Below I list my publications and preprints, together with a brief summary of the contents. A concise list of publications can be found in my CV.

Publications and preprints

 Work coming from instanton gauge theory

Instanton gauge theory's connection to flat connections and therefore the fundamental group of a 3- and 4-manifold makes it especially suited for tackling questions related to simply connected 4-manifolds. My research in this area has covered both applications and foundational aspects of the equivariant instanton theory.

Equivariant instanton homology ~ available on arXiv, to appear in Mem. AMS ~ changes from v1 to v2 here

This paper develops some analysis of instanton moduli spaces, and presents algebra related to the equivariant (co)homology of differential graded algebras and their modules. This is used to define four invariants of rational homology spheres that deserve to be called "equivariant instanton homology" which have some formal properties similar to those seen in Heegaard Floer and monopole Floer homology. 

Instantons and rational homology spheres (joint with Ali Daemi) ~ available on arXiv 

We promote equivariant instanton homology with all coefficients to a functor on the negative-definite cobordism category (with rational coefficients to a functor under arbitrary cobordisms). As a first example, we use this to give a extension of Taubes' Casson invariant to rational homology spheres. 

Filtered instanton homology and cosmetic surgery (joint with Ali Daemi and Tye Lidman) ~ available on arXiv

The cosmetic surgery conjecture asserts that distinct surgeries on the same knot are inequivalent. There has been a great deal of progress on this conjecture. In particular, if (K, r) has surgery oriented diffeomorphic to (K, s), then s = +- r and r in {2, 1, 1/2, 1/3, ...} We show that the filtered instanton homology of S_{1/n}(K) and S_{1/m}(K) are never equivalent for any two integers n,m, so the manifolds are distinct. This reduces us to the final case of the cosmetic surgery conjecture: slopes +-2.

3-manifolds without any embedding in symplectic 4-manifolds (joint with Ali Daemi and Tye Lidman) ~ available on arXiv, Geom. Topol. 28 (2024), no. 7, 3357-3372.

We prove that there exist infinitely 3-manifolds which don't embed into any closed symplectic 4-manifold (the closed condition is crucial here). This follows from the fact that our manifolds do not bound positive- or negative-definite 4-manifolds. The most intricate tool used here is the Chern-Simons invariant of a flat connection; behind the scenes, what we really do is compute a generalization of Ali Daemi's Gamma invariant for rational homology spheres, which fits into the framework of our joint work above.

 Work coming from monopole gauge theory

For various reasons (compactness and existence of deep computations chief among them), the neighboring field of monopole Floer homology has been developed further than the instanton theory. 

Monopoles, twisted integral homology, and Hirsch algebras (joint with Francesco Lin) ~ available on arXiv, Geom. Topol. 28 (2024), no. 8, 3697-3778 ~ an exposition

Francesco Lin and I have recently given a complete calculation of the "bar flavor" of the monopole Floer homology groups of an arbitrary 3-manifold. We use the pioneering work of Kronheimer and Mrowka, which determines that these groups are a certain 'twisted simplicial homology group'. Our contribution is primarily homological algebra: we show how the study of 'Hirsch algebras' allows us to give a completely explicit computation of the relevant twisted homology groups.  

On integral rigidity in Seiberg-Witten theory (joint with Francesco Lin) ~ available on arXiv 

This is a sort of follow-up to our previous paper; among other things, we discuss the cobordism maps in HM-bar. Instead of attempting to determine HM-bar as a functor, which is interesting and probably possible, we pursue applications of the cobordism map formula. We find that there are some cases where Seiberg--Witten invariants are purely cohomological in nature, and I think it would be interesting to see if there are similar constraints in Heegaard Floer theory.

 Miscellany

Below is some work which doesn't quite fit into the gauge theory paradigm. Some of this was obtained as a result of joint work with undergraduate students.

A Lefschetz decomposition over Z, and applications (joint with Analisa Faulkner Valiente) ~ available on arXiv 

The Lefschetz decomposition of Lambda^*(C^2g) into primitive subspaces is generalized to a filtration of the algebra Lambda^*(Z^2g) with favorable properties. We give applications in group homology and Heegaard Floer homology; I am somewhat optimistic that the filtration has other applications.

Fourier transforms and integer homology cobordism ~ available on arXiv ~ Algebr. Geom. Topol. 24 (2024), no. 7, 4085-4101.

Inspired by joint work with Aiden Sagerman below, I prove an extension of the classical result that homology cobordant 3-dimensional lens spaces are oriented diffeomorphic by showing that the same is true for direct sums. The key observation is that Fourier transforms allow one to effectively recognize the invariants of the summands. This was inspired by one of Aiden's observations when we were investigating the much more intricate scenario of products.

Affine hyperplanes in abelian groups (joint with Aiden Sagerman) ~ available on arXiv ~ Topology Appl. 339 (2023), Paper No. 108692.

Inspired by invariants with a certain multiplicativity property, we study isomorphisms between products of finite cyclic groups with the following curious property: if f(x) has some coordinate zero, then x has some coordinate zero. With an appropriate definition of "hyperplane", this reduces to the question: when can a hyperplane in a product of finite cyclic groups be contained in the union of coordinate hyperplanes? 

We completely classify these isomorphisms: as a special case, when all the cyclic factors have order larger than 2, they must be diagonal. Applying this to topology, we show that there exist a family of 4-manifolds X(n) which can be recovered from their products, and that any homeomorphism between products must induce the diagonal map on first homology.