School of Mathematics
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.
We study some problems relating to polynomials evaluated either at random Gaussian or random Bernoulli inputs. We present some new work on a structure theorem for degree-d polynomials with Gaussian inputs. In particular, if p is a given degree-d polynomial, then p can be written in terms of some bounded number of other polynomials q_1,...,q_m so that the joint probability density function of q_1(G),...,q_m(G) is close to being bounded. This says essentially that any abnormalities in the distribution of p(G) can be explained by the way in which p decomposes into the q_i .
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.
We present some novel approaches to the instability problem of Hamiltonian systems (in particular, the Arnold Diffusion problem). We show that, under generic conditions, perturbations of geodesic flows by recurrent dynamics yield trajectories whose energy grows to infinity in time (at a linear rate, which is optimal). We also show that small, generic perturbations of integrable Hamiltonian systems yield trajectories that travel large distances in the phase space. The systems that we consider are very general.
In this general survey talk, we will describe an approach to doing homotopy theory within Univalent Foundations. Whereas classical homotopy theory may be described as "analytic", our approach is synthetic in the sense that, in ``homotopy type theory", homotopical concepts such as points, paths, and homotopies are basic notions.
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'.
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
We all know Shannon's entropy of a discrete probability distribution. Physicists define entropy in thermodynamics and in statistical mechanics (there are several competing schools), and want to prove the Second Law, but they didn't succeed yet (very roughly speaking, the Second Law claims that the entropy always increases). What I do is motivated by physics, but I ask a new, strictly combinatorial/geometric question. Assume that we have a large finite set of points in the unit square.
he Weil height measures the “complexity” of an algebraic number. It vanishes precisely at 0 and at the roots of unity. Moreover, a finite field extension of the rationals contains no elements of arbitrarily small, positive heights. Amoroso, Bombieri, David, Dvornicich, Schinzel, Zannier and others exhibited many infinite field extensions of the rationals with a height gap. For example, the maximal extension of any number field with abelian Galois group has this property. I will present the history of this problem and talk about a new non-abelian example and its application.