Algebra and Algebraic Geometry Seminar Fall 2023: Difference between revisions
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|[https://www.math.harvard.edu/people/bogdanova-ekaterina/ Ekaterina Bogdanova (Harvard)] | |[https://www.math.harvard.edu/people/bogdanova-ekaterina/ Ekaterina Bogdanova (Harvard)] | ||
|Sheaves of categories on the moduli stack of local systems on the formal punctured disk via factorization | |[[#Ekaterina Bogdanova|Sheaves of categories on the moduli stack of local systems on the formal punctured disk via factorization]] | ||
|Dima | |Dima | ||
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In order to answer basic questions in modular and geometric representation theory, Bezrukavnikov and Kaledin introduced quantizations of symplectic varieties in positive characteristic. These are certain noncommutative algebras over a field of positive characteristic which have a large center. I will discuss recent work describing how to construct an important class of modules over such algebras. | In order to answer basic questions in modular and geometric representation theory, Bezrukavnikov and Kaledin introduced quantizations of symplectic varieties in positive characteristic. These are certain noncommutative algebras over a field of positive characteristic which have a large center. I will discuss recent work describing how to construct an important class of modules over such algebras. | ||
===Andrei Negut=== | ===Andrei Negut === | ||
'''Computing K-HA's of quivers''' | '''Computing K-HA's of quivers''' | ||
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Many interesting moduli stacks M in geometric representation theory admit interesting K-theoretic Hall algebras (K-HAs), defined by endowing the algebraic K-theory of M with an appropriate convolution product. While these algebras are notoriously hard to compute, they have an interesting relative called the shuffle algebra S. When M is a moduli stack of quiver representations, S is given by a collection of ideals inside polynomial rings, and their study can be reduced to commutative algebra. Fortunately/unfortunately, the commutative algebra in question is challenging, and we do not yet know of a complete description for a general quiver. In this talk, I will explain the general framework behind this problem, and survey results for the following special cases of quivers: | Many interesting moduli stacks M in geometric representation theory admit interesting K-theoretic Hall algebras (K-HAs), defined by endowing the algebraic K-theory of M with an appropriate convolution product. While these algebras are notoriously hard to compute, they have an interesting relative called the shuffle algebra S. When M is a moduli stack of quiver representations, S is given by a collection of ideals inside polynomial rings, and their study can be reduced to commutative algebra. Fortunately/unfortunately, the commutative algebra in question is challenging, and we do not yet know of a complete description for a general quiver. In this talk, I will explain the general framework behind this problem, and survey results for the following special cases of quivers: | ||
* double quivers arising in the theory of Nakajima quiver varieties | *double quivers arising in the theory of Nakajima quiver varieties | ||
* quivers corresponding to symmetric Cartan matrices, yielding simply laced quantum loop groups | * quivers corresponding to symmetric Cartan matrices, yielding simply laced quantum loop groups | ||
* quivers associated to toric Calabi-Yau threefolds in mathematical physics | *quivers associated to toric Calabi-Yau threefolds in mathematical physics | ||
=== Daniel Bragg === | ===Daniel Bragg=== | ||
'''A Stacky Murphy’s Law for the Stack of Curves''' | '''A Stacky Murphy’s Law for the Stack of Curves''' | ||
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We show that every Deligne-Mumford gerbe over a field occurs as the residual gerbe of a point of the moduli stack of curves. Informally, this means that the moduli space of curves fails to be a fine moduli space in every possible way. We also show the same result for a list of other natural moduli problems. This is joint work with Max Lieblich. | We show that every Deligne-Mumford gerbe over a field occurs as the residual gerbe of a point of the moduli stack of curves. Informally, this means that the moduli space of curves fails to be a fine moduli space in every possible way. We also show the same result for a list of other natural moduli problems. This is joint work with Max Lieblich. | ||
=== Xinchun Ma === | ===Xinchun Ma=== | ||
'''Filtrations on the finite dimensional representations of rational Cherednik algebras''' | '''Filtrations on the finite dimensional representations of rational Cherednik algebras''' | ||
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Under the Gordon-Stafford functor, every filtered representation of the type A rational Cherednik algebra corresponds to an equivariant coherent sheaf on the Hilbert scheme of points on the plane. Under the decategorification of this functor, the images of the finite-dimensional representations are conjectured to be closely related to the torus knot superpolynomials (with some special cases proved). There are several candidates for the filtrations coming from algebraic or geometric formulations conjectured to coincide with each other. I'll talk about recent developments on these conjectures including my own work in progress. | Under the Gordon-Stafford functor, every filtered representation of the type A rational Cherednik algebra corresponds to an equivariant coherent sheaf on the Hilbert scheme of points on the plane. Under the decategorification of this functor, the images of the finite-dimensional representations are conjectured to be closely related to the torus knot superpolynomials (with some special cases proved). There are several candidates for the filtrations coming from algebraic or geometric formulations conjectured to coincide with each other. I'll talk about recent developments on these conjectures including my own work in progress. | ||
