Mon, 26 May 2008

10:00 - 11:00
L3

Computation in quotients of polynomial rings and enumerative geometry

Daniel Grayson
(UIUC)
Abstract
Abstract: I will describe how computations are done using "Groebner bases" in quotient rings of polynomial rings, and I will describe explicitly the form of a particular Groebner basis for the ideal defining the ring parametrizing all factorizations of a monic polynomial of degree a+b+...+e into monic factors of degree a,b,...,e. That can be and is used in practice to compute intersection numbers involving of algebraic cycles arising as Chern classes on flag bundles of vector bundles. Simplest example: how many lines in 3-space meet four fixed lines?
Mon, 09 Jun 2008

17:00 - 18:00
L3

Uniqueness of Lagrangian trajectories for weak solutions of the two- and three-dimensional Navier-Stokes equations

James Robinson
(Warwick)
Abstract

I will discuss recent results concerning the uniqueness of Lagrangian particle trajectories associated to weak solutions of the Navier-Stokes equations. In two dimensions, for which the weak solutions are unique, I will present a mcuh simpler argument than that of Chemin & Lerner that guarantees the uniqueness of these trajectories (this is joint work with Masoumeh Dashti, Warwick). In three dimensions, given a particular weak solution, Foias, Guillopé, & Temam showed that one can construct at leaset one trajectory mapping that respects the volume-preserving nature of the underlying flow. I will show that under the additional assumption that $u\in L^{6/5}(0,T;L^\infty)$ this trajectory mapping is in fact unique (joint work with Witek Sadowski, Warsaw).

Tue, 13 May 2008
14:30
L3

Killed Branching Random Walks

Louigi Addario-Berry
(Oxford)
Abstract
Joint work with Nicolas Broutin.

The problem is related to searching in trees.  Suppose we are given a complete binary tree (a rooted tree in which the root has degree 2 and every other vertex has degree 3) with independent, identically distributed random edge weights (say copies of some random variable X, which need not be non-negative). The depth d(v) of a vertex v is the number of edges on the path from v to the root. We give each vertex v the label S_v which is the sum of the edge weights on the path from v to the root. For positive integers n, we let M_n be the maximum label of any vertex at depth n, and let M^* = max {M_n: n =0,1,...}. It is of course possible that M^* is infinity.

Under suitable moment assumptions on X, it is known that there is a constant A such that M_n/n --> A almost surely and in expectation. We call the cases A>0, A=0, and A< 0 supercritical, critical, and subcritical, respectively. When A <= 0 it makes sense to try to find the vertex of maximum weight M* in the whole tree.  One possible strategy is to only explore the subtree T_0 containing the root consisting only of vertices of non-negative weight.  With probability bounded away from zero this strategy finds the vertex of maximum weight.  We derive precise information about the expected running time for this strategy. Equivalently, we derive precise information about the random variable |T_0|. In the process, we also derive rather precise information about M*. This answers a question of David Aldous.
Mon, 19 May 2008

12:00 - 13:00
L3

Generating Tree Amplitudes in N=4 SYM and N=8 SG

Dan Freedman
(Cambridge and MIT)
Abstract
Abstract: We study n-point tree amplitudes of N=4 super Yang-Mills theory and N=8 supergravity for general configurations of external particles of the two theories. We construct generating functions for n-point MHV and NMHV amplitudes with general external states. Amplitudes derived from them obey SUSY Ward identities, and the generating functions characterize and count amplitudes in the MHV and NMHV sectors. The MHV generating function provides an efficient way to perform the intermediate state helicity sums required to obtain loop amplitudes from trees. The NMHV generating functions rely on the MHV-vertex expansion obtained from recursion relations associated with a 3-line shift of external momenta involving a reference spinor |X]. The recursion relations remain valid for a subset of N=8 supergravity amplitudes although they do not vanish asymptotically for all |X]. The MHV-vertex expansion of the n-graviton NMHV amplitude for n=5,6,...,11 is independent of |X] and exhibits the asymptotic behavior z^{n-12}. This presages difficulties for n > 12. Generating functions show how the symmetries of supergravity can be implemented in the quadratic map between supergravity and gauge theory embodied in the KLT and other similar relations between amplitudes in the two theories.
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