# Difference between revisions of "Algebra and Algebraic Geometry Seminar Fall 2019"

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|September 20 | |September 20 | ||

− | | | + | |Michael Kemeny |

− | | | + | |The geometric syzygy conjecture |

− | | | + | |Local |

|- | |- | ||

|September 27 | |September 27 | ||

Line 47: | Line 47: | ||

|October 18 | |October 18 | ||

|Kevin Tucker (UIC) | |Kevin Tucker (UIC) | ||

− | | | + | |TBD |

− | | | + | |Daniel |

|- | |- | ||

|October 25 | |October 25 | ||

+ | |Reserved | ||

| | | | ||

− | | | + | |Dima |

− | |||

|- | |- | ||

|November 1 | |November 1 | ||

|Michael Brown | |Michael Brown | ||

− | | | + | |Standard Conjecture D for Matrix Factorizations |

|Local | |Local | ||

|- | |- | ||

|November 8 | |November 8 | ||

|Patricia Klein | |Patricia Klein | ||

− | | | + | |Geometric vertex decomposition and liaison |

− | | | + | |Daniel |

|- | |- | ||

|November 15 | |November 15 | ||

− | | | + | |Libby Taylor |

− | | | + | |<math>\mathbb{A}^1</math>-local degree via stacks |

− | | | + | |Daniel/Soumya |

|- | |- | ||

|November 22 | |November 22 | ||

− | | | + | |Daniel Corey |

− | | | + | |Topology of moduli spaces of tropical curves with low genus |

− | | | + | |Local |

|- | |- | ||

|November 29 | |November 29 | ||

− | | | + | | No Seminar |

| Thanksgiving Break | | Thanksgiving Break | ||

| | | | ||

|- | |- | ||

|December 6 | |December 6 | ||

− | | | + | |Cynthia Vinzant |

− | | | + | |Log-concave polynomials, matroids, and expanders |

− | | | + | | Matroids Day |

|- | |- | ||

|December 13 | |December 13 | ||

− | | | + | |Taylor Brysiewicz |

− | | | + | |The degrees of Stiefel manifolds |

− | | | + | | Jose |

|} | |} | ||

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I will discuss the asymptotic non-vanishing of syzygies for products of projective spaces, generalizing the monomial methods of Ein-Erman-Lazarsfeld. This provides the first example of how the asymptotic syzygies of a smooth projective variety whose embedding line bundle grows in a semi-ample fashion behave in nuanced and previously unseen ways. | I will discuss the asymptotic non-vanishing of syzygies for products of projective spaces, generalizing the monomial methods of Ein-Erman-Lazarsfeld. This provides the first example of how the asymptotic syzygies of a smooth projective variety whose embedding line bundle grows in a semi-ample fashion behave in nuanced and previously unseen ways. | ||

+ | |||

+ | |||

+ | ===Michael Kemeny=== | ||

+ | '''The geometric syzygy conjecture''' | ||

+ | |||

+ | A famous classical result of M. Green asserts that the ideal sheaf of a canonical curve is generated by quadrics of rank four. Extending this to higher relations, one arrives at the so-called | ||

+ | Geometric Syzygy Conjecture, stating that extremal linear syzygies are spanned by those of the lowest possible rank. This conjecture further provides a geometric interpretation of Green's conjecture | ||

+ | for canonical curves. In this talk, I will outline a proof of the Geometric Syzygy Conjecture in even genus, based on combining a construction of Ein-Lazarsfeld with Voisin's approach to the study of | ||

+ | syzygies of K3 surfaces. | ||

+ | |||

+ | ===Michael Brown=== | ||

+ | '''Standard Conjecture D for Matrix Factorizations''' | ||

+ | |||

+ | In 1968, Grothendieck posed a family of conjectures concerning algebraic cycles called the Standard Conjectures. They have been proven in some special cases, but they remain open in general. In 2011, Marcolli-Tabuada realized two of these conjectures as special cases of more general statements, involving differential graded categories, which they call Noncommutative Standard Conjectures C and D. The goal of this talk is to discuss a proof, joint with Mark Walker, of Noncommutative Standard Conjecture D in a special case which does not fall under the purview of Grothendieck's original conjectures: namely, in the setting of matrix factorizations. | ||

+ | |||

+ | |||

+ | ===Patricia Klein=== | ||

+ | '''Geometric vertex decomposition and liaison''' | ||

+ | |||

+ | Geometric vertex decomposition and liaison are two frameworks that can be used to study algebraic varieties. These approaches have been used historically by two distinct communities of mathematicians. In this talk, we will describe a connection between the two. In particular, we will see how each geometrically vertex decomposable ideal is linked by a sequence of ascending elementary G-biliaisons of height 1 to an ideal of indeterminates and, conversely, how every G-biliaison of a certain type gives rise to geometric vertex decomposition. As a consequence, we establish that several families of ideals are glicci, including Schubert determinantal ideals, defining ideals of varieties of complexes, and defining ideals of lower bound cluster algebras. This talk is based on joint work with Jenna Rajchgot. | ||

