Past Events
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…
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…
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…
The de Jong-Debarre Conjecture predicts that the space of lines on any smooth hypersurface of degree d <= n in P^n has dimension 2n-d-3. We prove this conjecture for n > 2d, improving on the previously-known exponential bounds. We prove an analogous result for k-planes, and use this…
In 1982, Hamilton pioneered the use of Ricci flow in the study of the geometry of 3-manifolds. Moreover, he established a conceptual program on how Ricci flow with surgery might be used to yield even deeper results. This program was famously completed by Perelman in 2003 and led to proofs of the…