Colloquia — Fall 2022

Friday, November 18, 2022

Title: Self-similar transformation and chirped waves in nonlinear wave models
Speaker: Jiefang Zhang, Communication University of Zhejiang, China
Time: 3:00pm–4:00pm
Place: CMC 130
Sponsor: W. Ma


Self-similarity is the phenomenon of similarity between the local area and the whole area. Self-similarity exists widely in nature, such as golden section, Fibonacci sequence, fractal and so on, and they are deeply related to each other. Self-similar dynamical effects have flourished into a research area of great importance and interest in different branches of physics including hydrodynamics, quantum field theory, plasma physics, and Bose-Einstein condensates, etc. In nonlinear optics, some important results on self-similarity have been reported. For example, the evolution of self-written waveguides, self-similarity in an optical fiber amplifier and a laser resonator, the spatial self-similar waves in graded index amplifiers, and the nonlinear compression of chirped solitary waves, and so on. These self-similar waves (or optical similaritons) may be useful in all-optical data-processing schemes and the design of pulse compressors and amplifiers, since they can maintain their overall shapes but with their amplitudes and widths changing with the modulation of system parameters such as dispersion or diffraction, nonlinearity, gain, and inhomogeneity. We will give a brief research overview and discuss about some of our recent relevant works on \((2+1)\)-dimensional nonlinear wave models.

Thursday, November 10, 2022

Title: Equipartition of energy in damped wave equations
Speaker: Guillermo Reyes Souto, University of Southern California
Time: 4:00pm–5:00pm
Place: MS Teams
Sponsor: D. Savchuk


We prove an asymptotic energy equipartition result for abstract damped wave equations of the form $$ u_{tt}+2F(S)\,u_t+S^2u=0, $$ where \(S\) is a positive definite selfadjoint operator with purely absolutely continuous spectrum and the damping operator \(F(S)\) is “small” in some sense. This means that under certain assumptions, we have $$ \frac{\tilde K(t)}{\tilde P(t)}\to 1\qquad\text{ as }\ t\to\infty, $$ where \(\tilde K(t)\) and \(\tilde P(t)\) are conveniently defined kinetic and potential energies of the given (nontrivial) solution. The latter are nothing but operator-weighted versions of the corresponding usual energies, associated to the undamped equation. Previous results, concerning the undamped case and the scalar-damped one, are particular cases of ours.

We propose an extension of the concepts of hyperbolicity and unitarity that allows one to consider the equipartition property in a more general setting.

Some examples involving PDEs, as well as pseudo-differential equations, are given. We show in an example that equipartition does not hold for the usual energies, thus justifying the introduction of their weighted counterparts.

This is a joint work with J. A. Goldstein, University of Memphis.

Friday, November 4, 2022

Title: Geometry of Discrete Integrable Systems: QRT Maps and Discrete Painlevé Equations
Speaker: Anton Dzhamay, Univ. of Northern Colorado
Time: 4:00pm–5:00pm
Place: CMC 130

Sponsor: R. Teodorescu


Many interesting examples of discrete integrable systems can be studied from the geometric point-of-view. In this talk we will consider two classes of examples of such systems: autonomous (QRT maps) and non-autonomous (discrete Painlevé equations). We introduce some geometric tools to study these systems, such as the blowup procedure to construct algebraic surfaces on which the mappings are regularized, linearization of the mapping on the Picard lattice of the surface and, for discrete Painlevé equations, the decomposition of the Picard lattice into complementary pairs of the surface and symmetry sub-lattices and construction of a birational representation of affine Weyl symmetry groups that gives a complete algebraic description of our non-linear dynamic. This talk is based on joint work with Stefan Carstea (Bucharest) and Tomoyuki Takenawa (Tokyo).

Title: The number of zeros of complex polynomials when complex conjugation is involved
Speaker: Erik Lundberg, Florida Atlantic University
Time: 3:00pm–4:00pm
Place: CMC 130
Sponsor: D. Khavinson


The zeros of complex (analytic) polynomials are subject to the fundamental theorem of algebra which tells us that the number of zeros (counted with multiplicity) is the same as the degree. A problem posed by T. Sheil-Small in the early 90s asks to investigate the number of zeros of harmonic polynomials \(p(z) +\operatorname{conjugate}\,(q(z))\) with \(\deg p > \deg q\). A related problem posed by W. Hengartner in 2000 asks to study the number of preimages of a complex number \(w\) under a so-called log-harmonic polynomial which has the form \(p(z)*\operatorname{conjugate}\,(q(z))\). In this talk, I will discuss the intricacy caused by complex conjugation, and I will review progress on these two problems including some ongoing joint work with Dmitry Khavinson and Sean Perry. Time permitting, I will also mention some parallel problems in the study of gravitational lensing.

Friday, October 21, 2022

Title: Collaborations Among Binary Operations
Speaker: Sergio López-Permouth, Ohio University
Time: 3:00pm–4:00pm
Place: CMC 130
Sponsor: M. Elhamdadi


Given two binary operations, \(\ast\) and \(\circ\), on a set \(S\), a third operation, \(\square\), on \(S\), is said to be a collaboration between \(\ast\) and \(\circ\) if, for all \(a\), \(b \in S\), \(\square(a,b)\in\{\ast(a,b),\circ(a,b)\}\). Collaborations have also been named two-option magmas earlier, in ortder to emphasize their similarity with previously studied concepts such as one-value magmas and two-value magmas.

The dichotomy inherent to the definition of a collaboration makes it clear that one can use graphs to represent such operations. Take \(S\) to be the vertices and connect \(a\) with \(b\) when \(\star\) is to be used (and not, otherwise). For that reason, the expression graph magmas has been associated to both one-value and two-value magmas.

Characterizations of associative one-value and two-value magmas are available in the literature. We ponder when a collaboration between two (not-necessarily associative) operations yield and associative operation. A lot of our discussion centers on the cases when \(S=\mathbb{Z}\) and the operations \(\ast\) and \(\circ\) are addition and subtraction, and when \(S=\mathbb{N}\) and the operations \(\ast\) and \(\circ\) are either addition and multiplication.

We report on an initial exploration of these concepts and will mention several problems that are suggested by them. If time allows, I will mention some connections between so-called group collaborations and skew braces.

This talk is a report on a collaboration (no pun intended) with Aaron Nicely and Majed Zailaee.

Friday, October 7, 2022

Title: On rolling of manifolds
Speaker: Irina Markina, University of Bergen, Bergen, Norway
Time: 3:00pm–4:00pm
Place: CMC 130 or MS Teams
Sponsor: C. Bénéteau


In the talk, we will introduce the notion of rolling one manifold over another. The idea of the rolling map originated as a simple mathematical model of rolling a ball over a plate with the constraints of no-slip and no-twist motion in the works of S. Chaplyging (1896) K. Nomizu (1978), R. Bryan and L. Hsu (1993). The geometric features are closely related to the distributions of E. Cartan type (1910). Later this idea was extended to the rolling of Riemannian manifolds of any dimension, as an isometry map preserving the parallelism of vector fields. After the historical overview and the necessary definitions, we also mention some applications in the interpolation and construction of stochastic processes on manifolds.