Calibrations of Degree Two and Regularity Issues

Constante Bellettini
Princeton University; Member, School of Mathematics
April 9, 2013

Calibrated currents naturally appear when dealing with several geometric questions, some aspects of which require a deep understanding of regularity properties of calibrated currents. We will review some of these issues, then focusing on the two-dimensional case where we will show a surprising connection with pseudo-holomorphic curves and an infinitesimal regularity result, namely the uniqueness of tangent cones

Solvability in Polynomials of Pell Equations in a Pencil and a Conjecture of Pink

Umberto Zannier
Scuola Normale Superiore de Pisa, Italy
April 10, 2013

The classical Pell equation $X^2-DY^2=1$, to be solved in integers $X,Y\neq 0$, has a variant for function fields (studied already by Abel), where now $D=D(t)$ is a complex polynomial of even degree and we seek solutions in nonzero complex polynomials $X(t),Y(t)$. In this context solvability is no longer ensured by simple conditions on $D$ and may be considered `exceptional'.

Construction of the Kuranishi Structure on the Moduli Space of Pseudo-Holomorphic Curves

Kenji Fukaya
Simons Center for Geometry and Physics
April 12, 2013

To apply the technique of virtual fundamental cycle (chain) in the study of pseudo-holomorphic curve, we need to construct certain structure, which we call Kuranishi strucuture, on its moduli space. In this talk I want to review certain points of its construction.

Analytical Approach to Parallel Repetition

Irit Dinur
Weizmann Institute; Radcliffe institute
April 15, 2013

We propose an “analytical” framework for studying parallel repetitions of one-round two-prover games. We define a new relaxation of the value of a game, val+, and prove that it is both multiplicative and a good approximation for the true value of the game. These two properties imply Raz's parallel repetition theorem as
$val(G^k) ~ val+(G^k) = val+(G)^k ~ val(G)^k$
Using this approach, we will describe a reasonably simple proof for the NP-hardness for $label-cover(1,delta)$, the starting point of many inapproximability results.

Uncertainty Principle

Klim Efremenko
Tel-Aviv University; Member, School of Mathematics
April 23, 2013

Informally, uncertainty principle says that function and its Fourier transform can not be both concentrated. Uncertainty principle has a lot of applications in areas like compressed sensing, error correcting codes, number theory and many others. In this talk we will try to survey different formulations of uncertainty principle. In this talk we will be mostly focused on the discreet analog of uncertainty principle.