Algebraic Geometry/Commutative Algebra Seminar, 2016–2017

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

Abstracts can be found below.

Spring Schedule

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

Date Speaker Title
Tuesday, Jan. 17, 4-5pm
118 Nieuwland Science Hall
Department Colloquium
David Smyth (Australian National University) Moduli Spaces in Algebraic Geometry
Thursday, Jan. 19, 4-5pm
118 Nieuwland Science Hall
Department Colloquium
Eric Riedl (UIC) Spaces of rational curves on varieties
Wednesday, Jan. 25 No seminar
Wednesday, Feb. 1 Giuseppe Favacchio (Catania) Points in multiprojective spaces and Hilbert functions of multigraded algebras
Abstract
Friday, Feb. 3, 4-5pm
129 Hayes-Healy Hall
Department Colloquium
David Eisenbud (Berkeley) Linear equations over polynomial rings
Wednesday, Feb. 8 No seminar
Friday, Feb. 17
4-5pm in 258 Hurley
Note special day/time
Jerzy Weyman (Connecticut) Finite free resolutions and Kac-Moody Lie algebras
Wednesday, Feb. 22 Claudia Polini (Notre Dame) Degree Bounds for Local Cohomology
Wednesday, Mar. 1 No seminar
Wednesday, Mar. 8 Ramin Takloo-Bighash (UIC) Rational points on zero loci of Brauer elements
Wednesday, Mar. 15 No seminar (Spring break)
Wednesday, Mar. 22 Robin Hartshorne (Berkeley) Duality for de Rham cohomology of algebraic D-modules
Wednesday, Mar. 29, 4-5pm
129 Hayes-Healy Hall
Department Colloquium
Robin Hartshorne (Berkeley) A short walk in the garden of algebraic curves
Wednesday, Apr. 5 Martha Precup (Northwestern) The singular locus of Semisimple Hessenberg varieties
Wednesday, Apr. 12 Steven Sam (Wisconsin) Boij-S\"oderberg theory for Grassmannians
Wednesday, Apr. 19 Giovanni Rosso (Cambridge) Eigenvarieties for (non-cuspidal) automorphic forms
Wednesday, Apr. 26

Fall Schedule

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

Date Speaker Title
Wednesday, Aug. 31 Dong Quan Ngoc Nguyen (Notre Dame) Certain sets over function fields are polynomial families
Wednesday, Sept. 7 Juan Migliore (Notre Dame) Lefschetz properties and a problem on fat points
Abstract
Wednesday, Sept. 14 Andrei Jorza (Notre Dame) Derivatives of p-adic L-functions
Wednesday, Sept. 21 No seminar
Wednesday, Sept. 28 Sam Evens (Notre Dame) Algebraic geometry of the complex Gelfand-Zeitlin system
Wednesday, Oct. 5 Robin Hartshorne (Berkeley) Smoothing singularities
Wednesday, Oct. 12 Vlad Matei (Wisconsin) Higher moments of arithmetic functions in short intervals: a geometric perspective
Wednesday, Oct. 19 No seminar (Fall break)
Wednesday, Oct. 26 Ana Balibanu (Chicago) The Peterson Variety and the Wonderful Compactifi cation
Abstract
Wednesday, Nov. 2 Alessio Sammartano (Purdue) Blowup algebras of rational normal scrolls
Wednesday, Nov. 9 Amber Russell (Butler University) The Generalized Springer Correspondence and Graham’s Variety
Wednesday, Nov. 16 Pablo Solis (Caltech) Compactifications and Gauged Gromov-Witten Theory
Friday, Nov. 18
11:30-12:30 in 312 DeBartolo
Note special day/time
Kuei-Nuan Lin (Penn State Greater Allegheny) LCM lattices and dual hypergraphs of square-free monomial ideals
Wednesday, Nov. 23 No seminar (Thanksgiving)
Wednesday, Nov. 30 Rob Eggermont (Michigan) Finiteness properties in infinite dimension
Friday, Dec. 2
4-5pm in 117 Hayes-Healy
Department Colloquium
Brandon Levin (Chicago) Serre's conjecture on modular forms

