Titles and abstracts for the conference talks
Overview Talk: An introduction (invitation? celebration?) to the work of Karen E. Smith (Notes) (Talk by Kevin Tucker)
Bhargav Bhatt (Princeton/IAS, virtual)
Title: A new construction of the Deligne-Du Bois complex
Abstract: The algebraic de Rham complex of a smooth algebraic variety is a fundamental tool across algebraic geometry (and beyond). In characteristic 0, for possibly singular varieties, Hodge theory provides a well-behaved (and useful) variant of the de Rham complex --- the Deligne-Du Bois complex. After recalling this story, I will report on some work in progress explaining why the latter complex arises naturally from non-archimedean geometry.
Rankeya Datta (Missouri)
Title: Uniform approximation of Abhyankar valuation ideals in mixed characteristic
Abstract: In joint work with Ein and Lazarsfeld, Karen Smith used asymptotic multiplier ideals to prove a surprising uniform containment result for valuation ideals associated with an Abhyankar valuation that is centered on a smooth point of a variety over a field of characteristic 0. A consequence of this result is that the topologies defined by the valuation ideals of two Abhyankar valuations sharing a common smooth center are comparable in a linear sense, that is, Izumi’s theorem holds for Abhyankar valuations and not just divisorial ones. Analogs of these results were proved by me in prime characteristic using asymptotic test ideals. In this talk we will highlight a mixed characteristic version of the results of Ein-Lazarsfeld-Smith using recent advances in mixed characteristic multiplier/test ideals.
Links: Slides
Daniel Duarte (UNAM)
Title: A counterexample to Nash's question on resolution of singularities.
Abstract: The Nash blowup of an algebraic variety is a modification that replaces singular points by limits of tangent spaces. It was proposed to resolve singularities by iterating Nash blowups. In this talk we present a counterexample to this question. This is a joint work with Federico Castillo, Maximiliano Leyton and Álvaro Liendo.
Lawrence Ein (UIC)
Title: Syzygies and singularities of secant varieties of curves
Abstract: We will describe my joint work with Wenbo Niu and Jinhyung Park on syzygies and singularities of secant varieties of curves.
Eleonore Faber (Graz)
Title: Noncommutative resolutions of normal toric varieties and beyond
Abstract: This talk is about some results on noncommutative resolutions of toric varieties: Let R be the coordinate ring of a normal affine toric variety over a field of arbitrary characteristic. Then the module R^{1/q} of q-th roots of R is a direct sum of conic modules and End_R(R^{1/q}) provides a so-called noncommutative resolution of singularities of R for q >> 0, in particular, it is a ring of finite global dimension. This has first been shown by Spela Spenko and Michel Van den Bergh, and in joint work with Greg Muller and Karen Smith we gave an explicit description of the combinatorial structure of these endomorphism rings. I will comment on recent progress and joint work in progress with Milena Hering and Kevin Tucker on noncommutative resolutions for seminormal monoid algebras.
Sara Faridi (Dalhousie)
Title: The Algebra and Combinatorics of Extremal Ideals
Abstract: Extremal ideals are square-free monomial ideals which provide a sharp bound for algebraic properties of all square-free monomial ideals, as well as all of their powers. Though they are rather simple to define and display many pleasing symmetries, they are difficult to study due to their size, as well as the algebraic information they encode.
This talk will be a survey of what extremal ideals and the work surrounding them. I will report on joint work(s) with Trung Chau, Susan Cooper, Art Duval, Sabine El-Khoury, Thiago Holleben, Hasan Mahmood, Sarah Mayes-Tang, Susan Morey, Liana Sega, and Sandra Spiroff.
Mel Hochster (Michigan, virtual)
Title: Research inspired by the work of Karen Smith
Abstract: The talk will focus on the research of Karen Smith over several decades, with special emphasis on several results that have both laid the foundations for and provided inspiration for work done by many other mathematicians. Part of the talk will discuss recent results of the speaker and Yongwei Yao that fit into this category.
