Colloquia — Spring 2022

Friday, April 29, 2022

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


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.

Friday, April 22, 2022

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


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


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


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


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


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


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


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.