20172018 Academic Year
Pretalks begin an hour earlier in LC 317R (unless otherwise indicated)
Date  Room  Speaker  Title  Host 
Aug 25 3:30pm 
LC 317R  Jesse Kass (University of South Carolina) 
How to count lines on the cubic surface arithmetically? (No pretalk) 
(local) 
Aug 28 3:30pm 
LC 407  Blake Farman (University of South Carolina) 
Kernels for Noncommutative Projective Schemes (No pretalk) 
(local) 
Oct 9 3:30pm 
LC 317R  Bastian Haase (Emory University) 
Gerbe patching over arithmetic curves with a view towards homogeneous spaces 
Duncan 
Oct 16 3:30pm 
LC 317R  Matthew Ballard (University of South Carolina) 
The derived geometry of birational maps 
(local) 
Oct 23 3:30pm 
LC 317R  Xiaofei Yi (University of South Carolina) 
Algebraic Linkage 
(local) 
Oct 30 3:30pm 
LC 317R  Alexander Duncan (University of South Carolina) 
Cubic surfaces in positive characteristic 
(local) 
Nov 6 3:30pm 
LC 317R  Dustin Cartwright (University of Tennessee) 
Dual complexes of uniruled varieties 
Duncan 
Nov 13 3:30pm 
LC 317R  Eloisa Grifo (University of Virginia) 
Symbolic powers and the Containment Problem 
Vraciu 
Nov 20 3:30pm 
LC 317R  Patrick McFaddin (University of South Carolina) 
Exceptional collections on toric varieties 
(local) 
Nov 27 3:30pm 
LC 317R  Andrew Kustin (University of South Carolina) 
Use a Macaulay inverse system to detect an embedded deformation 
(local) 
Jan 29 3:30pm 
LC 317R  Candace Bethea (University of South Carolina) 
Naive homotopy classes of endomorphisms of P^{1} 
(local) 
Feb 5 3:30pm 
LC 317R  Mohammed Alabbood (University of South Carolina) 
The 27 lines on a smooth cubic surface in P^{3} 
(local) 
Mar 2 3:30pm 
LC 303B  Michael Burr (Clemson University) 
Understanding Algorithmic Complexity Using Algebraic Geometry 
Duncan 
Mar 19 3:30pm 
LC 317R  Keller Vandebogert (University of South Carolina) 
Properties of Local Rings with Decomposable Maximal Ideals 
(local) 
Apr 13 2:10pm 
LC 310  Lola Thompson (Oberlin College) 
(Speaking in Number Theory Seminar) 

Apr 16 3:40pm 
LC 405  Pete Clark (University of Georgia) 
Densities of sets of integers represented by quadratic forms (Note unusual location and time. Talk is 80 minutes; no pretalk.) 
Thorne 
Apr 30 3:30pm 
LC 317R  Dori Bejleri (Brown University) 
TBA 
Kass 
Abstracts
Mohammed Alabbood  The 27 lines on a smooth cubic surface in P^{3}
In this seminar, we will discuss the problem of existence of a line on a nonsingular cubic surface S in P^{3} by using the resultant and the polar to construct a polynomial whose roots will be the lines in S. We will also present a way to label these lines in terms of two fixed disjoint lines on a cubic surface S in such a way that we can determine all the remaining 25 lines on S.
Matthew Ballard  The derived geometry of birational maps
We will discuss how to assign an integral transform of derived categories to any Dflip. This involves using a little homotopical machinery and yields a functor with pleasant homological properties. Some examples will be discussed.
Pretalk: I will talk about flips and flops and expectations for how they change the derived category.
Candace Bethea  Naive homotopy classes of endomorphisms of P^{1}
The aim of this talk is to explain Cazanave's computation the naive homotopy classes of rational maps P^{1} → P^{1} using symmetric bilinear forms. The pretalk will introduce rational functions on P^{1} and the naive direct sum operation. The talk will introduce the Bézout form, and I will discuss the isomorphism between naive homotopy classes of endomorphisms of P^{1} and the monoid of stable isomorphism classes of nondegenerate symmetric bilinear forms under direct sum.
Michael Burr  Understanding Algorithmic Complexity Using Algebraic Geometry
Many algorithms in computer science take a polynomial system as input. The algorithmic complexity of these algorithms is often expressed in terms of the complexity of the polynomial system (e.g., the degree and bitsize of the coefficients). In this talk, I will present an alternate way to understand the algorithmic complexity using the geometry of (the embedding of) the underlying algebraic variety using a technique called continuous amortization. This is a uniform technique that has led to stateoftheart algorithmic complexity bounds. I will not assume a computer science background, and I will try to provide some insight into how pure mathematicians can successfully translate their tools to applied problems.
