We introduce a conformally invariant nonlinear sigma model on the bulk of contact manifolds with boundary condition on the Legendrian links in any odd dimension. We call any finite energy solution a contact instanton. We also explain its Ha...

In recent years, artificial intelligence has made remarkable progress in developing algorithms that can learn from vast amounts of carefully labeled data. This paradigm of supervised learning has made great success in training specialist mo...

It is a fascinating and challenging problem to count number fields with bounded discriminant. It has so many applications in number theory. We give two examples. First, we compute the average of the smallest primes belonging to a conjugacy ...

Machine learning (ML) has achieved unprecedented empirical success in diverse applications. It now has been applied to solve scientific problems, which has become an emerging field, Scientific Machine Learning (SciML). Many ML techniques, h...

For an isolated singularity, the intersection with a small sphere forms a smooth manifold, called the link of a singularity. It admits a canonical contact structure, and this turns out to be a fine invariant of singularities and provides an...

I will tell two interrelated stories illustrating fruitful interactions between combinatorics and Hodge theory. The first is that of Lorentzian polynomials, based on my joint work with Petter Brändén. They link continuous convex...