A seminar, spoken by Tong Li, was held on December 23th, 2022, at Seoul National University. After the presentations, we took photos.

There will be an invited talk.

Title: Hyperbolic-parabolic PDE Models of Chemotaxis
Speaker: Tong Li (Univ. of Iowa)
Time: 2022 December 23th, 15:30 ~ 16:20
Place: 27-325
Abstract:
We study global existence and long-time behavior of solutions for hyperbolic-parabolic PDE models of chemotaxis. We show the existence and the stability of traveling wave solutions to a system of nonlinear conservation laws derived from the Keller-Segel model. Traveling wave solutions of chemotaxis models with growth are also investigated. Moreover, we find oscillatory traveling wave solutions to an attractive chemotaxis system which are biologically relevant.

There will be an invited talk.

Title: Fredholm Property of the Linearized Boltzmann Operator for a Mixture of Polyatomic Gases
Speaker: Marwa Shahine (Univ. of Bordeaux)
Time: 2022 November 11th, 15:00 ~ 16:00
Place: 27-325
Abstract:
In this talk, we consider the Boltzmann equation that models a mixture of polyatomic gases assuming the internal energy to be continuous. Under some convenient assumptions on the collision cross-section, we prove that the linearized Boltzmann operator L is a Fredholm operator. For this, we write L as a perturbation of the collision frequency multiplication operator. We prove that the collision frequency is coercive and that the perturbation operator is Hilbert-Schmidt integral operator.

There will be an invited talk.

Title: Reinforcement Hedging
Speaker: Kiseop Lee (Professor, Purdue University)
Time: 2022 December 2ed, 16:00 ~ 16:50
Place: 27-325
Abstract:
We investigate option hedging in an incomplete market with a reinforcement learning algorithm called double deep Q-network (DDQN). The agent of DDQN learns the optimal policy that generates replicating portfolios without prior knowledge of the stochastic representation of an underlying asset price process. First, we interpret a mean-variance approach in quadratic hedging in a reinforcement learning framework. This study includes three simulation studies for different underlying asset price processes: geometric Brownian motion (GBM), Heston, and GBM with compound Poisson jumps. For each study, a DDQN agent learns the optimal policy, and we compare the algorithm performance with delta hedging. Second, we discuss limitations that stem from the structure of reinforcement learning in finance.

There will be an invited talk.

Title: Approximation of the bitemperature Euler system in 2D
Speaker: Stephane Brull (Professor, Univ. of Bordeaux)
Time: 2022 July 12th, 10:30am~11:30am
Place: 27-325
Abstract:
This talk is devoted to the numerical approximation of the bidimensional bitemperature Euler system. This model is a nonconservative hyperbolic system describing an out of equilibrium plasma in a quasi-neutral regime. This system is non conservative because it involves products between velocity and pressure gradients that cannot be transformed into a divergential form. We develop a second order numerical scheme by using a discrete BGK relaxation model. The second order extension is based on a subdivision of each cartesian cell into four triangles to perform affine reconstructions of the solution. Such ideas have been developed in the litterature for systems of conservation laws. We show here how they can be used in our nonconservative setting. The numerical method is implemented and tested in the last part of the paper.

There will be an invited talk.

Title: Embedding Carnot groups into bounded dimensional Euclidean spaces with optimal distortion
Speaker: Sang Woo Ryoo (Princeton Univ.)
Time: 2020 August 5th & 7th, 09:30am~11:3am
Place: 27-116
Abstract:
In the first part of the talk, I will first introduce the concept of a Carnot group, which generalizes Euclidean spaces, and a differentiation theorem by Pansu which proves that noncommutative Carnot groups cannot be bi-Lipschitzly embedded into Euclidean spaces. I will then present Assouad’s theorem, which shows that the snowflakes of Carnot groups (and more generally doubling metric spaces) can be bi-Lipschitzly embedded into Euclidean spaces, although the target dimension may be very high. After that, I will state a result of Naor and Neiman which significantly reduces the target dimension at the cost of worsening the distortion, and a result of Tao that achieves both optimal distortion and optimal target dimension in the case of the Heisenberg group. My work, which has the above title, is a generalization of Tao’s result to the setting of Carnot groups. If time permits, I will introduce the Lovasz local lemma and some Littlewood-Paley theory on Carnot groups, which are tools used in the proof.

In the second part of the talk, I will explain the key conceptual difficulties in constructing the embedding, by analogy with the Nash embedding theorem. One difficulty is a loss of derivatives problems when solving for a certain linear equation (in the Nash embedding theorem one needs to solve for a quadratic form). This is overcome by a variant of the Nash-Moser iteration scheme pioneered by Tao, and it carries without difficulty to the setting of Carnot groups. The other difficulty is a construction of orthonormal fields which is used in the iterative construction of the embedding. In the Nash embedding theorem this is dealt with using basic fibre bundle topology, and in Tao’s proof this is done using a quantitative homotopy argument, but both seem to fail in the setting of Carnot groups. The main contribution of my paper is that this can be overcome even in the most general setting of doubling metric spaces by using the Lovasz local lemma and concentration of measure phenomenon on the Euclidean sphere.

There will be an invited talk.

Title: Regularity and Long-Time Behavior for Hydrodynamic Flocking Models
Speaker: Prof. Trevor Leslie (Univ. of Wisconsin-Madison)
Time: 2020 March 10th, 04:00pm~05:00pm
Place: 27-220
Abstract:
We consider the Euler Alignment model, a hydrodynamic analog of the discrete Cucker-Smale flocking ODE system. The salient feature of the Cucker-Smale model is the interaction of agents through a so-called “communication protocol” that tends to align the velocities of the agents; this alignment mechanism also drives the Euler Alignment PDEs. We discuss several of the different communication protocols treated in the literature and the corresponding wellposedness theory for the Euler Alignment model in each case. In the case of smooth, bounded protocols in one space dimension, it is possible to completely characterize the initial data leading to the existence of regular solutions, in terms of a certain pointwise balance between the initial velocity and the “total influence” from the density profile. This balance is captured by the sign of a certain quantity that we denote by e. Under the additional assumption of periodicity, a quantity closely related to e determines the long-time distortion of the density profile away from its average value.