=== Junyan Zhao === | ===Junyan Zhao=== | ||
==== Moduli of curves and K-stability ==== | ====Moduli of curves and K-stability ==== | ||
The K-moduli theory provides us with an approach to study moduli of curves. In this talk, I will introduce the K-moduli of certain log Fano pairs and how it relates to moduli of curves. We will see that the K-moduli spaces interpolate between different compactifications of moduli of curves. In particular, the K-moduli gives the last several Hassett-Keel models of moduli of curves of genus six. | The K-moduli theory provides us with an approach to study moduli of curves. In this talk, I will introduce the K-moduli of certain log Fano pairs and how it relates to moduli of curves. We will see that the K-moduli spaces interpolate between different compactifications of moduli of curves. In particular, the K-moduli gives the last several Hassett-Keel models of moduli of curves of genus six. | ||
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Syzygies of algebraic varieties have long been a topic of intense interest among algebraists and geometers alike. After the pioneering work of Mark Green on curves, numerous attempts have been made to extend some of these results to higher dimensions. It has been proposed that the syzygies of adjoint linear series L=K+mA, with A ample is a natural analogue for higher dimensions to explore. The very ampleness of adjoint linear series is not known for even threefolds. So the question that has been open for many years is the following (Question): If A is base point free and ample, does L satisfy property N_p for m>=n+1+p? Ein and Lazarsfeld proved this when A is very ample in 1991. In a joint work with Justin Lacini, we give a positive answer to the original question above. | Syzygies of algebraic varieties have long been a topic of intense interest among algebraists and geometers alike. After the pioneering work of Mark Green on curves, numerous attempts have been made to extend some of these results to higher dimensions. It has been proposed that the syzygies of adjoint linear series L=K+mA, with A ample is a natural analogue for higher dimensions to explore. The very ampleness of adjoint linear series is not known for even threefolds. So the question that has been open for many years is the following (Question): If A is base point free and ample, does L satisfy property N_p for m>=n+1+p? Ein and Lazarsfeld proved this when A is very ample in 1991. In a joint work with Justin Lacini, we give a positive answer to the original question above. | ||
===Ekaterina Bogdanova=== | |||
'''Sheaves of categories on the moduli stack of local systems on the formal punctured disk via factorization''' | |||
Given a DG category acted on by the category of quasi-coherent sheaves on ''LocSys''<sub>G</sub>(''D''{{sup|&circ}}) (the stack of G-local systems on the punctured formal disk <math>D^{\circ}</math>), one can define a factorization Rep(G)-module category. Following ideas of Beilinson and Drinfeld, Gaitsgory conjectured that this construction loses no information: that it gives a fully faithful 2-functor QCoh(LocSys_G(D^{\circ}))-mod (DGCat) —> Rep(G)-mod^fact(DGCat). I will give a quick introduction to the local Geometric Langlands program, discuss preliminaries, and the role of the above conjecture in this context. If time permits, we will discuss a partial result in the direction of the conjecture. Namely, the fully faithfulness for QCoh(LocSys_G(D^{\circ}))-modules set-theoretically supported over the stack of local systems with restricted variation on the formal punctured disk. | |||
Revision as of 23:49, 22 November 2023
The seminar normally meets 2:30-3:30pm on Fridays, in the room VV B135.
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Fall 2023 Schedule
| date | speaker | title | host/link to talk |
|---|---|---|---|
| September 15 | Joshua Mundinger | Quantization in positive characteristic | local |
| September 22 | Andrei Negut | Computing K-HA's of quivers | local |
| October 6 | Daniel Bragg (Utah) | A Stacky Murphy’s Law for the Stack of Curves | Josh |
| October 13 | Xinchun Ma (UChicago) | Filtrations on the finite dimensional representations of rational Cherednik algebras | Josh |
| November 3 | Junyan Zhao | Moduli of curves and K-stability | Peter W |
| November 17 | Purnaprajna Bangere | Syzygies of adjoint linear series on projective varieties | Michael K |
| December 1 | Ekaterina Bogdanova (Harvard) | Sheaves of categories on the moduli stack of local systems on the formal punctured disk via factorization | Dima |
| December 8 | Wanchun (Rosie) Shen (Harvard) | TBA | Andrei |
Abstracts
Joshua Mundinger
Quantization in positive characteristic
In order to answer basic questions in modular and geometric representation theory, Bezrukavnikov and Kaledin introduced quantizations of symplectic varieties in positive characteristic. These are certain noncommutative algebras over a field of positive characteristic which have a large center. I will discuss recent work describing how to construct an important class of modules over such algebras.
