Colloquium in Keller 303

The speaker is Dr. Hailun Zheng, from the University of Houston-Downtown.

Title: Polytope and spheres: the enumeration and reconstruction problems

Abstract: Consider a simplicial d-polytope P or a simplicial (d-1)-sphere P with n vertices. What are the possible numbers of faces in each dimension? What partial information about P is enough to reconstruct P up to certain equivalences?

In this talk, I will introduce the theory of stress spaces developed by Lee. I will report on recent progress on conjectures of Kalai asserting that under certain conditions one can reconstruct P from the space of affine stresses of P —- a higher-dimensional analog of the set of affine dependencies of vertices of P. This in turn leads to new results in the face enumeration of polytopes and spheres; in particular, a strengthening of (the numerical part of) the g-theorem.

Joint work with Satoshi Murai and Isabella Novik.

Title: Small-Scale Fluid Dynamics: From Microfluidics to Microfiltration

Abstract: Understanding microscale fluid and particle transport is critical to perfecting the manufacturing and use of microfluidic technologies in medical, industrial, and environmental engineering applications. In this talk, I will discuss two projects concerned with solute transport and diffusion at the microscale tackled via analytical and experimental approaches. 

Many wastewater management facilities aimed at water purification in the United States utilize hollow-fiber micro- or ultra-filtration. In these systems, pipes are split into thousands of micro or nanometer-scale capped tubes with permeable walls. As wastewater flows through the filter, foulants are captured by the membraned walls, allowing clean water to exit. I will discuss a first step towards understanding the fluid dynamics of these systems through the development of a 2D model for the flow of wastewater through a single hollow-fiber. Resolving the fluid dynamics details of filtration would allow for better control of the fouling process and could improve its efficiency.

In the latter part of the talk, I will focus on passive diffusion into microchannels with dead-end pores, which are ubiquitous in natural and industrial settings. I will describe a repeatable and accessible experimental protocol developed to study the passive diffusion process of a dissolved solute into dead-end pores of rectangular and trapezoidal geometries. The experimental data is compared directly to analytical solutions of an effective 1D diffusion model: the Fick-Jacobs equation. The role of the pore geometry on the passive diffusion process will be highlighted. Ongoing and future directions will be discussed.

Title: Stochastic maximum principle for weighted mean-field system

Abstract: We study the optimal control problem for a weighted mean-field system. A new feature of the control problem is that the coefficients  depend on the state  process as well as  its weighted measure and the control variable. By applying variational technique, we establish a stochastic maximum principle. Also, we establish a sufficient condition of optimality.  As an application, we investigate the optimal premium policy of an insurance firm for asset–liability management problem.

Title: modeling diseases 
Abstract: We will talk about my journey into mathematics and learn how to use math to model the spread of diseases.

Title: Maximum Likelihood Degree of Brownian motion tree models

Abstract: A Brownian motion tree model is a Gaussian model whose associated set of covariance matrices is linearly constrained according to common ancestry in a phylogenetic tree. In joint work with Jane Coons, Aida Maraj, and Ikenna Nometa, we study the complexity of inferring the maximum likelihood (ML) estimator for a Brownian motion tree model by computing its ML-degree. Our main result is that the ML-degree of the BMT model on a star tree with n + 1 leaves is 2^(n+1) – 2n – 3, which was previously conjectured by Amendola and Zwiernik. We also prove that the ML-degree of a Brownian motion tree model is independent of the choice of root and we find a combinatorial formula for the likelihood function. In this talk, I will introduce Brownian motion tree models and the tools from computational algebraic geometry that we use to compute the ML-degree.

Title: Fluid dynamics within symbiotic systems

Abstract: To understand the fluid dynamics of marine symbiotic systems, systems composed of multiple organisms, fluid-structure interaction problems must be solved. Challenges exist in developing numerical methods to solve these flow problems with boundary conditions at fluid-structure interfaces. We are studying the impact of fluid flow on symbiotic systems within two different biological systems. (1) Pulsing soft corals, Xeniidae corals, that have internal algal endosymbionts, zooxanthellae, which provide them with much of their energy and (2) the bobtail squid, Euprymna scolopes, that depend on the luminous bacteria, Vibrio fischeri, to protect them from predators through counterillumination. Both of these problems are motivated by field and experimental work in the marine sciences. I will discuss these related data, mathematical models and numerical methods developed to study these problems and provide comparisons with the modeling.