One hundred years ago, Ramanujan attached two conjectures to a function that he had defined when studying sums of integer squares. Time permitting I shall sweep over the twentieth century and indicate their ramifications in geometry, the automorphic world, classical and modern analytic and probabilistic number theory. Hopefully I shall reach recent results in which I, or Jonathan Kish and myself, have had a hand.
Ramanujan's tau-function and something of what it provoked
Sep. 13, 2016 1pm (MATH 220)
Jakub Oprsal (Charles University, Prague)
X
One way to measure term complexity of a finite algebra is to count how many symbols are needed to write down a term describing a particular n-ary term operation, and taking the maximum of these values among all n-ary term operations of the algebra. We will talk about algebras with a Mal'cev term for which there is a polynomial (in n) upper bound on these numbers.
Term complexity in Mal'cev algebras
Sep. 13, 2016 2pm (MATH 220)
Jakub Oprsal (University of Krakow, Poland)
X
One of the ways to formulate the famous Taylor's modularity conjecture is the following: Suppose that we have two sets of identities in disjoint languages such that the variety described by its union has modular congruence lattices. Then one of the varieties described by either of the sets has modular congruences. We will discuss some ideas of a proof that the above is true for sets of identities implying that all the operations are idempotent (i.e., satisfy f(x,…,x) = x).
Taylor's modularity conjecture
Sep. 13, 2016 2pm (MATH 350)
Lie Theory
Jon Lamar (CU)
X
A supercharacter theory, S, of a group G is called characteristic if the action of Aut(G) on the lattice of supercharacter theories SCT(G) fixes S. I will provide a classification of the sublattice of characteristic supercharacter theories of the dihedral groups as well as discuss possible ways to characterize the remainder of the full lattice.
Ramanujan's tau-function and something of what it provoked