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

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Line 25: | Line 25: | ||

|Local | |Local | ||

|- | |- | ||

− | |February 8 | + | |February 8 (B135) |

|[http://www.mit.edu/~ivogt/ Isabel Vogt (MIT)] | |[http://www.mit.edu/~ivogt/ Isabel Vogt (MIT)] | ||

| Low degree points on curves | | Low degree points on curves | ||

Line 32: | Line 32: | ||

|February 15 | |February 15 | ||

|Pavlo Pylyavskyy (U. Minn) | |Pavlo Pylyavskyy (U. Minn) | ||

− | | | + | |Zamolodchikov periodicity and integrability |

|Paul Terwilliger | |Paul Terwilliger | ||

|- | |- | ||

|February 22 | |February 22 | ||

− | |Michael Brown | + | |Michael Brown (Wisconsin) |

|Chern-Weil theory for matrix factorizations | |Chern-Weil theory for matrix factorizations | ||

|Local | |Local | ||

|- | |- | ||

|March 1 | |March 1 | ||

− | | | + | |[https://math.berkeley.edu/~ceur/ Chris Eur (UC Berkeley)] |

− | | | + | |Chow rings of matroids, ring of matroid quotients, and beyond |

− | | | + | |Daniel |

|- | |- | ||

|March 8 | |March 8 | ||

− | |Jay Kopper (UIC) | + | |[http://homepages.math.uic.edu/~kopper/ Jay Kopper (UIC)] |

− | | | + | |Stable restrictions of vector bundles on projective varieties |

|Daniel | |Daniel | ||

|- | |- | ||

|March 15 | |March 15 | ||

− | | | + | |[http://www.math.wisc.edu/~maxim/Sing19program.html Singularities] |

− | | | + | |No regular meeting |

− | | | + | |Max |

|- | |- | ||

|March 22 | |March 22 | ||

Line 61: | Line 61: | ||

|- | |- | ||

|March 29 | |March 29 | ||

− | | | + | | |

|TBD | |TBD | ||

|Daniel | |Daniel | ||

+ | |- | ||

+ | |April 5 (at 11:00 in B329) | ||

+ | |[http://www-personal.umich.edu/~ecanton/ Eric Canton (Michigan)] | ||

+ | |Log canonical thresholds, Kahler seminorms, and normalized volume | ||

+ | |Michael | ||

|- | |- | ||

|April 5 | |April 5 | ||

+ | |[http://www.math.tamu.edu/~ola/ Alexsandra Sobieska (Texas A&M)] | ||

|TBD | |TBD | ||

− | | | + | |Daniel |

− | |||

|- | |- | ||

|April 12 | |April 12 | ||

− | | | + | |Botong Wang |

− | | | + | |Lyubeznik numbers of irreducible projective varieties |

− | | | + | |local |

|- | |- | ||

|April 19 | |April 19 | ||

|[http://www-personal.umich.edu/~grifo/ Eloísa Grifo (Michigan)] | |[http://www-personal.umich.edu/~grifo/ Eloísa Grifo (Michigan)] | ||

− | | | + | |Symbolic Powers and the (Stable) Containment Problem |

− | | | + | |Daniel |

|- | |- | ||

|April 26 | |April 26 | ||

Line 86: | Line 91: | ||

|- | |- | ||

|May 3 | |May 3 | ||

− | | | + | |Hannah Larson (Stanford) |

− | | | + | |Vector bundles on P^1 bundles |

− | | | + | |Melanie |

|} | |} | ||

== Abstracts == | == Abstracts == | ||

+ | |||

+ | ===Daniel Smolkin=== | ||

+ | '''Symbolic Powers in Rings of Positive Characteristic''' | ||

+ | |||

+ | The n-th power of an ideal is easy to compute, though difficult to describe geometrically. In contrast, symbolic powers of ideals are difficult to compute while having a natural geometric description. In this talk, I will describe how to compare ordinary and symbolic powers of ideals using the techniques of positive-characteristic commutative algebra, especially in toric rings and Hibi rings. This is based on joint work with Javier Carvajal-Rojas, Janet Page, and Kevin Tucker. Graduate students are encouraged to attend! | ||

