Veranstaltungen

The amplituhedron is a geometric object discovered by Arkani-Hamed and Tranka (2013) in their study of planar N=4 Super Yang Mills scattering amplitudes. In my talk I will describe this object, its properties, and the recent resolutions of two major conjectures in the field: the BCFW triangulation conjecture and the cluster adjacency conjecture for BCFW tiles. Based on joint works with C. Even-Zohar, T. Lakrec, M. Parisi, M. Sherman-Bennett and L. Williams.

How to use group theory to make a neural network understand that a cat is a cat, no matter of its orientation.

More information: https://zucmap.ethz.ch/call_made

''Suppose that we are given a polynomial which can be expressed as a sum of squares of polynomials with real coefficients, for example f(X,Y,Z) = (X-4Y)^2 + (17Z^3 - 4XYZ)^2. Then it is clear that no matter what (real) inputs we put in, we will get something positive. Minkowski, in his PhD defence, asked about the converse - suppose we have a polynomial that takes only positive values, must this be because it is a sum of squares? Hilbert, who was sat in the audience, realised that the answer is no; but modified the question slightly, and famously included it as the 17th in a list of open problems presented at the 1900 ICM in Paris. The main part of this talk will be devoted to the solution to this problem, relying on some algebraic foundations of Artin, and a nice model-theoretic proof due to Robinson. Time permitting we will look at modern day generalisations of this question, partial solutions, and (depending on audience interest) their applications to the Connes Embedding Problem, Kazhdan's Property (T), norm computability, and more. No knowledge of any of these topics will be assumed.

We consider Anosov diffeomorphisms of the 3-torus $\mathbb{T}^3$ which admit a partially hyperbolic splitting $\mathbb{T}^3 = E^s \oplus E^c \oplus E^u$ whose central direction $E^c$ is uniformly expanded. We may consider the 2-dimensional unstable foliation $W^{cu}$ tangent to $E^c \oplus E^u$, but also the 1-dimensional strong unstable foliation $W^u$ tangent to $E^u$. The behavior of $W^{cu}$ is reasonably well understood; in particular, such systems have a unique invariant measure whose disintegrations along the leaves of $W^{cu}$ are absolutely continuous: the SRB measure. The behavior of $W^u$ is less understood; we can similarly consider the class of measures whose disintegrations along the leaves of $W^u$ are absolutely continuous, the so-called u-Gibbs measures. It is well-known that the SRB measure is u-Gibbs; conversely, in a joint work with S. Alvarez, D. Obata and B. Santiago, we show that in a neighborhood of conservative systems, if the strong bundles $E^s$ and $E^u$ are not jointly integrable, then there exists a unique u-Gibbs measure, which is the SRB measure.

We define the g=0 open FJRW theory for (W,G) where W is a Fermat polynomial and G is its maximal symmetry group. We calculate all disk invariants, and classify the wall crossing group. We prove mirror symmetry with Saito's B-model.Based on joint works with Mark Gross and Tyler Kelly.

In a joint work with Olga Bernardi and Martin Leguil, we study the dynamics of dissipative convex billiards. In these billiards, the usual elastic reflection law is replaced with a new law where the angle bends towards the normal after each collision. For such billiard dynamics there exists a global attractor; we are interested in the topological and dynamical complexity of an invariant subset of this attractor, the so-called Birkhoff attractor, whose study goes back to Birkhoff, Charpentier, and, more recently, Le Calvez. We show that for a generic convex table, on one hand, the Birkhoff attractor is simple, i.e., a normally contracted submanifold, when the dissipation is strong; while, on the other hand, the Birkhoff attractor is topologically complicated and presents a rich dynamics when the dissipation is mild.

The Gordian distance u(K,K') between two knots K and K' is defined as the minimal number of crossing changes needed to relate K and K'. The unknotting number of a knot K, a classical yet hard to compute knot invariant, arises as the Gordian distance between K and the trivial knot. Several lower bounds for both invariants exist. A well-known bound for the unknotting number is given by the Rasmussen invariant, which is extracted from Khovanov homology, a bigraded chain complex associated to a knot up to chain homotopy equivalence. In this talk, I will introduce a new lower bound for the Gordian distance, \lambda, coming from Khovanov homology. After introducing all the relevant ingredients I will present some results about \lambda. In particular, \lambda turns out to be sharper than the Rasmussen invariant as a lower bound for the unknotting number. This is based on joint work with Lukas Lewark and Claudius Zibrowius.

