A venue for invited and local speakers to present their research on topics surrounding algebraic geometry and number theory, broadly conceived. All meetings start at 14:10 sharp and end at 15:10. Meetings are held in the subterranean room -101. We expect to broadcast most meetings over Zoom at the URL https://us02web.zoom.us/j/85116542425?pwd=MzVDRmZJUVh2NXlObFVkM1N0MCt3Zz09

The seminar meets on Wednesdays, 14:10-15:10, in -101

Supersingular elliptic curves, quaternion algebras and some applications to cryptography

Part of the talk is expository: I will explain how supersingular isogeny graphs can be used to construct cryptographic hash functions and survey some of the rich mathematics involved. Then, with this motivation in mind, I will discuss two recent theorems by Jonathan Love and myself. The first concerns the generation of maximal orders by elements of particular norms. The second states that maximal orders of elliptic curves are determined by their theta functions.

Part of the talk is expository: I will explain how supersingular isogeny graphs can be used to construct cryptographic hash functions and survey some of the rich mathematics involved. Then, with this motivation in mind, I will discuss two recent theorems by Jonathan Love and myself. The first concerns the generation of maximal orders by elements of particular norms. The second states that maximal orders of elliptic curves are determined by their theta functions.

The lecture will start with a few useful (and probably new!) theorems on adic completion of commutative rings and modules. Then I will discuss derived adic completion, in its two flavors: the idealistic and the sequential. The weak proregularity (WPR) condition on an ideal \a in a ring A, which is a subtle generalization of the noetherian condition on the ring A, is a necessary and sufficient condition for the two flavors of derived completion to agree. WPR occurs often in the context of perfectoid theory, and I will finish the talk with theorems relating WPR to prisms.

Topological recursion is a remarkable universal recursive procedure that has been found in many enumerative geometry problems, from combinatorics of maps, to random matrices, Gromov-Witten invariants, Hurwitz numbers, Mirzakhani’s hyperbolic volumes of moduli spaces, knot polynomials. A recursion needs an initial data: a spectral curve, and the recursion defines the sequence of invariants of that spectral curve.
There is a duality in topological recursion which allows one to obtain closed formulas for the invariants of the recursion and which has implications in free probability theory and integrable hierarchies. In the talk I will survey recent progress in the topic with the examples from Hurwitz numbers theory, Hodge integrals and combinatorics of maps.

The talk is based on the joint works with A. Alexandrov, P. Dunin-Barkowski, M. Kazarian and S. Shadrin.

Lara—Srinivas—Stix, building on joint work with Esnalut, have recently shown that the etale fundamental group of a connected proper scheme over an algebraically closed field is topologically finitely presented, thus answering a question raised in SGA. The proof relies on a finite presentation criterion of Lubotzky for profinite groups, resolutions of singularities/alterations, a theorem of Deligne—Ilusie on the Euler characteristic, as well as other modern and classical results in (arithmetic) algebraic geometry.