===Juliette Bruce=== | ===Juliette Bruce=== | ||

Line 104: | Line 114: | ||

In this talk we will discuss an arithmetic analogue of the gonality of a curve over a number field: the smallest positive integer e such that the points of residue degree bounded by e are infinite. By work of Faltings, Harris--Silverman and Abramovich--Harris, it is well-understood when this invariant is 1, 2, or 3; by work of Debarre--Fahlaoui these criteria do not generalize to e at least 4. We will study this invariant using the auxiliary geometry of a surface containing the curve and devote particular attention to scenarios under which we can guarantee that this invariant is actually equal to the gonality . This is joint work with Geoffrey Smith. | In this talk we will discuss an arithmetic analogue of the gonality of a curve over a number field: the smallest positive integer e such that the points of residue degree bounded by e are infinite. By work of Faltings, Harris--Silverman and Abramovich--Harris, it is well-understood when this invariant is 1, 2, or 3; by work of Debarre--Fahlaoui these criteria do not generalize to e at least 4. We will study this invariant using the auxiliary geometry of a surface containing the curve and devote particular attention to scenarios under which we can guarantee that this invariant is actually equal to the gonality . This is joint work with Geoffrey Smith. | ||

+ | |||

+ | ===Pavlo Pylyavskyy=== | ||

+ | |||

+ | '''Zamolodchikov periodicity and integrability''' | ||

+ | |||

+ | T-systems are certain discrete dynamical systems associated with quivers. They appear in several different contexts: quantum affine algebras and Yangians, commuting transfer matrices of vertex models, character theory of quantum groups, analytic Bethe ansatz, Wronskian-Casoratian duality in ODE, gauge/string theories, etc. Periodicity of certain T-systems was the main conjecture in the area until it was proven by Keller in 2013 using cluster categories. In this work we completely classify periodic T-systems, which turn out to consist of 5 infinite families and 4 exceptional cases, only one of the infinite families being known previously. We then proceed to classify T-systems that exhibit two forms of integrability: linearization and zero algebraic entropy. All three classifications rely on reduction of the problem to study of commuting Cartan matrices, either of finite or affine types. The finite type classification was obtained by Stembridge in his study of Kazhdan-Lusztig theory for dihedral groups, the other two classifications are new. This is joint work with Pavel Galashin. | ||

+ | |||

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

+ | |||

+ | '''Chern-Weil theory for matrix factorizations''' | ||

+ | |||

+ | This is joint work with Mark Walker. Classical algebraic Chern-Weil theory provides a formula for the Chern character of a projective module P over a commutative ring in terms of a connection on P. In this talk, I will discuss an analogous formula for the Chern character of a matrix factorization. Along the way, I will provide background on matrix factorizations, and also on classical Chern-Weil theory. | ||

+ | |||

+ | ===Chris Eur=== | ||

+ | '''Chow rings of matroids, ring of matroid quotients, and beyond''' | ||

+ | |||

+ | We introduce a certain nef generating set for the Chow ring of the wonderful compactification of a hyperplane arrangement complement. This presentation yields a monomial basis of the Chow ring that admits a geometric and combinatorial interpretation with several applications. Geometrically, one can recover Poincare duality, compute the volume polynomial, and identify a portion of a polyhedral boundary of the nef cone. Combinatorially, one can generalize Postnikov's result on volumes of generalized permutohedra, prove Mason's conjecture on log-concavity of independent sets for certain matroids, and define a new valuative invariant of a matroid that measures its closeness to uniform matroids. This is an on-going joint work with Connor Simpson and Spencer Backman. | ||

+ | |||

+ | ===Jay Kopper=== | ||

+ | '''Stable restrictions of vector bundles on projective varieties''' | ||

+ | |||

+ | Stable vector bundles---and more generally, stable sheaves---play a role in the classification of algebraic vector bundles analogous to that of simple groups in group theory. Recent developments in this subject have extended the notion of stability to the entire derived category of sheaves. This broader perspective can be used to study the classical moduli space. In this talk I will discuss these ideas in the context of restriction theorems: situations in which a stable vector bundle remains stable when restricted to a subvariety. I will conclude with some applications to higher-rank Brill-Noether theory. This is joint work with S. Feyzbakhsh. | ||

