Algebraic Geometry/Commutative Algebra Seminar, 2015-2016

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

Abstracts can be found below.

Fall Schedule

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

Date Speaker Title
Wednesday, Sept. 2 No Seminar --
Wednesday, Sept. 9 No Seminar --
Wednesday, Sept. 9
4:00 - ACMS colloquium
Hoon Hong (NC State) Root separation bound
Abstract and room
Thursday, Sept. 10
3:30 - ACMS Applied Math Seminar
Hoon Hong (NC State) Subresultants in roots
Abstract and room
Wednesday, Sept. 16 John Lesieutre (UIC) Constraints on threefolds admitting positive entropy
Abstract: see below
Monday, Sept. 21
4:00 - Math Colloquium
Bernd Sturmfels (Berkeley) Exponential varieties
Abstract and room
Wednesday, Sept. 23 No Seminar --
Wednesday, Sept. 30 Asilata Bapat (U. Chicago) The Strong Monodromy Conjecture for finite Coxeter arrangements
Abstract
Friday, Oct. 9
2:30-3:30 in HAYE 231
Note special day/time
Mihai Fulger (Princeton) Hilbert functions, volumes, and cycles
Abstract: see below
Wednesday, Oct. 14 Adam Boocher (Utah) Deviations of Graded Algebras
Abstract: see below
Wednesday, Oct. 21 No seminar (fall break) --
Wednesday, Oct. 28 Claudia Polini (Notre Dame) Rees algebras of codimension three Gorenstein ideals
Abstract: see below
Wednesday, Nov. 4 Eric Riedl (UIC) Rational Curves on Hypersurfaces
Abstract: see below
Tuesday, Nov. 10
2:00 - Logic Seminar
Andrei Minchenko
(Weizmann Institute, Jerusalem)
The Galois group of a parametrized linear differential equation
Abstract and room
Wednesday, Nov. 11 Marc Chardin (Paris) CANCELLED
Wednesday, Nov. 18 Oliver Pechenik (UIUC) Puzzles and Equivariant K-theory of Grassmannians
Abstract
Wednesday, Nov. 25 No seminar (Thanksgiving) --
Monday, Nov. 30
4:00-5:00 in HAYE 127
ACMS colloquium
Frank Schreyer (Saarlandes) Refined algorithm to compute syzygies
Abstract
Wednesday, Dec. 2 Daniel Erman (Wisconsin) Kakeya problems over finite fields
Abstract: see below
Wednesday, Dec. 9 Robin Hartshorne Local Cohomology as D-modules
Abstract

Spring Schedule

Date Speaker Title
Wednesday, January 20 No Seminar --
Wednesday, January 27 Andras Lorincz (Connecticut) Free resolutions of orbit closures of Dynkin quivers
Wednesday, February 3 No Seminar --
Wednesday, February 10 Claudiu Raicu (Notre Dame) Bernstein-Sato polynomials for maximal minors and sub-maximal Pfaffians
Wednesday, February 17 No Seminar --
Wednesday, February 24 Robin Hartshorne (Berkeley) Gorenstein liaison of projective varieties: known results and open problems
Wednesday, March 2
Wednesday, March 9 No seminar (spring break) --
Wednesday, March 16 Steve Oloo (Kalamazoo College) Generalized Moment Graphs and the Equivariant Intersection Cohomology of the Wonderful Group Compactification
Abstract
Wednesday, March 23 Alberto Fernandez Boix (Barcelona) On some local cohomology filtrations
Abstract
Wednesday, March 30 Paul Reschke (Michigan) Complex dynamics of birational surface maps defined over number fields
Abstract
Wednesday, April 6 Eric Wawerczyk (Notre Dame) Level Lowering of Galois Representations via L-Invariants
(See below for abstract.)
Wednesday, April 13 Andrei Jorza (Notre Dame) The commutative algebra of Fermat's last theorem
(See below for abstract.)
Wednesday, April 20 Andrei Negut (MIT) Categories over schemes: connecting braids to sheaves
(See below for abstract.)
Wednesday, April 27 Jack Jeffries (Michigan) Separating sets for actions of tori
(See below for abstract.)


Abstracts

Sept. 16, 2015

Speaker
John Lesieutre (UIC)
Title
Constraints on threefolds admitting positive entropy
Abstract
There are numerous examples of smooth, projective surfaces which admit automorphisms of positive entropy. However, relatively few examples are known for algebraic varieties in higher dimensions. I will give some constraints on the geometry of threefolds which can admit such automorphisms. For example, I will show that if one starts with a smooth threefold with no positive entropy automorphisms, and performs a sequence of blow-ups, any automorphism of the resulting threefold must be imprimitive. If time allows, I'll also mention a related example of a non-uniruled, terminal threefold with infinitely many K_X-negative rays on the cone of curves.

