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Jack Morava

Publications and source records attributed to Jack Morava.

At least 19 recordsLinked to original sources
arXiv2024

Circular symmetry-breaking and topological Noether currents

We propose a toy model for the algebraic topology of bubbling as circular symmetry-breaking, in terms of asymptotic expressions for Noether currents and cobordism of manifolds with circle actions free along boundaries. This leads to an interpretation of Planck's radiation law as the loss of a Noether symmetry when a bubble $[S^2/\mathbb{T}] \to {\rm pt}$ (analogous to blowing up or down in projective geometry) collapses, much like the von K\'arm\'an street of sparks left when a candle flame wisps out.

math-ph↗
arXiv2024

Notes on $\delta$-algebras and prisms in homotopy theory

J McClure's Dyer-Lashof operation in $p$-adic $K$-theory defines, in particular, a prismatic structure on the complex representation ring of the circle group. Work of Ando, Rezk, Stapleton, and others generalizes this to define a canonical lift of Frobenius for structured Lubin-Tate spectra. We suggest that recent work of K Ito and S Marks on $L$-typical prisms may extend this to local neighborhoods of the topological prime points $K(n)$ of the category of spectra.

math.AT↗
arXiv2023

The universal cover of the real projective line

We construct an interesting topological cover of the multiplicative group of the real line, related to Tate's elliptic curve with $q = e^\pi$. We use the language of homological algebra, 2D Lorentz geometry and high-school trigonometry; the intent is expository but the suggested applications may be unusual.

math.GT↗
arXiv2023

A homotopy-theoretic context for CKM/Birkhoff renormalization

We propose a geometric object slightly subtler than a complex line bundle with connection, a two-sphere fibration with structure group $\Omega^2_e S^2$, to parametrize a space of dimensional regularizations in the metaphysics of renormalization theory. Comments, corrections, advice and suggestions are very welcome. To be continued.

math.AT↗
arXiv2023

Brauer-Wall Groups and Truncated Picard Spectra of $K$-theory

We compute the first two k-invariants of the Picard spectra of $KU$ and $KO$ by analyzing their Picard groupoids and constructing their unit spectra as global sections of sheaves on the category of manifolds. This allows us to determine the E_\infty-structures of their truncations Pic(KU)[0,3] and Pic(KO)[0,2]. It follows that these truncated Picard spaces represent: the Brauer groups of Z/2-graded algebra bundles of Donovan-Karoubi, Moutuou and Maycock; the Brauer groups of super 2-lines; and the K-theory twists of Freed, Hopkins and Teleman. Our results also imply that that these spaces represent twists of String and Spin structures on manifolds and can be used to twist tmf-cohomology. Finally, we are able to identify pic(KU)[0,3] with a cotruncation of the Anderson dual of the sphere spectrum.

math.KT↗
arXiv2023

Notes toward a Newtonian thermodynamics

We interpret the moment generating function ${\bf E}(e^{tX}):= {\rm exp}_F(t) \in {\bf R}[[t]]$ of a random variable $X$ as the exponential of an associated one-dimensional formal group law $F$ defined over ${\bf R}$.

math.PR↗
arXiv2023

Periods for topological circle actions

The language of Harvey-Lawson currents and sparks may be useful for the study of Hopkins-Singer Wu classes in geometric topology, via equivariant Tate $K$-theory of circle actions and distributional generalizations of classical zeta functions.

math.AT↗
arXiv2022

A locally conformally symplectic structure on Kerr space-time

R Kerr's Ricci-flat Lorentz 4-manifold was shown by B Carter in 1968 to support a classically completely integrable system of geodesics; here we interpret this system in terms of a locally conformally symplectic structure, which we identify (\S 2.2) using characteristic classes defined by Kerr-Schild Cartesian coordinates (regarded as a map to the Cayley-Penrose compactification of Minkowski space). This leads to the definition of an interesting cobordism category of contact 3-manifolds and 4-dimensional locally conformally symplectic cobordisms between them.

