Local Cohomology of Thickenings on Sequences of Rings
Jenny Kenkel – University of Kentucky
Let R be a standard graded polynomial ring and let I be a homogenous prime ideal of R. Bhatt, Blickle, Lyubeznik, Singh, and Zhang examined the local cohomology of R/I^t as t grows arbitrarily large. I will discuss their results and give an explicit description of the transition maps between these local cohomology modules in a particular example. I will also consider asymptotic structure in a different direction: as the number of variables of R grows. This study of families of modules over compatible varying rings hints at the existence of OI structures, which I will discuss if time permits.
The cryptosystem C*, first proposed and studied by Matsumoto and Imai and introduced in EUROCRYPT ’88, is the predecessor of all of the so called “big field” schemes of multivariate cryptography. This scheme has since been broken, which has led to the introduction of modifiers. The introduction of the numerous modifiers of multivariate schemes have produced several variants that stay faithful to the central structure of the original. From the tumultuous history of C* derivatives we now see only a very few survivors in the cryptonomy. In this work, we revisit the roots of multivariate cryptography, investigating the viability of C* schemes, in general, under the entire multidimensional array of the principal modifiers. We reveal that there is a nontrivial space of combinations of modifiers that produce viable schemes resistant to all known attacks. This solution space of seemingly secure C* variants offers trade-offs in multiple dimensions of performance, revealing a family that can be optimized for disparate applications.
In this talk, we discuss about methods for proving existence and uniqueness of a root of a square analytic system in a given region. For a regular root, Krawczyk method and Smale’s α-theory are used. On the other hand, when a system has a multiple root, there is a separation bound isolating the multiple root from other roots. We define a simple multiple root, a multiple root whose deflation process is terminated by one iteration, and establish its separation bound. We give a general framework to certify a root of a system using these concepts.
The main problem in phylogenetics is to reconstruct evolutionary relationships between collections of species, typically represented by a phylogenetic tree. In the statistical approach to phylogenetics, a probabilistic model of mutation is used to reconstruct the tree that best explains the data (the data consisting of DNA sequences from homologous genes of the extant species). In algebraic statistics, we interpret these statistical models of evolution as geometric objects in a high-dimensional probability simplex. This connection arises because the functions that parametrize these models are polynomials, and hence we can consider statistical models as algebraic varieties. The goal of the talk is to introduce this connection and explain how the algebraic perspective leads to new theoretical advances in phylogenetics, and also provides new research directions in algebraic geometry. The talk material will be kept at an introductory level, with background on phylogenetics and algebraic geometry.
Seth Sullivant received his PhD in 2005 from the University of California, Berkeley. After a Junior Fellowship in Harvard’s Society of Fellows, he joined the department of mathematics at North Carolina State University in 2008 as an assistant professor. He was promoted to full professor in 2014 and distinguished professor in 2018. Sullivant’s work has been honored with a Packard Foundation Fellowship and an NSF CAREER award and he was selected as a Fellow of the American Mathematical Society. He helped to found the SIAM activity group in Algebraic Geometry where he has served as both secretary and chair. Sullivant’s current research interests include algebraic statistics, mathematical phylogenetics, applied algebraic geometry, and combinatorics.
We will be looking at composition algebras that satisfy certain weaker laws of associativity. These algebras can then be used for representations of generalizations of groups. We will then see how one can create other algebras satisfying the same identities to use as modular representations. This includes both commutative algebras and non-commutative algebras.
The Frobenius complexity of a local ring R measures asymptotically the abundance of Frobenius actions of order e on the injective hull of the residue field of R. It is known that, for Stanley-Reisner rings, the Frobenius complexity is either -infinity or 0. However, a complete description of the Frobenius complexity sequence c_ e(R)for all values of e is not yet known when R is Stanley-Reisner. We will provide the answer for a large class of Stanley-Reisner rings, generalizing work of Alvarez Montaner, Boix and Zarzuela.
Two seemingly unrelated combinatorial objects and actions are increasing tableaux with K-promotion (arising in K-theory of the Grassmannian), and rowmotion on plane partitions (arising in Lie theory). In this talk, we will give some background to understand these combinatorial objects and actions, describe the connection between these two things (joint w/ J. Striker and O. Pechenik), and then discuss how the results can be extended to the more general setting of increasing labelings of any partially ordered set (joint w/ J. Striker and C. Vorland).
Every electrical power system can be modeled by a graph G whose vertices represent buses and whose edges represent power lines. A phasor measurement unit (PMU) is a device that can be placed at a bus to observe the voltage at that bus as well as the current and its phase through all incident power lines. The problem of monitoring the entire electric power system using the fewest number of PMUs is closely related to the well-known vertex covering and dominating set problems in graph theory.
In this talk, we will give an overview of the PMU placement problem and its connections to commutative ring theory. We define the edge ideal I^P_G of a graph G vertices in a polynomial ring R=k[X_1, . . . , X_n] and we describe some algebraic properties of the quotient R/I^P_G. In particular, we will show that, when we restrict to trees, the Cohen- Macaulay property is implied by the unmixed property and implies the complete intersection property. We will also give examples to show that for non-trees, these implications can fail.
In 2017, Nasseh and Sather-Wagstaff proved that if M and N are finitely generated modules over a non-trivial fiber product R such that Tor^i_R(M,N)=0 for i»0, then M or N has finite projective dimension. Their proof uses Moore’s explicit free resolution of the second syzygy of M. Recently, Avramov, Iyengar, Nasseh, and Sather-Wagstaff have proved that the same conclusion holds for other classes of rings using differential graded algebra techniques. Thus, it is natural to ask whether one can explicitly describe DG algebra resolutions of fiber products. We will present progress on this question.
For A→B a flat homomorphism of commutative noetherian rings and M an injective A-module, we can calculate the injective dimension of M⊗_AB as a B-module by calculating the injective dimensions of the fibers F(p)=B_p/pB_p for each p∈Ass(M) by the formula id_B(M⊗B) = sup_{p∈Ass(M)} id_{F(p)}F(p) (Foxby, 1975). This formula can be used to recover the fact that injective modules remain injective under certain flat base changes, such as a localization. We work towards a generalization of Foxby’s Theorem to calculate the Gorenstein injective dimension of the base change of a Gorenstein injective module.
