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Abstract TBA

The Math Picture Language seminar will be held at 9:30 a.m. Boston time.
Click the link for a Zoom Link for Tuesday Math Picture Language Seminars.
Recorded seminars can be viewed on the Mathematical Picture Language YouTube channel

In this talk I will give an overview of some recent work, joint with Samuel Grushevsky, Klaus Hulek, and Radu Laza, on the geometry and topology of compactifications of the moduli spaces of cubic threefolds and cubic surfaces. A focus of the talk will be on some results regarding non-isomorphic smooth compactifications of the moduli space of cubic surfaces, showing that two natural desingularizations of the moduli space have the same cohomology, and are both blow-ups of the moduli space at the same point, but are nevertheless, not isomorphic, and in fact, not even K-equivalent.

 Generative models have attracted intense interests recently. In this talk, I will discuss one class of generative models, Generative Adversarial Networks (GANs).  I will first provide a gentle review of the mathematical framework behind GANs. I will then proceed to discuss a few challenges in GANs training from an analytical perspective. I will finally report some recent progress for GANs training in terms of its stability and convergence analysis.

A great tool in the arsenal of combinatorialists is modeling problems as counting  the lattice points of some convex polytope. Let $P\subseteq\R^d$ be a rational  convex polytope, that is, a polytope with vertices in $\mathbb{Q}^d$, then the Ehrhart function of the polytope $i(P,n)$ counts the number of integer lattice of the dilation $nP$ (here $nP$ denotes the polytope obtained from dilating $P$ by a factor n). Ehrhart functions have a rich history and many wonderful properties (e.g., Ehrhart himself proved that when $P$ is a lattice polytope, then $i(P,n)$ is a polynomial of degree $dim(P)$. The connections to Hilbert series are legendary). This topic has appeared in algebraic combinatorics, representation theory, algebraic geometry and others areas. But what if we count the integer lattice points with *weights*? Say  $w: \R^d \to \R$   a  function, often called a  \emph{weight function}. We can consider the, \emph{weighted Ehrhart} function: \[ i(P,w,n)=\sum_ {x\in nP \cap \Z^d} w(x).  \] (Here $w(x) := w(x_1,\dots,x_d)$ runs over the set of integer points belonging to $P$)

In this lecture I review what we know about weighted Ehrhart functions.

Some basic things remain true, other classical results have delicate variations and extensions. I will discuss several new theorems:

1) We generalized R. Stanley’s theorem that the $h^\ast$-polynomial of the Ehrhart series of a rational polytope has nonnegative coefficients and is monotone under containment of polytopes.  We show that these results continue to hold for \emph{weighted} Ehrhart series where lattice points are counted with polynomial weights, as long as the weights are homogeneous polynomials decomposable as sums of products of linear forms that are nonnegative on the polytope.

2) We also investigated  nonnegativity of the $h^\ast$-polynomial as a real-valued function for  a larger family of weights. In fact, discuss the case of  counting lattice points of a polytope that are weighted not by a simple polynomial, but by an Ehrhart quasi-polynomial of a family of parametric polytopes. As applications one can compute integrals and maximum values of such quasi-polynomials. We obtain new identities in representation theory and semigroup theory similar to RSK.

This work comprises 3 papers joint work with subsets of the following wonderful people: Esme Bajo, Rob Davis, Laura Escobar, Alexey Garber, Katharina Jochemko, Nathan Kaplan, Sofia Garzon-Mora, Josephine Yu, Rafael Villarreal, and Chengyang Wang.

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For information about the Combinatorics Seminar, please visit:

http://math.mit.edu/seminars/combin/

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One of the most remarkable possibilities of General Relativity concerns gravitational collapse to black holes, leaving behind a geometry with light rings, ergoregions and horizons. These peculiarities are responsible for uniqueness properties and energy extraction mechanisms that turn black holes into ideal laboratories of strong gravity, of particle physics (yes!) and of possible quantum-gravity effects. I will discuss some of the latest progress in tests of General Relativity with black holes.

Zoom: https://harvard.zoom.us/j/7855806609

For more information, please see:  Algebraic Dynamics Seminar at Harvard

This is joint work, the forward direction with Slaven Jabuka and the partial reverse direction with Brendan Owens.