Links: Slides
Jack Jeffries (Nebraska)
Title: Singularities in Algebra and D-simplicity
Abstract: Given a commutative ring R, there is a (typically noncommutative) ring D_R of differential operators that acts on R. When R is a polynomial ring over a field of characteristic zero, this is just the ring generated by elements of R and partial derivatives; for other rings R, the ring of differential operators can be much harder to compute. In this talk, we will discuss some fundamental results of Karen E. Smith relating the action of differential operators D_R on R to the singularities of R, and a number of more recent results building on this theme.
Sándor Kovács (University of Washington)
Title: The injectivity theorem for m-Du Bois singularities
Abstract: Du Bois singularities may be considered a generalization of rational singularities which include important non-rational singularities that appear in the theory of compact moduli spaces of higher dimensional varieties. Recently this class was further generalized to the classes of m-Du Bois singularities for any natural number m. In this new realm, 0-Du Bois singularities agree with the original Du Bois singularities.
In this talk I will discuss a theorem and some of its consequences that states that for varieties with (m-1)-Du Bois singularities (for completeness note that everything is (-1)-Du Bois), the natural morphism from the Grothendieck dual of the m-th graded Du Bois complex to the Grothendieck dual of its zero-th cohomology sheaf is injective on cohomology.
This was proved by Karl Schwede and myself in the m=0 case and Mustata and Popa for arbitrary m for lc singularities and Popa, Shen, and Vo for arbitrary m for isolated singularities. This general version confirms a conjecture of Popa, Shen, and Vo and several of its consequences.
Shiji Lyu (UIC)
Title: Approximation of complete local rings
Abstract: For simplicity, let R be a power series ring over a field k. 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. The method works in other situations as well, giving other consequences, for example in étale cohomology. This is joint work in preparation with Shizhang Li and Bogdan Zavyalov.
Linquan Ma (Purdue)
Title: Lech's inequality and stability of local rings
Abstract: We explore Lech's inequality relating the colength and multiplicity of m-primary ideals in a Noetherian local ring. We introduce an invariant that measures the sharpness of Lech's inequality and show its connections with singularities of asymptotically semi-stable varieties and singularities arising from the minimal model program. We compute this invariant in various examples. This is a joint work in progress with Ilya Smirnov.
Greg Muller (Oklahoma)
Title: An invitation to cluster algebraic geometry
Abstract: Cluster algebras were introduced by Fomin and Zelevinsky to axiomatize patterns found in the canonical basis of functions on certain Lie groups. In this talk, I will give a whirlwind survey of the basic definitions, big theorems, and unexpected applications, all through the lens of the geometry of the associated variety/scheme. From the unexpectedly good (like the totally positive part and the friezes), to the unexpectedly bad (like non-Noetherian singularities), the geometry of cluster algebras is still producing surprises 25 years later.
Links: Slides
Peter McDonald (UIC)
Title: Ideal closure operations via resolution of singularities in characteristic zero
Abstract: An important story in commutative algebra over the past 45 years has been a connection between singularities defined using the Frobenius in characteristic p with those coming out of birational algebraic geometry in characteristic zero via resolution of singularities. However, while these singularities in characteristic p can be studied via tight closure theory, there was no known “closure operation” in characteristic zero whose tight-closure-like properties were derived from such resolutions and their associated vanishing theorems. In this talk, I will introduce such an operation, which we call Koszul-Hironaka (KH) closure, and discuss it’s relationship to tight closure, highlighting their similarities (colon-capturing, a version of the Briancon-Skoda property, etc.) and differences. This is joint work with Neil Epstein, Rebecca R.G., and Karl Schwede.
Alapan Mukhopadhyay (EPFL)
Title: Direct summands of Frobenius pushforwards
Abstract: Given two finitely generated modules over a prime characteristic local ring, we will discuss the problem of (asymptotically) computing the maximum number of copies of one module appearing as summands of the iterated Frobenius pushforwards of the other one. The results will extend the F-signature theory. Time permitting, we will show that the F-signature function has a second coefficient for rings with finite F-representation type. The talk will report results from an ongoing joint work with Ilya Smirnov.
