There is a unique line through 2 points in the plane, and a unique conic through 5. These counts generalize to a count of degree d rational curves in the plane passing through 3d-1 points. Surprisingly, the problem of determining these numbers is connected to mathematical physics, and it was not until the 1990's that it was completely solved. For example, Kontsevich determined them with a celebrated recursive formula. Such formulas are valid when you allow your curves to be defined with complex coefficients.  Some of the solutions may be real, or integral, or defined over Q[i], but the fixed count does not see the difference. Homotopy theory on the other hand, studies continuous deformations of maps. In its modern form, it provides a framework to study shape in many contexts, including the motivic homotopy of algebraic varieties. This talk will introduce some interactions of homotopy theory with the arithmetic of solutions to enumerative problems in geometry. We will use this to completely determine an enriched "count" of degree d rational curves passing through 3d-1 points over an arbitrary field of characteristic not 2 or 3. This enumeration is joint work with Erwan Brugallé and Johannes Rau.

The famous Calderón problem consists in determining the conductivity of a medium by making voltage and current measurements at
the boundary. We consider in this talk a nonlocal analog of this problem. Nonlocal operators arise in many situations where long term
interactions play a role.  The fractional Laplace is a prototype of a nonlocal operator. We will survey some of the main results on inverse problems associated with the fractional Laplacian showing that the nonlocality helps for the inverse problems.

Hessenberg varieties are subvarieties of flag varieties invented in the early 1990s by de Mari and Shayman.  De Mari and Shayman were motivated by questions in applied linear algebra, but, very quickly, people realized that Hessenberg varieties are very interesting objects of study from the point of view of algebraic groups.  

I got interested in Hessenberg varieties because of their connection to questions in combinatorics, in particular, the Stanley-Stembridge conjecture.  I'll explain this conjecture, now a theorem due to Hikita, and I will explain some of my work with Tim Chow, which resolved a conjecture of Shareshian and Wachs connecting Hessenberg varieties directly to Stanley-Stembridge. (I'll also try to say a few words about Hikita's work and the very exciting state the field is in now.)  Then I'll explain joint work with Escobar, Hong, Lee, Lee, Mellit and Sommers on the moduli of Hessenberg varieties. 

Who doesn't like one of these three: geometry, topology, and combinatorics? Sperner's lemma, a combinatorial statement that is equivalent to the Brouwer fixed point theorem in topology, is amazing and powerful. I'll explain why, give heartwarming old and new proofs, and present some generalizations to polytopes that has surprised me with diverse applications: to the study of triangulations, to fair division problems, and the Game of Hex.

Holomorphic Dynamics (in a narrow sense) is the theory of the iteration of rational maps on the Riemann sphere. It was founded in the classical work by Fatou and Julia around 1918. After about 60 years of stagnation, it was revived in the 1980s, bringing together deep ideas from Conformal and Hyperbolic Geometry, Teichmüller Theory, the Theory of Kleinian Groups, Hyperbolic Dynamics and Ergodic Theory, and Renormalization Theory from physics, illustrated with beautiful computer-generated pictures of fractal sets (such as various Julia sets and the Mandelbrot set). We will highlight some landmarks of this story.