Spring 2024
Time & Location: All talks are on Tuesdays in Norman Mayer G106 at 3:30PM unless otherwise noted.
Organizer: Sang-Eun Lee
September 10
Title: Transition of Equilibria in the Dynamics of Chemically Active Particles
Sang-Eun Lee - Tulane University
Abstract: Motivated by autophoretic droplet swimmers, we present a dynamical transition to the chemically active swimmers. With the periodic boundary condition imposed to the reaction-diffusion system, the particles secrete the chemicals with un unlimited supply and avoid the chemical so have anti-chemotactic. Due to the periodic boundary condition, there is an interesting transition of equilibrium of particles due to a dimensionless parameter in both one and two dimensions. In addition, we present an effect of fluid that affects the convection of both chemicals and the particles.
Time: 3:30 pm
Location: Norman Mayer G106
September 10
Title: A Kirchhoff Rod Model to Study Dynamics of Flexible Fibers and Rotating Helical Flagella in Stokes Flow
Rubaiyat Islam - Tulane University
Abstract: At the microscale, fluid motion is governed by Stokes equations. To study how microfibers behave in an ambient flow or how bacteria propel themselves with their rotating helical flagellum, we need a model to compute forces and torques along the fiber body and a way to include fluid interactions. Our computational framework is a Kirchhoff rod model coupled to regularized Stokeslet segments. This model takes advantage of the slenderness of fibers or flagella and uses a set of orthonormal triads to compute forces and torques. Passive filaments show rich shape deformations depending on their length and stiffness when subject to a background flow. Using a system of images for Stokeslet segments, we also show flagellated bacteria swimming in circles near a rigid wall.
Time: 3:30 pm
Location: Norman Mayer G106
September 24
Title: Modular Forms and Orders of Vanishing
Peter Marcus - Tulane University
Abstract: Modular forms are holomorphic functions with specific symmetry properties. Their Fourier expansions are generating functions for various sequences of interest, such as partition numbers and divisor sums. These functions live in finite-dimensional vector spaces, so by studying these spaces we can learn about these number-theoretic sequences. The dimensions of these spaces have well-known formulas, but there is no known formula for the maximal order of vanishing. In other words, if you write a row-reduced basis of Fourier expansions, when will the first nonzero Fourier coefficient of the last function occur? I will give an overview of this subject and progress on this problem.
Time: 3:30 pm
Location: Norman Mayer G106
October 1
Title: Tevelev Degrees of $\mathbb{P}^1$
Naufil Sakran - Tulane University
Abstract: In this talk, I will introduce the field of enumerative geometry and discuss recent developments in this area. Let $C$ be a general curve with $n$ marked points, and consider a degree $d$ map $\pi: C \to \mathbb{P}^1$, subject to specific incidence conditions.
Tevelev degree is defined as the number of such maps $\pi: C \to \mathbb{P}^1$. Interestingly, the computation of Tevelev degree presents intriguing connections with combinatorics, particularly through Dyck path counting and Schubert calculus. I will explore these topics, discussing their role in computing Tevelev degrees and presenting the latest formulas, as featured in recent work by R. Pandharipande, A. Cela, C. Lian, and others.
Time: 3:30 pm
Location: Norman Mayer G106