Janet Page (North Dakota)
Title: Smooth Surfaces with Maximally Many Lines
Abstract: How many lines can lie on a smooth surface of degree d? This classical question in algebraic geometry has been studied since at least the mid 1800s, when Clebsch gave an upper bound of d(11d-24) for the number of lines on a smooth surface of degree d over the complex numbers. Since then, Segre and then Bauer and Rams have given sharper upper bounds, the latter of which also holds over fields of characteristic p > d. However, over a field of characteristic p < d, there are smooth projective surfaces of degree d which break these upper bounds. In this talk, I’ll give a new upper bound for the number of lines which can lie on a smooth surface of degree d which holds over any field. In addition, we’ll fully classify those surfaces which attain this upper bound. It turns out that these smooth surfaces are distinguished by their affine cones from the perspective of F-singularities, and we will discuss this connection. This talk is based on joint work with Tim Ryan and Karen Smith and will touch on joint work Anna Brosowsky and with Zhibek Kadyrsizova, Jennifer Kenkel, Jyoti Singh, Adela Vraciu, and Emily Witt.
Eamon Quinlan-Gallego (Utah)
Title: F-jumping numbers and monodromy eigenvalues for homogeneous nondegenerate polynomials in positive characteristic
Abstract: Let F be a homogeneous polynomial over a field of characteristic p>0 with an isolated singularity, and let X be (the compactification of) its fiber away from the origin. We show that there is a connection between the p-integral F-jumping numbers of F and the monodromy eigenvalues on the slope zero part of the crystalline cohomology of X. This is joint work in progress with Hiroki Kato and Daichi Takeuchi.
Kenta Sato (Chiba University)
Title: Extending one-forms on F-regular singularities
Abstract: For a normal variety X, we say that X satisfies the logarithmic extension theorem for i-forms if, for every proper birational morphism f : Y to X, every i-form on the regular locus of X extends to a logarithmic i-form on Y. In this talk, we explain the logarithmic extension theorem for one-forms on strongly F-regular singularities. This is joint work with Tatsuro Kawakami.
Irena Swanson (Purdue)
Title: Primary decompositions and powers of ideals
Abstract: This is an expository talk about properties of associated primes and of primary decompositions of ideals in commutative Noetherian rings, especially in the context of ordinary and Frobenius powers of fixed ideals. The focus will be on the underlying motivations and methods. Karen Smith's work features prominently in this area.
Shunsuke Takagi (Tokyo)
Title: Uniform positivity of F-signature under reduction modulo p
Abstract: F-signature is an important numeric invariant of singularities in positive characteristic, which detects strong F-regularity. Carvajal-Rojas, Schwede and Tucker asked whether the mod p reductions of a complex klt type singularity have uniformly positive F-signature for almost all primes p. In this talk, we discuss this question. In particular, we explain that this is indeed the case when the singularity is a reductive quotient singularity. This talk is based on joint work with Tatsuki Yamaguchi.
Michel Van den Bergh (Vrije Universiteit Brussel / Hasselt University)
Title: TBA
Abstract: TBA
Adela Vraciu (South Carolina)
Title: F-pure thresholds of homogeneous polynomials
Abstract: The log canonical threshold is an invariant that measures how singular a hypersurface over an algebraically closed field of characteristic zero is. The F-pure threshold is the positive characteristic analog. Hypersurfaces with smaller F-pure thresholds are more singular. In this talk, we explore the possible values that can be achieved as F-pure thresholds of homogeneous polynomials. This is joint work with Karen Smith.
Tatsuki Yamaguchi (Institute of Science Tokyo)
Title: F-pure singularities in equal characteristic zero
Abstract: We introduce the notion of ultra-F-pure singularities, a characteristic zero analogue of F-pure singularities. We show that this notion is closely related to singularities of dense F-pure type when the ring in question is Q-Gorenstein. As an application, we prove that dense F-pure type descends under pure morphisms between Q-Gorenstein rings.