Algebra and Algebraic Geometry Seminar Fall 2021
The Seminar will take place on Fridays at 2:30 pm, either virtually (via Zoom) or in person, in room B235 Van Vleck.
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As a result of Covid-19, the seminar for this semester will be a mix of virtual and in-person talks. The default Zoom link for the seminar is https://uwmadison.zoom.us/j/9502605167 (sometimes we will have to use a different meeting link, if Michael K cannot host that day).
Fall 2021 Schedule
|date||speaker||title||host/link to talk|
|September 24||Michael Kemeny (local, in person)||The Rank of Syzygies|
|October 1||Michael K Brown (Auburn University)||Tate resolutions as noncommutative Fourier-Mukai transforms||Daniel|
|October 8||Yi (Peter) Wei (local)||Geometric Syzygy Conjecture in char p, with reveries from Ogus’ result on a versal deformation of K3 surfaces||Michael|
|October 15||Michael Perlman (Minnesota; virtual)||Mixed Hodge structure on local cohomology with support in determinantal varieties||Daniel|
|October 22||Ritvik Ramkumar (UC Berkeley)||Something about Hilbert schemes, probably||Daniel|
|October 29||CA+ meeting [ https://www-users.cse.umn.edu/~cberkesc/CA/CA2021.html]|
|November 12 -- TALK AT NONSTANDARD TIME||Jinhyung Park at 9:00am (Zoom)||Asymptotic vanishing of syzygies of algebraic varieties|
|November 12||Daniel Erman at usual time (2:30pm)||The geometry of virtual syzygies|
|November 19||Ritvik Ramkumar (UC Berkeley; Zoom)||Rational singularities of nested Hilbert schemes.||Daniel|
|December 3||Eric Ramos||Equivariant log-concavity|
|December 10||Federico Barbacovi (University College London; Zoom)||Categorical dynamical systems and Gromov—Yomdin type theorems||Andrei|
|April 8||Haydee Lindo||Daniel|
Title: The Rank of Syzygies
Abstract: I will explain a notion of rank for the relations amongst the equations of a projective variety. This notion generalizes the classical notion of rank of a quadric and is just as interesting! I will spend most of the talk developing this notion but will also explain one result which tells us that, for a randomly chosen canonical curve, you expect all the linear syzygies to have the lowest possible rank. This is a sweeping generalization of old results of Andreotti-Mayer, Harris-Arbarello and Green, which tell us that canonical curves are defined by quadrics of rank four.
Title: Tate resolutions as noncommutative Fourier-Mukai transforms
Abstract: This is joint work with Daniel Erman. The classical Bernstein-Gel'fand-Gel'fand (or BGG) correspondence gives an equivalence between the derived categories of a polynomial ring and an exterior algebra. It was shown by Eisenbud-Fløystad-Schreyer in 2003 that the BGG correspondence admits a geometric refinement, which sends a sheaf on projective space to a complex of modules over an exterior algebra called a Tate resolution. The goal of this talk is to reinterpret Tate resolutions as noncommutative analogues of Fourier-Mukai transforms, and to discuss some applications.
Title: Geometric Syzygy Conjecture in char p, with reveries from Ogus’ result on a versal deformation of K3 surfaces
Abstract: We aim to study syzygies of canonical curves in char p. I will briefly introduce how to translate the questions on curves to questions on K3 surfaces, where the Lazarsfeld-Mukai bundle plays a great role. I will show how to use Ogus’ result on a versal deformation of K3 surfaces, to help us resolve the case for a general K3 surface. And finally, I will sketch the proof of Geometric Syzygy Conjecture for even genus curve assuming an effective lower bound on the characteristics.
Title: Mixed Hodge structure on local cohomology with support in determinantal varieties
Abstract: Given a closed subvariety Z in a smooth complex variety, the local cohomology modules with support in Z are functorially endowed with structures as mixed Hodge modules, implying that they are equipped with Hodge and weight filtrations that subtly measure the singularities of Z. We will discuss new calculations of these filtrations in the case when Z is a generic determinantal variety. As an application, we obtain the Hodge ideals for the determinant hypersurface. Joint work with Claudiu Raicu.
Title: Asymptotic vanishing of syzygies of algebraic varieties
Abstract: In this talk, we show Ein-Lazarsfeld's conjecture on asymptotic vanishing of syzygies of algebraic varieties. This result, together with Ein-Lazarsfeld's asymptotic nonvanishing theorem, describes the overall picture of asymptotic behaviors of the minimal free resolutions of the graded section rings of line bundles on a projective variety as the positivity of the line bundles grows.
Title: The geometry of virtual syzygies
Abstract: One of the foundational results connecting syzygies with algebraic geometry properties was Mark Green’s result on N_p conditions for smooth curves of high degree. A modern and streamlined proof of this result comes via Green’s Linear Syzygy Theorem. I will discuss very recent work with Michael Brown which proves a Multigraded Linear Syzygy Theorem and uses this to obtain the first known examples of “virtual" N_p conditions for smooth curves of high degree in other toric varieties. This is joint work with Michael Brown.
Title: Rational singularities of nested Hilbert schemes.
Abstract: For a smooth surface S the Hilbert scheme of points S^(n) is a well studied smooth parameter space. In this talk I will consider a natural generalization, the nested Hilbert scheme of points S^(n,m) which parameterizes pairs of 0-dimensional subschemes X \supseteq Y of S with deg(X) = n and deg(Y) = m. In contrast to the usual Hilbert scheme of points, S^(n,m) is almost always singular and it is known that S(n,1) has rational singularities. I will discuss some general techniques to study S^(n,m) and apply them to show that S^(n,2) also has rational singularities. This relies on a connection between S^(n,2) and a certain variety of matrices, and involves square-free Gröbner degenerations as well as the Kempf-Weyman geometric technique. This is joint work with Alessio Sammartano.
Tite: Equivariant log-concavity
Abstract: Log concave sequences have been ubiquitous in combinatorics for decades. For instance, June Huh famously proved that the Betti numbers of the complement of a complex hyperplane arrangement always form a log concave sequence. In this talk I will introduce an equivariant version of log concave sequences for representations of groups, and present a conjecture of Nick Proudfoot on such sequences arising from hyperplane arrangements. I will then show that one can use a numerical version of representation stability to prove infinitely many cases of this conjecture for configuration spaces. This is joint work with Jacob Matherne, Dane Miyata, and Nick Proudfoot.
Title: Categorical dynamical systems and Gromov—Yomdin type theorems
Abstract: Categorical dynamical systems were introduced by Dimitrov—Haiden—Katzarkov—Kontsevich to give a categorification of topological dynamical systems. To every categorical dynamical system one can associate its entropy, which measures the complexity of the system. While in the topological world this measure is given by a number, in the categorical world the entropy is a function. The value at zero of this function takes the name of categorical entropy and mirrors the role of the topological entropy. In this talk I will report on joint work with Jongmyeong Kim in which we investigate a categorical version of a theorem of Gromov and Yomdin and we propose a categorical interpretation of one of the properties of holomorphic functions. Such interpretation allows us to give a sufficient condition for a categorified version of Gromov—Yomdin’s theorem to hold.