Veranstaltungen

The study of random phenomena governed by Levy processes, such as stock market fluctuations, brings us to integrodifferential elliptic equations. These equations also play a significant role in diverse fields like fluid mechanics and linear elasticity. In this talk, I will present a series of selected results that have contributed to the advancement of these equations. The focus will range from the exploration of boundary regularity and the development of specialized integration by parts formulas, to advancements in regularity for obstacle problems and insights into the regularity of phase transitions. A common feature of these works is the introduction of profound conceptual novelties in the study of these equations, especially in comparison to the traditional second-order elliptic operators. These innovations not only provide a deeper understanding of the equations themselves but also pave the way for new applications and methodologies for classical problems.

A disordered quantum system is mathematically described by a large Hermitian random matrix. One of the most striking phenomena expected to occur in such systems is a localization-delocalization transition for the eigenvectors. Originally proposed in the 1950s to model conduction in semiconductors with random impurities, this phenomenon is now recognized as a general feature of wave transport in disordered media, and is one of the most influential ideas in modern condensed matter physics. A simple and natural model of a disordered quantum system is given by the adjacency matrix of a random graph. I report on recent progress in analysing the phase diagram for the Erdös-Renyi model of random graphs. In particular, I explain the emergence of a fully localized and a fully delocalized phase, which are separated by a mobility edge.

https://www.math.uzh.ch/mat074

Spin glasses are models of tatistical mechanics in which a large number of elementary units interact with each other in a disordered manner. In the simplest case, there are interactions between any two units in the system, and I will start by reviewing some of the key mathematical results in this context. For modelling purposes, it is also desirable to consider models with more structure, such as when the units are split into two groups, and the interactions only go from one group to the other one. I will then discuss some of the technical challenges that arise in this case, as well as recent progress.

Let a < b be multiplicatively independent integers, both at least 2. Let A,B be closed subsets of [0,1] that are forward invariant under multiplication by a, b respectively, and let C be A x B. An old conjecture of Furstenberg asserted that any line not parallel to either axis must intersect C in Hausdorff dimension at most max(dim C,1)-1. He was able to prove a partial result in this direction using a new class of measure-valued processes, now referred to as "CP chains". A few years ago, Shmerkin and Wu independently gave two different proofs of Furstenberg's conjecture. In this talk I will sketch a more recent third proof that builds on some of Furstenberg's original results. In addition to those, the main ingredients are a version of the Shannon--McMillan--Breiman theorem relative to a factor and some standard calculations with entropy and Hausdorff dimension.

One of the main goals of statistical physics is to observe how spins displayed along a lattice Z^d interact together and fluctuate. When the spins belong to a discrete set (for example the celebrated Ising model where spins \sigma_x belong to {-1,+1}), the nature of the phase transitions which arise as one varies the temperature is now rather well understood. When the spins belong instead to a continuous space (for example the unit circle S^1 for the so-called XY model, the unit sphere S^2 for the classical Heisenberg model etc.), the nature of the phase transitions differs drastically from the discrete symmetry setting. The case where the (continuous) symmetry is non-Abelian is currently more mysterious than when the symmetry is Abelian. In the later case, phase transitions are caused by a change of behaviour of certain monodromies in the system called "vortices". They are called topological phase transitions for this reason. In this talk, after an introduction to the mathematics of spin systems with a continuous symmetry, I will present some recent results on these spins systems as well as on models naturally associated with them, such as Coulomb gases and random integer-valued interfaces. The talk is based on joint works with Juhan Aru, Paul Dario, Avelio Sepúlveda and Tom Spencer.

I shall discuss how one can define in a natural way the notion of typical rotation numbers for families of circle maps with singularities. This problem is related to a well known fact that in the case of maps with singularities the set of parameters, corresponding to irrational rotation numbers, has zero Lebesgue measure. Our approach is based on the hyperbolicity of renormalizations. I shall also discuss a natural setting for the Kesten theorem in the case of maps with singularities.

