# Algebra & Combinatorics Seminar

The Algebra & Combinatorics Seminar meets on Fridays from 3–4 pm, in Lecture Hall LH-1 of the IISc Mathematics Department. The organizers are Apoorva Khare and R. Venkatesh.

### Autumn 2018

 Apoorva Khare (IISc Mathematics) Sep 7, 2018 The Diamond Lemma in ring theory

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

 R. Venkatesh (IISc Mathematics) Sep 14, 2018 Graphs vs. 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.

 Bharatram Rangarajan (Hebrew University, Jerusalem, Israel) Sep 26, 2018 Expanders: From Graphs to Complexes (note: unusual day)

Abstract. The aim of this talk is to give a high-level 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 Lubotzky-Phillips-Sarnak construction of Ramanujan graphs. Finally, I hope to motivate high-dimensional expanders using two interesting topics- the overlapping problem, and the threshold problem.

 Amritanshu Prasad (IMSc, Chennai) Sep 28, 2018 The words that describe symmetric polynomials (speaking in / subsumed by the Eigenfunctions Seminar)

 G.V.K. Teja (IISc Mathematics) Oct 5, 2018 Patterns in recurring decimals

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.

 Pooja Singla (IISc Mathematics) Oct 12, 2018 Gelfand's criterion and multiplicity one results

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.

 Arvind Ayyer (IISc Mathematics) Oct 19, 2018 Factorization theorems for classical group characters

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.

 Dhruv Ranganathan (IAS (Princeton), MIT (Boston), USA; CMI (Chennai); Cambridge, UK) Oct 31, 2018 Tropical geometry of moduli spaces (speaking in / subsumed by the Eigenfunctions Seminar)

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

 Dhruv Ranganathan (IAS (Princeton), MIT (Boston), USA; CMI (Chennai); Cambridge, UK) Nov 2, 2018 Curve counting and tropical geometry

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.)

 K.N. Raghavan (IMSc, Chennai) Nov 9, 2018 TBA (speaking in / subsumed by the Eigenfunctions Seminar)

 Abhishek Banerjee (IISc Mathematics) Nov 16, 2018 TBA

Abstract. TBA