Weekly Bulletin

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

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.

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.

Ax-Schanuel theorems are statements about the Zariski closure of a formal (non-constant) curve lying on a leaf of a foliation of a complex algebraic variety. Here are two instances of this family of theorems:<BR> Theorem (Ax): For $t in (C[[s]]-C)^n$, if $trdeg C(t_1,..., t_n, exp(t_1),..., exp(t_n))/C < 1+n$ then a $Z$-linear combination of $t_i$'s is constant. <BR> Theorem (Pila-Tsimerman): For $t in (C[[s]]-C)^n$, if $trdeg C(t_1,...,t_n, j(t_1),..., j(t_n), ..., j''(t_n))/C < 1+3n$ then there exist $k\leq l-1$ and $h$ in $PGL_2^+(Q)$ such that $t_k = h(t_l)$.<BR> The natural extension of these theorems to developing maps of a rational $(G, G/H)$-structure on a algebraic variety splits in two different parts: 1/ prove that $j(t_k)$ is algebraic over $C(j(t_l))$; 2/ prove the existence of $h$. I will explain how the first part can be obtained from a general result on principal connexions using elementary differential Galois theory and how the second part is obtained in the special case of product of projective structures on curves using model theory of differentially closed fields.

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).

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.

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.)