Georgia Institute of Technology College of Sciences

TBA

This is joint work with Jen Hom and Tye Lidman. Attaching a Casson handle to a slice disk complement gives a smooth manifold homeomorphic to R^4. In the 90's De Michelis and Freedman asked how these choices affect the smooth type of the resulting manifold. This problem has seen some progress since then but is still not well understood. We show that if two slice knots have sufficiently different knot Floer homology, then the resulting exotic R^4's made with the simplest Casson handle are distinct. This gives a countably infinite family of exotic R^4's made with different slice disk complements. We then produce exotic R^4's with various phenomena, and re-prove a theorem of Bizaca-Etnyre on smoothings of product manifolds Y x R. Our main tool is Gadgil's end Floer homology, which we show how to compute effectively by analyzing a certain cobordism map. Time permitting, I'll discuss an upcoming result on exotic planes in R^4 and branched covers, and plans to study more noncompact exotic phenomena.

We give an overview of Teichmuller theory, the deformation theory of Riemann surfaces. The richness of the subject comes from all the perspectives one can take on Riemann surfaces: complex analytic for sure, but also Riemannian, topological, dynamical and algebraic.  In the past 40 years or so, interest has erupted in an extension of Teichmuller theory, here thought of as a component of the character variety of surface group representations into PSL(2,\R), to the study of the character variety of surface group representations into higher rank Lie groups (e.g. SL(n, \R)). We give a even breezy  discussion of that.

The Chebotarev density theorem is a powerful tool in number theory, in part because it guarantees the existence of primes whose Frobenius lies in a given conjugacy class in a fixed Galois extension of number fields.  However, for some applications, it is necessary to know not just that such primes exist, but to additionally know something about their size, say in terms of the degree and discriminant of the extension.  In this talk, I'll discuss recent work with Peter Cho and Asif Zaman on a closely related problem, namely determining the least prime with a given cycle type.  We develop a new, comparatively elementary approach for thinking about this problem that nevertheless frequently yields the strongest known results.  We obtain particularly strong results in the case that the Galois group is the symmetric group $S_n$ for some $n$, where determining the cycle type of a prime is equivalent to Chebotarev.