## Schedule

All talks are located in lecture hall -101 at the Alte Universität, Rheinsprung 9 in Basel.

### Friday, June 1st

Time | Speaker | Title | Abstract |
---|---|---|---|

2:30pm - 3:30pm | Euler classes transgressions and Eisenstein cohomology of $GL(N)$ | In work-in-progress with Pierre Charollois, Luis Garcia and Akshay Venkatesh we give a new construction of some Eisenstein classes for $GL_N (\mathbb{Z})$ that were first considered by Nori and Sczech. The starting point of this construction is a theorem of Sullivan on the vanishing of the Euler class of $SL_N (\mathbb{Z})$-vector bundles and the explicit transgression of this Euler class by Bismut and Cheeger. Their proof indeed produces a universal form that can be thought of as a kernel for a regularized theta lift for the reductive dual pair ($GL_1$ , $GL_N$ ). This suggests looking to reductive dual pairs ($GL_k$ , $GL_N$) with $k >1$ for possible generalizations of the Eisenstein cocycle. This leads to interesting arithmetic lifts. | |

3:30pm - 4:00pm | |||

4:00pm - 5:00pm | Differential equations and Diophantine geometry | Over the past decade there has been a great surge of activity around the interface between diophantine geometry and tame geometry (o-minimality) following the discovery of the Pila-Wilkie counting theorem and it's numerous applications around problems of unlikely intersections and functional transcendence. I will briefly review these developments, and then discuss possible refinements of the Pila-Wilkie theorem in the directions of effectivity and sharper asymptotics. For both of these directions, the generality of the o-minimal framework is too great to expect any serious improvements. As the main subject of the talk, I will discuss classes of sets defined using solutions of various types of differential equations. I will explain how these classes arise as natural candidates which are, on the one hand, general enough to encompass the sets important for diophantine applications; and on the other hand, structured enough to give hope of establishing substantial refinements of the Pila-Wilkie theorem. If time permits I will also discuss some applications of these results to effectivity questions in problems of unlikely intersections. | |

5:00pm - 5:30pm | |||

5:30pm - 6:30pm | TBA | TBA |

### Saturday, June 2nd

Time | Speaker | Title | Abstract |
---|---|---|---|

08:45am - 09:15am | |||

09:15am - 10:15am | Shimura varieties, torsion classes, and Galois representations | In this talk, I will describe joint work with Peter Scholze on torsion in the cohomology of certain unitary Shimura varieties. I will explain how the theory of perfectoid spaces and p-adic Hodge theory give new insights into the geometry and cohomology of Shimura varieties. I will also mention applications of these results to elliptic curves over imaginary quadratic fields, joint with Allen, Calegari, Gee, Helm, Le Hung, Newton, Scholze, Taylor, and Thorne. | |

10:15am - 10:45am | |||

10.45am - 11.45am | Long and small gaps in the spectrum of rectangular tori | We study various statistics on gaps between the first $N$ eigenvalues of the Laplacian on a rectangular billiard in comparison to the corresponding quantity for a Poissonian sequence. The proofs use various tools from diophantine analysis and analytic number theory. This is joint work with Radziwill and also with Bourgain and Rudnick. | |

12:00pm - 13:00pm |
Height pairings and common zeroes on $\mathbb{P}^1$ | Given height functions on $\mathbb{P}^1(\bar{\mathbb{Q}})$, the Arakelov-Zhang intersection number defines a non-degenerate pairing between them. In this talk, I will discuss families of heights where the pairing is uniformly bounded from below by some positive quantity. This gives uniform upper bounds on their numbers of common zeroes. Natural examples arise from dynamical systems on P^1 and from elliptic curves defined over number fields. This is joint work with Holly Krieger and Hexi Ye. |