+ | |||

+ | ===Libby Taylor=== | ||

+ | '''<math>\mathbb{A}^1</math>-local degree via stacks''' | ||

+ | |||

+ | We extend results of Kass-Wickelgren to define an Euler class for a non-orientable vector bundle on a smooth scheme, valued in the Grothendieck-Witt group of the ground field. We use a root stack construction to produce this Euler class and discuss its relation to <math>\mathbb{A}^1</math>-homotopy-theoretic definitions of an Euler class. This allows one to apply Kass-Wickelgren's technique for arithmetic enrichments of enumerative geometry to a larger class of problems; as an example, we use our construction to give an arithmetic count of the number of lines meeting $6$ planes in <math>\mathbb{P}^4</math>. | ||

+ | |||

+ | ===Daniel Corey=== | ||

+ | '''Topology of moduli spaces of tropical curves with low genus''' | ||

+ | This talk is joint work with Sam Payne and Daniel Allcock. We develop basic techniques for studying fundamental groups and singular homology of generalized CW-complexes. As an application, we show that the moduli spaces of tropical curves Delta_g and Delta_{g,n} are simply connected, for g at least 1. We also show that Delta_3 is homotopy equivalent to the 5-sphere, and that Delta_4 has 3-torsion in H_5. | ||

+ | |||

+ | ===Cynthia Vinzant=== | ||

+ | '''Log-concave polynomials, matroids, and expanders''' | ||

+ | |||

+ | Complete log-concavity is a functional property of real multivariate polynomials that translates to strong and useful conditions on its coefficients. I will introduce the class of completely log-concave polynomials in elementary terms, discuss the beautiful real and combinatorial geometry underlying these polynomials, and describe applications to random walks on simplicial complexes. Consequences include proofs of Mason's conjecture that the sequence of numbers of independent sets is ultra log-concave and the Mihail-Vazirani conjecture that the basis exchange graph of a matroid has expansion at least one. This is based on joint work with Nima Anari, Kuikui Liu, and Shayan Oveis Gharan. | ||

+ | |||

+ | ===Taylor Brysiewicz=== | ||

+ | '''The degrees of Stiefel manifolds''' | ||

+ | |||

+ | The Stiefel manifold of orthonormal bases for k-planes in an n-dimensional space goes by a different name in the world of frame theory: the space of Parseval n-frames for a k-dimensional space. The Stiefel manifold, and many other spaces of finite frames, can be viewed as an algebraic variety embedded in the space of k by n matrices. This viewpoint is known as algebraic frame theory and still many algebro-geometric properties of these varieties remain unknown, in particular, their degrees. As a first step in understanding these varieties, we compute a formula for the degrees of Stiefel manifolds using techniques from classical algebraic geometry, representation theory, and combinatorics. This is joint work with Fulvio Gesmundo. | ||

+ | |||

+ | == Notes == | ||

+ | Because of exams and/or travel, Daniel is unable to attend seminars on Oct 11, Oct 18, Nov 15, and Dec 13. |

## Latest revision as of 11:16, 11 December 2019

The seminar meets on Fridays at 2:25 pm in room B235 Van Vleck.

Here is the schedule for the previous semester, for the next semester, and for this semester.

## Contents

## Algebra and Algebraic Geometry Mailing List

- Please join the AGS Mailing List 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).

## Fall 2019 Schedule

date | speaker | title | host(s) |
---|---|---|---|

September 6 | Yuki Matsubara | On the cohomology of the moduli space of parabolic connections | Dima |

September 13 | Juliette Bruce | Semi-Ample Asymptotic Syzygies | Local |

September 20 | Michael Kemeny | The geometric syzygy conjecture | Local |

September 27 | |||

October 4 | |||

October 11 | |||

October 18 | Kevin Tucker (UIC) | TBD | Daniel |

October 25 | Reserved | Dima | |

November 1 | Michael Brown | Standard Conjecture D for Matrix Factorizations | Local |

November 8 | Patricia Klein | Geometric vertex decomposition and liaison | Daniel |

November 15 | Libby Taylor | [math]\mathbb{A}^1[/math]-local degree via stacks | Daniel/Soumya |

November 22 | Daniel Corey | Topology of moduli spaces of tropical curves with low genus | Local |

November 29 | No Seminar | Thanksgiving Break | |

December 6 | Cynthia Vinzant | Log-concave polynomials, matroids, and expanders | Matroids Day |

December 13 | Taylor Brysiewicz | The degrees of Stiefel manifolds | Jose |

## Abstracts

### Yuki Matsubara

**On the cohomology of the moduli space of parabolic connections**

We consider the moduli space of logarithmic connections of rank 2 on the projective line minus 5 points with fixed spectral data. We compute the cohomology of such moduli space, and this computation will be used to extend the results of Geometric Langlands correspondence due to D. Arinkin to the case where the this type of connections have five simple poles on ${\mathbb P}^1$.