FRIDAY Oct. 9, 2015

Speaker
Mihai Fulger (Princeton)
Title
Hilbert functions, volumes, and cycles
Abstract
Hilbert functions appear naturally in algebraic geometry when one considers section rings of ample divisors A on projective varieties. By allowing big divisors (D=A+E, A ample and E effective) as well, one obtains Hilbert functions with more interesting behavior, e.g. their asymptotic behavior can be controlled by an irrational number. In general, the asymptotic measure of growth of the section ring of D is the volume vol(D). If F is an effective divisor, then it is easy to check that vol(D-F)<=vol(D)<=vol(D+F). It is interesting to ask when either equality holds. In joint work with Koll\'ar and Lehmann, I showed that an equality vol(D\pm F)=vol(D) implies surprisingly that the Hilbert functions of D\pm F and D coincide (in all degrees). The result has applications to moduli problems.

Oct. 14, 2015

Speaker
Adam Boocher (Utah)
Title
Deviations of Graded Algebras
Abstract
The deviations of a graded algebra are a sequence of integers that determine the Poincare series of its residue fi eld and arise as the number of generators of certain DG algebras. In a sense, deviations measure how far the ring is from being a complete intersection. We study extremal deviations among those of algebras with a fi xed Hilbert series. We prove that, like the Betti numbers, deviations do not decrease when passing to initial and lex-segment ideals. We also prove that deviations grow exponentially for Golod rings and for algebras presented by certain edge ideals. Combinatorial considerations, including some open questions will be discussed.

October 28, 2015

Speaker
Claudia Polini (Notre Dame)
Title
Rees algebras of codimension three Gorenstein ideals
Abstract
The Rees ring and the special fiber ring of an ideal arise in the process of blowing up a variety along a subvariety. Rees rings and special fiber rings also describe, respectively, the graph and the images of rational maps between projective spaces. A classical open problem in commutative algebra, algebraic geometry, elimination theory, and geometric modeling is to compute explicitly the equations defining the images of rational maps and therefore of such rings. In this talk we describe the solution to this problem for linearly presented grade three Gorenstein ideals. This is joint work with Andrew Kustin and Bernd Ulrich.

Nov. 4, 2015

Speaker
Eric Riedl (UIC)
Title
Rational Curves on Hypersurfaces
Abstract
One way to understand the geometry of a variety is to understand its rational curves. Even for some relatively simple varieties, little is known about their spaces of rational curves. Many people have made previous progress on these questions, but there remain many open cases. In joint work with David Yang, we investigate the dimensions of the spaces of rational curves on very general hypersurfaces, and prove that for n > d+1 or d > (3n+1)/2, the spaces of rational curves have the expected dimension, as conjectured (in various cases) by several people, including Coskun, Harris and Starr, and Voisin. In this talk, we focus our attention particularly on the n > d+1 case and try to motivate some of the ideas used to attack this problem.

Dec. 2, 2015

Speaker
Dan Erman (Wisconsin)
Title
Kakeya problems over finite fields
Abstract
Abstract: The Kakeya Needle Problem has its origins in harmonic analysis, but it has led to a number of interesting related questions about algebra and geometry over finite fields. I will first give background on this famous problem. Then I will talk about recent work of myself and Jordan Ellenberg which uses degeneration techniques to make progress on some of the related algebraic questions.

Jan. 27, 2016

Speaker
Andras Lorincz (Connecticut)
Title
Free resolutions of orbit closures of Dynkin quivers
Abstract
In this talk, we show how one can construct the minimal free resolutions of orbit closures of Dynkin quivers. These can be viewed as generalizations of Lascoux's resolution for determinantal ideals. We use the resolutions to prove that such orbit closures are normal, Cohen-Macaulay and have rational singularities. They also allow us to read off explicitly the minimal set of generators of their defining ideals. This is joint work with Jerzy Weyman.

Feb. 10, 2016

Speaker
Claudiu Raicu (Notre Dame)
Title
Bernstein-Sato polynomials for maximal minors and sub-maximal Pfaffians
Abstract
Bernstein-Sato polynomials are important invariants in the study of singularities: they are associated classically to hypersurfaces, and in work of Budur- Mustaţă-Saito to more general algebraic varieties. Despite their simple definition, Bernstein-Sato polynomials are notoriously difficult to compute, and they have been described explicitly in only a few cases. In my talk I will explain how basic results on local cohomology modules and invariant differential operators can be used to describe the Bernstein-Sato polynomials for the varieties of general (resp. skew-symmetric) matrices of sub-maximal rank. This can then be used to verify the Strong Monodromy Conjecture for the said varieties. Joint work with András Lőrincz, Uli Walther and Jerzy Weyman.

