Algebra and Algebraic Geometry Seminar Fall 2021: Difference between revisions
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This is the schedule for Fall 2021; here is [[Algebra and Algebraic Geometry Seminar|the current schedule]]. | |||
The Seminar will take place on Fridays at 2:30 pm, either virtually (via Zoom) or in person, in room B235 Van Vleck. | |||
==Algebra and Algebraic Geometry Mailing List== | ==Algebra and Algebraic Geometry Mailing List== | ||
*Please join the AGS mailing list by sending an email to ags+join@g-groups.wisc.edu to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link). | *Please join the AGS mailing list by sending an email to ags+join@g-groups.wisc.edu to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link). | ||
| Line 18: | Line 19: | ||
|- | |- | ||
|September 24 | |September 24 | ||
|Michael Kemeny (local) | |Michael Kemeny (local, in person) | ||
|The Rank of Syzygies | |The Rank of Syzygies | ||
| | | | ||
| Line 30: | Line 31: | ||
|October 1 | |October 1 | ||
|Michael K Brown (Auburn University) | |Michael K Brown (Auburn University) | ||
| | |Tate resolutions as noncommutative Fourier-Mukai transforms | ||
| | | | ||
|Daniel | |Daniel | ||
|- | |- | ||
|October 8 | |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 | |October 15 | ||
|Michael Perlman (Minnesota; virtual) | |||
|Mixed Hodge structure on local cohomology with support in determinantal varieties | |||
| | | | ||
| | |Daniel | ||
|- | |- | ||
|October 22 | |October 22 | ||
| Line 53: | Line 54: | ||
|- | |- | ||
|October 29 | |October 29 | ||
| | |CA+ meeting [ https://www-users.cse.umn.edu/~cberkesc/CA/CA2021.html] | ||
| | | | ||
| | | | ||
| Line 59: | Line 60: | ||
|- | |- | ||
|November 5 | |November 5 | ||
| | | | ||
|- | |- | ||
|November 12 | |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 | |November 19 | ||
|Ritvik Ramkumar (UC Berkeley; Zoom) | |||
|Rational singularities of nested Hilbert schemes. | |||
| | | | ||
| | |Daniel | ||
|- | |- | ||
|November 26 | |November 26 | ||
| Line 82: | Line 90: | ||
|- | |- | ||
|December 3 | |December 3 | ||
| | |Eric Ramos | ||
|Equivariant log-concavity | |||
| | | | ||
| | | | ||
| Line 88: | Line 97: | ||
|- | |- | ||
|December 10 | |December 10 | ||
|Federico Barbacovi (University College London; Zoom) | |||
|Categorical dynamical systems and Gromov—Yomdin type theorems | |||
| | | | ||
|Andrei | |||
|- | |||
|April 8 | |||
|Haydee Lindo | |||
| | | | ||
| | | | ||
| | |Daniel | ||
|- | |- | ||
|} | |} | ||
| Line 98: | Line 113: | ||
===Speaker Name=== | ===Speaker Name=== | ||
Title: | ===Michael Kemeny=== | ||
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''. | |||
===Michael Brown=== | |||
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. | |||
===Peter Wei=== | |||
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. | |||
===Michael Perlman=== | |||
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. | |||
===Jinhyung Park=== | |||
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. | |||
===Daniel Erman=== | |||
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. | |||
===Ritvik Ramkumar=== | |||
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. | |||
===Eric Ramos=== | |||
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. | |||
===Federico Barbacovi=== | |||
Title: Categorical dynamical systems and Gromov—Yomdin type theorems | |||
Abstract: | 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. | ||
Latest revision as of 00:07, 16 February 2022
This is the schedule for Fall 2021; here is the current schedule.
The Seminar will take place on Fridays at 2:30 pm, either virtually (via Zoom) or in person, in room B235 Van Vleck.
Algebra and Algebraic Geometry Mailing List
- Please join the AGS mailing list by sending an email to ags+join@g-groups.wisc.edu to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).
COVID-19 Update
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 5
|
|||||
| 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 | ||
| November 26 | Thanksgiving | ||||
| 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 |
Abstracts
Speaker Name
Michael Kemeny
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.
Michael Brown
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.
Peter Wei
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.
Michael Perlman
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.
Jinhyung Park
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.
Daniel Erman
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.
Ritvik Ramkumar
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.
Eric Ramos
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.
Federico Barbacovi
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.