Omar Antolín Camarena

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About me

I’m a graduate student at the Harvard math department. You can email me at .

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

Expository writing

Something to read right here in the browser:

• Notes and mathematical short stories.

• Things I was surprised to learn are false.

• Iteration of rational maps, also available in PDF. This is my minor thesis¹, 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):

• A Whirlwind Tour of the World of (∞,1)-categories. [show abstract]

  This introduction to higher category theory is intended to a give the reader an intuition for what -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.

• On a Theorem of Kas and Schlessinger with Sarah C. Koch [show abstract]

  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.

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.

Research papers

• Nilspaces, nilmanifolds and their morphisms with Balazs Szegedy. [show abstract]

  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.

• Computing arithmetic invariants for hyperbolic reflection groups with Gregory R. Maloney and Roland K. W. Roeder. [show abstract]

  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.

Links

Topologists, category theorists and derived algebraic geometers²: David Ayala, John Baez, Clark Barwick, David Ben-Zvi, Dan Dugger, John Francis, Dennis Gaitsgory, Moritz Groth, André Joyal, William Lawvere, Tom Leinster, Jacob Lurie, Peter May, Ieke Moerdijk, David Nadler, Charles Rezk, Emily Riehl, Chris Schommer-Pries, Urs Schreiber, Michael Shulman, Carlos Simpson, David Spivak, Ross Street, Bertrand Toën.

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

(Limits of combinatorial structures)-ists^{3, 4}: Tim Austin, Christian Borgs, Jennifer Chayes, Persi Diaconis, Gábor Elek, Tim Gowers, Ben Green, Svante Janson, Yoshiharu Kohayakawa, László Lovász, Jaroslav Nešetřil, Patrice Ossona de Mendez, Alexander Schrijver, Vera Sós, Bálazs Szegedy, Terence Tao, Katalin Vesztergombi.

Top of the line lecture notes on various topics: Akhil Mathew, Anton Geraschenko, Chris Schommer-Pries, Chao Li