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Let Col(x) = 3x+1 if x is odd, and x/2 if x is even. The infamous Collatz conjecture states that for every positive integer x, Col^n(x) = 1 for some n. Recently, Tao proved that for any function f(x) increasing to infinity, and almost all positive integers x (in the sense of logarithmic density…

Abstract: In the early 1990s Ribet observed that the classical mod l multiplicity one results for modular curves, which are a consequence of the q-expansion principle, fail to generalize to Shimura curves. Specifically he found examples of Galois representations which occur with multiplicity 2…

Abstract

I will discuss a method to define Khovanov and Lee homology for links in connected sums of copies of S1 times S2.  From here we can define an s-invariant that gives genus bounds on oriented cobordisms between links.  I will discuss some applications to surfaces in…