Algebraic Geometry/Commutative Algebra Seminar, 2014-2015

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

Fall Schedule

The seminar will meet on Wednesdays, 3:00-4:00 in 258 Hurley unless otherwise noted.

Spring Schedule

The seminar will meet on Wednesdays, 3:00-4:00 in 258 Hurley unless otherwise noted.

Abstracts

Sept. 3, 2014

Speaker

Claudiu Raicu (Notre Dame)

Title

Characters of equivariant D-modules on Veronese cones

Abstract

Equivariant local systems on orbits of a group action give rise via the Riemann-Hilbert correspondence to equivariant D-modules, which are typically hard to describe. I will explain how to compute explicitly the characters of the GL-equivariant D-modules supported on the Veronese cones. In particular, I will appeal to recent results of de Cataldo, Migliorini and Mustaţă on the Decomposition Theorem for toric maps, and show how representation theoretic stabilization results are used in a crucial way in the calculation.

Sept. 10, 2014

Speaker

Anand Pillay (Notre Dame)

Title

Diophantine geometry and model theory; old and new perspectives

Abstract

Diophantine geometry is roughly speaking about rational points on algebraic varieties. I will discuss results and problems of "Manin-Mumford" type; the structure of varieties containing many "special" (such as rational) points. I will also two generations of interaction with model theory: via stability theory and via o-minimality. If I have time I will say something about a new recent account of function field Mordell-Lang in positive characteristic.

Sept. 17, 2014, Special time: 2:30-3:30

Speaker

Greg Smith (Queen's University)

Title

Parliaments of polytopes and toric vector bundles

Abstract

How do the properties of line bundles extend to vector bundles? After reviewing the basic features of toric vector bundles, we will introduce a collection of rational convex polytopes associated to a toric vector bundle. Lattice points in these polytopes correspond to generators for the space of global sections and edges are related to jets. These polyhedral tools also lead to new bundles with an intriguing mix of positivity properties.

Sept. 17, 2014 Colloquium (4:00-5:00)

Speaker

Greg Smith (Queen's University)

Title

Nonnegative sections and sums of squares

Abstract

A polynomial with real coefficients is nonnegative if it takes on only nonnegative values. For example, any sum of squares is obviously nonnegative. For a homogeneous polynomial with respect to the standard grading, Hilbert famously characterized when the converse holds, that is when every nonnegative homogeneous polynomial is a sum of squares. After reviewing some history of this problem, we will examine this converse in more general settings such as global sections of a line bundle. This line of inquiry has unexpected connections to classical algebraic geometry and leads to new examples in which every nonnegative homogeneous polynomial is a sum of squares. This talk is based on joint work with Grigoriy Blekherman and Mauricio Velasco.

Sept. 24, 2014

Speaker

Andy Kustin (South Carolina)

Title

The structure of Gorenstein-linear resolutions of Artinian algebras

Abstract

Click here.

Oct. 1, 2014

Speaker

Andrei Jorza (Notre Dame)

Title

Moduli spaces of modular forms

Abstract

This talk is an introduction to the state-of-the-art perspectives on congruences between modular forms, before a later talk on current research. The idea is that congruences between modular forms can be thought of in terms of Banach norms on certain Banach modules. Fredholm theory of compact operators allows one to construct an analytic variety acting as a moduli space of modular forms. These geometric objects have revolutionized number theory when they appeared in the 80s and again, in a more general form, in the late 90s. Their geometric properties (smoothness, connected components) are related to hard problems in number theory.

Oct. 8, 2014

Speaker

Ching-Jui Lai (Purdue)

Title

Exceptional collection on fake projective planes

Abstract

In this talk, I present the joint work with S.K. Yeung on a geometric approach to the existence of a sequence of special type exceptional collection on fake projective planes. This answers partially a question of Galkin, Katzarkov, Mellit, and Shinder, [GKMS]. We will also discuss some motivations and related problems about phantoms and minifolds.

