One of the major successes of tropical geometry is Mikhalkin's correspondence theorem, which relates complex and tropical curves in toric surfaces. In non-toric surfaces Mikhalkin's correspondence does not hold; there are tropical curves not arising from complex curves or algebraic curves over any field. We will explain some local obstructions to approximating tropical curves coming from a tropical intersection product. This product is related to the intersection product defined by Kaveh and Khovanskii on divisors in more general spaces. We will also see examples of tropical curves which do not satisfy some classical theorems of geometry.

Resolution of singularities consists in constructing a non-singular model of an algebraic variety. This is done by applying a proper birational map that is a local isomorphism at the smooth points. Often too much information is lost about the original variety if the smooth points are the only ones where the desingularization map is a local isomorphism. In these cases, a desingularization preserving some minimal singularities is necessary. This suggests the question of whether, given a class of singularity types S, it is possible to remove with a birational map all singularities not in S while still having a local isomorphism over the singularities of type S. We will study this problem when S consists of all stable simple normal crossings.

After giving the final details of the heat equation proof, I shall give some application. Most notably Hirzebruch's Signature theorem and the Riemann-Roch theorem for compact complex manifolds.

SPECIAL FIELDS POSTDOCTORAL FELLOW TALK

I will first give a brief introduction to the metric aspects of noncommutative geometry and ideas from spectral geometry that have played an important role in their development. Noncommutative tori $\mathbb{T}_\theta^n$ are important $C^*$-algebras that have been studied vastly in noncommutative geometry due to their importance, among which is their role in the study of foliated manifolds. In a recent seminal paper, A. Connes and P. Tretkoff proved the Gauss-Bonnet theorem for the noncommutative two torus $\mathbb{T}_\theta^2$ equipped with its canonical conformal structure. In a series of joint works with M. Khalkhali, we extended this result to general translation invariant conformal structures, computed the scalar curvature, and proved the analog of Weyl's law and Connes' trace theorem for $\mathbb{T}_\theta^2$. Our final formula for the curvature of $\mathbb{T}_\theta^2$ precisely matches with the one computed independently by A. Connes and H. Moscovici. A purely noncommutative feature is the appearance of the modular automorphism from Tomita-Takesaki theory in the computations and the final formula for the curvature. In this talk I will review these results and will then turn to part of our recent work on the curved geometry of noncommutative four tori $\mathbb{T}_\theta^4$. That is, I will explain the computation of scalar curvature and the analog of the Einstein-Hilbert action for $\mathbb{T}_\theta^4$, and show that metrics with constant curvature are critical points of this action.