Abstract: Introduced by Chi Li, the local volume of a klt singularity plays an important role in the study of K-stability of Fano varieties, their moduli spaces and boundedness of singularities. The notion is closely related to graded sequences of ideals, or filtrations, on the singularity by the work of Yuchen Liu. In this talk I will introduce a metric on the space of filtrations on a general local ring and discuss some basic properties. On a klt singulatiry, the geometry of spaces of filtrations under this metric is related to the stable degeneration theorem, which is the fundamental problem of local volumes. Such relations depict a picture similar to the question of existence and uniqueness of Kähler-Einstein metrics on a Fano variety. This talk is based on joint work of Harold Blum and Yuchen Liu and some ongoing work.

Abstract: Let $k$ be a field and $R$ be the power series ring $k[[T_1,\ldots,T_n]]$. Finite type schemes over $R$ were used in, for example, Hironaka's resolution of singularities and the works of de Fernex-Ein-Mustata on ACC of log canonical thresholds. In this talk, we discuss a systematic way of approximating finite type schemes over $R$ using schemes essentially of finite type over $k$, preserving various types of singularities and homological properties. This allows us to extend known results and constructions for varieties to finite type schemes over $R$, including formulas for multiplier ideals, deformation of singularities, and big Cohen-Macaulay algebras.

Abstract: Let S be a Shimura variety such that every connected component of the space of complex points of S arises as the quotient of a Hermitian symmetric domain by a torsion-free arithmetic group. In the 1970s, Borel proved that any holomorphic map from a complex algebraic variety V into such a Shimura variety S is algebraic! In this talk, I'll discuss joint work with Anand Patel, Ananth Shankar, and Xinwen Zhu on a p-adic version of this result.

Abstract: I will talk about two results. The first is a new case of the BSD conjecture, contained in a joint work with Hamacher and Zhao. Namely, we prove the conjecture for elliptic curves of height 1 over a global function field of genus 1 under a mild assumption. This is obtained by specializing a more general theorem on the Tate conjecture. The key geometric idea is an application of rigidity properties of the variations of Hodge structures to study deformation of line bundles in positive and mixed characteristic. Then I will talk about a generalization of such deformation results recently obtained with Urbanik. Namely, we show that for a sufficiently big arithmetic family of smooth projective varieties, there is an open dense subscheme of the base over which all line bundles in positive characteristics can be obtained by specializing those in characteristic 0.

Abstract: Sir Simon Donaldson conjectured that there should exist a virtual fundamental class on the moduli space of surfaces of general type inspired by the geometry of complex structures on the general type surfaces. In this talk I will present a method to construct the virtual fundamental class on the moduli stack of lci (locally complete intersection) covers over the moduli stack of general type surfaces with only semi-log-canonical singularities. A tautological invariant is defined by taking the integration of the power of the first Chern class of the CM line bundle over the virtual fundamental class. This can be taken as a generalization of the tautological invariants on the moduli space of stable curves to the moduli space of stable surfaces. If time permits, we also talk about the possible methods to construct a virtual fundamental class on the Alexeev moduli space of stable maps from semi-log-canonical surfaces to projective varieties.

Abstract: A Tevelev degree is a type of Gromov-Witten invariant where the domain curve is fixed in moduli. After reviewing the basic definitions and the previously known results, I will discuss enumerativity of Tevelev degrees for several classes of Fano varieties. This is based on joint work with Lehmann, Lian, Riedl, Starr, and Tanimoto.

Abstract: The most basic algebraic varieties are projective spaces, and their closest relatives are rational varieties. These are varieties that admit a 1-to-1 parametrization by projective space on a dense open subset; hence, rational varieties are the easiest varieties to understand. Historically, rationality problems have been of great importance in algebraic geometry; for example, Severi was interested in finding rational parametrizations for moduli spaces of Riemann surfaces (algebraic curves), and the classical Lüroth problem was concerned with determining the rationality of certain varieties. Over fields that are not algebraically closed (such as the rational numbers), the arithmetic of the field adds additional subtleties to the rationality problem. When the dimension of the variety is at most 2, there are effective criteria to determine rationality. However, in higher dimensions, there are no such known criteria, even after restricting to threefolds that become rational over the algebraic closure of the ground field. In this talk, I will first give a survey of some results on rationality of algebraic varieties. Then I will explain results studying a rationality criteria for 3-dimensional varieties over non-algebraically-closed fields.

Abstract: In my talk, I will discuss various methods of measuring the mildness of complex and arithmetic singularities defined by polynomial equations, with a particular focus on the newly established connections between birational geometry, commutative algebra, and arithmetic geometry.

Abstract: A major pursuit in algebraic geometry for many decades has been syzygies, which is the study of the qualitative features of equations defining algebraic varieties. This topic is by now fairly well developed for curves, where we now have several theorems describing when exactly the syzygy groups vanish, assuming the curve is embedded by a line bundle which is either canonical or sufficiently positive. But we know very little in the higher dimensional case, embarrassingly even in the simplest possible case of projective space embedded by a positive degree line bundle (Veronese varieties). In this case, we do have a good conjecture by Ein-Lazarsfeld predicting exactly when the syzygy groups vanish. I will explain a proof of this conjecture in the case of the two extremal ends of it.

