We give a deterministic polynomial-time $2^{O(r)}$-approximation algorithm for the number of bases of a given matroid of rank r and the number of common bases of any two matroids of rank r. To the best of our knowledge, this is the first nontrivial deterministic approximation algorithm that works for arbitrary matroids. Based on a lower bound of Azar, Broder, and Frieze, this is almost the best possible result assuming oracle access to independent sets of the matroid.

There are two main ingredients in our result. For the first, we build upon recent results of Adiprasito, Huh, Katz, and Wang on combinatorial Hodge theory to show that the basis generating polynomial of any matroid is a (completely) log-concave polynomial. Formally, we prove that the multivariate generating polynomial of the bases of any matroid is (and all of its directional derivatives along the positive orthant are) log-concave as functions over the positive orthant. For the second ingredient, we develop a general framework for approximate counting in discrete problems, based on convex optimization. The connection goes through subadditivity of the entropy. For matroids, we prove that an approximate superadditivity of the entropy holds by relying on the log-concavity of the basis generating polynomial.

We study irreducible odd mod p Galois representations $\overline{\mathit{\rho }}:\mathrm{Gal}(\bar{F}∕F)\to G({\bar{\mathbb{F}}}_p)$, for F a totally real number field and G a general reductive group. For $p{\gg }_{G,F}0$, we show that any $\overline{\mathit{\rho }}$ that lifts locally, and at places above p to de Rham and Hodge–Tate regular representations, has a geometric p-adic lift. We also prove non-geometric lifting results without any oddness assumption.

Singularities of the mean curvature flow of an embedded surface in ${\mathbb{R}}^3$ are expected to be modeled on self-shrinkers that are compact, cylindrical, or asymptotically conical. In order to understand the flow before and after the singular time, it is crucial to know the uniqueness of tangent flows at the singularity.

In all dimensions, assuming that the singularity is of multiplicity 1, uniqueness in the compact case has been established by the second-named author, and in the cylindrical case by Colding and Minicozzi. We show here the uniqueness of multiplicity-1 asymptotically conical tangent flows for mean curvature flow of hypersurfaces.

In particular, this implies that when a mean curvature flow has a multiplicity-1 conical singularity model, the evolving surface at the singular time has an (isolated) regular conical singularity at the singular point. This should lead to a complete understanding of how to “flow through” such a singularity.

We introduce an elementary method to study the border rank of polynomials and tensors, analogous to the apolarity lemma. This can be used to describe the border rank of all cases uniformly, including those very special ones that resisted a systematic approach. We also define a border rank version of the variety of sums of powers and analyze its usefulness in studying tensors and polynomials with large symmetries. In particular, it can be applied to provide lower bounds for the border rank of some interesting tensors, such as the matrix multiplication tensor. We work in a general setting, where the base variety is not necessarily a Segre or Veronese variety, but an arbitrary smooth toric projective variety. A critical ingredient of our work is an irreducible component of a multigraded Hilbert scheme related to the toric variety in question.