March 30, 2016

Speaker
Paul Reschke (Michigan)
Title
Complex dynamics of birational surface maps defined over number fields
Abstract
This work is joint with Mattias Jonsson. For a birational self-map with non-trivial first dynamical degree on a complex surface, Bedford and Diller defined an energy condition which when satisfied guarantees nice dynamical properties for the map (with regard, in particular, to a naturally defined invariant measure). However, Favre and Buff showed that the energy condition can fail and that in fact maps without the nice dynamical properties do exist. We show that the energy condition is always satisfied when the map is defined over a number field. The proof uses the dynamics of local height functions associated to an expanding eigen-class in the real Neron-Severi group for the surface. I will explain the use of local height functions in detail for the special case when the surface is the projective plane. I will then describe the difficulties and solutions arising in the general case where the eigen-class is big and nef but not ample; here we obtain an intermediate result of independent interest which describes the dynamics of curves on the surface that are distinguished by being orthogonal to the expanding direction.

April 6, 2016

Speaker
Eric Wawerczyk (Notre Dame)
Title
Level Lowering of Galois Representations via L-Invariants
Abstract
In their 1993 paper, Greenberg and Stevens proved the Mazur-Tate-Tietelbaum conjecture for an elliptic curve E/Q with split multiplicative reduction at a prime p. This result relates the derivative of the p-adic L-function of E to the classical Hasse-Weil L-function of the curve with a multiplicative factor called the L-invariant. To do so they put the elliptic curve E in a universal deformation ring of p-ordinary modular forms, R, called a Hida Family, and proved a fomula relating the L-invariant to a logarithmic derivative of p-adic analytic functions, defined on Spec(R), which interpolate the p-th Hecke eigenvalue of the modular forms in the family. Using this formula, relating the L-invariant to logarithmic derivatives, Greenberg-Stevens (1994) gave a new proof of a special case of Ribets Theorem (1990), on level-lowering of modular Galois representations, by assuming a condition on the L-invariant.

We establish the analogous results of level-lowering for Galois representations attached to p-ordinary Siegel modular forms with the same condition on their L-invariants. We place the Siegel modular form in a Hida Family and use the analogous formula of Harron-Jorza (2016) relating logarithmic derivatives of p-adic functions interpolating Hecke eigenvalues to the L-invariant.


April 13, 2016

Speaker
Andrei Jorza (Notre Dame)
Title
The commutative algebra of Fermat's last theorem
Abstract
The proof of Fermat's last theorem uses wide ranging results from number theory, algebraic geometry, arithmetic geometry, analysis and commutative algebra. It is precisely the commutative algebra part that was most innovative and exactly the ingredient that was missing in Wiles' initial, incorrect, proof. In this expository talk I will describe how one proves Fermat's last theorem by explaining the strategy in terms of commutative algebra. The talk is aimed at the usual audience of the seminar so I will describe clearly as a black box anything I won't explain.

April 20, 2016

Speaker
Andrei Negut (MIT)
Title
Categories over schemes: connecting braids to sheaves
Abstract
We will present a paradigm which takes a monoidal category and endows it with functors to/from a projective scheme. Our main application is when the category in question consists of Soergel bimodules associated to braids. Then our functor produces sheaves on the flag Hilbert (dg) scheme. We conjecture that the sheaf one associates to a braid has certain invariance properties under conjugation and stabilization moves, thus giving rise to a geometric knot invariant. Join work with Eugene Gorsky and Jacob Rasmussen.

April 27, 2016

Speaker
Jack Jeffries (Michigan)
Title
Separating sets for actions of tori
Abstract
One modern notion in invariant theory is that of a separating set. A separating set for a group action G on a variety X is a set of invariants such that if there is some f ∈ k[X] G such that f(v) = f(w), then there is an h ∈ S such that h(v) = h(w). This notion has attracted interest because it may be much easier to compute a separating set than to compute the whole ring of invariants, but separating sets still reflect much of the geometry of the group action like invariant rings do. In this talk, I will discuss some new results on separating sets for actions of tori, focusing on connections with local cohomology and secant varieties. This is based on joint work with Emilie Dufresne.

Math Department - University of Notre Dame