Please think about giving a talk in the Algebra Seminar, or inviting an outside speaker.

Abstract: This is an expository talk on how the theory of symmetric functions, such as Schur functions, ties in with the theory of plane partitions and the representation theory of $S_n$. We will conclude with some open problems that have connections with number theory and discuss current attempts at extensions of the above relations.

Abstract: This is a continuation of the first talk on Aug. 31.

Abstract: I will give an introduction to Clifford algebras and Lie algebras, and give examples showing how their representation theory is connected in certain cases to combinatorics. In particular, I will discuss the bosonic representation of an infinite dimensional Heisenberg Lie algebra whose weight spaces have dimensions given by the classical partition function. In contrast, Clifford algebras have fermionic representations whose weight spaces have dimensions given by certain restricted partitions (into distinct parts). These provide simple models of two kinds of fundamental particles in physics, bosons and fermions.

Abstract: In this talk we will introduce the theory of cellular automata, beginning with a simple explanation of the most famous cellular automaton, John Conway's Game of Life. We will then generalize the Game of Life, putting the theory on more rigorous mathematical footing so we can discuss important classes of groups related to the theory, namely surjunctive groups, residually finite groups, and Hopfian groups. Our talk will conclude with a discussion of some of the open problems related to them.

Abstract: Finitely generated infinite nilpotent groups, such as the discrete Heisenberg group, possess rich geometric structure while having algebraic structure amenable to a variety of computational techniques. We will review some established results about word growth and subgroup growth in nilpotent groups due to Milnor, Bass, Guivarc'h, Gromov, Grunewald, du Sautoy, and others. We then introduce a metric space structure on the collection of finite-index subgroups, and state results on growth of metric balls in this space. This covers work joint with Khalid Bou-Rabee of CCNY.

Abstract: Moduli spaces parametrizing objects associated with a given space are a rich source of spaces with interesting structures. A Hilbert scheme of n points of a projective/quasi-projective variety is the moduli space parametrizing 0-dimensional subschemes of length n. It parametrizes “n points” on the space and is highly related to the n-th symmetric product. But it contains more information than that. The local structure is also related to partition functions and Young diagrams. We will use concrete examples (e.g. the space being $\mathbb{C}$ or $\mathbb{C}^{2}$) and give an expository talk on the topic.

Abstract: Rings and groups are among the many classes of (universal) algebras, called varieties, which contain constant(s) as part of their structure. In fact, the ring with unity $\textbf{R}=\langle\{0,1\};+,\cdot,0,1\rangle$ whose only elements are constants, generates a variety interesting in its own right, namely Boolean rings. Notably, $\textbf{R}$ is 0-generated as it is required to contain both 0 and 1. This talk aims to provide a small, general theory for those varieties which are generated by a single 0-generated algebra.

Abstract: Given a finite group $G$, we can consider the influence the set of character degrees has on the structure of the group. In this talk, we will restrict our view to the characters which take their values over $\mathbb{R}$, and examine groups whose real-valued character degrees satisfy certain patterns.

Abstract: There is a strong but mysterious connection between the complex character degrees and conjugacy class sizes of finite groups. Many important results on character degrees admit a dual version for conjugacy class sizes and vice versa. Both of these numerical invariants have strong influence on the structure of the groups. In this expository talk, I will survey some known results and open problems concerning these two invariants.

Abstract: A fruitful line of research in the representation theory of finite groups is to relate group structure and the degrees of ordinary irreducible characters (that is, the dimensions of the irreducible complex representations). For example, the celebrated It${\hat o}$-Michler theorem says that a finite group has normal, abelian Sylow p-subgroups if and only if all its ordinary irreducible character degrees are coprime to p. In this talk, we discuss a more general setting in which only the relative degrees of characters over a normal subgroup are considered.

Abstract: Circulants matrices provide a nontrivial, beautiful, and simple set of objects in matrix theory. They appear quite naturally in many problems in network theory and non-linear dynamics. The circulant diagonalization theorem describes the eigenspectrum and eigenspaces of a circulant matrix explicitly via the fast Fourier transform. Consequently, many problems involving circulant matrices have closed-form or analytical solutions. In this talk, we generalize the circulant diagonalization theorem to the join of several circulant matrices. We then discuss some applications of our theorems to computational neuroscience and spectral graph theory.

A panopto recording of the talk by Tung T. Nguyen is available through the following link: Algebra Seminar talk by Tung T. Nguyen

Abstract: We define $f(n)$ to be the number of groups of order $n$ up to isomorphism. As $n$ increases, the problem of classifying groups of order $n$ becomes hard. Charles Sims [86] proved in 1965 that there is a upper bound for the number of groups of order $n$ up to isomorphism. This talk will show that the error term in the Sims bound for the number of $p$-groups of order $p^m$ may be improved if we restrict our enumeration to $p$-groups of nilpotency class at most $3$. Indeed, our aim is to show that the number of such groups is at most $p^{(2m^3/27+O(m^2))}$.