Weekly Bulletin

In a pioneering work of 1998, Eliashberg and Thurston proved that, in ambient dimension 3, C^2 foliations can always (besides one example) be C^0-approximated by contact structures in the space of plane fields. To do so, they introduce what they called ``confoliations'', that are geometric structures which are mid-way between foliations and contact structures. While these have been studied extensively in dimension 3 by many authors, the situation in higher dimensions is less clear, and there is not even a consensus on what their definition should be. In this talk, I will introduce a new notion of high-dimensional symplectic confoliation, and describe how on one hand it naturally leads to a notion of approximation by contact structures that encompasses all previously known ad-hoc examples, and on the other hand how it also constitute a good class of structures, generalizing contact ones, of which one can study symplectic fillability questions. This is based on work joint with Robert Cardona and on work in progress joint with Seungook Yu.

<p><span data-olk-copy-source="MessageBody">Cryptography is an essential part of today's privacy focused world. But how does Cryptography guarantee that your data is indeed 'encrypted'? In this talk, we will take a look at how hard mathematical problems give rise to useful cryptographic tools going beyond encryption. We will also delve into what it means for a problem to be called hard, and finally we will look at a particular example (code equivalence), explore its hardness, and see how it's utilised to give us aforementioned cryptographic tools.</span></p>

This is a report of joint work with Bergström-Diaconu-Westerland and Miller-Patzt-Randal-Williams. There is a "recipe" due to Conrey-Farmer-Keating-Rubinstein-Snaith which allows for precise predictions for the asymptotics of moments of many different families of L-functions. Our work concerns the CFKRS predictions in the case of the quadratic family over function fields, i.e. the family of all L-functions attached to hyperelliptic curves over some fixed finite field. One can relate this problem to understanding the homology of the moduli space of hyperelliptic curves with symplectic coefficients. With Bergström-Diaconu-Westerland we compute the stable homology groups of the moduli space of hyperelliptic curves with these coefficients, together with their structure as Galois representations. We moreover show that the answer matches the number-theoretic predictions. With Miller-Patzt-Randal-Williams we prove a uniform range for homological stability with these coefficients. Together, these results imply the CFKRS predictions for all moments in the function field case, for all sufficiently large (but fixed) q.

Polynomial or birational transformations of ℂⁿ form huge groups as soon as n > 1, and the following basic questions are open in general: <ul> <li>Do they admit normal subgroups (apart from the obvious subgroup of Jacobian 1 automorphisms in the polynomial case)? </li> <li>Do they satisfy a Tits alternative, as in the linear case?</li> <li>Is any finite subgroup conjugate to a subgroup of GL(n)?</li></ul> After explaining the context, I will discuss some particular cases in dimensions 3 and 4 where we can answer these questions, using some actions on some metric spaces with negative curvature. (based on several joint works with P. Przytycki)

For an ergodic dynamical system, the cutoff describes an abrupt transition to equilibrium. Historically introduced in seminal work by Diaconis, Shahshahani and Aldous for card shuffling and other random walks on finite groups, there are now numerous examples of Markov chains and Markov processes where the cutoff has been established. Most of the current examples are on finite spaces. In this talk, we study cutoff for classical processes -- namely Brownian motion and geodesic paths -- on compact hyperbolic manifolds, and we develop a spectral strategy introduced by Lubetzky and Peres in 2016 for Ramanujan graphs and further developed in different geometric contexts. In particular, we extend results obtained by Golubev and Kamber in 2019 to any dimension and still are able to obtain cutoff under weaker hypothesis. Joint work with C. Bordenave

<p><span style="caret-color: #000000; color: #000000; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; display: inline !important; float: none;">I will discuss how to obtain Lipschitz estimates for regularised optimal transport problems using a variational approach. In particular, this </span><br style="caret-color: #000000; color: #000000; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;"><span style="caret-color: #000000; color: #000000; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; display: inline !important; float: none;">gives Lipschitz regularity for entropic optimal transport independent of the regularisation parameter. A crucial step in the approach are local </span><br style="caret-color: #000000; color: #000000; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;"><span style="caret-color: #000000; color: #000000; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; display: inline !important; float: none;">L^\infty-estimates, which are of independent interest. The talk is based on joint work with Rishabh Gvalani (ETH).</span></p>

Spaces of unramified maps are a higher-dimensional generalization of spaces of admissible covers, introduced by Kim, Kresch, and Oh. I will discuss unramified maps to abelian varieties and how they can be used to compute cycles associated with spaces of stable maps. Joint work with Jeremy Feusi and Aitor Iribar López.