Oct. 15, 2014

Speaker

Andrew Snowden (Michigan)

Title

Grobner bases for representations of categories

Abstract

I will explain how one can use ideas from commutative algebra, such as Grobner bases, to study representations of categories. (I will also explain what a representation of a category is!) I will then give some applications of this theory, including the resolution of Schwartz's artinian conjecture in the generic representation theory of finite fields.

Oct. 29, 2014

Speaker

Mihai Fulger (Princeton University)

Title

Positivity for higher (co)dimensional numerical cycle classes

Abstract

It is classical to study the geometry of a projective variety by looking at positive cones of numerical Cartier divisor or curve classes. The advancement of the minimal model program through the study of extremal rays of the Mori cone is one of the success stories here. In higher (co)dimension, many counterexamples and very few positive results are known. We recover the expected properties of the pseudoeffective cone, the natural generalization of the Mori cone of curves. This is joint work with Brian Lehmann.

Nov. 5, 2014

Speaker

Kangjin Han (KIAS)

Title

Syzygy bound on the cubic strand of a projective variety and 3-linear resolutions

Abstract

Let X be any projective variety in P^N over an algebraically closed field. Suppose that X is nondegenerate, i.e. it is not contained in any hyperplane of P^N. A few years ago, K. Han and S. Kwak developed a technique to compare syzygies under projections, and as applications they proved sharp upper bounds on the ranks of higher linear syzygies, and characterized the extremal and next-to-extremal cases.

In this talk, we report on generalizations of these results, which are part of an on-going project with S. Kwak and J. Ahn. First, let us consider any variety X such that the defining ideal I_X has no generators of degree less than 3. Since I_X has no generators of degree ≤ 2, the first non-vanishing strand of the resolution comes from linear syzygies of minimal generators of degree 3. We consider a basic degree bound and sharp bounds for generators and syzygies in this cubic strand. Furthermore, we discuss the extremal cases at the end.

Nov. 12, 2014 Colloquium (4:00-5:00)

Speaker

Robin Hartshorne (Berkeley)

Title

Colloquium: Duality in Topology, Algebraic Geometry, and Commutative Algebra

Abstract

To show where modern duality theorems come from, I will give a semi-historical talk starting with Poincaré duality in topology, going on to duality theorems in complex manifolds. Then Serre's introduction of sheaf theory and cohomology into abstract algebraic geometry, with his duality theorem, followed by Grothendieck's generalizations. This leads to the introduction of local cohomology in commutative algebra, and the corresponding local duality theorems.

Nov. 19, 2014

Speaker

Robin Hartshorne (Berkeley)

Title

D-modules and local cohomology

Abstract

I will introduce some of the basic ideas and results about D-modules, that is modules over a ring having also an action of the derivations of that ring. Then I hope to explain some recent work of Lyubeznik showing that local cohomology modules, while in general not finitely generated over the ring, are nevertheless finitely generated as D-modules. This allows one to prove finiteness results such as the finiteness of the number of associated primes of a local cohomolgy module.

Dec. 3, 2014

Speaker

Linquan Ma (Purdue)

Title

On Lech's conjecture

Abstract

I will talk about a long-standing conjecture of Lech on the multiplicities of a faithfully flat extension of local rings. I will discuss several attempts to attack this conjecture. I will show how this conjecture is related to some questions on modules of finite length and projective dimension and discuss some recent progress.

Dec. 10, 2014

Speaker

Dominic Searles (UIUC)

Title

Deformed cohomology of generalized flag varieties

Abstract

In 2006, P. Belkale-S. Kumar introduced a new product on cohomology of generalized flag varieties. We present a new rule for the Belkale-Kumar product for flag varieties of type A, after the puzzle rule of A. Knutson-K. Purbhoo. Our rule uses the combinatorial model of root-theoretic Young diagrams: the pictures of the inversion sets of Weyl group elements.

Inspired by recent work of S. Evens-W. Graham, in joint work with O. Pechenik we also introduce a deformation of the cohomology of generalized flag varieties. A special case gives the Belkale-Kumar deformation. This construction yields a new, short proof that the Belkale-Kumar product is well-defined. Another special case gives a different product structure, picking out triples of Schubert varieties that behave nicely under projections.

