Free resolutions over the polynomial ring have a storied and active record of study. However, resolutions over other rings are much more mysterious; even simple examples can be infinite! In these cases, we look to any combinatorics of the ring to glean information. This talk will present a minimal free resolution of the ground field over the semigroup ring arising from rational normal 2-scrolls, and (if time permits) a computation of the Betti numbers of the ground field for all rational normal k-scrolls.
The problem of finding minimal free resolutions of monomial ideals in polynomial rings has been central to commutative algebra ever since Kaplansky raised the problem in the 1960s and his student, Diana Taylor, produced the first general construction in 1966. The ultimate goal along these lines is a construction of free resolutions that is universal — that is, valid for arbitrary monomial ideals — canonical, combinatorial, and minimal. This talk describes a solution to the problem valid in characteristic 0 and almost all positive characteristics. The differential involves a sum over lattice paths of weights associated to higher-dimensional analogues of spanning trees in appropriate simplicial complexes.
In 2013, Tao et al. introduced the ABC Simple Matrix Encryption Scheme, a multivariate public key encryption scheme. The scheme boasts great efficiency in encryption and decryption, though it suffers from very large public keys. It was quickly noted that the original proposal, utilizing square matrices, suffered from a very bad decryption failure rate. As a consequence, the designers later published updated parameters, replacing the square matrices with rectangular matrices and altering other parameters to avoid the cryptanalysis of the original scheme presented in 2014 by Moody et al.
In this talk, we show that making the matrices rectangular, while decreasing the decryption failure rate, actually, and ironically, diminishes security. We show that the combinatorial rank methods employed in the original attack of Moody et al. can be enhanced by the same added degrees of freedom that reduce the decryption failure rate. Moreover, and quite interestingly, if the decryption failure rate is still reasonably high, as exhibited by the proposed parameters, we are able to mount a reaction attack to further enhance the combinatorial rank methods. To our knowledge this is the first instance of a reaction attack creating a significant advantage in this context.
The units in a number ring are arithmetic gems that are poorly understood from the standpoint of arithmetic statistics. For example, what is the probability that the ring of integers of a number field contains a unit of mixed signature, i.e., a unit that has both a positive and a negative image in two distinct real embeddings?
Absent theorems, we present Cohen-Lenstra style heuristics for unit signatures of odd abelian number fields. In addition, we analyze the equation x^3 – ax^2 + bx – 1 = 0 and prove that there are infinitely many cyclic cubic fields with no units of mixed signature. This is joint work with Noam Elkies, Ila Varma, and John Voight.
A differential graded (DG) algebra is an algebra equipped with a differential satisfying the graded Leibniz (product) rule. The existence of associative DG structures on minimal free resolutions implies many desirable properties of the module being resolved. Moreover, any such DG structure induces an algebra structure on so-called Tor modules. In this talk, we will examine the Tor algebra structure of certain classes of compressed rings. In particular, we will encounter a new class of counterexamples to a conjecture of Avramov and show that there exist ideals defining rings of arbitrarily large type whose Tor algebra has class G.
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.
