Algebraic Geometry/Commutative Algebra Seminar, 2017–2018

To volunteer to give a talk, or for anything else regarding the seminar, contact Claudiu Raicu.

Spring Schedule

The seminar will meet on Wednesdays, 3:00–4:00pm in 258 Hurley unless otherwise noted. Related events are also listed below.

Fall Schedule

The seminar will meet on Wednesdays, 3:00–4:00pm in 258 Hurley unless otherwise noted. Related events are also listed below.

Abstracts

Aug. 30, 2017

Speaker

Liubomir Chiriac (University of Massachusetts Amherst)

Title

Distribution of the Fourier coefficients in pairs of newforms

Abstract

Given two distinct newforms, I will present several statistical results concerning the joint distribution of their Fourier coefficients, with special emphasis on questions about dominance. The talk will touch upon refined multiplicity one problems in a few different contexts, as well as some applications of a natural generalization of the Sato-Tate conjecture for pairs of newforms.

Sept. 6, 2017

Speaker

Daniele Rosso (Indiana University Northwest)

Title

Irreducible components of exotic Springer fibers

Abstract

The Springer resolution is a resolution of singularities of the variety of nilpotent elements in a reductive Lie algebra. It is an important geometric construction in representation theory, but some of its features are not as nice if we are working in Type C (Symplectic group). To make the symplectic case look more like the Type A case, Kato introduced the exotic nilpotent cone and its resolution, whose fibers are called the exotic Springer fibers. We give a combinatorial description of the irreducible components of these fibers in terms of standard Young bitableaux and obtain an exotic Robinson-Schensted correspondence. This is joint work with Vinoth Nandakumar and Neil Saunders.

Sept. 27, 2017

Speaker

Mike Perlman (Notre Dame)

Title

Regularity of Pfaffian Thickenings

Abstract

Let $S$ be the coordinate ring on the spaces of $n\times n$ complex skew-symmetric matrices. This ring has a natural action of the general linear group $\textnormal{GL}_n(\mathbb{C})$, and we study the Castelnuovo-Mumford regularity of ideals $I\subseteq S$ that are invariant under this action. In particular, we compute the regularity of basic invariant ideals and large powers of ideals of Pfaffians. As a consequence, we characterize when these ideals have linear minimal free resolution. This work is inspired by the recent results of Raicu, who solved the analogous problem in the case of generic $n\times m$ matrices with a $\textnormal{GL}_n(\mathbb{C})\times \textnormal{GL}_m(\mathbb{C})$-action.

Oct. 11, 2017

Speaker

Anurag Singh (University of Utah)

Title

The number of equations defining an algebraic set

Abstract

Given an algebraic set, i.e., the solution set of a family of polynomial equations, what is the minimal number of polynomials needed to define this set? The question is surprisingly difficult, with a rich history. We will give a partial survey, and discuss results and questions coming from local cohomology theory.

Oct. 12, 2017

Speaker

Anurag Singh (University of Utah)

Title

Hankel determinantal rings

Abstract

We will discuss various aspects of rings defined by minors of Hankel matrices of indeterminates, and sketch a proof that these have rational singularities. This is joint work with Aldo Conca, Maral Mostafazadehfard, and Matteo Varbaro.

Oct. 25, 2017

Speaker

Robin Hartshorne (U.C. Berkeley)

Title

Grothendieck duality

Abstract

I will explain the general idea of Grothendieck’s duality for coherent sheaf cohomology, with emphasis on his vision of the functorial nature of the theorem.

Nov. 1, 2017

Speaker

Carl Wang-Erickson (Imperial College)

Title

Massey products in cohomology and a number-theoretic application

Abstract

We will explain how an appropriate infinity-algebra structure on cohomology allows for an explicit presentation of a deformation problem. This expands on the better-known mantra that deformations and obstructions arise from first and second degree cohomology, respectively. Deformations of group representations will be featured, in preparation for an application. The application will be drawn from joint work with Preston Wake, where we answer a question of Barry Mazur about congruences between Eisenstein series and cusp forms in terms of Massey products in Galois cohomology.

