semidefinite programming (sdp) forms a class of convex optimization problems with remarkable modeling power. apart from its classical applications in combinatorics and control, it also enjoys a range of applications in data science. this talk first discusses various concrete sdps in data science and their conditioning. in particular, we show that even though slater’s constraint qualification condition may fail, these sdps satisfy an important regularity, strict complementarity, which ensures the good conditioning of the problem. in the second part of the talk, based on the regularity and computational structure shared by these problems, we design time- and space-efficient algorithms to solve these sdps.

we describe a principled approach for constructing non-vacuum initial data sets for the cauchy problem in general relativity. the core idea has an interesting history of having been known in the '70s,
forgotten by the mathematical relativity community for decades, and now
independently rediscovered and rigorously demonstrated. we show how
it explains why certain techniques for generating initial data
worked well in the past, but also how it leads to new equations
with applealing physical properties when generating initial data containing
fluids. the talk will be targeted at a broad audience.

i’ll discuss recent progress on the existence theory for harmonic maps, in particular the existence of harmonic maps of optimal regularity from manifolds of dimension n>2 to every non- aspherical closed manifold containing no stable minimal two-spheres. as an application, we’ll see that every manifold carries a canonical family of sphere-valued harmonic maps, which (in dimension<6) stabilize at a solution of a spectral isoperimetric problem generalizing the conformal maximization of laplace eigenvalues on surfaces. based on joint work with mikhail karpukhin.

 i will discuss recent advances in our understanding of the stable homotopy groups of spheres using formal groups, modular forms, and  (time permitting) motives.

syzygies of algebraic varieties have long been a topic of intense interest among algebraists and geometers alike. starting with the pioneering work of mark green on curves, numerous attempts have been made to extend these results to higher dimensions. ein and lazarsfeld proved that if a is a very ample line bundle, then k_x + ma satisfies property n_p for any m>=n+1+p. it has ever since been an open question if the same holds true for a ample and basepoint free. in joint work with purnaprajna bangere we give a positive answer to this question.