Title: Wedge Product and Naturality in Discrete Exterior Calculus

Abstract: In exterior calculus on smooth manifolds, the exterior derivative and wedge product are natural with respect to smooth maps between manifolds, that is, these operations commute with pullback. In discrete exterior calculus (DEC), simplicial cochains play the role of discrete forms, the coboundary operator serves as the discrete exterior derivative, and the antisymmetrized cup product provides a discrete wedge product. We show that these discrete operations in DEC are natural with respect to abstract simplicial maps. A second contribution is a new averaging interpretation of the discrete wedge product in DEC. We also show that this wedge product is the same as Wilson’s cochain product defined using Whitney and de Rham maps. This may lead to existence of more accurate combinatorial wedge products that are associative in the limit. In any case, the combinatorial product may be useful in creating an A-infinity or C-infinity algebra. Joint work with Mark Schubel and Daniel Berwick-Evans.

**Keller 303**

Speaker: James Hyde, Copenhagen

Title: Groups, Orders and Dynamics.

Abstract: I will describe some of the history of combinatorial group theory, and its connections with algorithmic questions such as the decidability of the word problem.

I will then define ordered groups and describe their connection to group actions on the real line. Then I will state some of my recent results (in part joint with Lodha) resolving two central questions in the area.

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.