We will consider Bayesian nonparametric settings with functional unknowns and we will be interested in evaluating the asymptotic performance of the posterior in the infinitely informative data limit, in terms of rates of contraction. We will be especially interested in priors which are adaptive to the smoothness of the unknown function. In the last decade, certain hierarchical and empirical Bayes procedures based on Gaussian process priors, have been shown to achieve adaptation to spatially homogenous smoothness. However, we have recently shown that Gaussian priors are suboptimal for spatially inhomogeneous unknowns, that is, functions which are smooth in some areas and rough or even discontinuous in other areas of their domain. In contrast, we have shown that (similar) hierarchical and empirical Bayes procedures based on Laplace (series) priors, achieve adaptation to both homogeneously and inhomogeneously smooth functions. All of these procedures involve the tuning of a hyperparameter of the Gaussian or Laplace prior. After reviewing the above results, we will present a new strategy for adaptation to smoothness based on heavy-tailed priors. We will illustrate it in a variety of nonparametric settings, showing in particular that adaptive rates of contraction in the minimax sense (up to logarithmic factors) are achieved without tuning of any hyperparameters and for both homogeneously and inhomogeneously smooth unknowns. We will also present numerical simulations corroborating the theory. This is joint work with Masoumeh Dashti, Tapio Helin, Aimilia Savva and Sven Wang (Laplace priors) and Ismaël Castillo (heavy-tailed priors)

In the 70's Itô settled the excursion theory of Markov processes, which is nowadays a fundamental tool for analyzing path properties of Markov processes. In his theory, Itô also introduced a method for building Markov processes using the excursion data, or by gluing excursions together, the resulting process is known as the recurrent extension of a given process. Since Itô's pioneering work the method of recurrent extensions has been added to the toolbox for building processes, which of course includes the martingale problem and stochastic differential equations. The latter are among the most popular tools for building and describing stochastic processes, in particular in applied models as they allow to physically describe the infinitesimal variations of the studied phenomena. In this work we answer the following natural question. Assume X is a Markov process taking values in R that dies at the first time it hits a distinguished point of the state space, say 0, which happens in a finite time a.s., that X satisfies a stochastic differential equation, and finally that X admits a recurrent extension, say Z, is a processes that behaves like Z up to the first hitting time of 0, and for which 0 is a recurrent and regular state. If any, what is the SDE satisfied by Z? Our answer to this question allows us to describe the SDE satisfied by many Feller processes. We analyze various particular examples, as for instance the so-called Feller brownian motions and diffusions, which include their sticky and skewed versions, and also continuous state branching processes and spectrally positive Levy processes.

The seminar aims to discuss the following question: given a Lagrangian homotopy 2-class $\alpha$ in a Kähler-Einstein manifold, can we identify a distinguished representative of $\alpha$ by minimizing the area among Lagrangian surfaces in $\alpha$? We will introduce the concepts of Lagrangian surface, Lagrangian angle and Hamiltonian variations and present some of their properties. Subsequently we will address some existence and regularity aspects of the question above, drawing connections to the theories of minimal surfaces and harmonic maps.

I will review the mathematical analysis behind two classes of new exciting materials: Twisted bilayer graphene (TBG) and twisted semiconductors (TMDs) with an emphasis on their mathematical properties and differences. If time permits I will discuss results on these models under disorder. Joint work with Maciej Zworski Izak Oltman Martin Vogel and Mengxuan Yang

We regard the optimal reinsurance problem as an iterated optimal transport problem between a (known) initial distribution and an (unknown) resulting risk exposure of the insurer. We also provide conditions that allow to characterize the support of optimal treaties, and show how this can be used to deduce the shape of the optimal contract, reducing the task to a finite-dimensional optimization problem, for which standard techniques can be applied. The proposed approach provides a general framework that encompasses many reinsurance problems, which we illustrate in several concrete examples, providing alternative proofs of classical optimal reinsurance results as well as establishing new optimality results, some of which contain optimal treaties that involve external randomness. Finally, we explain how in the current framework one can approach the problem of moral hazard in reinsurance and provide characterizations that avoid it.

Del Pezzo surfaces are classified by their degree d, and integer between 1and 9. The lower the degree, the more arithmetically complex these surfaces are. It is generally believed that, if a del Pezzo surface has one rational point, then it has many, and that they are well-distributed. After giving an overview of different notions of ‘many’ rational points and what is known so far for del Pezzo surfaces, I will focus on joint work with Julian Demeio and Sam Streeter where we prove weak weak approximation for del Pezzo surfaces of degree 2 with a general point.

We discuss techniques to calculate symplectic invariants on CY 3-folds M, namely Gromov-Witten (GW) invariants, Pandharipande-Thomas (PT) invariants, and Donaldson-Thomas (DT) invariants. Physically the latter are closely related to BPS brane bound states in type IIA string compactifications on M. We focus on the rank r_6=1 DT invariants that count D6-D2-D0 brane bound states related to PT- and high genus GW invariants, which are calculable by mirror symmetry and topological string B-model methods modulo certain boundary conditions, and the rank zero DT invariants that count rank r_4=1 D4-D2-D0 brane bound states. It has been conjectured by Maldacena, Strominger, Witten and Yin that the latter are governed by an index that has modularity properties to due S-duality in string theory and extends to a mock modularity index of higher depth for r_4>1. Again the modularity allows to fix the at least the r_4=1 index up to boundary conditions fixing their polar terms. Using Wall crossing formulas obtained by Feyzbakhsh certain PT invariants close to the Castelnouvo bound can be related to the r_4=1,2 D4-D2-D0 invariants. This provides further boundary conditions for topological string B-model approach as well as for the D4-D2-D0 brane indices. The approach allows to prove the Castenouvo bound and calculate the r_6=1 DT- invariants or the GW invariants to higher genus than hitherto possible.