Abstracts


Aug. 31, 2016

Speaker
Dong Quan Ngoc Nguyen (Notre Dame)
Title
Certain sets over function fields are polynomial families
Abstract
Let A be a commutative ring with 1. A subset X of A^n is a polynomial family with d parameters if it is the range of a polynomial map from A^d to A^n. It is an old question of Skolem (1938) whether the group SL_2(A) with A being the set of integers is a polynomial family. Only recently, Vaserstein (2010) answered Skolem's question in the affirmative. Along the way, he also shows that many arithmetic groups including the symplectic groups Sp_{2n}(\mathbb{Z}), the orthogonal groups SO_n(\mathbb{Z}), and the corresponding spinor groups Spin_n(\mathbb{Z}) are polynomial families. In this talk, I will discuss my result proving that SL_n(A) with n > 1 is a polynomial family, where A is the polynomial ring over a finite field of q elements. This is a function field analogue of Vaserstein's theorem.

Sept. 14, 2016

Speaker
Andrei Jorza (Notre Dame)
Title
Derivatives of p-adic L-functions
Abstract
L-functions are analytic objects attached to a number of arithmetic and analytic objects and are the core identifier of modular forms and Galois representations. For example L-functions uniquely determine the Langlands correspondences and their analytic properties are part of a sweeping series of conjectures that include the Birch and Swinnerton-Dyer conjecture. I will explain my recent work, in collaboration with Daniel Barrera and Mladen Dimitrov, on how to compute derivatives of L-functions in a p-adic analytic setting for (Hilbert) modular forms. The main idea is to express the value of an L-function as an integral over certain geometric cycles and then to prove, geometrically, a factorization formula.

Sept. 28, 2016

Speaker
Sam Evens (Notre Dame)
Title
Algebraic geometry of the complex Gelfand-Zeitlin system
Abstract
I will discuss some joint work with Colarusso on the complex algebraic version of the Gelfand-Zeitlin integrable system for gl(n) and so(n). In particular, I will explain how using notions from invariant theory facilitates extension of known results from gl(n) to so(n). In particular, the invariant theory quotients gl(n) -> gl(n)//GL(n-1) and so(n) -> so(n)//SO(n-1) are flat morphisms, and current questions focus on the structure and degenerations of the fibres of this morphism.

Oct. 5, 2016

Speaker
Robin Hartshorne (Berkeley)
Title
Smoothing singularities
Abstract
A singular point on an algebraic variety is smoothable if there exists a flat family in which the nearby varieties are nonsingular. I will give examples and some recent results from my book on deformation theory. This is algebraic geometry, but it would be interesting to find the analogous problem in commutative algebra and study it there.

Oct. 12, 2016

Speaker
Vlad Matei (Wisconsin)
Title
Higher moments of arithmetic functions in short intervals: a geometric perspective
Abstract
In joint work with Daniel Hast, we recast the paper of John Keating and Zeev Rudnick "The variance of the number of prime polynomials in short intervals and in residue classes" by studying the geometry of these short intervals through an associated highly singular variety. We manage to recover their results for a a general class of arithmetic functions up to a constant and also obtain information about the higher moments. Recently work of Brad Rodgers in "Arithmetic functions in short intervals and the symmetric group" gives new insight into the geometry of our variety.

Nov. 2, 2016

Speaker
Alessio Sammartano (Purdue)
Title
Blowup algebras of rational normal scrolls
Abstract
The Rees ring and the special fiber ring of a polynomial ideal I, also known as the blowup algebras of I, play an important role in commutative algebra and algebraic geometry. A central problem is to describe the defining equations of these algebras. I will discuss the solution of this problem when I is the homogeneous ideal of a rational normal scroll.

