Triangle groups, automorphic forms, and torus knots

Discontinuous Groups and Automorphic Forms

Automorphic Forms and Metaplectic Groups

In 1952, Gelfand and Fomin noticed that classical modular forms were related to representations of SL2(R). As a result of this realization, Gelfand later defined GLr automorphic forms via representation theory. A metaplectic form is just an automorphic form defined on a cover of GLr, called a metaplectic group. In this talk, we will carefully construct the metaplectic covers of GL2(F) where F i...

On iterated torus knots and transversal knots

A knot type is exchange reducible if an arbitrary closed n{braid representative K of K can be changed to a closed braid of minimum braid index nmin(K) by a nite sequence of braid isotopies, exchange moves and {destabilizations. (See Figure 1). In the manuscript [6] a transversal knot in the standard contact structure for S3 is de ned to be transversally simple if it is characterized up to trans...

Cohomology of Arithmetic Groups and Periods of Automorphic Forms

(1) Firstly, I gave a brief introduction to the cohomology of arithmetic groups, in particular the fact that the “tempered” (informally: “near the middle dimension”’) part of the cohomology looks like the cohomology of a torus. This corresponds to §2, §3, §4 of these notes. For the reader totally unfamiliar with the cohomology of arithmetic groups, §3 might be the best thing to focus on. (2) Ne...

Automorphic Forms

We apply a theorem of J. Lurie to produce cohomology theories associated to certain Shimura varieties of type U(1, n−1). These cohomology theories of topological automorphic forms (TAF ) are related to Shimura varieties in the same way that TMF is related to the moduli space of elliptic curves. We study the cohomology operations on these theories, and relate them to certain Hecke algebras. We c...