Veranstaltungen

Few years ago, Morier-Genoud and Ovsienko introduced a new interesting “quantisation” of the real numbers generalising the classical q-version of integers due to Euler and Gauss. I will explain a link between the corresponding q-deformed rational numbers and the classical Burau representation of the braid group B_3. This will be applied to the open problem of classification of faithful complex specializations of this representation. The talk is based on a joint work with Sophie Morier-Genoud and Valentin Ovsienko.

I’ll discuss the connection between (sub)leading asymptotics of the Floer theoretic link spectral invariants and the classical Calabi and Ruelle invariants along with their connections to the algebraic structure of conservative homeomorphism groups. Based on joint joint works with Cristofaro-Gardiner, Humilière, Mak and Smith.

The idea of successively refining a discrete model to obtain a limit is ubiquitous in mathematics and its application to physics. In this talk, we will better understand what this means for stochastic processes, in particular for some random grid curves in the plane. We will discuss conformal invariance, a certain geometric Markov property and how these lead us to consider Schramm-Loewner-Evolution.

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

We develop the first pure node-differentially-private algorithms for learning stochastic block models and for graphon estimation with polynomial running time for any constant number of blocks. The statistical utility guarantees match those of the previous best information-theoretic (exponential-time) node-private mechanisms for these problems. The algorithm is based on an exponential mechanism for a score function defined in terms of a sum-of-squares relaxation whose level depends on the number of blocks. The key ingredients of our results are (1) a characterization of the distance between the block graphons in terms of a quadratic optimization over the polytope of doubly stochastic matrices, (2) a general sum-of-squares convergence result for polynomial optimization over arbitrary polytopes, and (3) a general approach to perform Lipschitz extensions of score functions as part of the sum-of-squares algorithmic paradigm.

For hyperbolic 3-manifolds we study two quantifications of the property of having vanishing first Betti number, one analytic and one topological: the spectral gap for the Laplacian on coclosed 1-forms and the size of the first torsion homology group. First we construct a sequence of closed hyperbolic integer homology spheres with volume tending to infinity and a uniform coclosed 1-form spectral gap, which answers a question asked by Lin-Lipnowski in connection with the existence of irreducible solutions to the Seiberg-Witten equations. Interestingly, the proof uses a topological property of the sequence in order to deduce a uniform gap. We'll then explore the connection between a sequence of 3-manifolds having such a uniform gap and exponential in volume torsion homology growth. In the context of towers of arithmetic 3-manifolds, such a connection was suggested by the work of Bergeron-Şengün-Venkatesh. We show that for any sequence of closed hyperbolic 3-manifolds with uniformly bounded rank, if it has a uniform coclosed 1-form spectral gap, then it must have exponential torsion homology growth. Based on joint work with Amina Abdurrahman, Anshul Adve, Ben Lowe, and Jonathan Zung.

As part of a mini-course hosted by the FIM, the speaker will give a complete proof of the Majorizing Measures Theorem (roughly 4h of lecture). In the occasion of the announcement of the awarding of the Abel prize to Michel Talagrand, the organisers suggested, and the speaker kindly agreed, to isolate this part of the course as a 4h event, where the proof (and some of the motivation) will be made accessible also to audience that is not attending the course. This will take place on 10.04.24 split in two two-hour sessions, one in the morning (10:15) and another in the afternoon (14:15), both at G43. All are welcome and we hope to see many of you there!

Let U be a non-conical strictly convex divisible set. Even though the boundary S of U is not C^2, Benoist showed that S is C^1+ and Crampon established that S has a sort of anisotropic Holder regularity -- described by a list L of real numbers -- at almost all of its points. In this talk, we discuss our joint work with P. Foulon and P. Hubert showing that S is maximally anisotropic in the sense that the list L contains no repetitions thanks to the features of the Hilbert flow.

As part of a mini-course hosted by the FIM, the speaker will give a complete proof of the Majorizing Measures Theorem (roughly 4h of lecture). In the occasion of the announcement of the awarding of the Abel prize to Michel Talagrand, the organisers suggested, and the speaker kindly agreed, to isolate this part of the course as a 4h event, where the proof (and some of the motivation) will be made accessible also to audience that is not attending the course. This will take place on 10.04.24 split in two two-hour sessions, one in the morning (10:15) and another in the afternoon (14:15), both at G43. All are welcome and we hope to see many of you there!

Finite Element Exterior Calculus (FEEC) provides a cohomology framework for structure-preserving discretisation of a large class of PDEs. Differential complexes are important tools in FEEC. The de Rham complex is a basic example, with applications in curl-div related problems such as the Maxwell equations. There is a canonical finite element discretisation of the de Rham complex, which in the lowest order case coincides with discrete differential forms (Whitney forms). Different problems involve different complexes. In this talk, we provide an overview of some efforts towards Finite Element Tensor Calculus, inspired by tensor-valued problems from continuum mechanics and general relativity. On the continuous level, we systematically derive new complexes from the de Rham complexes. On the discrete level, We review the idea of distributional finite elements, and use them to obtain analogies of the Whitney forms for these new complexes. A special case is Christiansen’s finite element interpretation of Regge calculus, a discrete geometric scheme for metric and curvature.

