Weekly Bulletin

A translation surface is given by polygons in the plane, with sides identified by translations to create a closed Riemann surface with a flat structure away from finitely many singular points. Understanding geodesic flow on a surface involves understanding saddle connections. Saddle connections are the geodesics starting and ending at these singular points and are associated to a discrete subset of the plane. To measure the behavior of saddle connections of length at most R, we obtain precise decay rates as R goes to infinity for the difference in angle between two almost horizontal saddle connections. This is based on joint work with Jon Chaika.

Why is it not straightforward to get Fukaya categories to be filtered? And why do we care about this? This will be a introductory talk, in which we will try to answer the questions above, and outline some partials solutions. Our talk will serve as a warm up for the talk planned on 24.10 by Gionvanni Ambrosioni which will present a more general approach to these questions.

What sets optimization on Euclidean spaces apart from optimization on Riemannian manifolds? Not that much. By and large, classical algorithms for smooth optimization on Euclidean spaces carry over to optimize on Riemannian manifolds, often with essentially the same (adequately rephrased) theoretical guarantees. As a notable exception, it turns out that such is not the case for Nesterov's accelerated gradient descent. Building on several ideas by Hamilton and Moitra (2021), we construct a resisting oracle that shows it is not possible to minimize smooth, strongly geodesically convex functions on Hadamard manifolds at an accelerated rate. I will give some geometric intuition as to why negative curvature makes life harder for optimizers, and why this may not be the end of the story. This is work with Chris Criscitiello, COLT 2022.

Quasiconvexity is a fundamental notion in the vectorial Calculus of Variations and is essentially equivalent to the applicability of the Direct Method which ensures the existence of minimizers. A fundamental problem, considered by Morrey in the 50s and 60s, is whether quasiconvexity is equivalent to ellipticity (in the sense of Legendre-​Hadamard). In 1992, Šverák showed that in 3 or higher dimensions they are not equivalent, but the two-​dimensional case remains open. In this case one can hope for a "complex analysis miracle", and we will discuss deep connections of Morrey's problem to old questions in Complex Analysis.

More information: https://eth-its.ethz.ch/activities/its-fellows--seminar.htmlcall_made

In recent work of Lopez, Malmskog, Matthews, Pinero-Gonzales, and Wootters, we constructed codes for local recovery of erasures with high availability and constant-bounded rate from the Hermitian curve. These new codes, called Hermitian-lifted codes, are evaluation codes with evaluation set being the set of $F_(q^2)$-rational points on the affine curve. The novelty is in terms of the functions to be evaluated; they are a special set of monomials which restrict to low degree polynomials on lines intersected with the Hermitian curve. The resulting codes are neither punctured traditional lifted codes, nor subcodes of previously defined locally recoverable codes on the Hermitian curve. This talk will introduce the codes, bounds on their parameters, and discuss questions for further research. <BR> <BR> (**This eSeminar will also be live-streamed on Zoom, using the same meeting details as previous seminars. If you do not have meeting details, please contact simran.tinani@math.uzh.ch **)

We prove the macroscopic cousins of three conjectures: <br> 1) a conjectural bound of the simplicial volume of a Riemannian manifold in the presence of a lower scalar curvature bound, <br> 2) the conjecture that rationally essential manifolds do not admit metrics of positive scalar curvature, <br> 3) a conjectural bound of ℓ²-Betti numbers of a spherical Riemannian manifolds in the presence of a lower scalar curvature bound. <br> The macroscopic cousin is the statement one obtains by replacing a lower scalar curvature bound by an upper bound on the volumes of 1-balls in the universal cover. Group actions on Cantor spaces surprisingly play an important role in the proof. <br>The talk is based on joint work with Sabine Braun.

In this presentation, we develop an innovative approach - X-MESH - to overcome a major difficulty associated with numerical simulation in engineering: we aim to provide a revolutionary way to track physical interfaces in finite element simulations. The idea is to use so-called extreme mesh deformations. This new approach should allow low computational cost simulations as well as high robustness and accuracy. X-MESH is designed to avoid the pitfalls of current ALE methods by allowing topological changes on fixed mesh. The key idea of X-MESH is to allow elements to deform until they reach a zero measure. For example, a triangle can deform into an edge or even a point. This idea is rather extreme and completely revisits the interaction between the meshing community and the computational community, which for decades have been trying to interact through beautiful meshes. <br> In this talk, we will focus on both the mathematical issues related to the use of zero-measure elements and the X-MESH resolution scheme. Several applications will be targeted: the Stefan model of phase change, two-phase flows and contact between deformable solids.