Jan. 21, 2015

Speaker

Aaron Silberstein (Chicago)

Title

Geometric Reconstruction of Function Fields

Abstract

This talk will be an introduction to Bogomolov's Program of Anabelian Geometry, as developed by Bogomolov, Tschinkel, Pop, Topaz, and the speaker. We will apply Bogomolov's program - and in particular the technique of geometric reconstruction - to geometric Galois actions and Grothendieck-Teichmüller theory.

Jan. 28, 2015

Speaker

Kevin Tucker (UIC)

Title

On the Limit of the F-signature Function in Characteristic Zero

Abstract

The F-signature of a local ring in positive characteristic gives a measure of singularities by analyzing the asymptotic behavior of the number of splittings (F-splittings) of large iterates of the Frobenius endomorphism. One can also incorporate ideal pairs by restricting the set of "allowable" splittings, and varying the coefficient of the ideal gives rise to the F-signature function of the pair. While for each fixed characteristic p > 0 these functions tend to be extremely complicated, in the few examples that have been computed they tend to limit to a piecewise polynomial function as p tends to infinity. In this talk I will discuss what is known about these functions and their limits, and present a number of new computations for diagonal hypersurfaces. The new computations (joint with Shideler) build on the techniques of Han and Monsky used to compute the Hilbert-Kunz multiplicities of diagonal hypersurfaces.

FRIDAY, Feb. 6, 2015

Speaker

Jerzy Weyman (Connecticut)

Title

Semi-invariants of quivers, cluster algebras and the hive model

Abstract

The saturation theorem for Littlewood-Richardson coefficients was a fashionable subject about a decade ago. There are two completely different proofs of the theorem: the original one by Knutson-Tao based on their hive model, and a proof based on quiver representations given by Harm Derksen and myself. So far there was no link between these two proofs.

Recently Jiarui Fei discovered a remarkable cluster algebra structure on the ring $SI(T_{n,n,n},\beta(n))$ of semi-invariants of a triple flag quiver, whose weight spaces have dimensions that are Littlewood-Richardson coefficients.

In proving his result he uses both the hive model and the quiver representations. It turns out that the link between the two approaches is the quiver with potential underlying the cluster algebra structure. The combinatorics of g-vectors for this quiver with potential turns out to be identical to the hive model.

In my talk I will explain the notions involved and basic ideas behind Jiarui Fei's proof.

Feb. 11, 2015

Speaker

Vivek Mukundan and Jacob Boswell (Purdue)

Title

Rees algebras and almost linearly presented ideals

Abstract

Consider a grade 2 perfect ideal $I$ in $R=k[x_1,\cdots,x_d]$ which is generated by forms of the same degree. Assume that the presentation matrix $\varphi$ is almost linear, that is, all but the last column of $\varphi$ consist of entries which are linear. For such ideals, we find explicit forms of the defining equations of the Rees algebra $\mathcal{R}(I)$. We also introduce the notion of iterated Jacobian duals.

Feb. 25, 2015

Speaker

Jose Rodriguez (Notre Dame)

Title

Numerical irreducible decomposition of multiprojective varieties

Abstract

Numerical algebraic geometry is a growing area of algebraic geometry that involves describing solution sets of systems of polynomial equations. This area has already had an impact in kinematics, statistics, PDE's, and pure math.

This talk will introduce key concepts in numerical algebraic geometry that are used to describe positive dimensional projective varieties. In particular, witness sets will be defined and the classic "regeneration procedure" will be described. The second part of the talk will describe a new "Multi-Regeneration Procedure". This technique gives an effective way of describing multiprojective varieties and determining their multidegrees.

Throughout the talk motivating examples will be provided, and no previous knowledge of numerical algebraic geometry will be assumed. This is joint work with Jonathan Hauenstein.

Monday, March 16, 2015, 3:00-4:00 in Hurley 258

Speaker

Luke Oeding (Auburn)

Title

Are all secant varieties to Segre products arithmetically Cohen-Macaulay?

