Research

My research interests are in geometry, topology, and their applications. In particular, I am focused on inverse problems for topological and geometric invariants, topological optimization and approximation, learning with topological invariants, and the connections between them. I have also been working in the area of data fusion, using techniques from geometry and topology to analyze and synthesize time series data.

Publications & Preprints

A Convolutional Persistence Transform
(with Paul Bendich)
[arXiv]
We combine convolutions and persistent homology to develop a new injective topological transform. We demonstrate experimentally that this pipeline greatly increases the power of topological features, even using random filters and vectorizing diagrams using only total persistence.

Improving Metric Dimensionality Reduction with Distributed Topology
(with Alex Wagner and Paul Bendich)
[arXiv]
We use distributed persistence and local geometry to define a new dimensionality-reduction pipeline, DIPOLE.

From Geometry to Topology: Inverse Theorems for Distributed Persistence (with Alex Wagner and Paul Bendich) [SoCG]
We propose a distributed persistence invariant which provably interpolates between geometry and topology.
Here is a recording of a talk I gave about this paper at ATiA.

A Fast and Robust Method for Global Topological Functional Optimization (with Alex Wagner and Paul Bendich) [AISTATS]
We propose a new framework for optimizing topological functionals on simplicial complexes that is faster, and produces more robust optima, than prior methods.

Geometric Fusion via Joint Delay Embeddings (with Paul Bendich) [Fusion2020]
We use geometric and topological methods to fuse time series.
Won 2nd runner up in the general category of the Fusion 2020 Best Paper Award!

Intrinsic Topological Transforms via the Distance Kernel Embedding (with Clément Maria, Steve Oudot)
[SOCG]
We use spectral geomety to define a novel topological transform.

Inverse Problems in Topological Persistence (with Steve Oudot)
[Proceedings of the Abel Symposium]
A survey of inverse problems in applied topology.

Barcode Embeddings for Metric Graphs (with Steve Oudot)
[Algebraic and Geometric Topology]
We study the inverse problem for the intrinsic persistent homology transform on metric graphs.

Relaxing the Integral Text: A Challenge for the Advanced Calculus Student (with Paul Carter)
[College Mathematics Journal]

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