By a tiled order we mean a right Noetherian prime semiperfect and semidistributive ring with the nonzero Jacobson radical. For example, a serial tiled order $A$ is a Noetherian (but non-Artinian) serial indecomposable ring. A ring $A$ is decomposable if $A = A_1 \times A_2$, otherwise $A$ is indecomposable. Every serial tiled order $A$ is hereditary and Gorenstein, i.e., $inj.dim_A A_a = inj.dim_A {_A} A = 1$.

Let $R(A)$ be the Jacobson radical of a tiled order $A$. For any tiled order $A$ there exists a countable set of two sided ideals $I_1 \supset I_2 \supset \dots$, where $R^2 (A) \supset I_1, I_{k+1} \neq I_k$ and all quotient rings $A/I_k$ are Frobenius.

For any permutation $\sigma \in S_n$ there exists a Frobenius ring $B$ with the Nakayama permutaion $\sigma$. We consider the exponent matrices of tiled orders, in particular, Gorenstein matrices. We discuss the relations between exponent matrices and quivers of tiled orders, cyclic Gorenstein orders and doubly stochastic matrices, Gorentstein matrices and tiled orders of injective dimension one.

The Schur functions can be decomposed into "nonsymmetric Schur
functions" obtained through a certain specialization of Macdonald
polynomials. We explore several combinatorial properties of these
polynomials and a connection to crystal graphs.

How many times can a prime number be expressed as the sum of n squares? What is the asymptotic distribution of integer points on a family of ellipsoids $ax^2 + by^2 + cz^2 = n$ as $n$ tends to infinity?
I will explain how modular forms can be used to address these questions and others.

In this talk we will survey the development of the Green's
function for the Boltzmann equation. The talk will include
the motivation from the field of hyperbolic conservation
laws, the connection between the Boltzmann equation and the
hyperbolic conservation laws, and the particle-like and the
wave-like duality in the Boltzmann equation. With all these
components one can realize a clear layout of the Green's
function for the Boltzmann equation. Finally we will present
the application of the Green's function to an
initial-boundary value problem in the half space domain.