Nov. 9, 2016

Speaker
Amber Russell (Butler University)
Title
The Generalized Springer Correspondence and Graham’s Variety
Abstract
The Springer correspondence relates irreducible representations of the Weyl group for a reductive Lie algebra to a subset of simple perverse sheaves on the nilpotent cone for that Lie algebra. Essential to this result is the Springer resolution and its fibers. In his generalization of the Springer correspondence, Lusztig relates each simple perverse sheaf on the nilpotent cone with an irreducible representation of a relative Weyl group. In this talk, I will discuss a map defined by William Graham in Type A which plays a role in Lusztig's generalized setting that is similar to that of the Springer resolution in the classical version. I will focus on a combinatorial description of the irreducible components of the fibers of Graham's map, and the connection to Lusztig’s generalized Springer correspondence. This is joint work with William Graham and Martha Precup.

Nov. 16, 2016

Speaker
Pablo Solis (Caltech)
Title
Compactifications and Gauged Gromov-Witten Theory
Abstract
I will give an introduction to gauged Gromov-Witten theory. The theory naturally leads to studying compactifications of the moduli space of G bundles on nodal curves, which I'll discuss briefly. Then I'll focus on a version of gauged Gromov-Witten theory developed by Woodward and Gonzalez and I'll present a theorem which is joint work with Woodward and Gonzalez on the properness of the moduli of scaled gauged maps satisfying a stability condition introduced by Mundet and Schmitt.

Nov. 18, 2016

Speaker
Kuei-Nuan Lin (Penn State Greater Allegheny)
Title
LCM lattices and dual hypergraphs of square-free monomial ideals
Abstract
Given a square-free monomial ideal I in a polynomial ring R over a field k, we would like to know the projective dimension of I. We recall the definition of LCM lattice of a monomial ideal introduced by Gasharov, Peeva and Welker, and the definition of the dual hypergraph of a square-free monomial ideal introduced by Kimura, Terai and Yoshida. We describe the relationship between the LCM lattice and the dual hypergraph of a given square-free monomial ideal. In the joint work with Mantero, we show two square-free monomial ideals have the same projective dimension if they have the same dual hypergraph. We use the properties of LCM lattice to find whether two different dual hypergraphs have the same projective dimension. This is joint work with Sonja Mapes.

Nov. 30, 2016

Speaker
Rob Eggermont (Michigan)
Title
Finiteness properties in infinite dimension
Abstract
Is any ideal in a polynomial ring over a field finitely generated? By Hilbert’s Basis Theorem, the answer is obviously yes provided that the ring has only finitely many variables. Equally obvious is the fact that the answer is no provided that the ring has infinitely many variables. Equivalent statements are true if we replace the word ideal by the word variety. However, if we pose that the ideals (respectively varieties) satisfy the constraint that they are stable under the action of some group of symmetries, we can sometimes show that only finitely many equations are required up to symmetry. In this talk, I will discuss a few examples in which we know whether there are finiteness properties like this, as well as introduce some of the many examples in which we still don’t know anything for sure. I will also say a few words about our recent results in this area (joint work with Harm Derksen and Andrew Snowden).

Feb. 17, 2017

Speaker
Jerzy Weyman (Connecticut)
Title
Finite free resolutions and Kac-Moody Lie algebras
Abstract
Let us recall that a format (r_n,...,r_1) of a free complex 0-->F_n-->F_{n-1}-->...--> F_0 over a commutative Noetherian ring is the sequence of ranks r_i of the i-th differential d_i. We will assume that rank F_i =r_i+r_{i+1}. We say that an acyclic complex F_{gen} of a given format over a given ring R_{gen} is generic if for every complex G of this format over a Noetherian ring S there exists a homomorphism f:R_{gen}--> S such that G=F_{gen}\otimes_{R_{gen}} S. For complexes of length 2 the existence of the generic acyclic complex was established by Hochster and Huneke in the 1980's. It is a normalization of the ring giving a generic complex (two matrices with composition zero and rank conditions). I will discuss the ideas going into the proof of the following result:

Associate to a triple of ranks (r_3, r_2, r_1) a triple (p,q,r)=(r_3+1, r_2-1, r_1+1). Associate to (p,q,r) the graph T_{p,q,r} (three arms of lengths p-1, q-1, r-1 attached to the central vertex). Then there exists a Noetherian generic ring for this format if and only if T_{p,q,r} is a Dynkin graph. In other cases one can construct in a uniform way a non-Noetherian generic ring, which deforms to a ring carrying an action of the Kac-Moody Lie algebra corresponding to the graph T_{p,q,r}.

