I’m a graduate student at the Harvard math department. You can email me at . This is my CV and these1 are my articles on the arXiv.

I’m interested in homotopy theory, higher category theory, derived algebraic geometry and combinatorics.

## Research

• We give a simple universal property of the multiplicative structure on the Thom spectrum of an $$n$$-fold loop map, obtained as a special case of a characterization of the algebra structure on the colimit of a lax $$\mathcal{O}$$-monoidal functor. This allows us to relate Thom spectra to $$\mathbb{E}_n$$-algebras of a given characteristic in the sense of Szymik. As applications, we recover the Hopkins–Mahowald theorem realizing $$H\mathbb{F}_p$$ as a Thom spectrum, and compute the topological Hochschild homology and the cotangent complex of various Thom spectra.

• In this note, we construct a general form of the chromatic fracture cube, using a convenient characterization of the total homotopy fiber, and deduce a decomposition of the E(n)-local stable homotopy category.

• Omar Antolín Camarena pointed out a gap in the proofs in [BT&G, Bro06] of a condition for the Phragmen–Brouwer Property not to hold; this note gives the correction in terms of a result on a pushout of groupoids, and some additional background.

• Recent developments in ergodic theory, additive combinatorics, higher order Fourier analysis and number theory give a central role to a class of algebraic structures called nilmanifolds. In the present paper we continue a program started by Host and Kra. We introduce nilspaces as structures satisfying a variant of the Host-Kra axiom system for parallelepiped structures. We give a detailed structural analysis of abstract and compact topological nilspaces. Among various results it will be proved that compact nilspaces are inverse limits of finite dimensional ones. Then we show that finite dimensional compact connected nilspaces are nilmanifolds. The theory of compact nilspaces is a generalization of the theory of compact abelian groups. This paper is the main algebraic tool in the second authors approach to Gowers’s uniformity norms and higher order Fourier analysis.

• Positive graphs with Endre Csóka, Tamás Hubai, Gábor Lippner, László Lovász. [show abstract]

We study “positive” graphs that have a nonnegative homomorphism number into every edge-weighted graph (where the edgeweights may be negative). We conjecture that all positive graphs can be obtained by taking two copies of an arbitrary simple graph and gluing them together along an independent set of nodes. We prove the conjecture for various classes of graphs including all trees. We prove a number of properties of positive graphs, including the fact that they have a homomorphic image which has at least half the original number of nodes but in which every edge has an even number of pre-images. The results, combined with a computer program, imply that the conjecture is true for all graphs up to 9 nodes.

• We describe a collection of computer scripts written in PARI/GP to compute, for reflection groups determined by finite-volume polyhedra in $$\mathbb{H}^3$$, the commensurability invariants known as the invariant trace field and invariant quaternion algebra. Our scripts also allow one to determine arithmeticity of such groups and the isomorphism class of the invariant quaternion algebra by analyzing its ramification. We present many computed examples of these invariants. This is enough to show that most of the groups that we consider are pairwise incommensurable. For pairs of groups with identical invariants, not all is lost: when both groups are arithmetic, having identical invariants guarantees commensurability. We discover many “unexpected” commensurable pairs this way. We also present a non-arithmetic pair with identical invariants for which we cannot determine commensurability.

## Expository writing

Something to read right here in the browser:

• Iteration of rational maps, also available in PDF. This is my minor thesis2, it includes fairly complete proofs of Sullivan’s No Wandering Domains theorem and the classification of the periodic Fatou components and is, I hope, fairly light reading.

### Expository papers (PDFs):

• This introduction to higher category theory is intended to a give the reader an intuition for what (∞,1)-categories are, when they are an appropriate tool, how they fit into the landscape of higher category, how concepts from ordinary category theory generalize to this new setting, and what uses people have put the theory to. It is a rough guide to a vast terrain, focuses on ideas and motivation, omits almost all proofs and technical details, and provides many references.

• This is an expository paper about Kas and Schlessinger’s construction of a versal deformation space for an analytic space which is locally a complete intersection. This result has a distinct algebro-geometric flavor, but we do not assume any familiarity with concepts from algebraic geometry such as flatness or nonreducedness. In fact, we hope this paper can serve as an introduction to these ideas for geometers dealing with analytic spaces.

• The van Kampen theorem. The version of the van Kampen theorem for the fundamental groupoid and applications, including Ronnie Brown’s neat proof of the Jordan curve theorem.

## Teaching

In the spring of 2012, I taught a tutorial about the fundamental groupoid called Groupoids in Topology.

For a combined precalculus/calculus course called Math Ma, I wrote a couple of interactive webpages using the brilliant JXSGraph library:

• Bottle calibrator. You can draw the profile of a flask by dragging or adding points on a curve and see in real-time the graph of the height reached by some liquid poured in the flask as a function its volume.

• Secant line animation. Yet another version of the classical secant line animation. You can specify your own function.

Topologists, category theorists and derived algebraic geometers3:

You can find some of these people at the nLab or the n-Category Café.

(Limits of combinatorial structures)-ists4, 5:

Top of the line lecture notes on various topics:

1. For some reason a direct search doesn’t turn up all of them.

2. This an excellent program requirement: a chance to make a fool of yourself in writing by producing an exposition of a topic outside your field of expertise in a limited amount of time. I loved it! All math departments should have this.

3. Separating them into the three topics named is left as an exercise for the reader.

4. Possibly not the standard name for workers in this field.

5. The people on this list do lots of other stuff too.