+ | |||

+ | ===Shamgar Gurevich=== | ||

+ | |||

+ | '''Harmonic Analysis on GLn over finite fields, and Random Walks''' | ||

+ | |||

+ | There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the {\it character ratio}: | ||

+ | |||

+ | $$ | ||

+ | trace(\rho(g))/dim(\rho), | ||

+ | $$ | ||

+ | |||

+ | for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant {\it rank}. This talk will discuss the notion of rank for GLn over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM). | ||

+ | |||

+ | ===Eric Canton=== | ||

+ | '''Log canonical thresholds, Kahler seminorms, and normalized volume''' | ||

+ | |||

+ | The log canonical threshold of a closed subscheme Y of an algebraic variety X gives some real number that measures the singularities of Y. This is, in turn, defined in terms of the "amount" of a given divisor that must be inserted to make X\Y a smooth variety relatively compact (i.e. proper) over X; this "amount" goes by the name of the log discrepancy of that divisor on Y. Already, the study of log discrepancies is subtle when X is a complex variety, but without the guarantee of a smooth compactification in positive characteristics, effective results can seem remote. In this talk, I present an approach to effective results in positive characteristics from my thesis. This approach is described in terms of the Berkovich analytic space associated to X, realizing the log discrepancy as a natural seminorm to put on the sheaf of Kahler differentials of X, when X is normal. I'll finish by discussing new directions related to K-stability. | ||

+ | |||

+ | ===Botong Wang=== | ||

+ | |||

+ | '''Lyubeznik numbers of irreducible projective varieties''' | ||

+ | |||

+ | Lyubeznik numbers are invariants of singularities that are defined algebraically, but has topological interpretations. In positive characteristics, it is a theorem of Wenliang Zhang that the Lyubeznik numbers of the cone of a projective variety do not depend on the choice of the projective embedding. Recently, Thomas Reichelt, Morihiko Saito and Uli Walther constructed examples of reducible complex projective varieties whose Lyubeznik numbers depend on the choice of projective embeddings. I will discuss their works and a generalization to irreducible projective varieties. | ||

+ | |||

+ | ===Eloísa Grifo=== | ||

+ | '''Symbolic powers and the (stable) containment problem''' | ||

+ | |||

+ | Given a variety X in C^d, corresponding to an ideal I, which polynomials vanish up to order n along X? The polynomials in the n-th symbolic power of I, which are often not the same as the polynomials in the n-th ordinary power of I. | ||

+ | |||

+ | In trying to compare symbolic and ordinary powers, Harbourne conjectured that a famous containment by Ein--Lazersfeld--Smith, Hochster--Huneke and Ma--Schwede could be improved. Harbourne's Conjecture is a statement depending on n that unfortunately has been disproved for particular values of n. However, recent evidence points towards a stable version of Harbourne's conjecture, where we substitute all n by all n large enough. Some of that evidence is joint work with Craig Huneke and Vivek Mukundan. | ||

+ | |||

+ | ===Hannah Larson=== | ||

+ | |||

+ | '''Vector bundles on P^1 bundles''' | ||

+ | |||

+ | Every vector bundle on P^1 splits as a direct sum of line bundles. Given a vector bundle E on a P^1 bundle PW --> B, the base B is stratified by subvarieties defined by the condition that the restriction of E to the fibers has a certain splitting type. It is natural to ask for the classes of the closures of these strata in the Chow ring of B. We answer this question through a study of the moduli stack of vector bundles on P^1 bundles. In describing this moduli space, we also discover an algebraic version of Bott periodicity. This is joint work with Ravi Vakil. |

## Latest revision as of 11:41, 26 April 2019

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

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).

