On a Hilbert modular surface over $\mathbb Z$, there are two
families of arithmetic cycles. One family consists of the
Hirzebruch-Zagier divisors
$\mathcal T_m$ of codimension $1$, indexed by positive integers $m$, and
another consists of the CM cycles $CM(K)$ of codimension 2, indexed by
quartic CM number fields $K$. When $K$ is not biquadratic, $\mathcal T_m$
and $CM(K)$ intersect properly, and a natural
question is, what is the intersection number? In this talk, we present a
conjectural formula for the intersection number of
Bruinier and myself. We give two partial results in this talk. If time
permits, I will also briefly describe two applications:
one of the consequences is
a generalization of the Chowla-Selberg formula, and another is a
conjecture of Lauter on Igusa invariants.

Informally speaking, a compact set in the plane is computable if
there exists an algorithm to draw it on a screen with an arbitrarily
high magnification. We investigate the question of computability
of a Julia set of a quadratic polynomial and obtain some surprising
answers.
(joint work with Mark Braverman)

Our main goal is to study the behavior of the error in finite element approximations of partial differential equations. The error typically has two components -- local error and global (pollution) error. We also will discuss the very interesting phenomena of superconvergence, and in particular, how to determine superconvergence points, and what advantages can be derived from them.

An r-uniform hypergraph is a generalization of graphs, we consider a
subset of all r-element subsets of a given vertex set. A Hamiltionian
chain is a generalization of hamiltonian cycles for hypergraphs, it is a
"cycle" that contains all vertices. Among the several possible ways of
generalizations this is probably the most strong one, it requires the
strongest structure. Since there are many interesting questions about
hamiltonian cycles in graphs, we can try to answer these questions
for hypergraphs, too. I give a survey on
results about such questions.

Usually, a limit theorem of Probability Theory is a theorem that concerns convergence of a sequence of distributions $P_n$ to a distribution $P$. However, there is a number of works where the traditional setup is modified, and the object of study is two sequences of distributions, $P_n$ and $Q_n$, and the goal consists in establishing conditions implying the convergence
$P_n - Q_n ->0 (1)$
In particular problems,$P_n$ and $Q_n$ are, as a rule, the distributions of the r.v.'s $f(X_1,...,X_n)$ and $f(Y_1,...,Y_n)$, where $f(.)$ is a function, and $X_1,X_2$,... and $Y_1,Y_2$,... are two sequences of r.v.'s. The aim here is rather to show that different random arguments $X_1,...,X_n$ may generate close distributions of $f(X_1,...,X_n)$ , than to prove that the distribution of $f(X_1,...,X_n)$ is close to some fixed distribution (which, above else, may be not true). Clearly, such a framework is more general than the traditional one. First, as was mentioned, the distributions $P_n$ and $Q_n$, themselves do not have to converge. Secondly, the sequences $P_n$ and $Q_n$ are not assumed to be tight, and the convergence in $(1)$ covers situations when a part of the probability mass or the whole distributions "move away to infinity'", while the distributions $P_n$ and $Q_n$, are approaching each other.
We consider a theory on this point, including the very definition of convergence $(1)$, and a particular example of the invariance principle in the general non-classical setup.

In this talk we will develop the concept of arbitrage and discuss the theory of options pricing. Time permitting, we will present the famous Black-Scholes option pricing formula.

We will present an integral representation for automorphic
representations on $U(3) x GL(2)$. It involves parabolic induction to $U(4)$.
The resulting formula can be applied to determine the cuspidal
automorphic representations of $U(3)$ that occur in the restriction of
the Siegel induced residual spectrum of $U(4)$.