Pretalk: This will be a "what is"style talk on homotopy continuation. Homotopy continuation is the main tool in the (relatively new) field of numerical algebraic geometry. Homotopy continuation represents an application of tools from analysis and computation to approximate roots of polynomial systems. This tool has been successful in solving many applied problems including those when more algebraic methods (such as Groebner bases) fail. In this talk, I will provide an introduction to the theory and practice of homotopy continuation with minimal prerequisites.
Pete Clark  Densities of sets of integers represented by quadratic forms
Given an integral quadratic form, one seeks to determine the set of integers that it represents. Though classical  sums of squares were handled by Fermat, Lagrange and LegendreGauss  in general the problem remains open, notwithstanding the algebraic, arithmetic and analytic theories developed to address it.
In this talk we investigate the size of the set of integers represented. The main result is a "Near Hasse Principle"  an integral quadratic form represents 100% of the integers that it locally represents. This leads to an asymptotic for the number of integers of absolute value at most X represented. When there are at least three variables, this amounts to determining the density of the set of represented integers. We then address  but do not fully resolve  the inverse problem of which densities arise.
This is joint work with Paul Pollack, Jeremy Rouse and Kate Thompson.
Dustin Cartwright  Dual complexes of uniruled varieties
The dual complex of a semistable degeneration records the combinatorics of the intersections in the special fiber. The dual complex is homotopy equivalent to the Berkovich analytification, which is a nonArchimedean analogue of the analytic topology on a complex algebraic variety. I will talk about the following relation between the geometry of the general fiber and the topology of the dual complex, and thus the analytification: in arbitrary characteristic, the dual complex of an ndimensional uniruled variety has the homotopy type of an (n1)dimensional simplicial complex.
Pretalk  Various properties of varieties containing the word "rational": A rational variety is on that contains a dense open set isomorphic to an open subset of projective space. Although this definition is easy to state, it turns out to be difficult to verify in examples. Instead, algebraic geometers have noticed that it's easier to classify varieties not in terms of their rationality, but in terms of the existence of rational curves in the varieties. This leads to the notion of rationally connected varieties, ruled varieties, and other variations on these concepts.
Alexander Duncan  Cubic surfaces in positive characteristic
I describe the possible automorphism groups of a cubic surface over an algebraically closed field of arbitrary characteristic. We will see that, while the actual groups that appear in different characteristics can vary considerably, the differences can all be explained by a handful of simple geometric observations.
Pretalk: Given two curves in the complex projective plane, Bezout's theorem provides a simple formula for the number of intersection points (counting multiplicities). The goal of intersection theory is to generalize this idea to intersections of subvarieties of more general varieties. I will discuss the intersection theory of surfaces with an eye towards describing the configuration of 27 lines on a cubic surface.
Blake Farman  Noncommutative projective schemes and DG categories
In their 1994 paper, Noncommutative Projective Schemes, Michael Artin and J.J. Zhang introduce a generalization of usual projective schemes to the setting of not necessarily commutative algebras over a commutative ring. Gonçalo Tabuada in 2005 endows the category of differential graded categories with the structure of a model category and in 2007 Toën shows that its homotopy category is symmetric monoidal closed. In this talk, we’ll give a brief overview of these results, adapting Artin and Zhang’s noncommutative projective schemes for the language of DG categories, and discuss a “geometric” description of this internal Hom for two noncommutative projective schemes.
Eloisa Grifo  Symbolic powers and the Containment Problem
Symbolic powers consist of the functions that vanish up to a certain order at each point in a given variety. The Containment Problem for symbolic and ordinary powers of ideals asks when the containment I^{(a)} ⊆ I^{b} holds. If I is a radical ideal in a regular ring, a famous result by EinLazersfeldSmith, HochsterHuneke and MaSchwede partially answers this question. Harbourne proposed an improvement on this result, which unfortunately does not hold in full generality. However, in this talk we will discuss versions of Harbourne’s Conjecture that do hold. In particular, we will discuss some joint work with Craig Huneke on the case when R/I is an Fpure ring, and give evidence that Harbourne’s conjecture may always hold eventually.
Bastian Haase  Gerbe patching over arithmetic curves with a view towards homogeneous spaces
We will discuss patching techniques and localglobal principles for gerbes over arithmetic curves. The patching setup is the one introduced by Harbater, Hartmann and Krashen. The results obtained for gerbes can be viewed as a 2categorical analogue on their results for torsors. Along the way, we also discuss bitorsor patching and local global principles for bitorsors. As an application of these results, we will study localglobal principles for homogeneous spaces through their quotient stacks.
