Boston College Algebraic Geometry Seminar

Abstract: In 1966, Tate proposed the Artin–Tate conjectures, which describe the special values of zeta functions associated to surfaces over finite fields. Building on this, and assuming the Tate conjecture, Milne and Ramachandran formulated and proved analogous conjectures for smooth proper schemes over finite fields. However, the formulation of these conjectures already relied on other unproven conjectures. In this talk, I will present an unconditional formulation and proof of these conjectures. The approach relies on the theory of F-Gauges, a notion introduced by Fontaine–Jannsen and further developed by Bhatt–Lurie and Drinfeld, which has been proposed as a candidate for a theory of p-adic motives. A central role is also played by the notion of stable Bockstein characteristics, which will be introduced in the talk.    

Abstract: In this talk, I apply Koszul duality to the Fukaya A-infinity algebra of a compact Lagrangian $L$ to study questions in mirror symmetry. 

The focus is on a localized mirror constructed intrinsically from the Floer theory of $L$. This construction leads to a natural equivalence between the local Fukaya category generated by $L$ and the category of finite-dimensional representations of the quiver algebra Koszul dual to $CF(L,L)$. Under suitable finiteness conditions, I explain how this local mirror can be globalized to recover the full mirror.