A summary for non-experts
My research lies within the fields of gauge theory, low-dimensional topology, and homological algebra. Using partial differential equations coming from physics, the goal of this field is to assign 'invariants' to 3- and 4-dimensional shapes which are unchanged upon wiggling the shape a bit. To show that two shapes are topologically the same (may be wiggled into another, in an abstract sense), one usually gives an explicit construction showing how to go from one to the other. It is much harder to show that one thing can't be transformed into another.
To do so, imagine we can assign some gadget (maybe a number, maybe a group, maybe something much more intricate) which is unchanged under wiggling - an invariant. If the invariants of two shapes are different, then you cannot transform one shape into another. So the game is to construct invariants which are intricate enough that they can distinguish lots of shapes and rule out lots of different kinds of behavior of these shapes, but not so intricate that it's impossible to compute them!
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
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 thusfar been about developing foundational aspects, to bring these tools up to speed with more well-developed invariants from monopole and Heegaard Floer theory.
Equivariant instanton homology ~ available on arXiv ~ changes from v1 to v2 detailed here
In this paper (which constituted my thesis), I develop some analysis of instanton moduli spaces, and present algebra related to the equivariant (co)homology of differential graded algebras and their modules. Using these tools, I define four invariants of rational homology spheres that deserve to be called "equivariant instanton homology" and have some formal properties similar to those seen in Heegaard Floer and monopole Floer homology.
Gard Olav Helle's recent thesis gives computations of these invariants for Y an orbit of the Lie group SU(2). I hope to give more explicit computations in future work, especially over Z/2 (where the structure is especially intricate, comparable to involutive Heegaard Floer homology).
Functoriality and invariance of equivariant instanton homology (joint with Ali Daemi) ~ available on arXiv ~ comments welcome!
We substantially improve the invariance results in my thesis, and promote equivariant instanton homology with all coefficients to a functor on the negative-definite cobordism category (and 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.
In future work we will use this to study certain F_2-homology cobordism invariants of F_2-homology spheres which have interesting new properties.
3-manifolds without any embedding in symplectic 4-manifolds (joint with Ali Daemi and Tye Lidman) ~ available on arXiv, to appear in Geometry & Topology ~ comments welcome!
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.
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, to appear in Geometry & Topology ~ comments welcome! ~ 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.
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.
Fourier transforms and integer homology cobordism ~ available on arXiv ~ to appear in Applied and Geometric Topology ~ comments welcome!
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 ~ To appear in Topology & its Applications ~ comments welcome!
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.
