Greg Blekherman, Georgia Institute of Technology
Do Sums of Squares Dream of Free Resolutions?
A real polynomial is called nonnegative if it takes only nonnegative values. Sums of squares of polynomials (and rational functions) are obviously nonnegative. I will briefly review the rich history of this area of real algebraic geometry. I will explain how this topic is inextricably linked to classical topics in algebraic geometry and commutative algebra, such as free resolutions. I will discuss a specific example of square-free monomial ideals, which connects this area to positive semidefinite matrix completion problems.
Alessandro De Stefani, University of Nebraska
Symbolic Powers in Mixed Characteristic
The Zariski-Nagata theorem in one of its classical versions states that, if P is a prime ideal in a polynomial ring over the complex numbers, then the n-th symbolic power of P consists of all the polynomial functions that vanish of order at least n at all points of the variety defined by P. We prove analogous results in mixed characteristic, combining the properties of differential operators and p-derivations. This talk is based on joint work with Eloísa Grifo and Jack Jeffries.
Daniel Hernández, University of Kansas
Frobenius Powers of Ideals in Regular Rings
In this talk, we introduce the notion of the generalized Frobenius powers of an ideal in a regular ring of prime characteristic. As an application of this theory, we establish a prime characteristic analog of Howald's result relating the log canonical threshold of a polynomial with that of its term ideal. Time permitting, we will also discuss applications of this theory to that of Bernstein-Sato polynomials, and other invariants of singularities. This is joint work with Emily Witt and Pedro Teixeira.
Sam Payne, Yale University
A Tropical Motivic Fubini Theorem with Applications to Donaldson-Thomas Theory
I will present a new tool for the calculation of Denef and Loeser’s motivic nearby fiber and motivic Milnor fiber: a motivic Fubini theorem for the tropicalization map, based on Hrushovski and Kazhdan’s theory of motivic volumes of semi-algebraic sets. As time permits, I will discuss applications of this method, which include the solution to a conjecture of Davison and Meinhardt on motivic nearby fibers of weighted homogeneous polynomials, and a very short and conceptual new proof of the integral identity conjecture of Kontsevich and Soibelman, first proved by Lê Quy Thuong. Both of these conjectures emerged in the context of motivic Donaldson-Thomas theory.
Ilya Smirnov, University of Michigan
Lech's Inequality and its Improvements
In 1960 Lech found a simple inequality that relates the colength and the multiplicity of a primary ideal in a local ring. Unfortunately, his proof also shows that the inequality is never sharp if dimension is at least two. The goal of this talk is to present a stronger form of Lech's inequality and an even stronger conjecture that will make the inequality sharp. I will also discuss a Lech-type inequality on the multiplicity and the minimal number of generators of an integrally closed ideal.
Mark Walker, University of Nebraska
On Complexes of Free Modules
Let R be local ring of dimension d and F a minimal complex of free R-modules. We consider the Conjecture: If F is bounded and has non-zero finite length homology, then the total rank of F must be at least 2d. When F is the minimal resolution of a module of finite length and finite projective dimension, this conjecture is a weak form of the well-known Buchsbaum-Eisenbud-Horrocks Conjecture. A variant of this conjecture is well-known in topology as the Toral Rank Conjecture.
In this talk I will discuss recent progress, both positive and negative, toward settling this conjecture. In particular, I will discuss examples, constructed in joint work with Srikanth Iyengar, showing that the conjecture fails in general. Finally, I will describe how these counter-examples lead also to counter-examples of some related conjectures.
Dana Weston, University of Missouri
Descent and Ascent of Perinormality in Some Ring Extension Settings
An integral domain A with quotient field K is defined as perinormal if the only local rings, lying between A and K, to satisfy the going-down theorem over A are of the type Ap for p ∈ Spec(A). Let B stand for A[X] or Aˆ (if A is local). We investigate conditions under which perinormality of one of A or B implies the perinormality of the other. (joint work with Andrew McCrady)
Eric Canton, University of Nebraska
Asymptotic Invariants of Ideal Sequences in Positive Characteristic via Berkovich Spaces
I will describe my recent work extending results of Jonsson and Mustata regarding log canonical thresholds of graded sequences of ideals to the positive characteristic setting. The techniques are a blend of F-singularity machinery and non-Archimedean geometry.
Justin Chen, University of California, Berkeley
Flat Maps to and from Noetherian Rings
We investigate flat maps where the source or target is a Noetherian ring, giving necessary and/or sufficient conditions on a ring for such maps to exist. Along the way, we develop some general facts about flat ring maps, and exhibit many examples, including a new class of zero-dimensional local rings with nice properties.
