Seminars & Events for 2008-2009
Feynman's path integral formulation of equilibrium statistical mechanics allows us to include nuclear quantum effects in computer simulations. This is particularly important when dealing with light particles like the protons participating in hydrogen bonds.
Counting Faces of Randomly-Projected Polytopes, with applications to Compressed Sensing, Error-Correcting Codes, and Statistical Data Mining
Grushko's Theorem states that $rank(G*H)=rank(G)+rank(H)$, where $rank$ is the minimum number of generators for a group, and $*$ denotes the free product. We will present Stallings' (topological) proof of Grushko's Theorem.
Hadwiger's conjecture states that if a graph is not $t$-colorable then it contains the complete graph on $t+1$ vertices as a minor. The case $t=4$ is equivalent to the four color theorem and the case $t=5$ was proved by Robertson, Seymour, and Thomas with the use of the four color theorem. For $t>5$, the conjecture remains open.
We will discuss the problem of effectiveness of Iitaka fibrations for surfaces and threefolds.
This talk will discuss recent advances in regards to some of the main open problems about hyperbolic 3-manifolds in the context of arithmetic hyperbolic 3-manifolds.
Let $X,Y$ be measurable spaces and $\eta : X \to Y$ be a measurable function. Under what conditions on $\eta$ is the composition with $f : Y \to C$ a well defined operation when $f$ is only specified almost everywhere?
We show that if S is a finite type hyperbolic surface which is not the 3 or 4-holed sphere or 1-holed torus, then the Ending lamination space of S is connected, locally path connected and cyclic. Using Klarrich's theorem this implies that the boundary of a curve complex associated to any such space is connected, locally path connected and cyclic.
In a first part, we shall prove a quantitative nonvanishing result conjectured by Ph.Michel and A.Venkatesh which concerns the special derivatives occuring in the Gross-Zagier formula. Our method relies on two classical equidistribution Theorems in arithmetic geometry. In a second part, we shall explain how to refine these results with methods from analytic number theory.
I will discuss the construction of a new example with positive sectional curvature on a 7-dimensional manifold homeomorphic to the unit tangent bundle of the 4-sphere. The metric is of Kaluza Klein type on an orbifold principle bundle over the 4-sphere and is closely related to the geometry of self dual Einstein and 3- Sasakian metrics.
In many applications, the main goal is to obtain a global low dimensional representation of the data, given some local noisy geometric constraints. In this talk we will show how the problems listed below can be efficiently solved by constructing suitable operators on their data and computing a few eigenvectors of sparse matrices corresponding to the data operators.
We prove well-posedness for compressible flow with free-boundary in physical vacuum, modeled by the 3D compressible Euler equations. The vanishing of the density at the vacuum boundary induces degenerate hyperbolic equations that become characteristic, requiring a separate analysis of time, normal, and tangential derivatives to handle the manifest 1/2-derivative loss.
I will discuss a set-theoretic analog of the classical Torelli theorem for curves.
The first half of the talk will be devoted to probabilistic models of "spatial" random permutations, that involve points in $R^d$. Permutations are weighed according to the length of the jumps. The main question deals with the occurrence of infinite cycles. The second part of the talk will be devoted to the quantum Bose gas in the path-integral representation.
For a long time, a famous open problem was to figure out whether a number was prime quickly (in polynomial time). It's interesting to see how under the generalized riemann hypothesis, the problem becomes completely straightforward. I will introduce the relevant concept and present the simple proof. During the second half, I will present a provably polynomial time test, without reliance on GRH.
We extend to the setting of Lie groupoids the notion of the cardinality of a finite groupoid (a rational number, equal to the Euler characteristic of the correspondingdiscrete orbifold). Since this quantity is an invariant under equivalence of groupoids, we call it the volume of the associatedstack rather than of the groupoid itself.
This talk will start with a brief status report on magnetic fusion energy research. One of the key challenges in fusion has been the occurrence of fine-scale turbulent fluctuations, which cause plasma to leak out of a magnetic trap, so we would like to be able to predict and reduce this turbulence.
We discuss multi-parameter Carnot-Carathéadory balls. In particular, we discuss questions motivated by multi-parameter singular integrals. These results generalize results due to Nagel, Stein, and Wainger in the single parameter setting.