## Fitting manifolds to data.

The problems come in two flavors.

Extrinsic Flavor: Given a point cloud in R^N sampled from an unknown probability density, how can we decide whether that probability density is concentrated near a low-dimensional manifold M with reasonable geometry? If such an M exists, how can we find it? (Joint work with S. Mitter and H. Narayanan)

## Studying Fluid Flows with Persistent Homology

## Protein Folding Characterization via Persistent Homology

We use persistent homology to analyze predictions of protein folding by trying to identify global geometric structures that contribute to the error when the protein is misfolded. The goal is to find correlations between global geometric structures, as measured by persistent homology, and the failure to predict the correct folding. This technique could be useful in guiding the energy minimization techniques to the correct minimum corresponding to the desired folding.

## Exceptional holonomy and related geometric structures: Dimension reduction and boundary value problems.

## Friends Lunch with a Member

## Exceptional holonomy and related geometric structures: Examples and moduli theory.

We will discuss the constructions of compact manifolds with exceptional holonomy (in fact, holonomy $G_{2}$), due to Joyce and Kovalev. These both use “gluing constructions”. The first involves de-singularising quotient spaces and the second constructs a 7-manifold from “building blocks” derived from Fano threefolds. We will explain how the local moduli theory is determined by a period map and discuss connections between the global moduli problem and Riemannian convergence theory (for manifolds with bounded Ricci curvature).

## Exceptional holonomy and related geometric structures: Basic theory.

## On Expressiveness and Optimization in Deep Learning

## Summation formulae and speculations on period integrals attached to triples of automorphic representations

Braverman and Kazhdan have conjectured the existence of summation formulae that are essentially equivalent to the analytic continuation and functional equation of Langlands L-functions in great generality. Motivated by their conjectures and related conjectures of L. Lafforgue, Ngo, and Sakellaridis, Baiying Liu and I have proven a summation formula analogous to the Poisson summation formula for the subscheme cut out of three quadratic spaces $(V_i,Q_i)$ of even dimension by the equation

$Q_1(v_1)=Q_2(v_2)=Q_3(v_3)$.