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We prove that in any dimension n there exists an origin-symmetric ellipsoid of volume c n^2 that contains no points of Z^n other than the origin. Here c > 0 is a universal constant. Equivalently, there exists a lattice sphere packing in R^n whose density is at least c n^2 / 2^n. Previously known constructions of sphere packings in R^n had densities of the order of magnitude of at most n log n / 2^n. Our proof utilizes a stochastically evolving ellipsoid that accumulates at least  c n^2 lattice points on its boundary, while containing no lattice points in its interior except for the origin.

https://yale.zoom.us/j/99948057179

We give an algorithm to reduce the number of generators of the Khovanov chain complex of torus braids $(\sigma_1\sigma_2 \dots\sigma_{n−1})^k$ on $n$ strands. I will begin the talk with context on the stable Khovanov homology of torus links leading to the open question of the structure of their homology theory, as well as potential applications to open questions concerning the colored Jones polynomial. Next I will discuss our work, joint with Carmen Caprau, Nicolle Gonzalez, and Radmila Sazdanovic, using Bar-Natan Gaussian elimination, that gives our whittled complex $\mathcal{FT}_n^k$. The whittled complex is homotopy-equivalent to the original Khovanov chain complex but with a reduced number of generators. After sketching the proof, I will end the talk discussing related future projects.