Alessandra Costantini, Purdue University
Cohen-Macaulay Property of Rees Algebras of Modules
We provide a class of modules having Cohen-Macaulay Rees algebra. Our result generalizes a well-known result by Johnson and Ulrich, where the Rees algebra of an ideal I was shown to be Cohen-Macaulay under suitable assumptions on the reduction number of I and on the depths of powers of I.
Rankeya Datta, University of Michigan
Uniform Approximation of Abhyankar Valuation Ideals in Functions Fields of Prime C
Using the theory of asymptotic test ideals, we prove the prime characteristic analogue of a characteristic 0 result of Ein, Lazarsfeld and Smith on uniform approximation of valuation ideals associated to real-valued Abhyankar valuations centered on regular varieties over perfect fields.
Benjamin Drabkin, University of Nebraska
Symbolic Defect of Squarefree Monomial Ideals
A central area of interest in the study of symbolic powers of ideals is the relationship between symbolic and ordinary powers. The symbolic defect of an ideal is a numerical invariant which measures the difference between the m-th ordinary and symbolic powers of an ideal. More specifically, the m-th symbolic defect of an ideal, I, is the number of minimal generators of the m-th symbolic power of I quotiented by the m-th ordinary power of I. The growth of the symbolic defect of a monomial ideal can be studied through Presburger counting functions and various combinatorial constructions.
Zachary Flores, Colorado State University
The Weak Lefschetz for a Graded Module
Over a field of characteristic zero, it is well known that complete intersections in codimension 3 have the Weak Lefschetz Property. This result follows from a beautiful blend of commutative algebra and algebraic geometry. We discuss a modest generalization of this result by extending these techniques to investigate when the cokernel of a linear map between free modules over a polynomial ring in three variables has the Weak Lefschetz Property.
Brent Holmes, University of Kansas
Generalized Nerves and Depth
We introduce and study generalized notions of the nerve complex for a collection of subsets of a topological space. For simplicial complexes, we show that these notions afford an elegant and efficient characterization of the depth of the complex. We also prove several results relating these new complexes and Serre conditions.
Hang Huang, University of Wisconsin
Equations of Kalman Varieties
Given a subspace L of a vector space V, the Kalman variety consists of all matrices of V that have a nonzero eigenvector in L. We applied Kempf Vanishing technique with some more explicit constructions to get a long exact sequence involving coordinate ring of Kalman variety, its normalization and some other related varieties in characteristic zero. We also extracted more information from the long exact sequence including the minimal defining equations for Kalman varieties.
Takumi Murayama, University of Michigan
Frobenius-Seshadri Constants: Limits in Algebraic Geometry
We introduce higher-order variants of the Frobenius-Seshadri constant due to Mustata and Schwede. These are constants defined as a limit supremum involving both ordinary and Frobenius powers of ideals defining closed points on a projective variety, and seem to be related to Hilbert-Kunz multiplicity and F-signature. As applications, we use these constants to give geometric information about ample line bundles on projective varieties, and give a characterization of projective space. Both applications were previously known only in characteristic zero.
Josh Pollitz, University of Nebraska
Koszul Varieties and an Application in the Derived Category
The support varieties of Avramov and Buchweitz were used to show certain (co)homological properties of a complete intersection hold. One example is that, over a complete intersection, one can can show the eventual vanishing of Ext modules for a pair of modules is equivalent to the eventual vanishing of Tor modules for that same pair of modules using these varieties. I will discuss how to define a generalization of these varieties when your ring is not necessarily a complete intersection. Moreover, these varieties will be the same as the classical support varieties when the underlying ring is a complete intersection. I will also give a sketch of a result that recovers the aforementioned result of Avramov and Buchweitz over complete intersections, and I will present an application in the derived category of non-complete intersections.
Tony Se, University of Mississippi
Semidualizing Modules of Ladder Determinantal Rings
A semidualizing module is a generalization of the canonical module of a Cohen-Macaulay local ring. We aim to determine the semidualizing modules of so-called ladder determinantal rings.
William Taylor, University of Arkansas
Interpolating Between Hilbert-Samuel and Hilbert-Kunz Multiplicity
We present a function that interpolates continuously between the Hilbert-Samuel and Hilbert-Kunz multiplicities of ideals in a ring of positive characteristic, and show how it can be calculated using volumes of polyhedra in Euclidean space.