Algebra Seminar

Algebra Seminar

Title. Smooth limits of plane curves and Markov numbers
 
Abstract. When can we guarantee that smooth proper limits of plane curves are still plane curves? Said a different way --- When is the locus of degree d plane curves closed in the (noncompact) moduli space of smooth genus g curves? It is relatively easy to see that if d>1, then d must be prime. Mori suggests that this may be enough in higher dimensions. Interestingly, in low dimensions, this is not sufficient. For example, Griffin constructed explicit families of quintic plane curves with a smooth limit that is not a quintic plane curve. In this talk we propose the following conjecture: Smooth proper limits of plane curves of degree d are always plane curves if d is prime and d is not a Markov number. We discuss the motivation and evidence for this conjecture which come from Hacking and Prokhorov's work on Q-Gorenstein limits of the projective plane.
Date:
-
Location:
POT 745
Tags/Keywords:
algebra seminar
Type of Event (for grouping events):
Algebra Seminar

Algebra Seminar

Title: Tropical Matroid Homology

 
Abstract: The intersection ring of matroids arises from the restriction of tropical intersection theory to Bergman fans. As an algebraic object, the intersection ring encodes many matroid invariants as homomorphisms and is naturally bigraded by the rank and ground set of the matroids which generate it. The matroid operations of deletion and contraction give rise to boundary maps, producing a tropical analog of the Kontsevich homology. We give an affirmative answer to the conjecture that these homology groups are trivial on the full intersection ring and show that these groups be used to measure whether a class of matroids is closed under certain extensions. We further extend this notion of homology to Chow rings of arbitrary matroids.
Date:
-
Location:
CB 336
Type of Event (for grouping events):
Algebra Seminar

Algebra Seminar

Title: The Birational Geometry of K-Moduli Spaces

Abstract: K-stability is a rapidly developing theory that allows one to construct moduli spaces for Fano varieties. In all known examples, K-moduli spaces are uniruled, so their Kodaira dimensions are negative infinity. In this talk we will describe components of K-Moduli spaces which are birational to M_g, in particular they have maximal Kodaira dimension when g is sufficiently large. This component parameterizes certain moduli spaces of vector bundles on smooth curves, and the main difficulty is to show that these moduli spaces are K-stable. To establish this we require good understanding of their toric degenerations.

Date:
-
Location:
745 Patterson Office Tower
Type of Event (for grouping events):
Algebra Seminar

Algebra Seminar

Title: Computing Free Resolutions of OI-Modules

Abstract: Free resolutions are a powerful tool in commutative and homological algebra. Much of the structure of a module can be encoded in a free resolution. For example, in the case of graded modules, free resolutions can be used to study Betti numbers and Hilbert functions. Certain homological constructions such as the Ext and Tor functors can be computed with free resolutions as well. In this talk we show how to compute free resolutions in the case of OI-modules over a Noetherian polynomial OI-algebra, where OI denotes the category whose objects are totally ordered finite sets and whose morphisms are strictly increasing functions.

Date:
-
Location:
745 Patterson Office Tower
Type of Event (for grouping events):
Algebra Seminar

Algebra Seminar

Title: Computing the Algebra of Conformal Blocks for sl_4

Abstract: Conformal blocks are finite-dimensional vector spaces that arise from the WZNW model of conformal field theory. These have applications in algebraic geometry, particularly in describing the moduli of principal bundles and the moduli of curves. In this talk, we will discuss recent progress on computing a presentation of the algebra of conformal blocks for sl_4. We also describe equations, the tropical variety, and a large family of toric degenerations for the case of a cone with genus 0 and 3 marked points.

Date:
-
Location:
CB 237
Type of Event (for grouping events):
Algebra Seminar

Algebra Seminar

Title: Mori Dream Spaces: What, Where, and Why You Would Care

Abstract: If you've asked me in the past two years about my research, I've likely launched into an explanation of Mori dream spaces, their properties, and/or my implementation of an algorithm in Macaulay2. However, my hallway explanation likely left you with even more questions: What are these objects? Where can I find them? Why would I care? Well, you're in luck! In this talk, I plan to answer all of these questions, give plenty of examples, and list some questions motivating future work. 

Date:
-
Location:
745 Patterson Office Tower
Type of Event (for grouping events):
Algebra Seminar

Algebra Seminar

Title: Pairs of quadratic forms over p-adic fields
 
Abstract: The Hasse-Minkowski theorem implies that if a quadratic form over a number field k has a nontrivial zero over every achimedean and non-achimedean completion of k, then the form will have a nontrivial zero over k. It's natural to ask whether this is true for common nontrivial zeros of pairs of quadratic forms. In an effort to answer this question, new results about pairs of quadratic forms over p-adic fields have been proven. One such result, by Heath-Brown, deals with finding forms in the pencil generated by a pair of quadratic forms over a p-adic field in 8 variables that split off 3 hyperbolic planes. In this talk, we will examine this result by Heath-Brown, and we will discuss the ongoing effort of generalizing Heath-Brown's hyperbolic plane result to pairs of quadratic forms over a p-adic field in an arbitrary number of variables.
Date:
-
Location:
745 Patterson Office Tower
Type of Event (for grouping events):
Algebra Seminar

Algebra Seminar

Title: The Gonality of Rook Graphs

Abstract: 

The gonality game is a type of chip-firing game which is played on graphs which has interesting applications to algebraic geometry. The objective of this game is to ensure that no vertex on our graph has a negative number of chips after one is stolen. In this talk we will learn optimal strategies for winning on 2-dimensional rook graphs.

Date:
-
Location:
CB 345
Type of Event (for grouping events):
Algebra Seminar

Algebra Seminar

Title:  Tropical Linear Series
 

Abstract:  A linear series is a subspace of the space of functions on an algebraic curve.  In this talk, we will discuss tropical analogues of linear series, with focus on a few key examples.  Time permitting, we will close with a number of open questions about these newly defined objects.

 

Date:
-
Location:
POT 745
Type of Event (for grouping events):
Algebra Seminar

Algebra Seminar

Title: Bundles over toric varieties and combinatorics
 
Abstract: Toric varieties are an accessible class of algebraic varieties because they can be studied via polyhedral geometry.  In this talk I'll explain how the addition of a little more combinatorics, in the form of representable matroids, can help us study vector bundles over toric varieties.  After introducing these ideas, I'll explain some classical results that can be proved with these techniques, and state some recent results and open questions.  
Date:
-
Location:
POT 745
Type of Event (for grouping events):
Algebra Seminar
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