# School of Mathematics

## 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.

## 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

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

## Circuit Lower Bounds for Nondeterministic Quasi-Polytime: An Easy Witness Lemma for NP and NQP

We prove that if every problem in NP has n^k-size circuits for a fixed constant k, then for every NP-verifier and every yes-instance x of length n for that verifier, the verifier's search space has a witness for x that can be encoded with a circuit of only n^O(k^3) size. An analogous statement is proved for nondeterministic quasi-polynomial time, i.e., NQP = NTIME[n^(log n)^O(1)]. This significantly extends the Easy Witness Lemma of Impagliazzo, Kabanets, and Wigderson [JCSS'02] which only held for larger nondeterministic classes such as NEXP.

## Operator Scaling via Geodesically Convex Optimization, Invariant Theory and Polynomial Identity Testing (Continued)

We propose a new second-order method for geodesically convex optimization on the natural hyperbolic metric over positive definite matrices. We apply it to solve the operator scaling problem in time polynomial in the input size and logarithmic in the error. This is an exponential improvement over previous algorithms which were analyzed in the usual Euclidean, “commutative” metric (for which the above problem is not convex).