This talk consists of two aspects about solving the radiative transport through the integral formulation. The radiative transport equation has been numerically studied for many years, the equation is difficult to solve due to its high dimensionality and its hyperbolic nature, in recent decades, the computers are equipped with larger memories so it is possible to deal with the full-discretization in phase space, however, the numerical efficiency is quite limited because of many issues, such as iterative scheme, preconditioning, discretization, etc. In this talk, we first discuss about the special case of isotropic scattering and its integral formulation, then walk through the corresponding fast algorithm for it. In the second part, we try to trivially extend the method to anisotropic case, and talk about the method’s limitation and some perspectives in both theory and numerics.

We will begin with the idea of General Relativity, which Einstein called his "happiest thought," and then proceed with a qualitative and quantitative discussion of the curvature of space-time. We will describe the central role of differential geometry in the subject and the important role that mathematicians have played proving the conjectures of the physicists, as well as making a few conjectures of our own. Finally, we will describe the geometry of black holes and their relationship to soap bubbles.

Locally Linear Embedding(LLE), is a well known manifold learning algorithm published in Science by S. T. Roweis and L. K. Saul in 2000. In this talk, we provide an asymptotic analysis of the LLE algorithm under the manifold setup. We establish the kernel function associated with the LLE and show that the asymptotic behavior of the LLE depends on the regularization parameter in the algorithm. We show that on a closed manifold, asymptotically we may not obtain the Laplace-Beltrami operator, and the result may depend on the non-uniform sampling, unless a correct regularization is chosen. Moreover, we study the behavior of the algorithm on a compact manifold with boundary. This talk is based on the joint work with Hau-tieng Wu.

Our brains are composed of networks of cells, including neurons and glial cells. While the significance of neurons has been established by biologists, the role of glial cells is less understood. One hypothesis is that glial cells facilitate neural communication in nearby neurons, while suppressing communication among more distant neurons via a reaction-diffusion process. I consider this proposed mechanism using partial and ordinary differential equation models. By analyzing the ordinary differential equation model, I can determine conditions for this hypothesis to hold. I then compare the results of this analysis with simulations of the partial differential equation model and discuss the biological implications.

At some point in every mathematician's life they have seen the paradoxical 'proof' that 1=0 obtained by different groupings of the infinite sum 1-1+1-1+... As we learn, the issue is that this series does not converge. The Eilenberg-Mazur swindle is a twist on this argument which shows that A+B+A+B+... = 0 implies that A=0=B in certain situations where we can make sense of the infinite sum. In this talk, we will explore these swindles, touching on many interesting areas of mathematics along the way.

We describe a geometric theory of dark matter called "wave dark matter" whose underlying equations are the Einstein-Klein-Gordon system of PDEs. In spherical symmetry this system has simple static state solutions which we use to model dark galactic halos. We outline some scaling properties of these states including two new boundary conditions which might account for the existence of an astrophysical scaling relation called the baryonic Tully-Fisher relation.

This talk will introduce parts of algebraic geometry and complex analytic geometry which are closely related to each other. These are important areas of pure mathematics. The presentation will start from scratch and hopefully reach the statement of a famous conjecture by the end. The first half of the talk should be accessible to incoming graduate students who have worked an exercise or two on projective space. Familiarity with complex analysis at the undergraduate level would be helpful. The second half of the talk will make use of cohomology. Students who have taken a semester of algebraic topology and are now taking a second semester together with a course on Riemann surfaces should be able to follow many of the details. Those who have completed a course on Riemann Surfaces and on Algebraic Geometry should be able to follow all the details. Professional mathematicians working in this general area will likely find new insights few and far between if present at all.

We study the dynamics of waves in media with a local periodic structure which varies adiabatically (over many periods of the periodic lattice) across the medium. We focus in particular on the case where symmetries of the periodic structure lead to degeneracies in the Bloch band dispersion surface. An example of such symmetry-induced degeneracies are the `Dirac pointsÂ? of media with `honeycomb latticeÂ? symmetry, such as graphene. Our results are as follows: (1) A systematic and rigorous derivation of the `anomalous velocityÂ? of wave-packets due to the Bloch bandÂ?s Berry curvature. The Berry curvature is large near to degeneracies, where it takes the form of a monopole. We also derive terms which do not appear in the works of Niu et al. which describe a `field-particleÂ? coupling effect between the evolution of observables associated with the wave-packet and the evolution of the wave packet envelope. These terms are of the same order as the anomalous velocity. (2) Restricting to one spatial dimension, the derivation of the precise dynamics when a wave-packet is incident on a Bloch band degeneracy. In particular we derive the probability of an inter-band transition and show that our result is consistent with an appropriately interpreted Landau-Zener formula. I will present these results for solutions of a model Schr\Â?{o}dinger equation; extending our results to systems described by Maxwell's equations is the subject of ongoing work. This is joint work with Michael Weinstein and Jianfeng Lu.