Let X be a subvariety of the n-dimensional complex torus. It is well-known in tropical geometry that the ``amoebas'' of X limit to the tropicalization of X. It is natural to ask whether there is a non-abelian analogue of this phenomenon. We take the point of view that an answer should be phrased in the setting of (spherical homogeneous spaces and) spherical varieties, which can be thought of as non-abelian analogues of toric varieties. Indeed, in 2016, Kaveh and Manon defined a ``spherical tropicalization map'' for spherical homogeneous spaces. Thus we can ask: for a subvariety Y of a spherical homogeneous space, is there a (family of) spherical amoebas which limit to the image of Y under the spherical tropicalization map, i.e. strop(Y)? I will explain the definitions necessary to state the question precisely, and discuss some preliminary results pointing to an answer.

We study the space of conjugacy classes of representations of the fundamental group of a punctured genus 2 surface into $SU(2)$, with the constraint that the loop around the puncture is sent to $-I$ (minus the identity matrix). In other words $A = M/SU(2)$ where $M$ is the space of representations of $\pi$ to SU(2) which send the loop around the puncture to $-I$, where $\pi$ is the fundamental group of a punctured genus 2 surface. We recover the Betti numbers of $A$ (a special case of the results found by Atiyah and Bott in their landmark 1982 paper). In this special case, we recover their result by much more elementary methods: a Mayer-Vietoris sequence using a decomposition of the space as the union of two subspaces, each of which retracts to $\mathcal{T}$, the space of commuting pairs in SU(2). Our main results include a new computation of the cohomology ring of A by elementary methods, and a computation of the cohomology groups of M. We also compute the ring structure of $\mathcal{T}$. We construct a retraction of two open dense subsets of $A$ to $\mathcal{T}$.

With a compact smooth toric variety $M$ and a fixed positive 0-cycle $A$ (a fixed finite set of points equipped with positive integral multiplicities) belonging to the union $ M^1$ of one-dimensional orbits of $M$ one can associate the following problem: {\it find all hypersurfaces $H\subset M$ such that $H$ does not pass through null-dimensional orbits and the intersection of $H$ with $ M^1$ is the 0-cycle $A$}.

This problem was solved in the case when $\dim M=2$ in [1]. Let $M_O$ be the closure in $M$ of an orbit $O$. Let $A_O$ be the 0-cycle $A\cap M_O$.

{\bf Theorem.} {\it The problem has at least one solution $H$ if and only if for each two-dimensional orbit $O$ the problem for the toric surface $M_O$ and the 0-cycles $A_O$ has at least one solution.

Moreover the intersection of any solution $H$ with the torus $(\Bbb C^*)^n$ can be defined by equation $Q=0$ where $Q$ is a Laurent polynomial whose Newton polyhedron $\Delta$ and coefficients at monomials belonging to edges of $\Delta$ can be found explicitly and whose coefficients at all other monomials in $\Delta$ are arbitrary complex numbers.}

{\bf References}

1. A. Khovanskii. Newton polygons, curves on torus surfaces, and the converse Weil theorem, Russian Math. Surveys 52 (1997), no. 6, 1251-1279.

This talk is based on collaboration with Anton Alekseev, Benjamin Hoffman, and Yanpeng Li, arXiv:1804.01504, arXiv:1808.06975, and work in progress.

The focus of this talk will be a map $\psi: \widetilde{\mathcal{M}} \to \mathcal{N}$ defined by Graham. The space $\widetilde{\mathcal{M}}$ is constructed using affine toric varieties, and the rich theory of these varieties can be used to obtain detailed information about $\widetilde{\mathcal{M}}$. We will discuss the relationship between Graham's construction and the generalized Springer correspondence for $SL_n(\mathbb{C})$, and describe the fibers of $\psi$ using the combinatorics of standard tableaux. This talk is based on joint work with William Graham and Amber Russell.

After motivating the study of this and related formulas, I will outline this geometric proof. Then I will explain the ingredients of a new proof of this formula which replaces Gr\"obner degenerations by combinatorics. This latter work is the outcome of an undergraduate summer research project with Aidan Lindberg.

We explain the connection of the lattice size of $P$ to its successive minima and show that in the case of polygons a reduced basis computes the lattice size, which leads to a very fast algorithm for computing ${\rm ls}(P)$.

We also provide an algorithm for finding a reduced basis is dimension 3, analyze its complexity, and explain the connection of the reduced basis to the successive minima. Although it is not true in dimension 3 that a reduced basis computes the lattice size, we show that for empty lattice tetrahedra there exists a reduced lattice basis that computes the lattice size. This is joint work with A. Harrison, A. Alajmi, and P. Tierney.