Seminar

Recently, the fractional Laplacian has received great attention in modeling complex phenomena that involve long-range interactions. However, the nonlocality of the fractional Laplacian introduces considerable challenges in both analysis and simulations. In this talk, I will present numerical methods to discretize the fractional Laplacian as well as error estimates. Compared to other existing methods, our methods are more accurate and simpler to implement, and moreover they closely resembles the central difference scheme for the classical Laplace operator.

Abstract:

Oscillatory and excitable cells are just two of many types of cells in your body. They interact with each other by sending pulses or voltage spikes. An excitable cell amplifies any input it receives, as long as the input is large enough while an oscillatory cell needs no input and generates a spontaneous rhythm. One can connect these cells and model how they interact using differential equations; in this talk, I will discuss what we have found. 

Thack 627

I will continue my talk from last week. I will quickly review material from last week and then discuss examples of equivariant cohomology of smooth toric vatieties and flag varieties as well as the localization theorem. Finally, will discuss an application of the Chinese remainder theorem in this setting. 

Abstract:

We will discuss applications of the Atiyah, Guillemin-Sternberg theorem, such as classification of toric manifolds, the Schur-Horn theorem, and a brief introduction about viewing Bernstein–Kushnirenko theorem via symplectic geometry.

In this talk I will briefly introduce Cartan decompositions of semisimple Lie groups/algebras with some applications to quantum systems. I will also discuss how Riemannian and sub-Riemannian geometry can be used to study the cut locus of control systems arising from these decompo- sitions, after passing to a singular quotient space.

Abstract:

An important result due to Sanderson and Ion says that characters of level one Demazure modules are specialized Macdonald polynomials. In this talk, I will introduce a new class of symmetric polynomials indexed by a pair of dominant weights of $sl_{n+1}$ which is expressed as linear combination of specialized symmetric Macdonald polynomials with coefficients defined recursively. These polynomials arose in my own work while investigating the characters of higher level Demazure modules.