UCSD Algebra seminar

The location is AP&M 7321 and the meeting time is 3PM unless specified otherwise.

Contact Steven Sam if you are interested in speaking.
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Spring 2020

DateSpeakerTitle and abstract
Mar. 30Thomas Grubb
An $FS^{op}$ structure on Fulton-MacPherson Compactifications

We will start by giving a brief introduction to representation stability and combinatorial categories. Then we will introduce Fulton and MacPherson's ``wonderful compactifications'' of configuration spaces, and describe how they may be studied through the lens of representation stability. In particular, we show that under a mild hypothesis we can approach the representation theory of these spaces using the combinatorial category $FS^{op}$. We end by discussing an attempt at showing that these spaces do exhibit representation stability, although to date the results of this approach have been fairly underwhelming. This talk can be accessed at https://ucsd.zoom.us/j/309940113 .

Winter 2020

DateSpeakerTitle and abstract
Feb. 3Alexander Sutherland
UC Irvine
On the Geometry of Solutions of the Sextic in Two Variables

Abel's theorem (1824) that the generic polynomial of degree n is solvable in radicals if and only if $n < 5$ is well-known. However, the classical works of Bring (1786) and Klein (1884) give solutions of the generic quintic polynomial by allowing certain other "nice" algebraic functions of one variable. For the sextic, it is conjectured that any solution requires algebraic functions of two variables. In this talk, we will examine and relate the intrinsic geometries of the known solutions of the sextic in two variables, extending the work of Green (1978).
Feb. 10Finley Mcglade
Hecke Modules for $\mathrm{SL}_3(\mathbb{Q}_p)$

From number theory to knots, Hecke algebras have applications within many areas of mathematics. In this talk we describe a pictorial calculus for computing convolution products in affine Hecke algebras over fields of charactistic zero. Convolution products of this type have been understood since the work of Iwahori and Matsumoto [1965]. However, using results of Parkinson, Ram and Schwer [2006], we can now draw pictures illustrating the rich combinatorial nature of these products. We describe this pictorial calculus in the example of $\mathrm{SL}_3(\mathbb{Q}_p)$. Its applicability is limited to characteristic zero.

In characteristic $p$, Schneider [2007] has shown that the representation theory of $\mathrm{SL}_3(\mathbb{Q}_p)$ is related to the derived category of modules over a differential graded algebra $\mathcal{H}_I^{\bullet}$. Here $\mathcal{H}_I^{\bullet}$ is a derived version of the usual Hecke algebra $\mathcal{H}_I$. Time permitting, I will explain some of Schneider's results. These include a computation of the cohomology algebra $h^{\ast}(\mathcal{H}^{\bullet}_I)$.

Fall 2019

DateSpeakerTitle and abstract
Nov. 4Daniela Amato
Universidade de Brasilia
Highly arc transitive and descendant-homogeneous digraphs with finite out-valency

We investigate infinite highly arc transitive digraphs with two additional properties, descendant-homogeneity and Property \(Z\). A digraph \(D\) is highly arc transitive if for each \(s \geq 0\) the automorphism group of \(D\) is transitive on the set of directed paths of length \(s\); and \(D\) is descendant-homogeneous if any isomorphism between finitely generated subdigraphs of \(D\) extends to an automorphism of \(D\). A digraph is said to have Property \(Z\) if it has a homomorphism onto a directed line.
We show that if \(D\) is a highly arc transitive descendant-homogeneous digraph with Property \(Z\) and \(F\) is the subdigraph spanned by the descendant set of a directed line in \(D\), then \(F\) is a locally finite 2-ended digraph with equal in- and out-valencies. If, moreover, \(D\) has prime out-valency then \(F\) is isomorphic to the digraph \(\Delta_p\). This knowledge is then used to classify the highly arc transitive descendant-homogeneous digraph of prime out-valency which have Property \(Z\).
Nov. 18David Ben-Ezra
Non-Linearity of Free Pro-$p$ Groups

It is a classical fact that free (discrete) groups can be embedded in $GL_{2}(\mathbb{Z})$. In 1987, Zubkov showed that for a free pro-$p$ group $F_{\hat{p}}$, the situation changes, and when $p>2$, $F_{\hat{p}}$ cannot be embedded in $GL_{2}(\Delta)$ when $\Delta$ is a profinite ring. In 2005, inspired by Kemer's solution to the Specht problem, Zelmanov sketched a proof for the following generalization: For every $d\in\mathbb{N}$ and large enough prime $p\gg d$, $F_{\hat{p}}$ cannot be embedded in $GL_{d}(\Delta)$.

The natural question then is: What can be said when $p$ is not large enough? What can be said in the case $d=p=2$ ? In the talk I am going to describe the proof of the following theorem: $F_{\hat{2}}$ cannot be embedded in $GL_{2}(\Delta)$ when $char(\Delta)=2$. The main idea of the proof is the use of trace identities in order to apply finiteness properties of a Noetherian trace ring through the Artin-Rees Lemma (Joint with E. Zelmanov).
Nov. 25Bakhrom Omirov
National University of Uzbekistan
On finite-dimensional Leibniz algebras

In this talk we will present several classical results on finite-dimensional Leibniz algebras. We give main examples of Leibniz algebras and show nilpotency of Leibniz algebras in terms of special kinds of derivations. Also, we present the structure of solvable Lie algebras with a given nilradical and with the maximality condition for the complementary subspace to the nilradical. Moreover, among such solvable Lie algebras we shall indicate a subclass of Lie algebras whose cohomology group is trivial. Finally, we provide some examples of infinite-dimensional Lie algebras with a similar structure.
Dec. 2Luigi Ferraro
Wake Forest University
Differential graded algebra over quotients of skew polynomial rings by normal elements

Differential graded algebra techniques have played a crucial role in the development of homological algebra, especially in the study of homological properties of commutative rings carried out by Serre, Tate, Gulliksen, Avramov, and others. In our work, we extend the construction of the Koszul complex and acyclic closure to a more general setting. As an application of our constructions, we show that the Ext algebra of quotients of skew polynomial rings by ideals generated by normal elements is the universal enveloping algebra of a color Lie algebra, and therefore a color Hopf algebra. As a consequence, we give a presentation of the Ext algebra when the elements generating the ideal form a regular sequence, this generalizes a theorem of Bergh and Oppermann. It follows that in this case the Ext algebra is noetherian, providing a partial answer to a question of Kirkman, Kuzmanovich and Zhang.