Andrei Negut
Computing K-HA's of quivers
Many interesting moduli stacks M in geometric representation theory admit interesting K-theoretic Hall algebras (K-HAs), defined by endowing the algebraic K-theory of M with an appropriate convolution product. While these algebras are notoriously hard to compute, they have an interesting relative called the shuffle algebra S. When M is a moduli stack of quiver representations, S is given by a collection of ideals inside polynomial rings, and their study can be reduced to commutative algebra. Fortunately/unfortunately, the commutative algebra in question is challenging, and we do not yet know of a complete description for a general quiver. In this talk, I will explain the general framework behind this problem, and survey results for the following special cases of quivers:
- double quivers arising in the theory of Nakajima quiver varieties
- quivers corresponding to symmetric Cartan matrices, yielding simply laced quantum loop groups
- quivers associated to toric Calabi-Yau threefolds in mathematical physics
Daniel Bragg
A Stacky Murphy’s Law for the Stack of Curves
We show that every Deligne-Mumford gerbe over a field occurs as the residual gerbe of a point of the moduli stack of curves. Informally, this means that the moduli space of curves fails to be a fine moduli space in every possible way. We also show the same result for a list of other natural moduli problems. This is joint work with Max Lieblich.
Xinchun Ma
Filtrations on the finite dimensional representations of rational Cherednik algebras
Under the Gordon-Stafford functor, every filtered representation of the type A rational Cherednik algebra corresponds to an equivariant coherent sheaf on the Hilbert scheme of points on the plane. Under the decategorification of this functor, the images of the finite-dimensional representations are conjectured to be closely related to the torus knot superpolynomials (with some special cases proved). There are several candidates for the filtrations coming from algebraic or geometric formulations conjectured to coincide with each other. I'll talk about recent developments on these conjectures including my own work in progress.
Junyan Zhao
Moduli of curves and K-stability
The K-moduli theory provides us with an approach to study moduli of curves. In this talk, I will introduce the K-moduli of certain log Fano pairs and how it relates to moduli of curves. We will see that the K-moduli spaces interpolate between different compactifications of moduli of curves. In particular, the K-moduli gives the last several Hassett-Keel models of moduli of curves of genus six.
Purnaprajna Bangere
Syzygies of adjoint linear series on projective varieties
Syzygies of algebraic varieties have long been a topic of intense interest among algebraists and geometers alike. After the pioneering work of Mark Green on curves, numerous attempts have been made to extend some of these results to higher dimensions. It has been proposed that the syzygies of adjoint linear series L=K+mA, with A ample is a natural analogue for higher dimensions to explore. The very ampleness of adjoint linear series is not known for even threefolds. So the question that has been open for many years is the following (Question): If A is base point free and ample, does L satisfy property N_p for m>=n+1+p? Ein and Lazarsfeld proved this when A is very ample in 1991. In a joint work with Justin Lacini, we give a positive answer to the original question above.
Ekaterina Bogdanova
Sheaves of categories on the moduli stack of local systems on the formal punctured disk via factorization
Given a DG category acted on by the category of quasi-coherent sheaves on LocSysG(DTemplate:Sup) (the stack of G-local systems on the punctured formal disk [math]\displaystyle{ D^{\circ} }[/math]), one can define a factorization Rep(G)-module category. Following ideas of Beilinson and Drinfeld, Gaitsgory conjectured that this construction loses no information: that it gives a fully faithful 2-functor QCoh(LocSys_G(D^{\circ}))-mod (DGCat) —> Rep(G)-mod^fact(DGCat). I will give a quick introduction to the local Geometric Langlands program, discuss preliminaries, and the role of the above conjecture in this context. If time permits, we will discuss a partial result in the direction of the conjecture. Namely, the fully faithfulness for QCoh(LocSys_G(D^{\circ}))-modules set-theoretically supported over the stack of local systems with restricted variation on the formal punctured disk.