In this talk we discuss the cubic wave equation in three dimensions. In three dimensions the critical Sobolev exponent is 1/2. There is no known conserved quantity that controls this norm. We prove unconditional global well-posedness for radial initial data in the critical Sobolev space.
Geodesic nets on Riemannian manifolds is a natural generalization of geodesics. Yet almost nothing is known about their classification or general properties even when the ambient Riemannian manifold is the Euclidean plane or the round 2-sphere.
In this talk I am going to survey some results and open questions (old
and new) about geodesic nets on Riemannian manifolds. Many of these
A matroid is an abstract combinatorial object which generalizes the notions of spanning trees,
and linearly independent sets of vectors. I will talk about an efficient algorithm based on the Markov Chain Monte Carlo technique
to approximately count the number of bases of any given matroid.
The proof is based on a new connections between high dimensional simplicial complexes, and a new class
of multivariate polynomials called completely log-concave polynomials. In particular, we exploit a fundamental fact from our
Positive geometries are real semialgebraic sets inside complex varieties characterized by the existence of a meromorphic top-form called the canonical form. The defining property of positive geometries and their canonical forms is that the residue structure of the canonical form matches the boundary structure of the positive geometry. A key example of a positive geometry is a projective polytope.
These capacities are defined in every dimension and give state of the art obstructions for various "stabilized" symplectic embedding problems such as one ellipsoid into another. They can also be described via symplectic cohomology and are related to counting pseudoholomorphic curves with tangency conditions. I will explain the basic idea of the construction and then give some computations, structural results, and applications.
Deep learning has led to rapid progress in open problems of artificial intelligence—recognizing images, playing Go, driving cars, automating translation between languages—and has triggered a new gold rush in the tech sector. But some scientists raise worries about slippage in scientific practices and rigor, likening the process to “alchemy.” How accurate is this perception? And what should the field do to combine rapid innovation with solid science and engineering?