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# Omar Antolín Camarena

Página web en español.

I’m a researcher at Instituto de Matemáticas, UNAM1 in Mexico City. You can email me at omar@matem.unam.mx. Before working here, I was a postdoc at UBC in Vancouver, and before that I studied a PhD at Harvard where my advisor was Jacob Lurie.

This is my CV and these2 are my articles on the arXiv. I’ve written a bunch of Emacs packages you can find on my GitHub account.

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

## Research

### Algebraic Topology

• To a compact Lie group $$G$$ one can associate a space $$E(2,G)$$ akin to the poset of cosets of abelian subgroups of a discrete group. The space $$E(2,G)$$ was introduced by Adem, F. Cohen and Torres-Giese, and subsequently studied by Adem and Gómez, and other authors. In this short note, we prove that $$G$$ is abelian if and only if $$\pi_i(E(2,G))=0$$ for $$i=1,2,4$$. This is a Lie group analogue of the fact that the poset of cosets of abelian subgroups of a discrete group is simply–connected if and only if the group is abelian.

• We show that for some classes of groups $$G$$, the homotopy fiber $$E_{\mathrm{com}} G$$ of the inclusion of the classifying space for commutativity $$E_{\mathrm{com}} G$$ into the classifying space $$BG$$, is contractible if and only if $$G$$ is abelian. We show this both for compact connected Lie groups and for discrete groups. To prove those results, we define an interesting map $$\mathfrak{c} \colon E_{\mathrm{com}} G \to B[G,G]$$ and show it is not nullhomotopic for the non-abelian groups in those classes. Additionally, we show that $$\mathfrak{c}$$ is 3-connected for $$G=O(n)$$ when $$n \ge 3$$.

• For each of the groups $$G = O(2), SU(2), U(2)$$, we compute the integral and $$\mathbb{F}_2$$-cohomology rings of $$B_\text{com} G$$ (the classifying space for commutativity of $$G$$), the action of the Steenrod algebra on the mod 2 cohomology, the homotopy type of $$E_\text{com} G$$ (the homotopy fiber of the inclusion $$B_\text{com} G \to BG$$), and some low-dimensional homotopy groups of $$B_\text{com} G$$.

• The Goodwillie derivatives of the identity functor on pointed spaces form an operad $$\partial_{\ast}(\mathrm{Id})$$ in spectra. Adapting a definition of Behrens, we introduce mod 2 homology operations for algebras over this operad and prove these operations account for all the mod 2 homology of free algebras on suspension spectra of simply-connected spaces.

• We describe the connected components of the space $$\mathrm{Hom}(\Gamma,SU(2))$$ of homomorphisms for a discrete nilpotent group $$\Gamma$$. The connected components arising from homomorphisms with non-abelian image turn out to be homeomorphic to $$\mathbb{RP}^3$$. We give explicit calculations when $$\Gamma$$ is a finitely generated \emph{free nilpotent} group. In the second part of the paper we study the filtration $$B_{\text{com}} SU(2)=B(2,SU(2))\subset\cdots \subset B(q,SU(2))\subset\cdots$$ of the classifying space $$BSU(2)$$ (introduced by Adem, Cohen and Torres-Giese), showing that for every $$q\geq 2$$, the inclusions induce a homology isomorphism with coefficients over a ring in which 2 is invertible. Most of the computations are done for $$SO(3)$$ and $$U(2)$$ as well.

• 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.

• I pointed out a small gap in Ronald Brown’s proof of the Jordan Curve Theorem, related to the Phragmen–Brouwer Property; this note gives the correction in terms of a result on a pushout of groupoids, and some additional background.

### Topology applied to Physics

• For symmorphic crystalline interacting gapped systems we derive a classification under adiabatic evolution. This classification is complete for non-degenerate ground states and only partial in the degenerate case. We do not assume an emergent relativistic field theory nor that phases form a topological spectrum. Using a slightly generalized Bloch decomposition (without quasi-particles) and Grassmanians made out of ground state spaces, we show that the $$P$$-equivariant cohomology of a $$d$$-dimensional torus gives rise to different interacting phases. We discuss the relation of our assumptions to those made for crystallographic SPT and SET phases.

• We derive a rigorous classification of topologically stable Fermi surfaces of non-interacting, discrete translation-invariant systems from electronic band theory, adiabatic evolution and their topological interpretations. For systems with Born-von Karman boundary conditions it is shown that there can only be topologically unstable Fermi surfaces. For systems on a half-space and with a gapped bulk, our derivation naturally yields a K-theory classification. Given the (d−1)-dimensional surface Brillouin zone $$X_s$$ of a d-dimensional half-space, our result implies that different classes of globally stable Fermi surfaces belong in $$K^{−1}(X_s)$$ for systems with only discrete translation-invariance. This result has a chiral anomaly inflow interpretation, as it reduces to the spectral flow for $$d=2$$. Through equivariant homotopy methods we extend these results for symmetry classes AI, AII, C and D and discuss their corresponding anomaly inflow interpretation.

### Combinatorics

• 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.

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.

### Computational Mathematics

• 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.

## Talks (most in Spanish)

### Slides

• Extensiones de grupos desde el punto de vista categórico. Coloquio del IMUNAM-Cuernava. PDF
• Higher generation of compact Lie groups by abelian subgroups. Pure Math Seminar, University of Leicester. PDF
• Generación superior de grupos de Lie compactos por subgrupos abelianos. Coloquio del CCM-UNAM. PDF
• Conjuntos afínmente conmutativos y clases laterales de subgrupos abelianos. Seminario de Álgebra y Geometría del IMATE Cuernavaca. PDF
• Las consecuencias de que las curvas tengan siempre cero o dos extremos: Una introducción a la Topología Diferencial. Coloquio de Orientación Matemática, FC-UNAM. PDF
• Espacios de Thom desde el punto de vista categórico, LI Congreso SMM, Villahermosa Tabasco. PDF
• Homotopía Motívica: Mezclando geometría algebraica con teoría de homotopía, talk for undergraduates, in Spanish. PDF
• Conmutatividad en Grupos de Lie, Coloquium style talk in Spanish. HTML PDF

## Expository writing

Something to read right here in the browser:

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

• My undegraduate thesis (in Spanish).

## Earlier Teaching Materials

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.