Weekly Bulletin

https://www.math.uzh.ch/ve-ps-det?key1=0&key2=1612&semId=47

In this talk, I will focus on some relationships between topology, analysis and geometry which are provided by studying the geodesic flow on non-compact manifolds with negative curvature. I will first recall some classical notions of entropy and then present entropies at infinity and entropy gap property. I will sketch various applications to counting closed orbits, Laplace spectrum and mixing properties of the geodesic flow.

Consider the scalar advection-diffusion equation. According to physical predictions, the advecting velocity field, if turbulent, may enhance diffusion so strongly that an artifact of the diffusivity remains in the inviscid limit. This phenomenon – the strict energy inequality in the transport equation obtained as an inviscid limit – is referred to as ‘anomalous dissipation’. I will present a recent joint result with László Székelyhidi and Bian Wu, proving that anomalous dissipation really occurs for scalars advected by a (typical) solution of Euler equation (with its regularity below the 1/3-Hölder continuity, the Onsager threshold). Consequently, we obtain non-uniqueness of the respective transport equations.

Wigner's surmise states that the spectrum of the Hamiltonian of heavy nuclei is distributed like that of a large random matrix. Since it was proposed by Wigner in 1956, the eigenvalue distribution of large random matrices has been used as a toy model to study the distribution of more complex mathematical objects such as random tiles or the longest increasing subsequence of a random perturbation. However, this universality phenomenon generally concerns distributions derived from Gaussian matrices, known as the Gaussian ensembles. In this talk, we will discuss more general universality classes that appear in the theory of random matrices, how they stand out and open questions.

We introduce the enumerative geometry of curves in the algebraic torus (C*)^2. We show that a certain class of invariants associated with moduli spaces of curves in (C*)^2 can be calculated explicitly using a refined tropical correspondence theorem. If time permits we will explain how the proof relies on higher double ramification cycles and work of Buryak-Rossi on integrable systems on the moduli space of curves. This is joint work with Patrick Kennedy-Hunt and Qaasim Shafi.

Isogenies are rational maps between elliptic curves that are also group morphisms with respect to the group structure of the elliptic curves. The kernel of an isogeny is always finite, and a natural way to describe an isogeny is to give a description of its kernel. Given the kernel of an isogeny, Velu formulas (or square root Velu formulas) allow to compute and evaluate the isogeny. These formulas are only efficient for small degree isogenies. Hence, in general, only smooth degrees isogenies can be computed and evaluated exploiting these formulas. The Deuring Correspondence allows to interpret supersingular isogenies as ideals. It enables the computation and the evaluation of isogenies of generic degree, provided that the endomorphism rings of the curves in play are known. The recent SIDH attacks have proven that the images of torsion points through an isogeny can be used to efficiently evaluate the isogeny if these points have (power)smooth order. This enables a brand new way to represent isogenies. This has been leveraged to design SQISignHD, a variant of the SQISign signature. SQISignHD is currently the most compact post-quantum signature scheme. In this talk, we will discuss this new isogeny representation and show how it is used in SQISignHD. We will then show how to adapt SQISignHD to obtain a signature scheme for the CSIDH group action setting.

A fundamental question in algebraic topology is the following: how much information on a space can be deduced from its cohomology. There is of course no hope to get more than the underlying homotopy type of the space. On the other hand, Sullivan has famously proved that the rational homotopy type of a space is determined by the algebra of rational cochains (in particular rationalized homotopy groups can be computed from this data). My goal in this talk is to explain an integral lift of this theorem based on the theory of binomial rings.

In this talk we generalize the concept of error sets beyond those defined by a metric and use the set-theoretic difference operator to characterize when these error sets are detectable or correctable by codes. We prove the existence of a general, metric-less form of the Gilbert-Varshamov bound, and show that - like in the Hamming setting - a random code corrects a generic error set with overwhelming probability. We define the generic error SDP (GE-SDP), which is contained in the complexity class of NP-hard problems, and use its hardness to demonstrate the security of generic error CVE (GE-CVE). This is a joint work with Freeman Slaughter.

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More information: https://eth-its.ethz.ch/activities/its-science-colloquium.htmlcall_made

Motivated by the paradigm of reservoir computing, we consider randomly initialised controlled ResNets defined as Euler-discretisations of neural controlled differential equations (Neural CDEs), a unified architecture which encompasses both RNNs and ResNets. We show that in the infinite-width-depth limit and under proper scaling, these architectures converge weakly to Gaussian processes indexed on some spaces of continuous paths and with kernels satisfying certain partial differential equations (PDEs) varying according to the choice of activation function, extending the results of Hayou (2022); Hayou & Yang (2023) to the controlled and homogeneous case. In the special, homogeneous, case where the activation is the identity, we show that the equation reduces to a linear PDE and the limiting kernel agrees with the signature kernel of Salvi et al. (2021a). We name this new family of limiting kernels neural signature kernels. Finally, we show that in the infinite-depth regime, finite-width controlled ResNets converge in distribution to Neural CDEs with random vector fields which, depending on whether the weights are shared across layers, are either time-independent and Gaussian or behave like a matrix-valued Brownian motion.

In the eighties, Kudla and Millson constructed a linear map between certain spaces of vector-valued Siegel modular cusp forms to the space of closed differential forms on some orthogonal Shimura variety. The injectivity of this map in genus 1 has been of great interest and has many applications, including the surjectivity of Borcherds' lift. The aim of this talk is to introduce orthogonal Shimura varieties and indicate why they might be of interest. We then proceed to explain the Kudla-Millson lift and its injectivity in genus 2. We end the talk with a cohomological application. This is joint work with Riccardo Zuffetti.

: I will make a proposal for a rigorous formulation of the so called refined topological string on a Calabi-Yau threefold X (admitting a torus action) in the framework of equivariant Gromov-Witten theory of X x C^2. After explaining how these GW invariants conjecturally relate to K-theoretic stable pair invariants, we will concentrate on the case when X is a local del Pezzo surface K_S. In this case the so called Nekrasov-Shatashvili limit can be identified with the GW theory of S relative a smooth anti-canonical curve which can be used to prove BPS integrality for S=P^2 in this limit. This is ongoing work with Andrea Brini.