Morse theoretic deformations for evaluating integrals.
Abstract: The goal is to evaluate Talyor coefficients of a rational function
via Cauchy’s integral
formula, these series arising as multivariate generating
functions. To do so, one would like to deform the chain of integration near a critical point for a log-linear height function
on the complex hypersurface . The chain must avoid the
surface , whence there is a topological obstruction at
the critical point, but it is important to push the rest of the
chain as far beyond this as possible (this makes the integrand
decay as rapidly as possible).
In this talk I will first give the combinatorial context in
which the problem arises. The attached pictures are examples
of combinatorial limit theorems resulting from such analyses.
Next, I will give the analytic reductions to a problem that
is essentially topological. Finally, I will discuss cases
where we know how to solve this problem. These rely on some
notions such as hyperbolicity and semi-continuity that have
been around for several decades, as well as on some new tricks
that work in low-dimensional cases.
Our second speaker is
of our very own Chemistry Department. Chris will be talking about:
Title: Toward Programmable Matter – Organic Molecules with Programmable Shapes and Functional Group Display
Over the past nine years, my laboratory has developed a radical new approach to creating large, complex molecules that can carry out complex functions in the way that biological proteins do. Our approach is to synthesize new stereochemically pure cyclic bis-amino acid building blocks (Buildomers) that we couple through pairs of amide bonds to create large molecules with programmed shapes. The oligomers are efficiently assembled on solid support using peptide synthesis techniques to first create a flexible oligomer that is then rigidified by the simultaneous formation of a second set of amide bonds between each adjacent pair of monomers. The structure of the resulting ladder-like molecules is controlled by the sequence and stereochemistry of the component monomers. The oligomer structures made accessible by this technology range from extended molecular rods to compact structures containing small-molecule sized cavities. These oligomers can be functionalized in a variety of ways to perform different functions. We are developing a software package that will enable anyone to construct oligomer sequences with designed shapes, through automated computer searches of sequence and conformational space. Our goal is to combine this software and molecular “hardware” to create a sort of “molecular compiler” that can take a high level specification of a function (eg: “catalyze a particular reaction” or “bind a target protein”) and convert it into a molecular implementation that will carry out that function.