Abstract

Implicitization problems are central in Applied Algebraic Geometry. Starting with an algebraic-statistical model for structured data (such as tensors with low rank) we often ask for the implicit defining equations for the associated algebraic variety. Usually some of these equations can be found (for example by linear algebra, ad hoc methods, or analyzing symmetry). A difficult problem is then to determine when the known equations suffice. Algebraic properties such as the arithmetically Cohen-Macaulay (aCM) property can be a big help, if it can be determined.

In this talk I will focus on tensors of restricted border rank, or secant varieties of Segre products. I will present what is known about the aCM question and how it can be used for the implicitization problem. I'll present recent computational experiments as well as a structural property of secant varieties that leads me to conjecture an affirmative answer to the aCM question.

March 25, 2015

Speaker

Uli Walther (Purdue)

Title

The logarithmic complex of a (locally homogeneous) divisor

Abstract

The Liouville form of a complex algebraic manifold X can be used to define a complex for each Euler homogeneous element f\in O_X on its domain. The complex relates to the Jacobian ideal and also to D-modules. It appears to be a resolution and in good circumstances resolves a prime Cohen--Macaulay ideal that arises as the annihilator of ann_D(f^s). We discuss applications to Bernstein--Sato polynomials of arrangements.

April 1, 2015

Speaker

Brian Harbourne (Nebraska)

Title

How singular can a reduced plane algebraic curve be?

Abstract

I will discuss what this means with a focus on the case of curves consisting of finite unions of lines in the plane. This question leads to several open problems in algebraic geometry and combinatorics related to some recent advances in current research. One such problem is: classify finite sets of complex lines in the complex plane such that every pair of lines cross somewhere but there are no points where exactly two lines cross.

April 8, 2015

Speaker

Morgan Brown (Michigan)

Title

Rational Connectivity and Analytic Contractibility

Abstract

Berkovich spaces are a natural setting for analysis on varieties over fields with non-archimedean valuation. They have been studied in a variety of contexts, including tropical geometry and number theory. I will give an introduction to Berkovich spaces, and explain recent connections between the theory of Berkovich spaces and the minimal model program. In particular, I will show that if $X$ is a rationally connected smooth projective variety over the Laurent series $\mathbb{C}((t))$, the Berkovich space is a contractible topological space. This is joint work with Tyler Foster.

April 15, 2015

Speaker

Wenbo Niu (Notre Dame)

Title

Mather-Jacobian singularities in generic linkage

Abstract

Mather-Jacobian (MJ) singularities are defined for arbitrary varieties without assuming normal Q-Gorenstein conditions. In this talk, we study the behavior of MJ singularities in generic linkage theory by showing that MJ-log canonical and MJ-canonical are preserved.

April 22, 2015

Speaker

Mattias Jonsson (Michigan)

Title

Degenerations of amoebae and Berkovich spaces

Abstract

A complex projective variety admits an analytification as a complex analytic variety and as a Berkovich (with respect to the trivial norm on the complex numbers). I will explain how a "hybrid" Berkovich space that contains both Archimedean and non-Archimedean data can be used to prove generalizations of a result of Mikhalkin and Rullgard about degenerations of amoebae onto tropical varieties

April 29, 2015

Speaker

Hadi Hedayatzadeh (Purdue)

Title

Exterior powers of p-divisible groups

Abstract

p-divisible groups are smooth formal group schemes which naturally arise as injective limits of p-power torsion in algebraic groups. A celebrated theorem of Serre and Tate states that the deformation theory of an abelian variety over perfect fields is equivalent to the deformation theory of its p-divisible group, and so there is a deep connection between modular forms, which live in the cohomology of moduli spaces of abelian varieties (Shimura varieties) and deformations of p-divisible groups (Lubin-Tate and Rapoport-Zink spaces). These deformation spaces appear naturally in the Langlands program.

In this talk I will talk about p-divisible groups and their deformation spaces. I will then discuss my recent proof of the existence of exterior powers of p-divisible groups and explain how their construction defines a natural map between certain deformation (Rapoport-Zink) spaces. This would imply the existence, e.g., of a determinant map between deformation spaces of p-divisible groups, with implications for recent work of Scholze and Weinstein.