In this survey talk I will describe how modular forms give invariants of manifolds, and how these invariants detect elements of the homotopy groups of spheres. These invariants pass through a cohomology theory of Topological Modular Forms (TMF). I will review the role that K-theory plays in detecting periodic families of elements in the homotopy groups of spheres (the image of the J homomorphism) in terms of denominators of Bernoulli numbers. I will then describe how certain higher families of elements (the divided beta family) are detected by certain congruences between q-expansions of modular forms.
I will review the definition of certain moduli spaces of abelian varieties (Shimura varieties) which generalize the role that the moduli space of elliptic curves plays in number theory. Associated to these Shimura varieties are cohomology theories of Topological Automorphic Forms (TAF) which generalize the manner in which Topological Modular Forms are associated to the moduli space of elliptic curves. These cohomology theories arise as a result of a theorem of Jacob Lurie.
In this sequence of three talks, I will aim to introduce the algebraic and geometric structure of Loop groups and their representations. We will begin with the basic structure of Affine Lie algebras. This will lead us to the algebraic theory of positive energy representations indexed by the level. On the geometric side, we will introduce the Affine Loop group and relate it to the central extension of the smooth loop group. We will also study the example of the special unitary group in some detail. In the remaining time, I will go into some of the deeper structure of Loop groups. This includes fusion in the representations of a given level (via the geometric notion of conformal blocks). Time permitting, I will also describe the homotopy type of the classifying space of Loop groups. No special background is required. It would be helpful to know the basic theory of root systems for semisimple Lie algebras, though this is not a strict requirement.
In this sequence of three talks, I will aim to introduce the algebraic and geometric structure of Loop groups and their representations. We will begin with the basic structure of Affine Lie algebras. This will lead us to the algebraic theory of positive energy representations indexed by the level. On the geometric side, we will introduce the Affine Loop group and relate it to the central extension of the smooth loop group. We will also study the example of the special unitary group in some detail. In the remaining time, I will go into some of the deeper structure of Loop groups. This includes fusion in the representations of a given level (via the geometric notion of conformal blocks). Time permitting, I will also describe the homotopy type of the classifying space of Loop groups. No special background is required. It would be helpful to know the basic theory of root systems for semisimple Lie algebras, though this is not a strict requirement.
Topological Automorphic Forms II: examples, problems, and applications
I will survey some known computations of Topological Automorphic Forms. K-theory and TMF will be shown to be special cases to TAF. Certain TAF spectra have been identified with $BP\langle 2\rangle$ by Hill and Lawson, showing these spectra admit $E_{oo}$ ring structures. $K(n)$-local TAF gives instances of the higher real K-theories $EO_n$, one of which shows up in the solution of the Kervaire invariant one problem. Associated to the TAF spectra are certain approximations of the $K(n)$-local sphere, which are expected to see "Greek letter elements" in the same manner that TMF sees the divided beta family. Finally, I will discuss some partial results and questions concerning an automorphic forms valued genus which is supposed to generalize the Witten genus.
In this sequence of three talks, I will aim to introduce the algebraic and geometric structure of Loop groups and their representations. We will begin with the basic structure of Affine Lie algebras. This will lead us to the algebraic theory of positive energy representations indexed by the level. On the geometric side, we will introduce the Affine Loop group and relate it to the central extension of the smooth loop group. We will also study the example of the special unitary group in some detail. In the remaining time, I will go into some of the deeper structure of Loop groups. This includes fusion in the representations of a given level (via the geometric notion of conformal blocks). Time permitting, I will also describe the homotopy type of the classifying space of Loop groups. No special background is required. It would be helpful to know the basic theory of root systems for semisimple Lie algebras, though this is not a strict requirement.