In the first part of the talk we will give an introductory explanation of the so-called bubble-tree convergence for critical points of conformally invariant Lagrangians on Riemann Surfaces. We will then proceed to discuss Morse Index Stability along these sequences, which relies on L2,1-quantization results. The talk is based on joint work with F. Da Lio and T. Rivière.

In the study of fluid dynamics, turbulence poses a significant challenge in predicting fluid behavior, and it remains a mystery for mathematicians and physicists alike. Recently, there has been some exciting progress in our understanding of ideal turbulence: starting from Onsager’s theorem to the resolution of strong Onsager's conjecture in L^3-framework. These developments have been accompanied by mathematical advances in Nash’s iteration. In this talk, I will provide an overview of turbulence and discuss these results.

Minimal surfaces are critical points of the area functional. I'll survey some classical and modern ideas related to their existence, regularity, and classification.

In this talk, I will describe the regularity theory for surfaces minimizing the prescribed mean curvature functional over isotopies in a closed Riemannian 3-manifold, which is a prescribed mean curvature counterpart of the celebrated regularity result of Meeks, Simon and Yau about minimizers of the area functional over isotopies. Whereas for the area functional minimizers over isotopies are smooth embedded minimal surfaces, minimizers of the prescribed mean curvature functional turn out to be C^{1,1} immersions which can have a large self-touching set where the mean curvature vanishes. Even though the proof broadly follows the same general strategy as in the case of the area functional, several new ideas are needed to deal with the lower regularity setting. This regularity theory plays an important role in Z. Wang-X. Zhou’s recent proof of the existence of 4 embedded minimal spheres in a generic metric on the 3-sphere. The results in this talk are joint work with Douglas Stryker (Princeton).

In probability theory, universality is the phenomenon where random processes converge to a common limit despite microscopic differences. For instance, the random walk, under mild conditions, converges to the same Brownian motion seen from afar, no matter the law of each independent step. This phenomenon underlies the appearance of the random simple curve, called SLE, as the universal scaling limit of interfaces in conformally invariant 2D systems. On the other hand, the family of simple curves has a Kähler structure, where we can study the geometric relations between curves. We will explain how these two worlds are tied together and show some applications of this link.

The gluing maps on the moduli spaces Mbar_{g,n} have played a crucial role in the intersection theory of Mbar_{g,n}, for example playing a key part in the definition of its tautological ring and of cohomological field theories. In the last few years, interpreting Mbar_{g,n} as a logarithmic space has also been an incredibly useful tool for understanding classical invariants, such as the double ramification (DR) cycle. However, joining these two concepts has been proven difficult, as the gluing maps are not logarithmic and hence the log structure and the gluing do not interact. In this talk I will explain the DR cycle, explain the difficulty in log gluing, and present a definition of log gluing. This also allows for the definition of log cohomological field theories, and in particular we find that the log DR cycle is a log cohomological field theory. This talk is based on joint work with David Holmes (arxiv:2308.01099). No previous knowledge of DR cycles or log geometry is assumed.

Given a topological embedding (i.e. injective continuous map), evaluation on finite subsets defines a map between configuration spaces which is coherent as we vary cardinalities. It turns out that, if the codimension is at least three, no homotopically information is lost in this process. This is in stark contrast to the situation in codimension zero, as shown by Krannich-Kupers. I will discuss some constructions and ideas involved in showing the high-codimension result, notably, a configuration space version of a torus trick from classical geometric topology. This is joint work with Michael Weiss.

The inverse problem of computational super-resolution is to recover fine features of a signal from bandlimited and noisy data. Despite long history of the question and its fundamental importance in science and engineering, relatively little is known regarding optimal accuracy of reconstructing the high resolution signal components, and how to attain it with tractable algorithms. In this talk I will describe recent progress on deriving optimal methods for super-resolving sparse sums of Dirac masses, a popular model in numerous applications such as spectral estimation, direction of arrival, imaging of point sources, and sampling signals below the Nyquist rate. Time permitting, I will also discuss generalizations of the theory and algorithms in several directions.

