Prime numbers: Techniques, results and questions
Friday 20 January, 4 pm, Lecture Theatre 1
The basic question in prime number theory is to try to understand the number of primes in some interesting set of integers. Unfortunately many of the most basic and natural examples are famous open problems which are over 100 years old!
If you haven't heard about Mathematrix before (no, it's not a trendy software startup) it's the Maths department's society for minorities in mathematics We run fortnightly discussion groups and social events with the aim of growing a friendly and supportive community, as well as tackling issues faced by minorities in maths (plus we have free sandwiches). Events are Wednesdays 1 - 2 pm in the Quillen Room (N3.12).
The basic question in prime number theory is to try to understand the number of primes in some interesting set of integers. Unfortunately many of the most basic and natural examples are famous open problems which are over 100 years old!
We aim to give an accessible survey of (a selection of) the main results and techniques in prime number theory. In particular we highlight progress on some of these famous problems, as well as a selection of our favourite problems for future progress.
Perverse sheaves are an indispensable tool in representation theory. Their stalks often encode important representation theoretic information such as composition multiplicities or canonical bases. For the nilpotent cone, there is an algorithm that computes these stalks, known as the Lusztig-Shoji algorithm. In this talk, we discuss how this algorithm can be modified to compute stalks of perverse sheaves on more general varieties. As an application, we obtain a new algorithm for computing canonical bases in certain quantum groups as well as composition multiplicities for standard modules of the affine Hecke algebra of $\mathrm{GL}_n$.