Weekly Bulletin

Algebraic geometry endows the cohomology groups of moduli spaces of curves with additional structures, such as (mixed) Hodge structures and Galois representations. Standard conjectures from arithmetic, regarding analytic continuations of L-functions attached to these Galois representations, lead to striking predictions, by Chenevier and Lannes, about which such structures can appear. I will survey recent results unconditionally confirming several of these predictions and studying patterns in the appearances of motives of low weight. The latter are governed by the operadic structures induced by tautological morphisms and the cohomology of graph complexes. Based on joint work with Jonas Bergström and Carel Faber; with Sam Canning and Hannah Larson; with Melody Chan and Søren Galatius; and with Thomas Willwacher.

Is mathematics an abstract expression of intelligence of the human species, or does it contribute to culture? What does the historical and archaeological evidence show? The beauty of mathematics lies in its precision, its unwavering logic, and its ability to solve “real world” problems.

More information: https://collegium.ethz.ch/veranstaltungen/?event=12635&cat=upcomingcall_made

We explain how to invariantly count holomorphic curves of all genera with Lagrangian boundary conditions in Calabi-Yau 3-folds. In the process we discover a natural geometric occurence of the HOMFLYPT skein relations, and prove the Ooguri-Vafa conjecture relating the HOMFLYPT invariant of a knot to the count of curves ending on a certain associated Lagrangian (the knot conormal transplanted to the resolved conifold).

I will present the possible Gromov-Hausdorff limits of geodesically complete, CAT(0)-spaces admitting a discrete group of isometries of bounded codiameter and the structure of the possible limit groups. The focus will be on the collapsing case: namely when the injectivity radius of the quotient space goes to zero along the sequence. Joint work with A.Sambusetti.

The Teukolsky equation is one of the fundamental equations governing linear gravitational perturbations of the Kerr black hole family as solutions to the vacuum Einstein equations. We show that solutions arising from suitably regular initial data decay inverse polynomially in time. Our proof holds for the entire subextremal range of Kerr black hole parameters, \(\vert a\vert < M\). This is joint work with Yakov Shlapentokh-Rothman (Toronto).

Consider critical site Bernoulli percolation on the triangular lattice, where each vertex is colored black or white with probability 1/2, independently of the other vertices. In 1999, Benjamini, Kalai and Schramm proved that crossing probabilities are noise sensitive: resampling a small proportion of the vertices lead to an independent percolation picture. Ten years later, Garban, Pete and Schramm obtained a sharp quantitative version of this result. These works rely on Fourier analysis, and are restricted to Bernoulli percolation (i.e. product measure) and the independent resampling dynamics. In this talk, we will first introduce and discuss the general question of noise sensitivity, focusing mostly on percolation applications. Then we will present a recent and robust approach that relies on geometrical arguments and not on spectral methods. Based on joint works with Hugo Vanneuville.

Polyharmonic Maass forms are generalizations of classical elliptic modular forms that obey weaker differential equations than the Cauchy-Riemann equations, and thus form a richer class of functions which accommodate, for instance, some generating series of period integrals. Their differential properties are captured by cyclic Harish-Chandra modules, which are in ono-to-one correspondence with cyclic representations of the two-cyclic and the Gelfand quiver. We show that all latter representations arise from polyharmonic Maass forms, and provide a explicit construction, which has a natural interpretation in terms of tensor products of Harish-Chandra modules.

We show how, at least in some class of examples ("Reeb-positive"), the all-genus skein-valued curve count on a non-compact Lagrangian (e.g. a Harvey-Lawson brane in a toric CY3) is annihilated by an operator equation which is a skein-valued quantization of the mirror curve. As an application we prove the all-color version of the Ooguri-Vafa conjecture mentioned above.