Algebra and Number Theory
Department of Mathematics and Statistics
Texas Tech University
The Hochschild cohomology is a tool for studying associative algebras that has a lot of structure: it is a Gerstenhaber algebra. This structure is useful because of its applications in deformation and representation theory, and recently in quantum symmetries. Unfortunately, computing it remains a notoriously difficult task. In this talk we will present techniques that give explicit formulas of the Gerstenhaber algebra structure for general twisted tensor product algebras. This will include an unpretentious introduction to this cohomology and to our objects of interest, as well as the unexpected generality of the techniques. This is joint work with Tekin Karadag, Dustin McPhate, Tolulope Oke, and Sarah Witherspoon.
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In local algebra, the quotient of the bounded derived category by the subcategory of perfect complexes is often referred to as the
singularity category. The quotient is trivial for a regular ring, and for a Gorenstein ring it is triangulated equivalent the
stable category of maximal Cohen--Macaulay modules or, from our point of view, to the homotopy category of totally acyclic complexes of
finitely generated projective modules. I will give an overview of a few recent papers in which we use the flat--cotorsion theory to extend these ideas to schemes.
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Let $(R,\mathfrak{m},k)$ be a commutative local ring and $\mathbf{K_{tac}}(R)$ the category of totally acyclic $R$-complexes. In this talk, I will define an extension of critical degree for $R$-modules to totally acyclic complexes. In addition, I will discuss how the critical and cocritical degrees of an $R$-complex change under certain operations, with a focus on operations involving tensor products of complexes.
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Let $R$ be a commutative local ring to which we associate the subcategory $\mathbf{K_{tac}}(R)$ of the homotopy category of $R$-complexes, consisting of the totally acyclic complexes. When $R$ is a hypersurface ring of finite CM type we can show that this category has only finitely many non-isomorphic, indecomposable objects. We will use these objects and the idea of approximation, to classify totally acyclic complexes over certain complete intersection rings using the defined notion of Arnold-tuples.
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We consider a key-exchange protocol proposed based on a free nilpotent p-group. We found that by using an algorithm to solve a discrete logarithm problem with matrices over a finite field, we can then recover an integer through a series of multiplication in the group which mimics an intercepted private parameter.
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In 1988, Avramov, Kustin, and Miller gave a complete classification of the possible Tor-algebra structures arising from the minimal free resolution of a quotient ring $R/I$ with projective dimension 3. Absent from this classification was a complete description of which Tor-algebra structures actually arise as the Tor-algebra of some quotient $R/I$ with some prescribed homological data. This problem of realizability has remained open, with recent progress by Christensen, Veliche, and Weyman, giving stronger restrictions on the set of Tor-algebra classes that may occur. In this talk, we will explore how the process of "trimming" an ideal can be used to preserve Tor-algebra class while altering homological data. We will see how an explicit algebra structure on a certain resolution of these ideals descends to multiplication on the Tor-algebra to explain why trimming an ideal often yields another ideal with the same Tor-algebra class. Finally, we discuss applications to the realizability problem for Tor-algebras.
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A construction of Tate shows that every algebra over a ring $R$ possesses a DG-algebra resolution over $R$. These resolutions are not always minimal and Avramov even shows that certain algebras cannot have a minimal resolution with a DG-algebra structure. In this talk, I will explicitly construct the minimal free resolution of $k [\![ \underline{x} ]\!] /\mathcal{I} \times_k k [\![ \underline{y} ]\!] / \mathcal{J}$ over $k [\![ \underline{x}, \underline{y} ]\!]$. From there, I will discuss when this minimal resolution exhibits a DG-structure.
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In this talk, we provide a Zagier-type formula for the multiple t-values (special Hurwitz zeta values),
\begin{align*}
\displaystyle t(k_{1}, k_{2}, \ldots, k_{r})
&=2^{-(k_{1}+k_{2}+\ldots +k_{r})}\zeta(k_{1}, k_{2}, \ldots, k_{r}; -\tfrac{1}{2}, -\tfrac{1}{2}, \ldots, -\tfrac{1}{2})\\
&=\sum_{1\leq n_{1}< n_{2}< \ldots < n_{r}}
\frac{1}{(2n_{1}-1)^{k_{1}} (2n_{2}-1)^{k_{2}} \ldots (2n_{r}-1)^{k_{r}}}.
\end{align*}
Our formula is similar to Zagier’s formulas for multiple zeta values $\zeta(2, \ldots, 2, 3)$ and will involve $\mathbb{Q}$-linear combinations of powers of $\pi$ and odd zeta values. The derivation of the formula for $t(2, \ldots, 2, 3)$ relies on a rational zeta series approach via a Gauss hypergeometric function argument.
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