This talk will explain a new construction of secure cryptographic hash functions from Ramanujan graphs. First we will explain cryptographic hash functions and the importance of the collision-resistance property. After a brief overview of expander graphs, we will give a construction of provable collision resistant hash functions from expander graphs in which finding cycles is hard.

As an example, we give a family of optimal expander graphs for provable collision resistant hash function constructions: the family of Ramanujan graphs constructed by Pizer. Pizer described a family of Ramanujan graphs, where the nodes of the graph are isomorphism classes of supersingular elliptic curves over $F_p^2$, and the edges are n-isogenies, n a prime different from $p$. When the hash function is constructed from one of Pizer's Ramanujan graphs, then collision resistance follows from hardness of computing isogenies between supersingular elliptic curves.

Joint work with Denis Charles and Eyal Goren

In order to function most proteins have to fold to a unique three
dimensional shape. Thus, the same protein sequence helps in both
biological function and protein folding. I will give a brief introduction
to protein folding, the priniciple of minimal frustration and the funneled
energy landscapes. I will then discuss the interplay of folding and function
through the structure of the protein in the context of the beta-trefoil
fold family of proteins.

An old conjecture of Erd\H os and Szemer\'edi states that if $A$
is a finite set of
integers then the sum-set or the product-set should be large. The sum-set of
$A$ is defined
as $A+A=\{a+b | a,b \in A\}$ and the product set is $A\cdot A=\{ab | a,b \in
A\}.$
Erd\H os and Szemer\'edi conjectured that the sum-set or the product set is
almost quadratic
in the size of $A,$ i.e. $\max (|A+A|,|A\cdot A|)\geq c|A|^{2-\delta}$ for
any positive $\delta$.
I proved earlier that $\max (|A+A|,|A\cdot A|)\geq c|A|^{14/11}/\log{|A|},$
for any finite set of complex numbers, $A.$
In this talk we improve the bound further for sets of real numbers.

Brownian motion is a stochastic process named after botanist Robert Brown,
who is credited with having discovered it in 1827 after observing the
erratic behavior of pollen grains. It has since become one of the most
important ideas in probability theory and has wide variety of applications
everywhere from physics to finance.

The talk will begin with a brief discussion of the necessary probability
background, including the notions of expectation, martingales, and random
walks. I will then define Brownian motion and discuss some of its more
interesting properties. If time permits, I will also discuss how Brownian
motion is used to define stochastic differential equations.