Pretalk: Field Patching
In this talk, we will give an introduction to the patching technique introduced by Harbater, Hartmann and Krashen. This technique allows to study algebraic objects such as quadratic forms or central simple algebras over a fixed field via studying the same objects over a finite system of overfields. After discussing the general framework, we will focus on the case of arithmetic curves and discuss a couple of recent results obtained via this technique.
Jesse Kass  How to count lines on the cubic surface arithmetically?
A celebrated 19th century result of Cayley and Salmon is that a smooth cubic surface over the complex numbers contains exactly 27 lines. Over the real numbers, the number of lines on a smooth cubic surface depends on the surface, but Segre showed that a certain signed count of lines is the same for such surfaces. In my talk, I will explain Segre’s result and then extend that result to an arbitrary field. This is an application of A1homotopy theory.
All work is joint with Kirsten Wickelgren.
Andrew Kustin  Use a Macaulay inverse system to detect an embedded deformation
Let k be a field of characteristic not equal to 2, P be a standardgraded polynomial ring in four variables over k, and A=P/I be an Artinian Gorenstein kalgebra of embedding dimension four which is defined by six homogeneous forms and has socle degree three. We use the homogeneous Macaulay inverse system that corresponds to I to prove that A has an embedded deformation. In other words, we prove that there is an ideal J in P and an element f in P with f regular on P/J and I=(J,f). (It quickly follows that f is a quadratic form in P and that J is a 5generated grade three Gorenstein ideal generated by the maximal order Pfaffians of a 5 by 5 alternating matrix of linear forms.)
The previous best results describing the structure of sixgenerated grade four Gorenstein ideals assume that P/I is a generic complete intersection. We treat the case when P/I is a graded Artinian algebra over a field at the expense of assuming that the socle degree of P/I is three.
The talk is about joint work with Sabine El Khoury from the American University of Beirut.
Pretalk: If one is working in characteristic zero, then the Macaulay Inverse System for an Artinian Gorenstein ring P/I, with socle degree 3, is a homogeneous form of degree three in the polynomial ring generated by the four partial derivative operators. So, in some sense, the talk is about cubic surfaces in projective 3space. In the pretalk I will explain how to use Divided Powers to make the notion of Macaulay Inverse System both coordinate free and characteristic free.
Patrick McFaddin  Exceptional collections on toric varieties
Exceptional collections of a variety X are effectively decompositions of the derived category of X, analogous to an (semi)orthonormal basis of a vector space with innerproduct. The existence of exceptional collections for smooth projective toric varieties defined over the complex numbers was settled affirmatively by Kawamata using the framework afforded by the toric Minimal Model Program. More recently, Ballard, Favero, and Katzarkov have given another proof using Variation of GIT. The study of toric varieties defined over arbitrary fields (socalled arithmetic toric varieties) has been taken up by a number of authors, although much less is known about their derived geometry. In this talk, we will discuss an effective Galois descent result for such collections and provide applications to arithmetic toric varieties of dimension two, three, and four. This is joint work with M. Ballard and A. Duncan.
Keller Vandebogert  Properties of Local Rings with Decomposable Maximal Ideals
This talk will start with two conditions due to a paper of Huneke and Jorgensen called the Auslander Condition and the Uniform Auslander Condition, and show that neither of these properties is necessarily preserved under localization. After some discussion of the constructions demonstrating this, we will move on to an examination of fiber products of finite CohenMacaulay type. We start with a relation between rings realized as nontrivial fiber products and induced isomorphisms between integral closures in the total fraction fields, and look more closely at how module structure over a nontrivial fiber product gives information about the structure over the individual pieces of that fiber product, and vice versa. Lastly, the main result for the talk will be stated and if time permits further discussion will ensue.
Pretalk: We will go over the necessary (nontrivial) terminology to digest the material of the main talk. This will include, but is not limited to: fiber products, Poincare/Bass series, hypersurfaces, depth/codepth/type/embedding codepth, HilbertSamuel multiplicity, semidualizing modules, and some examples for each of the aforementioned terms.
Xiaofei Yi  Algebraic Linkage
Two projective varieties in P^{3} are directly linked if their union is a complete intersection. Two projective varieties are linked if they can be connected by a sequence of direct links. Two ideals I and J are directly linked if there is a regular sequence α in the intersection such that I:(α)=J and J:(α)=I. And they are linked if they can be connected by a sequence of direct links. Some properties are invariant under linkage, but some properties require a "good" linkage class to be invariant. In the seminar, we will discuss some properties invariant under linkage class, and then discuss a technique to construct a good linkage class. If time allows, we will also discuss the linkage in the homogeneous case.