Abstract: Grothendieck’s foundational work shows that many geometric properties of fibers in a projective family, for example smoothness, define constructible subsets of the base. However, one natural property is absent from this list: given a fiber, is the locus of points whose fibers are isomorphic to it constructible? Surprisingly, this question is closely related to the Minimal Model Program and the Kawamata-Morrison cone conjecture, a longstanding central problem in birational geometry. In this talk, I will explain the relationship between the cone conjecture and this constructibility problem and show how it yields a partial answer to the Kawamata-Matsuki conjecture on the finiteness of isomorphism classes of Calabi-Yau varieties within a fixed birational class.

Abstract: One way to study a variety is through the curves it contains.  This paradigm has shaped our understanding of smooth Fano varieties, but far less is known in the singular setting.  This talk will discuss several aspects of the theory of curves on singular Fano varieties, including the existence of free curves and Mori's Bend-and-Break.  This covers joint work with Osamu Fujino, Brian Lehmann, and Eric Riedl.

Abstract: A discrete valuation of a linear combination of terms is greater than or equal to the minimum of the valuations of the terms. A fundamental fact in toric geometry says that if the terms are monomials and the valuation corresponds to a toric boundary divisor, then this inequality will actually be an equality. We call this valuative independence. I will discuss joint work with M.-W. Cheung, T. Magee, and G. Muller in which we show that theta functions constructed from a positive scattering diagram (e.g., the Gross-Hacking-Keel-Kontsevich theta functions on cluster varieties) also satisfy valuative independence.  Applications include finding theta function bases for global sections of line bundles, linear independence of theta functions with specialized coefficients, and a general gluing lemma for theta functions from moduli of local systems on marked surfaces.  We then prove theta reciprocity, a symmetry property between valuations and theta functions on mirror pairs: roughly, val_v(theta_u)=val_u(theta_v).

Abstract: We will discuss valuative aspects of an affine variety admitting complete Calabi--Yau metrics, and in particular a valuative criterion for K-semistability of negative valuations.

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: We introduce a new stability condition for log Calabi--Yau fibrations over curves. We prove that it gives rise to a proper Deligne--Mumford stack with a projective coarse moduli space, whose boundary still parametrizes flat fibrations over curves. This is joint work with Giovanni Inchiostro.

Abstract: From the perspective of the Gross-Siebert program, the mirror symmetry of Calabi-Yau varieties is governed by certain canonical bases for the sections of the powers of an ample line bundle on a degeneration. It is expected that the pairing between these basis elements and valuations in the essential skeleton satisfies an independence condition, similar to the pairing between monomials and monomial valuations on a toric variety. In this talk, I will explain how tools from birational geometry and K-stability can be used to construct bases that fulfill this condition. This is joint work with Yuchen Liu.

Abstract: Let $p$ be a prime number. In the study of algebraic or analytic varieties over $\mathbb{Q}_p$, a powerful and commonly used cohomology theory is the $\ell$-adic étale cohomology for a prime $\ell\neq p$. Less well known and less developed is its $p$-adic analogue, called Hyodo–Kato cohomology, which is a refinement of de Rham cohomology. I will present a new geometric approach to this theory using stacks, highlight its main features, and, time permitting, discuss some arithmetic applications. Much of this talk is based on joint work with Anschütz, Le Bras, Rodríguez Camargo, and Scholze.

Abstract: Degree bounds have a long history in invariant theory. For any representation of a finite group over a characteristic zero field, the algebra of invariant polynomials is generated by invariants of degree at most the group order—this result of Emmy Noether dates to 1915, and is now known as the Noether bound. There is a vast literature generalizing and sharpening it. In the last 20 years, a parallel, very active literature has developed on the invariants required to separate all orbits of the group action. Generators for the invariant algebra separate orbits, but orbit separators do not necessarily generate the algebra, so this latter research program is a relaxation of the former.

In recent years, motivated by applications in signal processing, a program has developed on degree bounds for invariant polynomial generators of the field of rational invariants. Under reasonable geometric hypotheses, generators for the invariant field separate generic orbits, so this program can be seen as a further relaxation. The degrees of field generators can be dramatically lower than those of generating or separating invariants. They also behave differently—for instance separating and algebra generating degree are monotone increasing along chains of containments between representations of the same group, whereas field generation degree tends to decrease. 

The young literature on degree bounds for rational invariants deploys tools from diverse sources: computational commutative algebra, semigroup rings, geometry of numbers, additive combinatorics, tensor decomposition, and Galois theory. We provide a high-level survey including main results achieved so far, open lines of inquiry, and one or two interesting proofs.

This talk is based on the author’s joint work with Sylvan Crane, Harm Derksen, Thays Garcia, Karla Guzman, Rawin Hidalgo, Alexis Meneses, Dustin Mixon, Fabian Reimers, Consuelo Rodriguez, Mufit Sezer, and Maxine Song-Hurewitz, and also reports on independent work of Dan Edidin, Josh Katz, Reimers and Sezer.