School of Mathematics

Reed-Muller codes for random erasures and errors

Amir Shpilka
Tel Aviv University
April 26, 2016
Reed-Muller codes encode an $m$-variate polynomial of degree $r$ by evaluating it on all points in $\{0,1\}^m$. Its distance is $2^{m-r}$ and so it cannot correct more than that many errors/erasures in the worst case. For random errors one may hope for a better result. In his seminal paper Shannon exactly determined the amount of errors and erasures one can hope to correct for codes of a given rate. Codes that achieve Shannon's bound are called capacity achieving codes.

Exponential convergence to the Maxwell distribution of solutions of spatially inhomogenous Boltzmann equations

Gang Zhou
California Institute of Technology
April 25, 2016
In this talk I will present a recent proof of a conjecture of C. Villani, namely the exponential convergence of solutions of spatially inhomogenous Boltzmann equations, with hard sphere potentials, to some equilibriums, called Maxwellians.

A Heegaard Floer analog of algebraic torsion

Cagatay Kutluhan
University at Buffalo, The State University of New York; von Neumann Fellow, School of Mathematics
April 21, 2016
The dichotomy between overtwisted and tight contact structures has been central to the classification of contact structures in dimension 3. Ozsvath-Szabo's contact invariant in Heegaard Floer homology proved to be an efficient tool to distinguish tight contact structures from overtwisted ones. In this talk, I will motivate, define, and discuss some properties of a refinement of the contact invariant in Heegaard Floer homology. This is joint work with Grodana Matic, Jeremy Van Horn-Morris, and Andy Wand.

Standard conjecture of Künneth type with torsion coefficients

Junecue Suh
University of California, Santa Cruz
April 21, 2016
A. Venkatesh asked us the question, in the context of torsion automorphic forms: Does the Standard Conjecture (of Grothendieck's) of Künneth type hold with mod p coefficients? We first review the geometric and number-theoretic contexts in which this question becomes interesting, and provide answers: No in general (even for Shimura varieties) but yes in special cases.

Meridional essential surfaces of unbounded Euler characteristics in knot complements

João Nogueira
University of Coimbra
April 20, 2016
In this talk we will discuss further the existence of knot complements with essential surfaces of unbounded Euler characteristics. More precisely, we show the existence of a knot with an essential tangle decomposition for any number of strings. We also show the existence of knots where each complement contains meridional essential surfaces of simultaneously unbounded genus and number of boundary components. In particular, we construct examples of knot complements each of which having all possible compact surfaces embedded as meridional essential surfaces.

A characterization of functions with vanishing averages over products of disjoint sets

Pooya Hatami
Member, School of Mathematics
April 19, 2016
We characterize all complex-valued (Lebesgue) integrable functions $f$ on $[0,1]^m$ such that $f$ vanishes when integrated over the product of $m$ measurable sets which partition $[0,1]$ and have prescribed Lebesgue measures $\alpha_1,\ldots,\alpha_m$. We characterize the Walsh expansion of such functions $f$ via a first variation argument. Janson and Sos asked this analytic question motivated by questions regarding quasi-randomness of graph sequences in the dense model. We use this characterization to answer a few conjectures from [S. Janson and V.

Spectral gaps via additive combinatorics

Semyon Dyatlov
Massachusetts Institute of Technology
April 19, 2016
A spectral gap on a noncompact Riemannian manifold is an asymptotic strip free of resonances (poles of the meromorphic continuation of the resolvent of the Laplacian). The existence of such gap implies exponential decay of linear waves, modulo a finite dimensional space; in a related case of Pollicott--Ruelle resonances, a spectral gap gives an exponential remainder in the prime geodesic theorem. We study spectralgaps in the classical setting of convex co-compact hyperbolic surfaces, where the trapped trajectories form a fractal set of dimension $2\delta + 1$.

On the number of nodal domains of toral eigenfunctions

Igor Wigman
King's College, London
April 19, 2016
We study the number of nodal domains of toral Laplace eigenfunctions. Following Nazarov-Sodin's results for random fields and Bourgain's de-randomisation procedure we establish a precise asymptotic result for "generic" eigenfunctions. Our main results in particular imply an optimal lower bound for the number of nodal domains of generic toral eigenfunctions. This work is joint with Jerry Buckley.

Symplectic embeddings and infinite staircases

Ana Rita Pires
Fordham University
April 15, 2016
McDuff and Schlenk studied an embedding capacity function, which describes when a 4-dimensional ellipsoid can symplectically embed into a 4-ball. The graph of this function includes an infinite staircase determined by the odd index Fibonacci numbers. Infinite staircases have also been shown to exist in the graphs of the embedding capacity functions when the target manifold is a polydisk or the ellipsoid $E(2,3)$.

Veering Dehn surgery

Saul Schleimer
University of Warwick
April 12, 2016
(Joint with Henry Segerman.) It is a theorem of Moise that every three-manifold admits a triangulation, and thus infinitely many. Thus, it can be difficult to learn anything really interesting about the three-manifold from any given triangulation. Thurston introduced ``ideal triangulations'' for studying manifolds with torus boundary; Lackenby introduced ``taut ideal triangulations'' for studying the Thurston norm ball; Agol introduced ``veering triangulations'' for studying punctured surface bundles over the circle.