An Optimal Lower Bound for File Maintenance

Michael Saks
Rutgers, The State University of New Jersey
January 23, 2012
In the file maintenance problem, n integer items from the set {1,....,r} are to be stored in an array of size m>=n . The items are presented sequentially in an arbitrary order and must be stored in the array in sorted order (but not necessarily in consecutive locations in the array). Each new item is stored before the next arrives. If rm then we may have to shift the location of stored items IN

Members Seminar: The Role of Symmetry in Phase Transitions

Tom Spencer
Professor, School of Mathematics, Institute for Advanced Study
January 23, 2012

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.

CSDM: A Tutorial on the Likely Worst-Case Complexities of NP-Complete Problems

Russell Impagliazzo
Institute for Advanced Study
January 24, 2012
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.

Symplectic Dynamics Seminar: On Conjugacy of Convex Billiards

Vadim Kaloshin
Pennsylvania State University; Member, School of Mathematics, Institute for Advanced Study
January 25, 2012
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.)

Symplectic Dynamics Seminar: Symplectic Structures and Dynamics on Vortex Membranes

Boris Khesin
University of Toronto; Member, School of Mathematics, Institute for Advanced Study
January 25, 2012
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.

CSDM: A Survey of Lower Bounds for the Resolution Proof System

Avi Wigderson
Herbert H. Maass Professor, School of Mathematics, Institute for Advanced Study
January 31, 2012
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.