School of Mathematics

Large-scale vision benchmarks have driven---and often even defined---progress in machine learning. However, these benchmarks are merely proxies for the real-world tasks we actually care about. How well do our benchmarks capture such tasks? In this...

A transverse link in a contact 3-manifold forces topological entropy if every Reeb flow possessing this link as a set of periodic orbits has positive topological entropy. We will explain how cylindrical contact homology on the complement of...

In this talk we will be addressing the question whether a given Lagrangian torus in a toric monotone symplectic manifold can be realized as the fixed point set of an anti-symplectic involution (in which case it is called "real"). In the case of...

Gromov nonsqueezing tells us that symplectic embeddings are governed by more complex obstructions than volume. In particular, in 2012, McDuff-Schlenk computed the embedding capacity function of the ball, whose value at a is the size of the smallest...

One of the fundamental challenges in number theory is to understand the intricate way in which the additive and multiplicative structures in the integers intertwine. We will explore a dynamical approach to this topic. After introducing a new...

Wave maps are harmonic maps from a Lorentzian domain to a Riemannian target. Like solutions to many energy critical PDE, wave maps can develop singularities where the energy concentrates on arbitrary small scales but the norm stays bounded. Zooming...

I will explain a duality theorem with products in Rabinowitz-Floer homology. This has a bearing on string topology and explains a number of dualities that have been observed in that setting. Joint work in progress with Kai Cieliebak and Nancy...

Homological mirror symmetry predicts that the derived category of coherent sheaves on a curve has a symplectic counterpart as the Fukaya category of a mirror space. However, with the exception of elliptic curves, this mirror is usually a symplectic...

The classical Linnik problems are concerned with the equidistribution of adelic torus orbits on the homogeneous spaces attached to inner forms of GL2, as the discriminant of the torus gets large. When specialized, these problems admit beautiful...

Epidemiological forecasting is critically needed for decision making by national and local governments, public health officials, healthcare institutions and the general public. The Delphi group at Carnegie Mellon University was founded in 2012 to...