Abstract: This work is part of a broader program to develop necessary commutative algebra tools for the semirings arising from tropicalization. We will discuss different notions of integrality which while equivalent for rings are not for idempotent semirings. We will define integral closure and give different characterizations. If time permits we will give some examples of normalization in tropical geometry. 

Abstract: We study asymptotic behavior of the stability thresholds of a big line bundle, and prove explicit bounds on the error terms. This improves a theorem of Jin--Rubinstein--Tian by removing the assumption on the existence of a divisorial minimizer. A key step in our proof is to show that the stability thresholds of a big line bundle can always be computed by quasi-monomial valuations. This generalizes Blum--Jonsson's result on the stability thresholds of an ample line bundle.

Abstract: Semirings naturally arise in various contexts, including tropical geometry and geometry over the field with one element. In this talk, I will introduce some basic aspects of semirings and explore their applications to scheme theory over semirings, with a particular focus on line bundles and vector bundles. This is joint work with James Borger.

Abstract: A branchvariety of a projective k-scheme X is a geometrically reduced scheme Y equipped with a finite map to X. Alexeev and Knutson showed the existence of a proper moduli space of branchvarieties with fixed numerical invariants, but the projectivity of this space remained an open question. In this talk, we will discuss positivity results for some line bundles related to the determinant line bundle on the moduli space of equidimensional branchvarieties. As a consequence, we establish that this moduli space is projective.

Abstract: Given a degeneration X of a complex algebraic variety over the punctured unit disk, one can associate to X,  a non-archimedean analytic space called its analytification X^an.  The analytification retains all the original information about X but in general can be very complicated. A beautiful aspect of the theory is that X^an contains a simple combinatorial subspace, called a skeleton, that X^an deformation retracts onto. In this talk, we’ll see how skeletons still capture the complex analytic periods of X and along the way prove a conjecture of Kontsevich-Soibelman in the case of log Calabi-Yau surfaces. This is joint work with Jonathan Lai. 

Abstract: In 2015, K. Fujita showed that for any n-dimensional K-semistable Fano manifold, the anti-canonical volume is always less than or equal to that of complex projective space (CP^n). In this talk, I will discuss my recent joint work with Chi Li on characterizing the second-largest volume. We prove that for any n-dimensional K-semistable Fano manifold X that is not isomorphic to CP^n, the volume is at most 2nⁿ, with the equality holds if and only if X is a smooth quadric hypersurface or CP^1 × CP^{n-1}. Our proof is based on a new connection between K-stability and minimal rational curves.

Abstract: In the algebraic theory of K-stability, one crucial problem is to show the graded algebra associated with certain valuations are finitely generated. For a class of "special valuations", such finite generation theorems are first proved by Liu-Xu-Zhuang for Fano varieties and by Xu-Zhuang for klt singularities. In this talk, I will explain how to extend the finite generation theorem to a larger class of valuations and graded algebras by studying an extended Rees algebra. This new method also provides some extra information on the graded algebras associated with valuations. 

Abstract: In joint work with S. Nakamura, I showed that the category of projective geometries and collineations can be fully and faithfully embedded into Connes and Consani's ΓSet model for modules over the field with one element. It follows that one can produce simplicial sets from projective geometries via Segal's "delooping" procedure. Set-representable orthomodular posets, a.k.a. Dynkin systems, generalize the σ-algebras of probability theory and are prima facie completely unrelated.

    In this talk I will outline a computation of the Γ-set associated to the trivial, or "thin," geometries on n points. This is equivalent to computing the Segal delooping of the n-fold coproduct of the Krasner hyperfield with itself. The output of the computation is a Γ-set whose value at the k₊= {✽,1,2,...,k} is the set of Dynkin systems on the power set of of k₊.

Abstract: The Enriques criterion for unirationality of conic bundles states that if X is a conic bundle over P^2, then X is unirational if and only if the conic bundle admits a rational multisection. Furthermore, if X is unirational but not rational, then such a rational multisection necessarily has even degree. In this talk, we report on joint work in progress with Jeffrey Diller and Eric Riedl studying when such a conic bundle X -> P^2 admits a degree 2 rational multisection.