School of Mathematics
I will discuss a middle-dimensional generalization of Gromov's Non-Squeezing Theorem.
One of the oldest subjects in mathematics is the study of Diophantine equations, i.e., the study of whole number (or fractional) solutions to polynomial equations. It remains one of the most active areas of mathematics today. Perhaps the most basic tool is the simple idea of “congruences,” particularly congruences modulo a prime number. In this talk, Richard Taylor, Professor in the School of Mathematics, introduces prime numbers and congruences and illustrates their connection to Diophantine equations. He also describes recent progress in this area, an application, and reciprocity laws, which lie at the heart of much recent progress on Diophantine equations, including Wiles’s proof of Fermat’s last theorem.
The Resolution proof system is among the most basic and popular for proving propositional tautologies, and underlies many of the automated theorem proving systems in use today. I'll start by defining the Resolution system, and its place in the proof-complexity picture.
We present a Hamiltonian framework for higher-dimensional vortex filaments (or membranes) and vortex sheets as singular 2-forms with support of codimensions 2 and 1, respectively, i.e. singular elements of the dual to the Lie algebra of divergence-free vector fields. It turns out that the localized induction approximation (LIA) of the hydrodynamical Euler equation describes the skew-mean-curvature flow on higher vortex filaments of codimension 2 in any any dimension, which generalizes the classical binormal, or vortex filament, equation in 3D.
There are indications that in the 80s Guillemin posed a question: If billiard maps are conjugate, can we say that domains are the same up to isometry?
On one side, we show that conjugacy of different domains can't be C^1 near the boundary. In particular, billiard maps of the circle and an ellipse are both analytically integrable, but not C^1 conjugate. On the other side, if conjugate near the boundary s smoother, then domains are the same up to isometry.
(This is joint work with A. Sorrentino.)
The P vs. NP problem has sometimes been unofficially paraphrased as asking whether
it is possible to improve on exhaustive search for such problems as Satisfiability, Clique,
Graph Coloring, etc. However, known algorithms for each of these problems indeed are
substantially better than exhaustive search, if still exponential. Furthermore, although a
polynomial-time algorithm for any one of these problems implies one for all of them, these
improved exponential algorithms are highly specific, and it is unclear what the limit of
improvement should be.
This talk will review some theorems and conjectures about phase transitions of interacting spin systems in statistical mechanics. A phase transition may be thought of as a change in a typical spin configuration from ordered state at low temperature to disordered state at high temperature. I will illustrate how the symmetry of a spin system plays a crucial role in its qualitative behavior. Of particular interest is the connection between supersymmetric statistical mechanics and the spectral theory of random band matrices.
Legendrian contact homology (LCH) is a powerful holomorphic curves invariant for Legendrian submanifolds in contact manifolds. It is defined via a differential graded algebra (DGA), but the computation of its homology is often too difficult. Augmentations of this DGA are then used to obtain a linearized complex; its homology is called linearized LCH. In this talk, we show how pairs of augmentations can be used to define a bilinearized version of LCH, which is a refinement of the linearized LCH.