2026-2027 DANTS schedule
All talks are held in Kemeny 343 on Tuesday, 2:30-3:30pm
Sep. 15 - Benjamin Singer (Dartmouth College)
Analytic Geometry via Relative Algebraic Geometry
Analytic geometry has long occupied a difficult position between differential and algebraic geometry, using ideas and techniques from both while failing to retain several of their formal properties. A primary issue is that no classical framework admits a theory of quasicoherent sheaves satisfying descent, a problem that has become pressing in $p$-adic and complex geometry and their applications to number theory. In this talk, I will explain how one can amend this by realizing analytic geometry as “derived algebraic geometry” relative to a suitable $\infty$-category of Ind-Banach spaces, in the sense of Toën–Vezzosi, and equipping the result with an explicit six-functor formalism. I will then construct derived enhancements of two theories of analytic geometry over $\mathbb{C}$: classical complex analytic spaces, and an archimedean incarnation of Große-Klönne’s dagger analytic spaces. If there is time, I may also discuss how one can prove derived enhancements of classical complex geometry theorems in this setting (e.g. GAGA, Tannaka duality, or Grothendieck duality), or how this framework relates to other approaches (e.g. Clausen–Scholze’s work using condensed mathematics or older work of Lurie/Porta–Yue Yu). This is joint work with Arun Soor, building on recent foundational work of Ben-Bassat–Kelly–Kremnizer.
Sep. 22 - David Mumford (Brown University)
Reese Prosser Memorial Lecture. See more details at: prosser F26.
Sep. 29 - Nazim Khelifa (Duke University)
Variations of Hodge structures of maximal dimension
In the 1970's, Griffiths discovered a differential constraint satisfied by the local system formed by primitive middle cohomology groups of the members of family of smooth projective complex algebraic varieties called Griffiths transversality and used it to abstract the additional structure that such local systems inherit from geometry, resulting in the notion of variations of Hodge structures. Associated to such a variation of Hodge structure is a classifying holomorphic map, called its period map, to some parameter space for Hodge structures, called period domains. Because of Griffiths transversality, most period domains cannot be the target of surjective period maps. Carlson, Toledo, Kasparian and many others gave quantitative versions of this result by deriving bounds on the dimension of the image of a period map using the differential constraint given by Griffiths transversality. However, these bounds were local in nature while variations of Hodge structures whose base is algebraic are known to have remarkable global rigidity properties. After recalling some history of the problem we will explain how to use a natural object associated to a global variation of Hodge structure, its Hodge locus, to derive bounds on the dimension of its period image which take into account the global structure of the variation.
Oct. 06 - Asher Auel (Dartmouth College)
Recognizing zeta functions of K3 surfaces over finite fields
How many solutions does a system of polynomial equations have over a finite field? This question is important, with direct applications to the construction of error-correcting codes. The Hasse-Weil zeta function of a variety over a finite field is a generating function for the number of points over all finite extensions. By the Weil conjectures, the zeta function is a rational function determined by an alternating product of characteristic polynomials of the action of Frobenius on the l-adic cohomology of the variety. Hence we gain information on the number of rational points by studying these characteristic polynomials. In this talk, I'll give an update on the Honda-Tate program for K3 surfaces, which aims to determine the possible zeta functions of K3 surfaces over a finite field. I'll describe a recent breakthrough by Sair Shaikh, a recently graduated math major, and its applications to joint work with Engel, Kedlaya, and Shaikh to the problem of identifying K3 surfaces with no rational points.
Oct. 13 - Zachary Couvillion (Cornell University)
Computing the nonsurjective primes for dimension 3 abelian varieties over the rationals
Let A be a three-dimensional abelian variety over the rational numbers. For each prime $\ell$, the action of the absolute Galois group on the $\ell$-torsion points of A gives rise to a six-dimensional Galois representation over the finite field with ell elements. If A is generic, a theorem of Serre implies that for all but finitely many primes, the image of this mod $\ell$ Galois representation is as large as possible (given the restraint of the Weil pairing). In this talk, we examine the following question: when given such an abelian variety, can one determine this finite list of primes where the image of the Galois action is smaller than what would be expected?
Oct. 20 - Juliette Bruce (Dartmouth College)
A Tropical Perspective on the Nagata-Biran Conjecture
I will talk about the Nagata-Biran conjecture.
Oct. 27 - No talk
Nov. 03 - Salim Tayou (Dartmouth College)
The Zilber-Pink conjecture: old and new
I will survey recent results on the Zilber--Pink conjecture for variations of Hodge structures. If time permits, I will also discuss the recent OpenAI announcements on this topic in the case of moduli space of abelian threefolds.
Nov. 10 - No talk
Nov. 17 - Haohua Deng (Dartmouth College)
Jan. 05 -
Jan. 12 -
Jan. 19 -
Jan. 26 - Naomi Sweeting (MIT)