Abstract: A famous conjecture of Manin predicts an asymptotic formula for counting the number of rational points of bounded height on a Fano variety defined over a number field. In the 1990s, Batyrev developed a heuristic argument for a version of Manin's Conjecture over finite fields that assumes irreducibility of certain spaces of embedded rational curves. Though Batyrev's heuristics and Manin's initial conjecture are false in general, Geometric Manin's Conjecture (GMC) translates Batyrev's heuristic for Manin's Conjecture to statements about free rational curves on Fano varieties. In this talk, I will first review this translation and motivate the framework of GMC with concrete examples. I will then describe a recent proof of GMC for smooth Fano threefolds over the complex numbers by appealing to relationships between this framework and the Mori structures of Fano threefolds.

Abstract: We introduce a new invariant of G-varieties, the dual complex, which roughly measures how divisors in the complement of the free locus intersect. We show that the top homology group of this complex is an equivariant birational invariant of G-varieties. As an application, we demonstrate the non-linearizability of certain large abelian group actions on smooth hypersurfaces in projective space of any dimension and degree at least 3.

Abstract: Some moduli spaces in algebraic geometry come with a Kaehler metric of constant negative holomorphic curvature or more precisely, can be identified (usually via a period map) with a Zariski open subset of a complex ball quotient. About 27 years ago Allcock found a ball quotient of dimension 13, which is intriguing for two reasons: its expected connections with some of the sporadic finite simple groups (such as the Monster group) and the likelihood of it having a modular interpretation. I shall first review the most important known examples of ball quotients with a modular interpretation (all of dimension at most 10) and then describe recent progress on the 'moonshine properties' of the Allcock ball quotient.

Abstract: In this talk we discuss how the consideration of $A_r$ singularities, singularities which are of the form $k[x,y]/(y^2-x^{r+1})$, arose from the historical development of $mathcal M_{g,n}$ and $\overline{\mathcal M}_{g,n}$, and how the inclusion of these singularities enables computations of the integral Chow ring of $\overline{\mathcal M}_{g,n}$ for new values of $(g,n)$.

Abstract: In this talk I will give an overview of some recent work, joint with Samuel Grushevsky, Klaus Hulek, and Radu Laza, on the geometry and topology of compactifications of the moduli spaces of cubic threefolds and cubic surfaces. A focus of the talk will be on some results regarding non-isomorphic smooth compactifications of the moduli space of cubic surfaces, showing that two natural desingularizations of the moduli space have the same cohomology, and are both blow-ups of the moduli space at the same point, but are nevertheless, not isomorphic, and in fact, not even K-equivalent. I will also discuss a related moduli space, the moduli space of cubic surfaces with a marked line.

Abstract: It is well-known that the Torelli map, that turns a smooth curve of genus g into its Jacobian (a principally polarized abelian variety of dimension g), extends to a map from the Deligne-Mumford moduli of stable curves to the moduli of semi-abelic varieties by Alexeev. Moreover, it is also known that the Torelli map does not extend over the alternative compactifications of the moduli of curves as described by the Hassett-Keel program, including the moduli of pseudostable curves (can have nodes and cusps but not elliptic tails). But it is not yet known whether the Torelli map extends over alternative compactifications of the moduli of curves described by Smyth; what about the moduli of curves of genus g with rational m-fold singularities, where m is a positive integer bounded above? As a joint work in progress with Jesse Kass and Matthew Satriano, I will describe moduli spaces of curves with m-fold singularities (with topological constraints) and describe how far the Torelli map extends over such spaces into the Alexeev compactifications.

Abstract: (Joint with Philippe Nadeau and Vasu Tewari) We describe a family of polynomial endomorphisms analogous to Schubert divided differences for working with quasisymmetric polynomials, where certain toric Richardson varieties play and "forest polynomials" play the role of Schubert cycles and Schubert polynomials. The operations satisfy the surprising commutation relation TiTj=TjT_{i+1} for i>j.

Abstract: For a smooth algebraic surface over complex numbers, the Picard number $\rho$ is bounded above by the Hodge number $h^{1,1}$. A surface has maximal Picard number if $\rho=h^{1,1}$. Examples of this kind are rare, particularly when the Kodaira dimension is at least zero. Shioda's modular surface S(N) is the universal family of elliptic curves with the level N structure. It is an elliptic surface with a maximal Picard number. When N is large, it has Kodaira dimensions one. However, such a surface has only torsion sections. The natural question is, is there an example of Kodaira dimension one elliptic surface with maximal Picard number and has nonzero Mordell-Weil rank? In joint work with Donu Arapura, we answer the question positively. We found an elliptic surface over an elliptic curve with maximal Picard number, together with a section of infinite order.

Abstract: A projective variety X is called Calabi-Yau if its canonical divisor is Q-linearly equivalent to zero. The smallest positive integer m with mK_X linearly equivalent to zero is called the index of X. Together with L. Esser and B. Totaro, we use ideas from mirror symmetry to construct Calabi-Yau varieties with index growing doubly exponentially with dimension. We conjecture they are the largest index in each dimension based on evidence in low dimensions. We also give Calabi-Yau varieties with large orbifold Betti numbers or small minimal log discrepancy.