Research
A recurring strategy of my work is to “geometrize” a problem: translate it into a question about points, convexity, maps, or spaces, often with symmetries. Once the problem has a geometric form, tools such as symmetry-preserving maps, fixed-point theorems, or algebraic invariants associated to spaces may be used to detect global structure and obstructions.
Research areas
How can points in Euclidean space be partitioned so that the convex hulls of the parts intersect? Which intersection patterns are forced, and which can be avoided? Which results admit continuous generalizations?
This includes classical questions such as whether one space embeds into another and their quantitative generalizations such as whether one space admits a particularly generic embedding into another.
Fixed point theorems can guarantee the existence of equilibria, fair divisions, or other balanced outcomes.
How to reconstruct a sufficiently nice metric space from a sample? How to measure the dissimilarity of two shapes?
Many geometric and combinatorial problems become questions about whether certain symmetric maps exist. Equivariant topology supplies obstructions.
Topological tools can detect coloring obstructions, covering phenomena, and combinatorial structures that are otherwise invisible.
Which subconfigurations are forced in sufficiently rich point sets? Which subsets of Euclidean space cannot avoid configurations that average to zero?
Topology can force analytic phenomena such as zeros, sign changes, orthogonality, and cubature formulas, often by turning questions about functions into questions about equivariant maps.
For students
I am happy to hear from students who are interested in geometric and topological methods. Because mentoring works best when regular interaction is possible, I generally cannot mentor students who are not physically in Pittsburgh. I make exceptions only for graduate students based in Berlin.
Students interested in this line of work can find various resources on this page and are welcome to contact me. In this section you can find a list of some problem areas that I have developed geometric and topological methods for to give an idea of the types of questions I think about.
Any object or problem that requires understanding non-local information and phenomena might benefit from the introduction of topological tools. I am interested in finding the underlying geometry of any such object or problem. I am motivated by the development of geometric-topological methods; the eventual application area is not constrained and can come from algebra, analysis, combinatorics, convex & discrete geometry, game theory, group theory, theoretical computer science, etc. I am by no means an expert on all of these, and I enjoy learning about new areas and possible applications of geometry and topology.
Students interested in working with me should either be interested in a type of problem I have worked on in the past or suggest a new problem area that might be amenable to the tools of topology. It is not necessary to have such a problem in mind, when discussing research mentorship with me. Motivation to learn at least one geometric-topological toolkit and enjoyment of creative problem-solving are prerequisites. A tendency to think geometrically is a plus. Substantial prior topological knowledge is not necessary (but is needed for some, though not all, problems I think about).
Recorded talks
A recent seminar recording on recovering metric information from samples using Vietoris–Rips thickenings and optimal transport.
A colloquium talk on quantitative versions of topological obstructions and their connections to discrete geometry.
A public lecture on geometric thinking in mathematics, science, economics, algorithms, and data science.
An eight-part lecture series recorded July 25–August 4, 2022.
A talk on spaces of embeddings, chirality, and quantitative obstructions.
A talk on the topology of Rips complexes and projective-space coding questions.
Surveys
Selected starting points