain how Dvornicich–Zannier's strategy works, and how it is possible to combine it with a height argument to yield unconditional evidence towards Andrews–Petsche's conjecture.

Friday, April 15, 2022

Title
Speaker

Time
Place
Sponsor

Residual finiteness of quandles
Mahender Singh
Indian Institute of Science Education and Research Mohali
2:00pm–3:00pm
CMC 130
M. Elhamdadi

Abstract

Quandles are algebraic objects modelled on the three Reidemeister moves of planar diagrams of knots and links in the Euclidean 3-space. Besides being fundamental to knot theory, these objects arise in a variety of contexts such as set-theoretic solutions to the Yang-Baxter equation, Riemannian symmetric spaces and mapping class groups, to name a few. After a brief introduction, we will present some recent results on residual finiteness of quandles. We will prove that free quandles and link quandles are residually finite, which as a consequence implies that the word problem is solvable for such quandles.

Friday, April 1, 2022

Title
Speaker

Time
Place
Sponsor

Stable polynomials and bounded rational functions
Alan Sola
University of Stockholm
3:00pm–4:00pm
CMC 130
C. Bénéteau

Abstract

A polynomial in \(d\) variables that does not vanish in a fixed domain in \(C^d\) is said to be stable. Stable polynomials are important in several areas of mathematics as well as in applications. I will review some situations where stable polynomials feature, and will then focus on the role stable polynomials play as denominators of rational functions with good properties. This aspect of the theory will be illustrated through numerous examples and images.

Friday, March 11, 2022

Title
Speaker

Time
Place
Sponsor

An unified tau function and bilinear structure for several important nonlinear wave equations
Baofeng Feng
The University of Texas Rio Grande Valley
3:00pm–4:00pm
CMC 130
W. Ma

Abstract

We are concerned with several integrable nonlinear wave equations, important in physics, which include the massive Thirring model in quantum field theory, the Fokas–Lenells and complex short pulse equations in nonlinear optics and the modified Camassa–Holm equation in water waves. We show that all these four equations can be derived from the same set of bilinear equations satisfied by one tau function in the KP-Toda hierarchy. Furthermore, we will show that the discrete KP equation, the most fundamental equation for integrable systems, can generate the above set of bilinear equations, which paves a way for constructing integrable discrete analogues of those nonlinear wave equations with potential applications in numerical algorithms.

Title
Speaker


Time
Place
Sponsor

Self-reciprocal polynomials and reversed Dickson polynomials
Neranga Fernando
Mathematics and Computer Science
College of the Holy Cross
2:00pm–3:00pm
CMC 108
X. Hou

Abstract

Consider the polynomial \(f(x)=1+2x+3x^2+2x^3+x^4\).

Can you see that the coefficients of the polynomial \(f(x)\) form a palindrome? Such polynomials are called self-reciprocal polynomials. They have important applications in coding theory. I will explain them during my talk.

Let \(p\) be a prime and \(q=p^e\), where \(e\) is a positive integer. Let \(\mathbb{F}_q\) be the finite field with \(q\) elements. For \(a\in\mathbb{F}_q\), the \(n\)-th reversed Dickson polynomial of the \((k+1)\)-th kind \(D_{n,k}(a,x)\) is defined by $$ D_{n,k}(a,x)=\sum_{i=0}^{\lfloor\frac n2\rfloor}\frac{n-ki}{n-i}\dbinom{n-i}{i}(-x)^{i}a^{n-2i}, $$ and \(D_{0,k}(a,x)=2-k\). When \(p\) is odd, \(D_{n,k}(1,x)\) can be written as $$ D_{n,k}(1,x)=\Big(\frac{1}{2}\Big)^{n}\,f_{n,k}(1-4x), $$ where \[ f_{n,k}(x)=k\,\,\displaystyle\sum_{j\ge 0}\,\binom{n-1}{2j+1}\,\left(x^j-x^{j+1}\right)+2\,\,\displaystyle\sum_{j\geq 0}\,\binom{n}{2j}\,x^j\in\mathbb{Z}[x] \] for \(n\ge 1\) and $$ f_{0,k}(x)=2-k. $$

I am primarily interested in the question: When is \(f_{n,k}(x)\) a self-reciprocal polynomial?

In this talk, I will first explain what motivated me to consider this problem. Then, I will explain a complete answer to the question above.

Friday, February 25, 2022

Title
Speaker


Time
Place
Sponsor

Quantum deformations and twisted algebraic structures
Abdenacer Makhlouf
Université Haute Alsace
France
3:00pm–4:00pm
CMC 130
M. Elhamdadi

Abstract

A quantum deformation or \(q\)-deformation of algebras of vector fields consists of replacing usual derivation by a sigma-derivation. The main example is given by Jackson derivative and lead for example to \(q\)-deformation of sl_2, Witt algebra, Virasoro algebra and also Heisenberg algebras (oscillator algebras). The description of the new structures gave rise to a structure generalizing Lie algebras, called Hom-Lie algebras or quasi-Lie algebras studied first by Larsson and Silvestrov. Since then various classical algebraic structures and properties were extended to the Hom-type setting. The main feature is that the classical identities are twisted by homomorphisms.

The purpose of my talk is to give an overview of recent developments and provide some key constructions and examples on Hom-algebras, BiHom-algebras and their dualization. I will show that they lead to new-type cohomologies.