Nov. 8, 2017

Speaker

Charles Doran (University of Alberta/ICERM)

Title

Calabi-Yau Manifolds, Mirrors, and Moduli

Abstract

Calabi-Yau manifolds are of central importance in both algebraic and differential geometry, and have a range of key applications in quantum field theory and string theory. Nevertheless, in dimensions three and higher our knowledge of their construction and effective moduli has historically been tied to the ambient spaces in which they live. Hodge-theoretic considerations suggest that a more natural way to build Calabi-Yau manifolds is “from the inside out” via internal fibrations. We will discuss a first-ever classification of such Calabi-Yau threefolds and implications for the structure of their moduli spaces. Furthermore, by identifying mirror partners, we are led to a new mirror symmetry conjecture which unifies the Calabi-Yau and Fano/Landau-Ginzburg mirror proposals (and answers a question of A. Tyurin).

Nov. 9, 2017

Speaker

Markus Hunziker (Baylor University)

Title

Classical invariant theory and Harish-Chandra modules

Abstract

Classical invariant theory has played a central role in the development of modern mathematics and continues to have important applications in numerous areas of current interest. In this expository talk, I will present a new approach to classical invariant theory via combinatorics and representation theory related to Hermitian symmetric spaces. I will then explain how this approach can be used to solve several long-standing problems in invariant theory and beyond.

Nov. 15, 2017

Speaker

Nick Switala (UIC)

Title

The injective dimension of F-finite F-modules and holonomic D-modules

Abstract

If M is an F-module over a regular Noetherian ring of positive characteristic, or a D-module over a formal power series ring with coefficients in a field of characteristic zero, Lyubeznik showed that the injective dimension of M is bounded above by the dimension of its support. If M is F-finite (respectively, holonomic), we show that the injective dimension of M is bounded below by one less than the dimension of its support, and therefore can take only two possible values. We do this by studying the last term in the minimal injective resolution of M. (This is joint work with Wenliang Zhang.)

Dec. 6, 2017

Speaker

Alex Constantinescu (University of Genova)

Title

Linear Syzygies and Hyperbolic Coxeter Groups

Abstract

We show that the virtual cohomological dimension of a Coxeter group is essentially the same as the Castelnuovo-Mumford regularity of the Stanley-Reisner ring of its nerve. Using this connection, we modify a construction of Osajda in group theory to find for every positive integer r a quadratic monomial ideal, with linear syzygies, and regularity of the quotient equal to r. This answers a question of Dao, Huneke and Schweig, and shows that Gromov asked essentially the same question about the virtual cohomological dimension of hyperbolic Coxeter groups. For monomial quadratic Gorenstein ideals with linear syzygies we prove that the regularity of their quotients can not exceed four, which implies that for d > 4 every triangulation of a d-manifold has an induced square or a hollow simplex. All results are in collaboration with Thomas Kahle and Matteo Varbaro.

Feb. 7, 2018

Speaker

Emre Sertoz (Max Planck Institute)

Title

Computing periods of hypersurfaces

Abstract

Given a complex manifold X, the periods of X are complex numbers which describe the complex structure of X upon the underlying topological manifold. The periods of a smooth algebraic variety reveal finer geometric data more readily than the defining equations alone. However, periods are typically very hard to compute. In the past 20 years, an algorithm for computing the periods existed only for plane curves. We will describe a different algorithm which can compute the periods of any smooth projective hypersurface. As an application, we will demonstrate how to reliably guess the Picard rank of a quartic K3 surface from its periods computed up to numerical error.

Feb. 14, 2018

Speaker

Dylan Rupel (Notre Dame)

Title

Cell Decompositions for Rank 2 Quiver Grassmannians

Abstract

A quiver Grassmannian is a variety parametrizing subrepresentations of a given quiver representation. Reineke has shown that all projective varieties can be realized as quiver Grassmannians. In this talk, I will study a class of smooth projective varieties arising as quiver Grassmannians for (truncated) preprojective representations of an $n$-Kronecker quiver, i.e. a quiver with two vertices and $n$ parallel arrows between them. The main result will be a recursive construction of cell decompositions for these quiver Grassmannians together with a combinatorial labeling of the cells by which their dimensions may be directly computed.