Because of Perelman‘s groundbreaking solution of the geometrization conjecture we know exactly which topological criteria a closed three-manifold needs to satisfy so that it admits a hyperbolic metric. However, one short-coming of Perelman’s Ricci flow approach is that even if the topology of the manifold is very well understood (e.g., if one knows how it is glued together from different pieces), it gives no information about geometric data (e.g., the hyperbolic volume) in terms of the topological data. The goal of this talk will be to explain how a lot of geometric information about the hyperbolic metric of the three-manifold can be read off a purely topological object called the curve graph. No prior knowledge about the geometrization conjecture or the curve graph will be assumed.

We consider the following data perturbation model, where the covariates incur multiplicative errors. For two random matrices U, X, we denote by (U \circ X) the Hadamard or Schur product, which is defined as (U \circ X)_{i,j} = (U_{i,j}) (X_{ij}). In this paper, we study the subgaussian matrix variate model, where we observe the matrix variate data through a random mask U: \mathcal{X} = U \circ X, where X = B^{1/2} Z A^{1/2}, where Z is a random matrix with independent subgaussian entries, and U is a mask matrix with either zero or positive entries, where $E[U_{ij}] \in [0,1]$ and all entries are mutually independent.Under the assumption of independence between X and U, we introduce componentwise unbiased estimators for estimating covariance A and B, and prove the concentration of measure bounds in the sense of guaranteeing the restricted eigenvalue(RE) conditions to hold on the unbiased estimator for B, when columns of data matrix are sampled with different rates. We further develop multiple regression methods for estimating the inverse of B and show statistical rate of convergence. Our results provide insight for sparse recovery for relationships among entities (samples, locations, items) when features (variables, time points, user ratings) are present in the observed data matrix X with heterogeneous rates. Our proof techniques can certainly be extended to other scenarios. We provide simulation evidence illuminating the theoretical predictions.

This paper introduces a non-Gaussian dynamic currency hedging strategy for globally diversified investors with ambiguity. It provides theoretical and empirical evidence that, under the stylized fact of non-Gaussianity of financial returns and for a given optimal portfolio, the investor-specific ambiguity can be estimated from historical asset returns without the need for additional exogenous information. Acknowledging non-Gaussianity, we compute an optimal ambiguity-adjusted mean-variance (dynamic) currency allocation. Next, we propose an extended filtered historical simulation that combines Monte Carlo simulation based on volatility clustering patterns with the semi-parametric non-normal return distribution from historical data. This simulation allows us to incorporate investor's ambiguity into a dynamic currency hedging strategy algorithm that can numerically optimize an arbitrary risk measure, such as the expected shortfall. The out-of-sample backtest demonstrates that, for globally diversified investors, the derived non-Gaussian dynamic currency hedging strategy is stable, robust, and highly risk reductive. It outperforms the benchmarks of constant hedging as well as static/dynamic hedging approaches with Gaussianity in terms of lower maximum drawdown and higher Sharpe and Sortino ratios, net of transaction costs.

Advertisers are interested in measuring the effectiveness of their online marketing campaigns on various platforms. While user-based experiments are efficient and well-understood, they are not always feasible because of technical and legal reasons. Geo-based experiments are an attractive and privacy-centric alternative, where experimental units are defined as geographical regions instead of individual users. One issue with this type of experiments, however, is the presence of contamination (or interference) between units, due to natural movement of people and imprecision in geo-localization. In this work we will try to quantify the amount of contamination in our experiments and propose possible solutions to mitigate its adverse effect, both during the estimation at the end of the expriment and upstream at the design phase.

Chan-Galatius-Payne have recently identified an enormous amount of nontrivial unstable cohomology classes on the moduli spaces of curves, via an identification of the "top weight" cohomology of the mapping class group with the cohomology of Kontsevich's graph complex. I will explain that all these classes restrict nontrivially to the handlebody subgroup of the mapping class group, i.e. those mapping classes which extend to a handlebody filling. In the process we obtain a geometrically meaningful classifying space for the handlebody group. (Joint with Louis Hainaut.)