Ever since I learned that a coffee cup accepts graph embeddings that a sphere cannot, I can’t stopped telling my students. Linking in the Gauss-Bonnet theorem, Morse theory, a little crochet, and some careful considerations of non-euclidean pizza, and you have a great conversation with a class. After several iterations, I have tested a few topics and can share what works for me and what draws yawns and crickets. I hope to describe ideas for other things I haven’t tried yet. Audience members are encouraged to share their experiences.

Dimension 4 is unlike the other dimensions. There are many simply-connected closed 4-manifolds that admit infinitely many distinct smooth structures, and surprisingly, there are no smooth 4-manifolds known to have only finitely many smooth structures. Also, classification problems for smooth, simply-connected 4-manifolds are far from fully understood. For example, the generalized Poincaré conjecture is true topologically in all dimensions, but unknown smoothly in dimension 4. Attempts to resolve this conjecture have led to constructions of 4-manifolds that are homeomorphic, but not diffeomorphic (such manifolds are called "exotic"), with the particular goal of constructing exotic 4-spheres. In this first talk, I will mostly focus on the simply-connected symplectic 4-manifolds, describe the topological invariants, and give examples.

Continuing from last time, we will get into more details about symplectic manifolds. Then, we will provide background about Lefschetz fibrations and mapping class groups, which have a very nice and useful interaction with symplectic manifolds. Finally, we will introduce the tools we use to construct exotic 4-manifolds, like the symplectic connected sum and Luttinger surgery.

In this part, we will talk about some results we obtained and what we have covered in the first two parts. First, we will present a new construction of symplectic 4-manifolds that are homeomorphic but not diffeomorphic to $(2h+2k-1)\mathbb{CP}^{2}\#(6h+2k+3)\overline{\mathbb{CP}}^{2}$ with $(h,k) \neq (0,1)$, via Lefschetz fibrations and Luttinger surgery on the product manifolds $\Sigma_g \times T^2$. In the second half, we will construct families of Lefschetz fibrations over $S^2$ using finite order cyclic group actions on $\Sigma_g \times \Sigma_g$. These are joint works with Anar Akhmedov.

Manifolds appear in classical physics as space of states of a physical system in classical mechanics. When we move from classical world to quantum world, points as states are replaced by operators. In our talk, we will discuss manifold structure on such operator spaces and its geometry.

Manifolds appear in classical physics as space of states of a physical system in classical mechanics. When we move from classical world to quantum world, points as states are replaced by operators. In our talk, we will discuss manifold structure on such operator spaces and its geometry.

Complex hyperbolic space is a fairly natural complex analog of the more familiar real hyperbolic space. However, despite the similarities in construction, the geometric features of these two spaces can be quite different, and questions of the form "If [statement] is true in real hyperbolic space, is it also true in complex hyperbolic space?" might require vastly different techniques to answer. In this talk, I plan to give a gentle introduction to (complex) hyperbolic geometry.

As we saw last time, the boundary of complex hyperbolic 2-space is topologically $S^3$, but due to the natural action of $\operatorname{PU}(2,1)$, it geometrically inherits the structure of the (1-point compactification of the) Heisenberg group. In the early 2000's, Schwartz discovered that one could actually find a real hyperbolic 3-manifold in this strange $S^3$, and so it seems natural to ask which other 3-manifolds can arise in the boundary of complex hyperbolic 2-space. In this talk, I will introduce Falbel's program for finding $\operatorname{PU}(2,1)$ representations of 3-manifold groups and summarize some known results about 3-manifolds.

I'll use an elementary discussion of the nineteenth-century Riemann-Roch theorem to motivate later developments - vector bundles, characteristic classes, K-theory, and index theory - in algebraic topology and in the connections of algebraic topology with analysis.

I'll use an elementary discussion of the nineteenth-century Riemann-Roch theorem to motivate later developments - vector bundles, characteristic classes, K-theory, and index theory - in algebraic topology and in the connections of algebraic topology with analysis.