Weekly Bulletin

A de Rham-Betti class on a smooth projective variety X over a number field K is a rational class in the Betti cohomology of the analytification of X that descends to a class in the algebraic de Rham cohomology of X via the period comparison isomorphism. These classes are the analogues of Hodge classes, except that one uses the K-structure on de Rham cohomology instead of the Hodge filtration. The period conjecture of Grothendieck implies that de Rham-Betti classes should be algebraic. I will report on joint work with Mingmin Shen, where we prove that any de Rham-Betti class on a product of elliptic curves is algebraic. As a key intermediate step in the proof, we show that certain codimension-2 de Rham-Betti classes on hyper-Kähler varieties are Hodge.

This talk will report on an ongoing joint work with Pablo Lessa (Montevideo). We consider a random walk on a group of matrices. Under suitable assumptions, Oseledets Theorem yields numbers (the Lyapunov exponents) and a random splitting into so-called Oseledets subspaces. This splitting defines a (random) point in a product of Grassmannians. Our Main result is that the distribution of this point is an exact-dimensional measure. The dimension has a geometric interpretation in terms of the exponents and some partial entropies. The talk will present the statement and the main partial results. As an example, we also discuss the case when the random walk is supported by an Anosov 3-dimensional representation of a surface group.

One of the most challenging problems in geometry and physics is to compute higher genus Gromov-Witten invariants of compact Calabi-Yau 3-folds, such as the famous quintic 3-fold. I will briefly describe how physicists compute Gromov-Witten invariants of the quintic 3-fold up to genus 53, using five mathematical conjectures. Three of them have been already proved, and one of the remaining two conjectures has been solved in some genus. I will explain how to prove the last open one, called the Castelnuovo bound, which predicts the vanishing of Gopakumar-Vafa invariants for a given degree at sufficiently high genus. This talk is based on the joint work with Yongbin Ruan.

In the presence of confinement, the Einstein field equations are expected to exhibit turbulent dynamics. One way to introduce confinement to the equations is by imposing a negative value for the cosmological constant. In this setting, the AdS instability conjecture claims the existence of arbitrarily small perturbations to the initial data of Anti-de Sitter spacetime which, under evolution by the vacuum Einstein equations with reflecting boundary conditions at conformal infinity, lead to the formation of black holes after sufficiently long time. In this talk, I will present a rigorous proof of the AdS instability conjecture in the setting of the spherically symmetric Einstein-scalar field system. The construction of the unstable initial data will require carefully designing a family of initial configurations of localized matter beams and estimating the exchange of energy taking place between interacting beams over long periods of time, as well as estimating the decoherence rate of those beams. I will also briefly discuss some other problems related to turbulent phenomena in the asymptotically AdS setting.

Abstract: The message-passing paradigm has been the “battle horse” of deep learning on graphs for several years, making graph neural networks a big success in a wide range of applications, from particle physics to protein design. From a theoretical viewpoint, it established the link to the Weisfeiler-Lehman hierarchy, allowing to analyse the expressive power of GNNs. We argue that the very “node-and-edge”-centric mindset of current graph deep learning schemes may hinder future progress in the field. As an alternative, we propose physics-inspired “continuous” learning models that open up a new trove of tools from the fields of differential geometry, algebraic topology, and differential equations so far largely unexplored in graph ML.

More information: https://math.ethz.ch/news-and-events/events/research-seminars/zurich-colloquium-in-mathematics.html?s=hs22call_made

It is a well-known consequence of mirror symmetry that curve counting invariants of Calabi-Yau varieties should be "modular" in some sense. However, finding the proper notion of modularity has proven difficult in general and so far can only be done in various special cases e.g. elliptic curve, K3, quintic threefold. One might suspect that modularity also holds if one does curve counting on a family of Calabi-Yau varieties instead. Indeed this seems to work for elliptic threefolds in which case PT theory seems to be the most natural thing to study. In this case we conjecture that one gets quasi-Jacobi forms of certain weight that satisfy certain holomorphic anomaly equations, which we can make precise but not yet prove. Further, we introduce pi-stable pairs which seem to be even better behaved and should be related to ordinary PT theory via a wall-crossing formula similar to the DT/PT correspondence. We also present several examples for which our conjectures hold. This is joint work with Georg Oberdieck.

