Classification results for two-dimensional Lagrangian tori

Georgios Dimitroglou-Rizell
University of Cambridge
April 7, 2016
We present several classification results for Lagrangian tori, all proven using the splitting construction from symplectic field theory. Notably, we classify Lagrangian tori in the symplectic vector space up to Hamiltonian isotopy; they are either product tori or rescalings of the Chekanov torus. The proof uses the following results established in a recent joint work with E. Goodman and A. Ivrii. First, there is a unique torus up to Lagrangian isotopy inside the symplectic vector space, the projective plane, as well as the monotone $S^2 \times S^2$.

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.

Categorically Not! at IAS - On The Edge

April 14, 2016
The Institute for Advanced Study and Nautilus are pleased to present a special edition of Categorically Not!, the conversation series exploring the common ground of art, science and politics, April 14 at Wolfensohn Hall on the Institute campus (1 Einstein Drive, Princeton, NJ 08540). Moderated by writer, Categorically Not! creator and current IAS Director’s Visitor K.C. Cole, the event will feature a diversity of thinkers discussing the limits and horizons of science, art and culture, including biologist Sean B.

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 $E(2,3)$.

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.

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.