## Spring 2019 Schedule

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

January 25 | Daniel Smolkin (Utah) | Symbolic Powers in Rings of Positive Characteristic | Daniel |

February 1 | Juliette Bruce | Asymptotic Syzgies for Products of Projective Spaces | Local |

February 8 (B135) | Isabel Vogt (MIT) | Low degree points on curves | Wanlin and Juliette |

February 15 | Pavlo Pylyavskyy (U. Minn) | Zamolodchikov periodicity and integrability | Paul Terwilliger |

February 22 | Michael Brown (Wisconsin) | Chern-Weil theory for matrix factorizations | Local |

March 1 | Chris Eur (UC Berkeley) | Chow rings of matroids, ring of matroid quotients, and beyond | Daniel |

March 8 | Jay Kopper (UIC) | Stable restrictions of vector bundles on projective varieties | Daniel |

March 15 | Singularities | No regular meeting | Max |

March 22 | No Meeting | Spring Break | TBD |

March 29 | TBD | Daniel | |

April 5 (at 11:00 in B329) | Eric Canton (Michigan) | Log canonical thresholds, Kahler seminorms, and normalized volume | Michael |

April 5 | Alexsandra Sobieska (Texas A&M) | TBD | Daniel |

April 12 | Botong Wang | Lyubeznik numbers of irreducible projective varieties | local |

April 19 | Eloísa Grifo (Michigan) | Symbolic Powers and the (Stable) Containment Problem | Daniel |

April 26 | TBD | TBD | TBD |

May 3 | Hannah Larson (Stanford) | Vector bundles on P^1 bundles | Melanie |

## Abstracts

### Daniel Smolkin

**Symbolic Powers in Rings of Positive Characteristic**

The n-th power of an ideal is easy to compute, though difficult to describe geometrically. In contrast, symbolic powers of ideals are difficult to compute while having a natural geometric description. In this talk, I will describe how to compare ordinary and symbolic powers of ideals using the techniques of positive-characteristic commutative algebra, especially in toric rings and Hibi rings. This is based on joint work with Javier Carvajal-Rojas, Janet Page, and Kevin Tucker. Graduate students are encouraged to attend!

### Juliette Bruce

**Title: Asymptotic Syzygies for Products of Projective Spaces**

I will discuss results describing the asymptotic syzygies of products of projective space, in the vein of the explicit methods of Ein, Erman, and Lazarsfeld’s non-vanishing results on projective space.

### Isabel Vogt

**Title: Low degree points on curves**

In this talk we will discuss an arithmetic analogue of the gonality of a curve over a number field: the smallest positive integer e such that the points of residue degree bounded by e are infinite. By work of Faltings, Harris--Silverman and Abramovich--Harris, it is well-understood when this invariant is 1, 2, or 3; by work of Debarre--Fahlaoui these criteria do not generalize to e at least 4. We will study this invariant using the auxiliary geometry of a surface containing the curve and devote particular attention to scenarios under which we can guarantee that this invariant is actually equal to the gonality . This is joint work with Geoffrey Smith.

### Pavlo Pylyavskyy

**Zamolodchikov periodicity and integrability**

T-systems are certain discrete dynamical systems associated with quivers. They appear in several different contexts: quantum affine algebras and Yangians, commuting transfer matrices of vertex models, character theory of quantum groups, analytic Bethe ansatz, Wronskian-Casoratian duality in ODE, gauge/string theories, etc. Periodicity of certain T-systems was the main conjecture in the area until it was proven by Keller in 2013 using cluster categories. In this work we completely classify periodic T-systems, which turn out to consist of 5 infinite families and 4 exceptional cases, only one of the infinite families being known previously. We then proceed to classify T-systems that exhibit two forms of integrability: linearization and zero algebraic entropy. All three classifications rely on reduction of the problem to study of commuting Cartan matrices, either of finite or affine types. The finite type classification was obtained by Stembridge in his study of Kazhdan-Lusztig theory for dihedral groups, the other two classifications are new. This is joint work with Pavel Galashin.

### Michael Brown

**Chern-Weil theory for matrix factorizations**

This is joint work with Mark Walker. Classical algebraic Chern-Weil theory provides a formula for the Chern character of a projective module P over a commutative ring in terms of a connection on P. In this talk, I will discuss an analogous formula for the Chern character of a matrix factorization. Along the way, I will provide background on matrix factorizations, and also on classical Chern-Weil theory.

