Oklahama State University
Title: Generalized Borel ideals.
Abstract: Borel ideals are a well-studied class of monomial ideals known to be minimally resolved by the Eliahou-Kervaire resolution. In 2013, Francisco, Mermin, and Schwieg introduced Q-Borel ideals, a generalization of Borel ideals defined by relations in a poset. We extend their work and describe some homological and combinatorial properties of these ideals.
University of Wisconsin-Madison
Title: Asymptotic Syzygies for Products of Projective Space
Abstract: I will discuss results describing the asymptotic syzygies of products of projective space, in the vein of the explicit methods of Ein, Erman, and Lazarsfeld’s non-vanishing results on projective space.
University of Minnesota-Twin Cities
Title: Cluster Algebras and $k$-Positivity Tests
Abstract: A $k$-positive matrix is a matrix where all minors of order $k$ or less are positive. Computing all such minors to test for $k$-positivity is inefficient, as there are {2n \choose n}-1 of them in an $n \times n$ matrix. However, there are minimal $k$-positivity tests which only require testing $n^2$ minors. These minimal tests can be related by a series of exchanges, and form a family of sub-cluster algebras of the cluster algebra of total positivity tests. We give a description of the sub-cluster algebras that give $k$-positivity tests and ways to move between them.
Univesity of Nebraksa- Lincoln
Title: Symbolic Defect and Cover Ideals
Abstract: Let R be a commutative Noetherian ring, and let I be an ideal in R. The symbolic defect is a numerical measurement of the difference between the symbolic and ordinary powers of I. In the case that I has sufficiently well-behaved symbolic powers (i.e. its symbolic Rees algebra is finitely generated) we prove that the symbolic defect grows eventually quasi-polynomially. Furthermore, we describe more specifically the growth of the symbolic defect in certain classes of ideals arising from graphs, termed cover ideals.
University of Texas-Arlington
Title: Ideals with free Koszul homology
Abstract:It is well known that an ideal is a complete intersection if and only if all its positive Koszul homologies are zero. Also, complete intersections are known for their nice (co)homological properties. More generally, ideals with free non-zero Koszul homologies may also have some nice properties. The maximal ideal of a local ring is one of such ideals whose homological properties are well understood. Another example of such ideals is quasi-complete intersections which have been studied recently in several manuscripts. In this talk, I will focus on homological properties of ideals whose Koszul homologies are free and show that if the base ring is nice, they also carry certain nice properties.
Univesity of Utah
University of Wisconsin-Madison
Title: Secant ideals of Plucker embedded Grassmannians
Abstract: Secant varieties have long been a topic of interest in algebraic geometry. Despite this, very little is known about their algebraic structure. In this talk, we will define what a secant variety is, and discuss some of their various applications. This will lead into a new result which shows we can generate the r-th secant ideal of any Plucker ideal in degree bounded by some constant C(r) which does not depend on the choice of Plucker ideal.
University of Utah
Title: Local to global principles for generation time over commutative rings
Abstract: In the derived category of modules over a commutative noetherian ring a complex G is said to generate a complex X if the latter can be obtained from the former by taking finitely many summands and cones. The number of cones needed in this process is the generation time of X. In this talk I will present some local to global type results for computing this invariant, and also discuss some applications of these results.
University of Kansas
Title: Cohen-Macaulay Rings of Finite $\operatorname{\mathsf{CM}}_+$-Representation Type.
Abstract: Let $(R,\mathfrak{m},k)$ be a Cohen-Macaulay local ring. We say $R$ has finite $\operatorname{\mathsf{CM}}_+$-representation type if $R$ admits only finitely many nonisomorphic indecomposable maximal Cohen-Macaulay modules that are not locally free on the punctured spectrum of $R$. We prove several necessary conditions for $R$ to have finite $\operatorname{\mathsf{CM}}_+$-representation type. In some cases, we prove Gorenstein local rings of finite $\operatorname{\mathsf{CM}}_+$-representation type must be hypersurfaces, and we provide a classification of these rings in dimension $1$; if $R$ is complete, equicharacteristic, and with some hypotheses on $k$, they are exactly the hypersurfaces of countable Cohen-Macaulay representation type.
University of Missouri-Columbia
Title: Finiteness of Frobenius Test Exponents
Abstract: The Frobenius Test exponent of a local ring of prime characteristic is the smallest Frobenius bracket power which forces every ideal generated by a full system of parameters to agree with its (usually strictly larger) Frobenius closure. In this talk, we will discuss some cases where finiteness of this exponent is known and techniques to reduce the study of this exponent to the study of a modified version of the Hartshorne-Speiser-Lyubeznik numbers of a ring, which are related to a canonical Frobenius action on local cohomology at the maximal ideal of the ring.
University of Michigan
Title: On the behavior of F-injectivity under flat homomorphisms
Abstract: Let (R,m) -> (S,n) be a flat local homomorphism of local rings. The relationship between properties of R and S is well understood in many cases. For example, if S is regular, then R is regular, and if both R and S/mS are regular, then S is regular. We present similar results for F-injectivity, which is a property of rings of positive characteristic defined using the Frobenius action on local cohomology. F-injectivity is the most general class of F-singularities usually considered. We also give some applications to other questions about F-injective rings. This work is joint with Rankeya Datta.
Michigan State University
Title: Log-Canonical Poisson Structures
Abstract: A Poisson bracket on a commutative associative algebra is a Lie bracket which is also a biderivation with respect to the associative multiplication. A Poisson bracket is called "log-canonical" if it is given by certain very simple homogeneous quadratic expressions in the generators of the algebra. These structures show up in various places, including the theory of cluster algebras. In joint work with John Machacek, we prove that log-canonical brackets are "as simple as possible", in the sense that they cannot be transformed by rational maps into brackets given by lower degree polynomials.
Univesity of Nebraksa- Lincoln
Title: Quasi-Polynomial Growth of Betti Numbers over Complete Intersections
Abstract: It is known that the Betti numbers for any finitely generated module over a local complete intersection ring grow on the order of a polynomial. Further, it can be shown that, for large enough degree, there are two polynomials of interest: one explicitly giving the even Betti numbers and one giving the odd Betti numbers. The aim of this talk is to show a bound on the discrepancy of these two polynomials for every finitely generated module over a complete intersection with respect to an invariant of the ring called its ''quadratic codimension". This is joint work with Lucho Avramov and Mark Walker.
University of Michigan
Title: Uniform Matroidal Ideals: A Curation of Cameo Applications
Abstract: After first recalling select asymptotic homological properties of the basis- and circuit (square-free monomial) ideals of matroids, we shift to surveying recent illustrious cameos of these constructions for uniform matroids in algebra, geometry, and combinatorics. The square-free monomial ideal I =(xy, xz, yz) will be a resurgent protagonist.