Seminar Talks
All talks are at 12pm Mountain time (Salt Lake City Time)
Links to all the videos so far are below.
May-31 -- Hélène Esnault (Freie Universität Berlin)
Title: Finite presentation of the tame fundamental group
Abstract: Recall that if X is smooth complex projective, its underlying complex topological space is a finite CW complex, thus its topological fundamental group is finitely presented, thus its profnite completion, that is by the Riemann existence theorem its étale fundamental group, is as a profinite group finitely presented as well. If X is a smooth variety over an algebraic closed char. p>0 field, which admits a good compactification, we prove that its tame fundamental group is finitely presented as a profinite group.
Joint work with Mark Schusterman and V. Srinivas.
Video: Link to the video
June-1 -- János Kollár (Princeton)
Title: Class group and Picard group of singularities
Abstract: I will discuss the class group and Picard group of singularities, focusing on their structure for dlt singularities in positive characteristic.
Video: Link to the video
June-2 -- Rankeya Datta (University of Illinois at Chicago)
Title: Splinters -- an introduction and open questions
Abstract: Thanks to the Direct Summand Theorem, splinters are now known to be a characteristic independent notion of singularity like complete intersection, Gorenstein and Cohen-Macaulay singularities. However, splinters exhibit varied behavior based on the characteristic of the underlying ring or scheme. This talk will survey this class of singularities that has seen recent applications to vanishing theorems and a mixed characteristic minimal model program. We will highlight basic open questions about splinters and mention partial progress toward these questions. The talk is based on joint work with Kevin Tucker.
Video: Link to the video
Notes: Link to the notes
June-3 -- Kristin DeVleming (UC San Diego)
Title: K moduli of quartic K3 surfaces
Abstract: We will discuss a family of compactifications of moduli spaces of log Fano pairs coming from K-stability, and discuss an application to moduli of quartic K3 surfaces, with a focus on the locus of hyperelliptic K3s that arise as double covers of P1xP1 branched over a (4,4) curve. We will show that K-stability provides a natural way to interpolate between the GIT moduli space and the Baily-Borel compactification and will relate this interpolation to VGIT wall crossings. This is joint work with Kenny Ascher and Yuchen Liu.
Video: Link to the video
June-4 -- Jack Jeffries (University of Nebraska)
Title: A Jacobian criterion for nonsingularity in mixed characteristic.
Abstract: In this talk we will discuss a version of the Jacobian criterion for the singular locus that holds in mixed characteristic. Our theorem uses Joyal and Buium's notion of p-derivation. This is based on joint work with Melvin Hochster.
Video: Link to the video
June-7 -- Eloísa Grifo (UC Riverside)
Title: Symbolic powers in mixed characteristic
Abstract: In a polynomial ring over a perfect field, the symbolic powers of a radical ideal consist of the polynomials that vanish to order n on the corresponding variety, and can be described via differential operators. If we replace the field with a DVR, we need both differential operators and Joyal and Buium's notion of a p-derivation to give an analogous result. In this talk, we will discuss this application of p-derivations to commutative algebra, a joint project with Alessandro De Stefani and Jack Jeffries, including more recent work on an explicit Chevalley lemma for direct summands of polynomial rings.
Video: Link to the video
June-8 -- Harold Blum (Stony Brook University)
Title: An algebraic analogue of the Hamilton-Tian Conjecture
Abstract: The Hamilton-Tian Conjecture implies that any smooth complex Fano variety admits a degeneration to a Fano variety admitting a Kahler-Ricci soliton, which is a type of canonical metric.
In this talk, I will discuss an algebraic analogue of this statement phrased in the language of K-stability. Applications of this result include
(1) any possibly singular Fano variety admits a two step degeneration to a Fano variety admitting a Kahler-Ricci soliton and
(2) the moduli theory of K-unstable Fano varieties.
This talk is based on joint work with Yuchen Liu, Chenyang Xu, and Ziquan Zhuang. The main results build on previous work of Han and Li and a recent finite generation result of Liu, Xu, and Zhuang.
Video: Link to the video
June-9 -- Joe Waldron (Michigan State University)
Title: Purely inseparable Galois theory
Abstract: Given a field $K$ of characteristic $p$, a classical result of Jacobson provides a Galois correspondence between finite purely inseparable subfields of exponent one (those with $K^p\subset L\subset K$), and sub-restricted Lie algebras of $\mathrm{Der}(K)$. I will discuss joint work with Lukas Brantner in which we extend this Galois correspondence to subfields of arbitrary exponent using methods from derived algebraic geometry.
Video: Link to the video
Notes: Link to the notes
June-10 -- Carolina Araujo (Instituto Nacional de Matemática Pura e Aplicada)
Title: The Calabi problem for Fano threefolds
Abstract: In this talk, I will discuss recent progress on the Calabi problem for Fano threefolds. There are 105 deformation families of smooth Fano threefolds, which have been classified by Iskovskikh, Mori and Mukai. We determine whether or not the general member of each of these 105 families admits a Kähler-Einstein metric. In some cases, the general member of the family admits a Kähler-Einstein metric, while at least one member does not. A finer problem consists in classifying, within each family, which smooth Fano threefolds admit a Kähler-Einstein metric. This is accomplished for most of the families, and there is a conjectural picture for the remaining ones. This is a joint project with Ana-Maria Castravet, Ivan Cheltsov, Kento Fujita, Anne-Sophie Kaloghiros, Jesus Martinez-Garcia, Constantin Shramov, Hendrik Süss and Nivedita Viswanathan.
Video: Link to the video
June-11 -- Chenyang Xu (Princeton)
Title: Local K-stability theory
Abstract: Paralleling to the study of K-stability of Fano varieties, there is also a local stability theory guided by the study of the minimizing valuations of the normalized volume function on the "non-archimedean link” of a klt singularity. In this talk, we will discuss the recent progress on this topic, with a focus on the uniqueness of the minimizer up to rescaling (joint with Ziquan Zhuang).
Video: Link to the video