Feb. 22, 2017

Speaker
Claudia Polini (Notre Dame)
Title
Degree Bounds for Local Cohomology
Abstract
In this talk I will show how to estimate degrees of generators of local cohomology modules. I will also survey several applications to Rees algebras, to hyperplane sections, to symbolic powers, and to ideals of Pfaffians. This is joint work with Andy Kustin and Bernd Ulrich.

Mar. 8, 2017

Speaker
Ramin Takloo-Bighash (UIC)
Title
Rational points on zero loci of Brauer elements
Abstract
We consider the problem of counting the number of rational points of bounded height in the zero-loci of Brauer group elements on semi-simple algebraic groups over number fields. We obtain asymptotic formulae for the counting problem for wonderful compactifications using the spectral theory of automorphic forms. Applications include asymptotic formulae for the number of matrices over Q whose determinant is a sum of two squares. These results provide a positive answer to some cases of a question of Serre concerning such counting problems. This is joint work with Daniel Loughran and Sho Tanimoto.

Mar. 22, 2017

Speaker
Robin Hartshorne (Berkeley)
Title
Duality for de Rham cohomology of algebraic D-modules
Abstract
I will report on some recent work of Nicholas Switala and Wenliang Zhang in which they prove a duality theorem for the cohomology groups of graded D-modules, under the assumption that these are finite-dimensional. This relates to Switala's thesis, and also provides alternative proofs for recent results of Hartshorne and Polini on projective varieties.

Mar. 29, 2017

Speaker
Robin Hartshorne (Berkeley)
Title
A short walk in the garden of algebraic curves
Abstract
I will start with some elementary examples and questions about curves, see what they lead to, and then report on some more recent approaches. At least 3/4 of the talk should be accessible to everyone.

Apr. 05, 2017

Speaker
Martha Precup (Northwestern)
Title
The singular locus of Semisimple Hessenberg varieties
Abstract
Semisimple Hessenberg varieties are subvarieties of the flag variety with important connections to representation theory, algebraic geometry, and combinatorics. Like Schubert varieties, the structure of semisimple Hessenberg varieties can be studied using the combinatorics of the Weyl group. In this talk, we will define these varieties and compute their GKM-graphs. Then we’ll give a combinatorial criterion for identifying singular points in certain semisimple Hessenberg varieties. These results are joint work with Erik Insko.

Apr. 12, 2017

Speaker
Steven Sam (Wisconsin)
Title
Boij-S\"oderberg theory for Grassmannians
Abstract
An explicit description for the cone of Betti tables of graded modules over a polynomial ring was given in work of Eisenbud and Schreyer, and a surprising connection was established with the cone of cohomology tables of vector bundles on projective space. I will explain some ongoing work of Ford and Levinson, some joint with me, which seeks to establish Grassmannian analogues of this where graded modules are replaced by GL_k-equivariant modules for k>1.

Apr. 19, 2017

Speaker
Giovanni Rosso (Cambridge)
Title
Eigenvarieties for (non-cuspidal) automorphic forms
Abstract
Since the seminal work of Serre and Swinnerton-Dyer, people have been interested in congruences between modular forms. After recalling the geometric interpretation of modular forms, we shall presents some results of Hida and Coleman on the existence of eigenvarieties, i.e. moduli spaces for systems of eigenforms. If time allows it, we shall explain how this can be generalised to more general automorphic forms. This is joint work with Riccardo Brasca.

Math Department - University of Notre Dame