I will survey recent progress in defining and computing categorical enumerative invariants, analogues of Gromov-Witten invariants defined directly from a cyclic A_infinity category and a choice of splitting of the Hodge filtration on its periodic cyclic homology. A proposed definition of such invariants appeared in 2005 in work of Costello, but the original approach had technical problems that made computations impossible.
Today, in our modern world, we perceive the physical universe in mathematical terms; whether degrees on longitude and latitude on earth, or in units of space-time beyond our earthly horizons. This talk will present two ancient cuneiform tablets from Babylonia which offer a geometric impression of the physical world as experienced by ancient Babylonians. Comparisons will be made with a range of other ancient mathematical, geographic, and astronomical materials from the cuneiform Ancient Near East.
Abstract: A subgroup is said to be almost normal if it is commensurable
to all of its conjugates. Even though there may not be a well-defined
quotient group, there is still a well-defined quotient space that admits
an isometric action by the ambient group. We can deduce many geometric
and algebraic properties of the ambient group by examining this action.
In particular, we will use quotient spaces to prove a relative version
of Stallings-Swan theorem on groups of cohomological dimension one. We
Title: New bounds on the covering radius of a lattice.
We obtain new upper bounds on the minimal density of lattice coverings of R^n by dilates of a convex body K. We also obtain bounds on the probability (with respect to the natural Haar-Siegel measure on the space of lattices) that a randomly chosen lattice L satisfies L + K = R^n. As a step in the proof, we utilize and strengthen results on the discrete Kakeya problem. This is joint work with Or Ordentlich and Oded Regev.