Mathematics Colloquium
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Speaker: Arindam Banerjee, IIT Kharagpur
Date: 30 November, 2022 at 2.30 pm
Venue: Ramanujan Hall

Title: Castelnuovo-Mumford regularity of edge Ideals of graphs and their powers

Abstract: Regularity of edge ideals of graphs and their powers have been a very popular area of research in commutative algebra for the last one decade. 
Edge ideals are one of the rare classes of ideals where linear resolution of the ideal implies linear resolution for higher powers i.e minimum possible regularity 
for an edge ideal implies minimum possible regularity for all its higher powers(proved by Herzog, Hibi and Zheng). It was also characterised (by Froberg) 
that an edge ideal has linear resolution (or minimum possible regularity) if and only if the underlying graph is chordal.This motivated people to 
take up various projects for finding sharp upper bounds for regularity for various powers of edge ideals. Two questions largely guided this research:

1. The Nevo-Peeva Question: Is it true that all higher powers (greater than equal to 2) have linear resolutions for edge ideals with 
a. linear presentation (that is edge ideals whose first differential matrix of minimal free resolution has linear entries)  and 
b. regularity less than equal to $3$?

2. The sharp upper bound Conjecture for regularity of powers in terms of the regularity of the ideal: It is believed that 
regularity of an edge ideal is r implies for the s th power the regularity is bounded above by 2s+r-1.

There has been much progress towards both of these but both remain open so far. Recently the second question has been 
solved for all bipartite graphs. Also for all graphs the bound has been proven for the second power (r+2). 
There has been some effort to study how regularity behaves "on an average for all graphs" using some probabilistic methods. 

In this talk we plan to discuss the history and current state of research in this area.

Prof MS Raghunathan will give a course of lectures aimed at beginning PhD students

on a topic of basic importance to all of mathematics. Title and abstract are given below.

The course will begin on Monday 17th October at 4pm in the room A1-A2 of CDEEP on the ground floor of the Math building. Each lecture will be of 90 minutes. The course will run

roughly through the middle of December, so about 8 lectures.  Since Monday 24th October is  Deepawali, the lecture will be organized on 26th afternoon for which I will separately announce the precise timing.

It will be a lecture course in hybrid mode so that others not in IIT can also benifit from this course. Feel free to tell your friends in case it may interest them. Here is the zoom link in case you cannot attend in person:

Hope to see you there!

Best wishes, Dipendra

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Title: Compact Lie groups and their representations

Abstract: In this course I will first talk about the structure theory of compact Lie groups, beginning with the fact that a compact connected Lie group is an almost direct product of the identity connected component of its centre and its commutator subgroup (which is closed subgroup) conjugacy of maximal tori and the fact that every element is contained in a maximal torus. In the course of proving these results, some results on the topology of compact Lie groups which will also be proved. I will then establish Weyl's theorem which asserts that if G is a compact connected Lie group and [G,G]=G, π1(G,e) is finite (and hence the universal covering of a compact group whose abelianisation is trivial is a compact. Then I will introduce roots and weights and the Dynkin diagram of the compact group and sketch a proof of the fact that the Dynkin diagram determines the group locally. The remaining lectures will be devoted to representation theory. I will establish the bijective correspondence between 'Dominant Weights' and irreducible representations. The course will end with the Weyl Character Formula for the character of an irreducible representation corresponding to a 'dominant' weight. The entire theory is essentially  the same as the representation theory of reductive algebraic groups. I will off and on indicate how the two are related.  
I will be assuming some familiarity with basic theory of Lie groups such as the correspondence between Lie sub-algebras of the Lie group and Lie subgroups of the Lie groups; also with some basic results from algebraic topology.