Veranstaltungen

A central difficulty in the study of the intersection theory of the moduli space of curves is that there may exist tautological classes that are nonzero but intersect trivially with all tautological classes of complementary codimension. I will explain some recent progress on studying such classes by intersecting with classes outside of the tautological ring.

We study the problem of estimating the means of well-separated mixtures when an adversary may add arbitrary outliers. While strong guarantees are available when the outlier fraction is significantly smaller than the minimum mixing weight, much less is known when outliers may crowd out low-weight clusters - a setting we refer to as list-decodable mixture learning (LD-ML). In this case, adversarial outliers can simulate additional spurious mixture components. Hence, if all means of the mixture must be recovered up to a small error in the output list, the list size needs to be larger than the number of (true) components. We propose an algorithm that obtains order-optimal error guarantees for each mixture mean with a minimal list-size overhead, significantly improving upon list-decodable mean estimation, the only existing method that is applicable for LD-ML. Although improvements are observed even when the mixture is non-separated, our algorithm achieves particularly strong guarantees when the mixture is separated: it can leverage the mixture structure to partially cluster the samples before carefully iterating a base learner for list-decodable mean estimation at different scales.

Given a compact hyperbolic surface together with a suitable choice of orthonormal basis of Laplace eigenforms, one can consider two natural spectral invariants: 1) the Laplace spectrum Lambda, and 2) the 3-tensor C_{ijk} representing pointwise multiplication (as a densely defined map L^2 x L^2 -> L^2) in the given basis. Which pairs (Lambda,C) arise this way? Both Lambda and C are highly transcendental objects. Nevertheless, we will give a concrete and almost completely algebraic answer to this question, by writing down necessary and sufficient conditions in the form of equations satisfied by the Laplace eigenvalues and the C_{ijk}. This answer was conjectured by physicists, who introduced these equations (in an equivalent form) as a rigorous model for the crossing equations in conformal field theory.

Let Y be an algebraic curve over a number field K of genus at least 1. The reduction behavior of Y to characteristic p>0 can be used to compute arithmetic invariants of the curve. In this talk we will introduce notions of good, bad, and semistable reduction and discuss their connection to the L-series and the conductor of the curve. In the special case of hyperelliptic curves and p = 2, we will study the minimal extension over which the curve attains semistable reduction and the automorphism groups of the special fiber.

Seminar lecture I Seminar goals: review the moduli of abelian varieties and how they degenerate, understand the construction of its toroidal compactifications, using inputs from tropical and logarithmic geometry, study different topics that arise naturally:intersection theory on these moduli spaces, the compactification interesting loci from, logarithmic abelian varieties and some arithmetic side of the whole story. See https://people.math.ethz.ch/~airibar/reading_seminar

Let Y be a curve defined over a p-adic field. The local Galois representation of Y is determined by a Weil-Deligne representation. We explain how to compute this representation from the stable reduction of Y to characteristic p>0. This is joint works with D.K. Do, R. Nowak, S. Wewers, and X. Zhang.

Enumerative geometry counts geometric objects satisfying a list of properties. An important method in the area is to obtain these counts as intersection numbers on a suitable moduli space Mbar. In the talk I explain how logarithmic intersection theory can be used in different examples to define intersection numbers which are independent of the precise choice of this space Mbar. After a discussion of (double) Hurwitz numbers, we'll also see a new class of invariants - called k-leaky double Hurwitz descendants - defined in joint work with Cavalieri and Markwig. I discuss some of their properties, show tropical formulas calculating them and present some questions on their enumerative interpretation.