Feb. 21, 2018

Speaker

Andrei Jorza (Notre Dame)

Title

Comparing Hecke coefficients of automorphic forms

Abstract

The distribution of Hecke/Fourier coefficients of classical and Hilbert modular forms is well understood and the Sato-Tate conjecture describes their limit distribution. Little information is known in more general contexts, including $\operatorname{GL}(2)$ over arbitrary number fields. In recent work with Liubomir Chiriac we prove a number of distributional results on Hecke coefficients of automorphic representations. Among our applications are: (a) we show that sums of Hecke coefficients are negative for a positive density of primes, (b) we show a conjecture of Serre that Hecke coefficients are at least 1 in absolute value for a positive density of primes, and (c) we show that Hecke coefficients can be negative for a positive proportion of primes congruent to, e.g., 1 mod 8. The latter result is new even for elliptic curves.

Feb. 28, 2018

Speaker

Claudiu Raicu (Notre Dame)

Title

Koszul modules

Abstract

The Cayley-Chow form of a projective variety X is an equation that detects when a given linear space intersects X non-trivially. I will explain how it can be computed in the case when X is the Grassmannian of lines in its Plücker embedding, by relating it to a fascinating class of modules called Koszul modules. Despite the elementary definition of Koszul modules, their study has close ties to that of syzygies of generic canonical curves, but also important implications to the structure of Alexander invariants of finitely presented groups. Joint work with M. Aprodu, G. Farkas, S. Papadima, and J. Weyman.

Mar. 9, 2018

Speaker

Brian Harbourne (Nebraska)

Title

Rational Amusements for a Winter Afternoon

Abstract

In 1821 John Jackson published "Rational amusement for winter evenings or, A collection of above 200 curious and interesting puzzles and paradoxes". At least one of these puzzles has led to open problems in combinatorics about line arrangements, open problems which have recently become relevant to a growing body of work in commutative algebra and algebraic geometry. I will describe some of this work and its history. Little background in algebraic geometry, commutative algebra or combinatorics will be assumed.

Apr. 4, 2018

Speaker

Robin Hartshorne (Berkeley)

Title

Curves on the cubic surface in projective space

Abstract

A good way to construct curves in the projective three space is on the cubic surface. I will explain the divisor class group on the surface, how to compute the degree and genus of a curve, and a criterion for a nonsingular curve. I will try to develop everything from first principles. Reference: GTM 52, chapter V section 4.

Apr. 18, 2018

Speaker

András Lőrincz (Purdue)

Title

On categories of equivariant D-modules

Abstract

Let X be a complex algebraic variety with the action of an algebraic group G. In this talk I will discuss various results regarding G-equivariant D-modules on X. When G acts on X with finitely many orbits, the category of G-equivariant coherent D-modules is isomorphic to the category of finite-dimensional representations of a quiver with relations. We describe explicitly these quivers in the case when X is an irreducible multiplicity-free representation of a reductive group G. At the end I will mention some applications to local cohomology. This is joint work with Uli Walther.

Apr. 25, 2018

Speaker

Eric Ramos (Michigan)

Title

Commutative algebra in the configuration spaces of graphs

Abstract

Let G be a graph, thought of as a 1-dimensional simplicial complex. Then the n-stranded configuration space of G is the space F_n(G) = \{(x_1,\ldots,x_n) \in G^n \mid x_i \neq x_j\} / S_n, where S_n is the symmetric group on n letters. While one would hope to say something meaningful about the homology groups H_i(F_n(G)) of these spaces, it is known that they can be quite chaotic. Following recent trends in algebra, we therefore shift our focus to studying all of these groups simultaneously +_n H_i(F_n(G)), sacrificing knowledge about individual homology groups for statements which are more asymptotic in nature. In particular, it can be shown that +_n H_i(F_n(G)) can be encoded as the additive group of some finitely generated graded module over an integral polynomial ring. Specializing to the case of trees, we compute the generating degree of this module, and show that it naturally decomposes as a direct sum of graded shifts of square-free monomial ideals. As an application we show that the homology groups of F_n(G), in the case of trees, are only dependent on the degree sequence of G. We conclude the talk by discussing what little is known in the general case, and provide a conjecture describing what the Hilbert polynomials of such modules look like.

May 2, 2018

Speaker

Bernd Ulrich (Purdue)

Title

Integral Closures of Ideals and Modules: a survey

Abstract

We will talk about various instances where integral dependence of ideals and modules arises. These include applications in equisingularity theory, and questions about symbolic powers and the asymptotic behavior of ideal powers. A common theme in these applications are differentials and multiplicity theory.