In this talk, I will review the Geometric Langlands Correspondence in the tamely ramified cases, and after that, I will explain how the cohomology of above moduli space will be used.

### Juliette Bruce

**Semi-Ample Asymptotic Syzygies**

I will discuss the asymptotic non-vanishing of syzygies for products of projective spaces, generalizing the monomial methods of Ein-Erman-Lazarsfeld. This provides the first example of how the asymptotic syzygies of a smooth projective variety whose embedding line bundle grows in a semi-ample fashion behave in nuanced and previously unseen ways.

### Michael Kemeny

**The geometric syzygy conjecture**

A famous classical result of M. Green asserts that the ideal sheaf of a canonical curve is generated by quadrics of rank four. Extending this to higher relations, one arrives at the so-called Geometric Syzygy Conjecture, stating that extremal linear syzygies are spanned by those of the lowest possible rank. This conjecture further provides a geometric interpretation of Green's conjecture for canonical curves. In this talk, I will outline a proof of the Geometric Syzygy Conjecture in even genus, based on combining a construction of Ein-Lazarsfeld with Voisin's approach to the study of syzygies of K3 surfaces.

### Michael Brown

**Standard Conjecture D for Matrix Factorizations**

In 1968, Grothendieck posed a family of conjectures concerning algebraic cycles called the Standard Conjectures. They have been proven in some special cases, but they remain open in general. In 2011, Marcolli-Tabuada realized two of these conjectures as special cases of more general statements, involving differential graded categories, which they call Noncommutative Standard Conjectures C and D. The goal of this talk is to discuss a proof, joint with Mark Walker, of Noncommutative Standard Conjecture D in a special case which does not fall under the purview of Grothendieck's original conjectures: namely, in the setting of matrix factorizations.

### Patricia Klein

**Geometric vertex decomposition and liaison**

Geometric vertex decomposition and liaison are two frameworks that can be used to study algebraic varieties. These approaches have been used historically by two distinct communities of mathematicians. In this talk, we will describe a connection between the two. In particular, we will see how each geometrically vertex decomposable ideal is linked by a sequence of ascending elementary G-biliaisons of height 1 to an ideal of indeterminates and, conversely, how every G-biliaison of a certain type gives rise to geometric vertex decomposition. As a consequence, we establish that several families of ideals are glicci, including Schubert determinantal ideals, defining ideals of varieties of complexes, and defining ideals of lower bound cluster algebras. This talk is based on joint work with Jenna Rajchgot.

### Libby Taylor

**[math]\mathbb{A}^1[/math]-local degree via stacks**

We extend results of Kass-Wickelgren to define an Euler class for a non-orientable vector bundle on a smooth scheme, valued in the Grothendieck-Witt group of the ground field. We use a root stack construction to produce this Euler class and discuss its relation to [math]\mathbb{A}^1[/math]-homotopy-theoretic definitions of an Euler class. This allows one to apply Kass-Wickelgren's technique for arithmetic enrichments of enumerative geometry to a larger class of problems; as an example, we use our construction to give an arithmetic count of the number of lines meeting $6$ planes in [math]\mathbb{P}^4[/math].

### Daniel Corey

**Topology of moduli spaces of tropical curves with low genus**
This talk is joint work with Sam Payne and Daniel Allcock. We develop basic techniques for studying fundamental groups and singular homology of generalized CW-complexes. As an application, we show that the moduli spaces of tropical curves Delta_g and Delta_{g,n} are simply connected, for g at least 1. We also show that Delta_3 is homotopy equivalent to the 5-sphere, and that Delta_4 has 3-torsion in H_5.

### Cynthia Vinzant

**Log-concave polynomials, matroids, and expanders**

Complete log-concavity is a functional property of real multivariate polynomials that translates to strong and useful conditions on its coefficients. I will introduce the class of completely log-concave polynomials in elementary terms, discuss the beautiful real and combinatorial geometry underlying these polynomials, and describe applications to random walks on simplicial complexes. Consequences include proofs of Mason's conjecture that the sequence of numbers of independent sets is ultra log-concave and the Mihail-Vazirani conjecture that the basis exchange graph of a matroid has expansion at least one. This is based on joint work with Nima Anari, Kuikui Liu, and Shayan Oveis Gharan.

### Taylor Brysiewicz

**The degrees of Stiefel manifolds**

The Stiefel manifold of orthonormal bases for k-planes in an n-dimensional space goes by a different name in the world of frame theory: the space of Parseval n-frames for a k-dimensional space. The Stiefel manifold, and many other spaces of finite frames, can be viewed as an algebraic variety embedded in the space of k by n matrices. This viewpoint is known as algebraic frame theory and still many algebro-geometric properties of these varieties remain unknown, in particular, their degrees. As a first step in understanding these varieties, we compute a formula for the degrees of Stiefel manifolds using techniques from classical algebraic geometry, representation theory, and combinatorics. This is joint work with Fulvio Gesmundo.

## Notes

Because of exams and/or travel, Daniel is unable to attend seminars on Oct 11, Oct 18, Nov 15, and Dec 13.