### Chris Eur

**Chow rings of matroids, ring of matroid quotients, and beyond**

We introduce a certain nef generating set for the Chow ring of the wonderful compactification of a hyperplane arrangement complement. This presentation yields a monomial basis of the Chow ring that admits a geometric and combinatorial interpretation with several applications. Geometrically, one can recover Poincare duality, compute the volume polynomial, and identify a portion of a polyhedral boundary of the nef cone. Combinatorially, one can generalize Postnikov's result on volumes of generalized permutohedra, prove Mason's conjecture on log-concavity of independent sets for certain matroids, and define a new valuative invariant of a matroid that measures its closeness to uniform matroids. This is an on-going joint work with Connor Simpson and Spencer Backman.

### Jay Kopper

**Stable restrictions of vector bundles on projective varieties**

Stable vector bundles---and more generally, stable sheaves---play a role in the classification of algebraic vector bundles analogous to that of simple groups in group theory. Recent developments in this subject have extended the notion of stability to the entire derived category of sheaves. This broader perspective can be used to study the classical moduli space. In this talk I will discuss these ideas in the context of restriction theorems: situations in which a stable vector bundle remains stable when restricted to a subvariety. I will conclude with some applications to higher-rank Brill-Noether theory. This is joint work with S. Feyzbakhsh.

### Shamgar Gurevich

**Harmonic Analysis on GLn over finite fields, and Random Walks**

There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the {\it character ratio}:

$$ trace(\rho(g))/dim(\rho), $$

for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant {\it rank}. This talk will discuss the notion of rank for GLn over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).

### Eric Canton

**Log canonical thresholds, Kahler seminorms, and normalized volume**

The log canonical threshold of a closed subscheme Y of an algebraic variety X gives some real number that measures the singularities of Y. This is, in turn, defined in terms of the "amount" of a given divisor that must be inserted to make X\Y a smooth variety relatively compact (i.e. proper) over X; this "amount" goes by the name of the log discrepancy of that divisor on Y. Already, the study of log discrepancies is subtle when X is a complex variety, but without the guarantee of a smooth compactification in positive characteristics, effective results can seem remote. In this talk, I present an approach to effective results in positive characteristics from my thesis. This approach is described in terms of the Berkovich analytic space associated to X, realizing the log discrepancy as a natural seminorm to put on the sheaf of Kahler differentials of X, when X is normal. I'll finish by discussing new directions related to K-stability.

### Botong Wang

**Lyubeznik numbers of irreducible projective varieties**

Lyubeznik numbers are invariants of singularities that are defined algebraically, but has topological interpretations. In positive characteristics, it is a theorem of Wenliang Zhang that the Lyubeznik numbers of the cone of a projective variety do not depend on the choice of the projective embedding. Recently, Thomas Reichelt, Morihiko Saito and Uli Walther constructed examples of reducible complex projective varieties whose Lyubeznik numbers depend on the choice of projective embeddings. I will discuss their works and a generalization to irreducible projective varieties.

### Eloísa Grifo

**Symbolic powers and the (stable) containment problem**

Given a variety X in C^d, corresponding to an ideal I, which polynomials vanish up to order n along X? The polynomials in the n-th symbolic power of I, which are often not the same as the polynomials in the n-th ordinary power of I.

In trying to compare symbolic and ordinary powers, Harbourne conjectured that a famous containment by Ein--Lazersfeld--Smith, Hochster--Huneke and Ma--Schwede could be improved. Harbourne's Conjecture is a statement depending on n that unfortunately has been disproved for particular values of n. However, recent evidence points towards a stable version of Harbourne's conjecture, where we substitute all n by all n large enough. Some of that evidence is joint work with Craig Huneke and Vivek Mukundan.

### Hannah Larson

**Vector bundles on P^1 bundles**

Every vector bundle on P^1 splits as a direct sum of line bundles. Given a vector bundle E on a P^1 bundle PW --> B, the base B is stratified by subvarieties defined by the condition that the restriction of E to the fibers has a certain splitting type. It is natural to ask for the classes of the closures of these strata in the Chow ring of B. We answer this question through a study of the moduli stack of vector bundles on P^1 bundles. In describing this moduli space, we also discover an algebraic version of Bott periodicity. This is joint work with Ravi Vakil.