The v-number is an algebraic invariant of graded ideals. The original motivation behind the study of this invariant comes from coding theory. In this context, Cooper, Seceleanu et al. introduced v-numbers to investigate the asymptotic behavior of the minimum distance function of certain evaluation codes. Although they have been introduced quite recently, v-numbers have already inspired several results in different fields, such as commutative algebra and combinatorics. In this talk, I will survey the state-of-the-art and I will present some new results on v-numbers of binomial edge ideals and their relation with Castelnuovo-Mumford regularity and Gröbner degeneration (this is an ongoing project with Jaramillo-Velez). <BR> <BR> (**This eSeminar will take place over Zoom, using the same meeting details as previous seminars. If you do not have meeting details, please contact zita.fiquelideabreu@math.uzh.ch **)

We consider a non-volume preserving Anosov flow generated by X on a compact 3-manifold M. The Ruelle Zeta Function (RZF) of this flow is a meromorphic function defined as a certain infinite product over closed orbits. Its behaviour at zero is conjectured to carry topological information of M and X, and is intimately related to certain resonant spaces of flow-invariant differential forms at zero. We introduce the notion of helicity (average self-linking) for X, and compute the dimension of resonant spaces as a function of helicity and the winding cycles of some natural flow-invariant measures. Finally, we illustrate this theory for thermostat flows associated to holomorphic quadratic differentials giving rise to quasi-Fuchsian flows of Ghys.

A multivariate cryptograpic instance in practice is a multivariate polynomial system. So the security of a protocol rely on the complexity of solving a multivariate polynomial system. During this talk I will explain the invariant to which this complexity depends on: the solving degree. Unfortunately this invariant is hard to compute. We will talk about another invariant, the degree of regularity, that under certain condition, give us an upper bound on the solving degree. Finally we will talk about random polynomial systems and in particular what "random" means to us. We will give an upper bound on both the degree of regularity and the solving degree of such random systems. <BR> <BR> (**This eSeminar will take place over Zoom, using the same meeting details as previous seminars. If you do not have meeting details, please contact zita.fiquelideabreu@math.uzh.ch **)

We consider the 2-dimensional Stochastic Heat Equation (SHE), which falls outside the scope of existing solution theories for singular stochastic PDEs. When we regularise the SHE by discretising space-time, the solution can be identified with the partition function of a statistical mechanics model, the so-called directed polymer in random environment. We prove that as the discretisation is removed and the noise strength is rescaled in a critical way, the solution converges to a unique continuum limit: a universal process of random measures on R^2, which we call the critical 2d Stochastic Heat Flow. We investigate its features, showing in particular that it cannot be the exponential of a generalised Gaussian field. Based on joint work with R. Sun and N. Zygouras.

We will examine the theory of replicating derivatives and in particular the Fundamental Theorem of Derivative Trading using the theory of rough paths. No model of stock prices will ever be completely accurate, so it is important that any theory of replication is robust to small perturbations of the stock price process. The fundamental theorem of derivative trading one such robustness result which applies to perturbations of diffusion models. In this talk we will consider perturbations of the stock price that go beyond diffusion models, and indeed, beyond probabilistic models using rough path theory which has excellent continuity properties. We will see that the gamma hedging strategy which is widely used by practitioners emerges naturally from this theory, and indeed is in some sense essential to make the replication strategy work. This and the corresponding robustness results provide a mathematical explanation for the popularity of gamma hedging.

Fourier coefficients of harmonic Maass forms contain interesting arithmetic information. However, their rationality is only known in a few cases, some of which are related to deep questions such as the vanishing of derivative of certain L-functions. In this talk, I will discuss recent joint work with Stephan Ehlen and Markus Schwagenscheidt, where we used theta lifts to prove sharp rationality results about Fourier coefficients of harmonic Maass forms associated to CM newforms.

Investigators often express interest in effects that quantify the mechanism by which a treatment (exposure) affects an outcome. In this presentation, I will discuss how to formulate and choose effects that quantify mechanisms, beyond conventional average causal effects. I will consider the perspective of a decision maker, such as a patient, doctor or drug developer. I will emphasize that a careful articulation of a practically useful research question should either map to decision making at this point in time or in the future. A common feature of effects that are practically useful is that they correspond to possibly hypothetical but well-defined interventions in identifiable (sub)populations. To illustrate my points, I will consider examples that were recently used to motivate consideration of mechanistic effects, e.g. in clinical trials. In all of these examples, I will suggest different causal effects that correspond to explicit research questions of practical interest. These proposed effects also require less stringent identification assumptions.

What is a sheaf on a log scheme X? If we take the ordinary etale topology, we ignore the log structure. Taking the log etale topology, even the structure "sheaf" O_X is not a sheaf! The same goes for M_X and bar M_X. We introduce a new "rhizomic" topology on log schemes coarser than the log etale topology. Will this be enough? Time permitting, we will also discuss log intersection theory and a riddle about log jet spaces.