The completion of the genomes of model organisms represents just the beginning of a long march toward in-depth understanding of biological systems. One challenge in post-genomic research is the detection of functional patterns from full-length genomic sequences. This talk focuses on statistical methods in finding patterns with functional or structural importance in biological sequences, in particular the identification of transcription factor binding sites (TFBSs). Some of the underlying mathematical theories will be discussed as well.TFBSs are often short and degenerate in sequence. Therefore they are often described by position- specific score matrices (PSSMs), which are used to score candidate TFBSs for their similarities to known binding sites. The similarity scores generated by PSSMs are essential to the computational prediction of single TFBSs or regulatory modules. We develop the Local Markov Method (LMM), which provides local p-values as a more reliable and rigorous alternative. Applying LMM to large-scale known human binding site sequences in situ, we show that compared to current popular methods, LMM can reduce false positive errors by more than 50% without compromising sensitivity.

After a quick introduction to the spectral theory of groups and graphswe take a more careful look at the so called lamplighter group Land show that the discrete Laplace operator on the Cayley graph of L(with respect to a certain generating set) is pure point spectrum andthe spectral measure is discrete (and explicitly computed). This is thefirst example of a group with discrete spectral measure.We take an unusual point of view and realize the lamplighter group L asa group generated by a 2-state automaton. This approach, along with someC* arguments, provides a crucial tool in our computations.The above result is applied to answer a question of Michael Atyiah onthe possible range of $L^2$ Betti numbers. Namely, we construct a 7dimensional closed manifold whose third $L^2$ Betti number is not aninteger (it is 1/3). The manifold also provides a counterexample to theso called strong Atiyah Conjecture concerning a relation between therange of $L^2$ Betti numbers and the orders of the finite subgroups of thefundamental group of the manifold.

Symmetric polynomials are the invariants of the classical action of the symmetric group Sn on the space Q[Xn] of polynomials by permutation of the variables. It is well known that the dimension of the quotient of Q[Xn] by the ideal generated by symmetric, constant-free polynomials is n!. When we consider other actions or other groups, we have different spaces of invariants, and different quotients (coinvariants). We will discuss some examples, in particular quasi-symmetrizing actions, whose coinvariants have dimensions given by the Catalan numbers. We shall give explicit Grobner bases for the ideals generated by the invariants .

The Nonlinear Schrodinger equation, is an example of a dispersive wave equation which has many different asymptotic states depending on the initial data. Such time dependent equations play a central role in many of the latestscientific advances,such as Bose-Einstein condensates and optical devices .I will discuss the solutions of such equations,including the large time behavior:first rigorous proof of the phenomena of ground state selection, asymptotic instability of the excited states and more.These results are obtained by deriving a novel Nonlinear Master equation and multitime scale analysis of its properties.The talk will be general.

We will discuss the properties of a 2-generator 2-relator group whichshares many properties with the famous Richard Thompson group F. Thegroup arises as an ascending HNN extension of a certain group which isgenerated by a 3-state automaton. The latter group will be in the focusof our considerations. In particular we will explain that it is theiterated monodromy group of the map z --> $z^2-1$.

We will define the analog of the Ihara zeta function for a Cayley graphof a finitely generated group. This will also be done for an infiniteregular graph which is a limit of a sequence of finite graphs.Interesting examples of computations based on spectral theory of fractalgroups will be considered. Some aspects of the computations are relatedto dynamical systems.

We will discuss flows of fluids for which the flow velocity is much lessthan the sound speed. For such flows we isolate that aspect of the flowwhich creates sound.As an application we study flow which is nearly steady-state and show howto compute the amplitude and frequencies of the sound that it generates.Using this computation we show how to construct a flow meter which canmeasure the rate of flow of a fluid through a given vessel.