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--- seminars:alge [2025/09/12 15:50]
tongviet
+++ seminars:alge [2025/10/02 11:33] (current)
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    * **September 9**\\  <html> <span style="color:blue;font-size:120%"> Chris Schroeder (Binghamton University) </span></html> \\      **//A topological quantum field theory and invariants of finite groups//** \\   \\  <WRAP center box 90%> **//Abstract//**: In this talk, we will discuss the properties of finite groups that are witnessed by the group invariants arising in the context of Dijkgraaf-Witten theory, a topological quantum field theory, as invariants of surfaces. Assuming the theory is derived from the complex group algebra of a finite group, these invariants are generalizations of the commuting probability, an invariant that has been well studied in the literature. The main goal of this talk is to construct these invariants from scratch, assuming no previous knowledge of quantum mechanics. </WRAP>
-   * **September 16**\\  <html> <span style="color:blue;font-size:120%"> Alex Feingold (Binghamton University) </span></html> \\      **//Title//** \\   \\  <WRAP center box 90%> **//Abstract//**: Text of Abstract
+   * **September 16**\\  <html> <span style="color:blue;font-size:120%"> Alex Feingold (Binghamton University) </span></html> \\      **//Lie Algebras, Representations, Roots, Weights, Weyl groups and Cliﬀord Algebras//** \\   \\  <WRAP center box 90%> **//Abstract//**: Lie algebras and their representations have been well-studied and have applications in mathematics and physics. The classification of finite dimensional Lie algebras over **C** by Killing and Cartan inspired the classification of finite simple groups. Geometry and combinatorics are both involved through root and weight systems of representations, with the Weyl group of symmetries playing a vital role. Infinite dimensional Kac-Moody Lie algebras have deeply enriched the subject and connected with string theory and conformal field theory. In a collaboration with Robert Bieri and Daniel Studenmund, we have been studying tessellations of Euclidean and hyperbolic spaces which arise from the action of affine and hyperbolic Weyl groups. Our goal has been to define and study piecewise isometry groups acting on such tessellations.
+Today I will present background material on Lie algebras, representations and examples which show the essential structures. I will present a construction of representations of the orthogonal Lie algebras, $so(2n,F)$, of type
+$D_n$ as matrices and also using Clifford algebras to get spinor representations.
 </WRAP>
@@ Line 43: / Line 46: @@
    * **October 7**\\  <html> <span style="color:blue;font-size:120%"> No Algebra Seminar </span></html> \\
-   * **October 14**\\  <html> <span style="color:blue;font-size:120%"> Hung Tong-Viet (Binghamton University) </span></html> \\      **//Title//** \\   \\  <WRAP center box 90%> **//Abstract//**: Text of Abstract </WRAP>
+   * **October 14**\\  <html> <span style="color:blue;font-size:120%"> Hung Tong-Viet (Binghamton University) </span></html> \\      **//Orders of commutators and Products of conjugacy classes in finite groups//** \\   \\  <WRAP center box 90%> **//Abstract//**: Let $G$ be a finite group, $x\in G$, and let $p$ be a prime. In this talk, we explore conditions that forces $x$ to lie in certain characteristic subgroups of $G$. In particular, we prove that the commutator $[x,g]$ is a $p$-element for all $g\in G$ if and only if $x$ is central modulo $O_p(G)$, the largest normal $p$-subgroup of $G$. This result unifies and generalizes aspects of both the Baer-Suzuki theorem and Glauberman's $Z_p^*$-theorem. Additionally, we show that if $x\in G$ is a $p$-element and there exists an integer $m\ge 1$ such that for every $g\in G$, the commutator $[x,g]$ is either trivial or has order $m$, then the subgroup generated by the conjugacy class of $x$ is solvable. As an application, we confirm a conjecture of Beltran, Felipe, and Melchor: if $K$ is a conjugacy class in $G$ such that the product $K^{-1}K=1\cup D\cup D^{-1}$ for some conjugacy class $D$, then the subgroup generated by $K$ is solvable. </WRAP>
    * **October 21**\\  <html> <span style="color:blue;font-size:120%"> Inna Sysoeva (Binghamton University) </span></html> \\      **//Title//** \\   \\  <WRAP center box 90%> **//Abstract//**: Text of Abstract </WRAP>
@@ Line 53: / Line 56: @@
    * **November 11**\\  <html> <span style="color:blue;font-size:120%"> Tae Young Lee (Binghamton University) </span></html> \\      **//Title//** \\   \\  <WRAP center box 90%> **//Abstract//**: Text of Abstract </WRAP>
-   * **November 18**\\  <html> <span style="color:blue;font-size:120%">  (? University) </span></html> \\      **//Title//** \\   \\  <WRAP center box 90%> **//Abstract//**: Text of Abstract </WRAP>
+   * **November 18**\\  <html> <span style="color:blue;font-size:120%"> Nguyen N. Hung (University of Akron) </span></html> \\      **//Title//** \\   \\  <WRAP center box 90%> **//Abstract//**: Text of Abstract </WRAP>
    * **November 25**\\  <html> <span style="color:blue;font-size:120%">  (? University) </span></html> \\      **//Title//** \\   \\  <WRAP center box 90%> **//Abstract//**: Text of Abstract </WRAP>

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