Unlike DNA, RNA is single stranded and folds via bonds between pairs of complementary nucleotides while it is still being synthesized from DNA. One central problem in molecular biology is understanding the specific shape into which an RNA molecule folds, as its shape encodes functional information. We will focus on RNA secondary structure in this talk, that is the 2D arrangement of the final RNA configuration. For an RNA sequence, we will consider a representative set of secondary structures and discuss combinatorial methods to mine the structural information from the given ensemble. In particular, we will present a graph algorithms approach based on dissimilarity scores and community detection.

Lexicographic shellability is a powerful and popular technique used to study the topology of the order complex of a poset. The most useful flavor of lexicographic shellability is CL-shellability, which was introduced by Björner and Wachs in the early 1980s and which was then reformulated by them using the notion of recursive atom orderings. In this talk, I will start by providing a survey of some of the key ideas behind lexicographic shellability, particularly CL-shellability, recursive atom orderings, and CC-shellability, the more general version of lexicographic shellability introduced by Kozlov. Then, inspired by Bjorner and Wachs’ formulation of recursive atom orderings, I will introduce a recursive formulation of CC-shellability called generalized recursive atom orderings. I will then discuss some of the new and surprising results provided by this new formulation of CC-shellability.

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Using combinatorial structures to obtain resolutions of monomial ideals is an idea that traces back to Diana Taylor’s thesis, where a simplex associated to the generators of a monomial ideal was used to construct a free resolution of the ideal. This concept has been expanded over the years, with various authors determining conditions under which simplicial or cellular complexes can be associated to monomial ideals in ways that produce a free resolution.  

In a research project initiated at a BIRS workshop “Women in Commutative Algebra” in Fall 2019, the authors studied simplicial and cellular structures that produced resolutions of powers of monomialideals. This talk will focus on powers of square-free monomial ideals of projective dimension one. Faridi and Hersey proved that a monomial ideal has projective dimension one if and only if there is an associated tree (one dimensional acyclic simplicial complex) that supports a free resolution of the ideal. The talk will show how, for each power r >1, to use the tree associated to a square-free monomial ideal I of projective dimension one to produce a cellular complex that supports a free resolution of I^r. Moreover, each of these resolutions will be minimal resolutions. These cellular resolutions can also be viewed as strands of the resolution of the Rees algebra of I. This talk will contain joint work with Susan Cooper, Sabine El Khoury, Sara Faridi, Sarah Mayes-Tang, Liana Sega, and Sandra Spiroff.

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We present a novel code-based Authenticated Key Exchange (AKE) protocol.

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This talk will be about locally complete intersection maps of commutative noetherian rings. Results of Dwyer, Greenlees and Iyengar, and Pollitz lead to a characterization of the complete intersection property for a noetherian ring in terms of the structure, as a triangulated category, of the bounded derived category of the ring. In my talk I will present a similar characterization for locally complete intersection maps. This is joint work with Briggs, Iyengar, and Pollitz.

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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.

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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.

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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.

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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.

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