Bernoulli percolation of parameter p on Z^d is defined by deleting each edge of Z^d with probability 1-p, independently of the other edges. The exponential decay theorem - proven in the 80's by Menshikov and, independently, by Aizenman and Barsky - can be stated as follows: If the cardinality of the cluster of 0 is a.s. finite at some parameter p, then it has an exponential moment at every parameter q<p. I like to state this theorem this way because it illustrates the fact that decreasing p infinitesimally has a regularising effect on the percolation clusters. A new - shorter - proof has been proposed by Duminil-Copin and Tassion in 2016. The goal of this talk is to propose yet a new proof of this theorem, inspired by Russo's work from the early 80s, which relies on stochastic comparison techniques.

Ranking data are ubiquitous: we rank items as citizens, as consumers, as scientists, and we are collectively characterised, individually classified and recommended, based on estimates of our preferences. Preference data occur when we express comparative opinions about a set of items, by rating, ranking, pair comparing, liking, choosing or clicking, usually in an incomplete and possibly inconsistent way. The purpose of preference learning is to i) infer on the shared consensus preference of a group of users, or ii) estimate for each user their individual ranking of the items, when the user indicates only incomplete preferences; the latter is an important part of recommender systems. I present a Bayesian preference-learning framework based on the Mallows rank model with any right-invariant distance, to infer on the consensus ranking of a group of users, and to estimate the complete ranking of the items for each user. MCMC based inference is possible, by importance-sampling approximation of the normalising function, but mixing can be slow. We propose a Variational Bayes approach to performing posterior inference, based on a pseudo-marginal approximating distribution on the set of permutations of the items. The approach scales well and has useful theoretical properties. Partial rankings and non-transitive pair-comparisons are solved by Bayesian augmentation. The Bayes-Mallows approach produces well-calibrated uncertainty quantification of estimated preferences, which are useful for recommendation, leading to excellent accuracy and increased diversity, compared for example to matrix factorisation. Simulations and real-world applications help illustrate the method. This talk is based on joint work with Elja Arjas, Marta Crispino, Qinghua Liu, Ida Scheel, Øystein Sørensen, and Valeria Vitelli.

A century ago, Elie Cartan showed how to construct a group from a Riemannian manifold via parallel transport: the Riemannian holonomy group. He then used this group to study symmetric spaces. In 1953, Marcel Berger classified all the possible groups that can appear as the Riemannian holonomy of a Riemannian manifold. This result gives a list of interesting geometrical structures compatible with the Riemannian metric, including the classical Kähler geometry, Calabi--Yau geometry and Hyperkähler geometry. In this talk, I will provide an overview of Riemannian holonomy, including their connection with Einstein manifolds and minimal submanifolds. If time permits, I will discuss open problems and standard techniques used in the field.

We consider a system of N bosons confined in a unit box with periodic boundary conditions. We assume that the particles interact through a repulsive two-boxy potential with scattering length of order 1/N, i.e. the Gross-Pitaevskii regime. We establish a precise bound for the ground state energy E(N) of the system. While the leading contribution, of order N, to the energy has been known since the pioneering works of Lieb-Seiringer-Yngvason in the early 2000s, and the second order (of order one) corrections were more recently first determined by Boccato-Brennecke-Cenatiempo-Schlein, our estimate also resolves the next term in the asymptotic expansion of E(N), which is of the order (log N)/N, confirming Wu's predictions in 1959. Based on a joint work with Alessandro Olgiati, Diane Saint Aubin and Benjamin Schlein.

We investigate the pricing of single-asset autocallable barrier reverse convertibles in the Heston local-stochastic volatility (LSV) model. Despite their complexity, autocallable structured notes are the most traded equity-linked exotic derivatives. The autocallable payoff embeds an early redemption feature generating strong path- and model-dependency. Consequently, the commonly-used local volatility (LV) model is overly simplified for pricing and risk management. Given its ability to match the implied volatility smile and reproduce its realistic dynamics, the LSV model is, in contrast, better suited for exotic derivatives such as autocallables. We use quasi-Monte Carlo methods to study the pricing given the Heston LSV model and compare it with the LV model. In particular, we establish the sensitivity of the valuation differences of autocallables between the two models with respect to payoff features, model parameters, underlying characteristics, and volatility regimes. We find that the improved spot-volatility dynamics captured by the Heston LSV model typically result in higher prices, demonstrating the dependence of autocallables on the forward-skew and vol-of-vol risk. Moreover, we show that the parameters of the stochastic component of LSV models enable controlling for the autocallables price while leaving the fit to European options unaffected.The presentation is based on a joint work with Urban Ulrych and Francesco Ferrari and has grown from the Master Thesis of the latter.

Theta lifts have long been known for constructing examples of automorphic forms in different settings. The Borcherds lift, for instance, gives rise to remarkable product expansions in the orthogonal context and allows to derive relations for special divisors. It is closely related to the Kudla–Millson lift and both have played a significant role in studying special divisors of Shimura varieties of orthogonal type. These lifts were related to each other by Bruinier and Funke who also proved surjectivity/injectivity in several cases. We present an approach to generalise these results, involving cycle integrals and Hecke theory, and discuss two new results, one of which was proven in collaboration with Riccardo Zuffetti.