Rank-metric codes are codes for which each codeword is a matrix and the distance between two codewords is the rank of their difference. Rank-metric codes over finite fields are used in space-time coding, public-key cryptosystems, and random linear network coding. Works on nested-lattice-based network coding suggest the construction of more efficient physical-layer network coding schemes with network coding over finite chain rings. So, finite rings can be used in network coding, but how to detect and correct errors. In this talk, we present the generalization of some results in rank-metric codes to finite commutative principal ideal rings. These results are then applied in network coding as in the case of finite fields. Specifically, two existing encoding schemes of random linear network coding are combined to improve the error correction. <BR> <BR> (**This eSeminar will also be live-streamed on Zoom, using the same meeting details as previous seminars. If you do not have meeting details, please contact simran.tinani@math.uzh.ch **)

The fragmentation of a tree is a process which consists in cutting the tree at random points, thus splitting it into smaller connected components as time passes. In the case of the so-called Brownian tree, it turns out that the sizes of these subtrees, known as the Aldous-Pitman fragmentation process, have the same distribution as the lengths of the excursions over its current infimum of a linearly drifted Brownian excursion, as proved by Bertoin. We provide a natural coupling between these two objects. To this end, we make use of the so-called cut-tree of the Brownian tree, which can be seen as the genealogical tree of the fragmentation process. Joint work with Igor Kortchemski.

In order to study physical effects of quantum fields on curved spacetimes, one needs appropriate Hadamard states to describe the fields. In this talk, we present a rigorous construction, including the proof of the Hadamard property, of the Unruh state for the free scalar field on slowly rotating Kerr-de Sitter spacetimes.

Functionals are omnipresent in finance, whether it be the payoff of a claim, a hedging strategy, or path-dependent volatility. However, problems involving functionals are often infinite-dimensional and thus challenging from a conceptual and computational perspective. In the first part of the talk, we present a simulation algorithm based on the Karhunen-Loève expansion to efficiently price exotic options. In the second, we compare static expansions (Volterra/Wiener series as well as the novel intrinsic value expansion) and promote the functional Taylor expansion (FTE). The latter combines the Functional Itô Calculus with the signature to quantify the effect in a functional when a “shock" path is concatenated with the source path. The notions of analytic functionals and radius of convergence in the path space are then defined. We finally apply the FTE to decompose exotic claims. This is joint work with Bruno Dupire (Bloomberg LP).

Consider the regression problem where the response Y∈ ℝ and the covariate X ∈ ℝ^d for d≥1 are \textit{unmatched}. Under this scenario, we do not have access to pairs of observations from the distribution of (X,Y), but instead, we have separate datasets {Yi}_ni=1 and {Xj}_mj=1, possibly collected from different sources. We study this problem assuming that the regression function is linear and the noise distribution is known or can be estimated. We introduce an estimator of the regression vector based on deconvolution and demonstrate its consistency and asymptotic normality under an identifiability assumption. In the general case, we show that our estimator (DLSE: Deconvolution Least Squared Estimator) is consistent in terms of an extended ℓ2 norm. Using this observation, we devise a method for semi-supervised learning, i.e., when we have access to a small sample of matched pairs (Xk,Yk). Several applications with synthetic and real datasets are considered to illustrate the theory.

We address a spin version of the GW/H correspondence. The conjecture has a well-defined statement and one of the motivations behind it is that spin curves contain useful information for the GW theory of Kähler surfaces. In the original GW/H correspondence, Hurwitz numbers provide substantial structure to the GW side. For instance, they enjoy integrability of the 2D Toda type, they have a clear formulation in the Fock space and the ELSV formula is known. Moreover, the ELSV formula is compatible via localisation with the GW side. One could investigate similar questions for spin Hurwitz numbers. We provide some of the answers and prove the correspondence in the case of spin equivariant P^1.Based on completed work and on work in progress with A. Giacchetto, R. Kramer, A. Sauvaget.