

Abstract.
I will give a gentle introduction to the Diamond Lemma. This is a useful
technique to prove that certain "PBWtype" bases exist of algebras given
by generators and relations. In particular, we will see the PBW theorem
for usual Lie algebras.


Abstract.
Let $G$ be a finite simple graph. The Lie algebra $\mathfrak{g}$ of $G$
is defined as follows: $\mathfrak{g}$ is generated by the vertices of $G$
modulo the relations $[u, v]=0$ if there is no edge between the vertices
$u$ and $v$. Many properties of $\mathfrak{g}$ can be obtained from the
properties of $G$ and vice versa. The Lie algebra $\mathfrak{g}$ of $G$
is naturally graded and the graded dimensions of the Lie algebra
$\mathfrak{g}$ of $G$ have some deep connections with the vertex
colorings of $G$. In this talk, I will explain how to get the generalized
chromatic polynomials of $G$ in terms of graded dimensions of the Lie
algebra of $G$. We will use this connection to give a Lie theoretic proof
of of Stanley's reciprocity theorem of chromatic polynomials.


Abstract.
The aim of this talk is to give a highlevel overview of the theory of
expander graphs and introduce motivations and possible approaches to
generalizing it to higher dimensions. I shall begin with three
perspectives on expansion in graphs discrepancy, isoperimetry and mixing
time, and show a qualitative equivalence of these notions in defining
expansion for graphs. Next I shall briefly discuss upper and lower bounds
on expansion, and sketch the LubotzkyPhillipsSarnak construction of
Ramanujan graphs. Finally, I hope to motivate highdimensional expanders
using two interesting topics the overlapping problem, and the threshold
problem.




Abstract.
We will study recurrence patterns in decimal expansions of rational
numbers (in any integer base for this talk). After making some initial
observations, we will compute the length of the repeating part of any
fraction. We conclude by explaining this result over a Euclidean
domain.


Abstract.
We will describe Gelfand's criterion for the commutativity of associative
algebras and discuss some of its applications towards the multiplicity
one theorems for the representations of finite groups.


Abstract.
Characters of classical groups appear in the enumeration of many
interesting combinatorial problems. We show that, for a wide class of
partitions, and for an even number of variables of which half are
reciprocals of the other half, Schur functions (i.e., characters of the
general linear group) factorize into a product of two characters of
other classical groups. Time permitting, we will present similar results
involving sums of two Schur functions. All the proofs will involve
elementary applications of ideas from linear algebra.
This is joint work with Roger Behrend.


(Note: The following
Seminar on Nov 2 may be of related interest to those attending this
talk.)


Abstract.
The counts of algebraic curves in projective space (and other toric
varieties) has been intensely studied for over a century. The subject saw
a major advance in the 1990s, due to groundbreaking work of Kontsevich
in the 1990s. Shortly after, considerations from high energy physics led
to an entirely combinatorial approach to these curve counts, via
piecewise linear embeddings of graphs, pioneered by Mikahlkin. I will
give an introduction to the surrounding ideas, outlining new results and
new proofs that the theory enables. Time permitting I will discuss
generalizations, difficulties, and future directions for the
subject.
(Organizer's note: It may help to attend the
preceding Eigenfunctions Seminar on Oct 31, before this talk.)




Abstract.
An ordinary ring may be seen as a preadditive category with just one
object. This leads to the powerful analogy, first formulated explicitly
by Mitchell in 1975, that a small preadditive category should be seen as
a "ring with several objects". We will trace the history and development
of the category of modules over a preadditive category.