math.DG↗
arXiv2021

At the boundary of Minkowski space

The Cayley transform compactifies Minkowski space $\M$, realized as self-adjoint $2\times2$ complex matrices following Penrose, as the unitary group $\U(2)$. Its complement is a compactification of a copy of a light-cone as it is usually drawn, constructed by adjoining a bubble or $\CP_1$ of unitary matrices with eigenvalue $\pm 1$ at the ends of a lightcone at infinity. The Brauer-Wall group of $\U(2)$ (i.e. of fields of certain kinds of graded $\Cs$-algebras, up to projective equivalence) is $\Z_2 \times \Z$, defining an interesting class of nontrivial examples of Araki-Haag-Kastler backgrounds for quantum field theories on compactified Minkowski space. The second part of this paper extends such models to link presentations of more general spin four-manifolds.

math-ph↗
arXiv2020

Renormalization groupoids in algebraic topology

Continuing work begin in arXiv:1910.12609, we interpret the Hurewicz homomorphism for Baker and Richter's noncommutative complex cobordism spectrum $M\xi$ in terms of characteristic numbers (indexed by quasi-symmetric functions) for complex-oriented quasitoric manifolds, and show that automorphisms or cohomology operations on this representation are defined by a `renormalization' Hopf algebra of formal diffeomorphisms at the origin of the noncommutative line, previously considered (over $Q$) in quantum electrodynamics. The resulting structure can be presented in purely algebraic terms, as a groupoid scheme over $Z$ defined by a coaction of this Hopf algebra on the ring of noncommutative symmetric functions. We sketch some applications to symplectic toric manifolds, combinatorics of simplicial spheres, and statistical mechanics.

math.AT↗
arXiv2020

What $Ell$ sees that $K$ doesn't (when $p >3$)

We use Andrew Baker's analysis of the cofiber of the endomorphism of the $p$-adic elliptic spectrum ($p>3$) defined by multiplication by the `Hasse invariant' $E_{p-1}$ to present its completion away from the locus of ordinary elliptic curves as a sum of roughly $p/12$ copies (indexed by supesingular elliptic curves) of $p$-adic lifts of the height two mod $p$ cohomology theory $K(2)$. See a recent paper of Zhu Yifei for a much deeper exploration of the topics considered in this note.

math.AT↗
arXiv2020

Topological invariants of some chemical reaction networks

Certain toric dynamical systems studied in physical chemistry have associated toric varieties which, when smooth, represent elements in the homotopy groups $Mξ_*B\T$ of a symplectic variant of the $A_\infty$ Baker-Richter spectrum $Mξ$. The noncommutative Hopf algebroid $(Mξ_*,Mξ_*Mξ)$ has interesting connections to the group of formal diffeomorphism of the noncommutative line, conjecturally defining a noncommutative analog Helmholtz free energy for systems of complex symplectic (completely integrable) Hamiltonian toric manifolds

math.AT↗
arXiv2020

On oriented planar trees with three leaves

This elementary note proposes candidates for interesting continuous piecewise-smooth `Riemannian' metrics on the moduli spaces of rooted geodesic trees embedded in the Poincaré disk. A related digression observes the existence of an apparently hitherto unrecognized abelian topological group structure on the real projective line.

math.GT↗
arXiv2020

Complex orientations for THH of some perfectoid fields

This sketch argues that work of Hesselholt on the topological Hochschild homology of $\Cp$ extends, using work of Scholze and others, to define complex orientations for a version of topological Hochschild homology for rings of integers in a natural class of generalized cyclotomic perfectoid fields; and that the resulting spectra provide geometrically interesting targets for analogs of the Chern character, defined for certain integral lifts of the extraordinary $K$-functors of chromatic homotopy theory.

math.AT↗
arXiv2020

On formal groups and geometric quantization

The complex projective spaces, considered as prequantized symplectic manifolds, are roughly to the complete symmetric functions as those projective spaces, regarded as complex-oriented manifolds, are to Newton's power sums.

math.AT↗
arXiv2019

Operations on integral lifts of K(n)

This very rough sketch is a sequel to arXiv:1808.08587; it presents evidence that operations on lifts of the functors K(n) to cohomology theories with values in modules over valuation rings of local number fields, indexed by Lubin-Tate groups of such fields, are extensions of the groups of automorphisms of the indexing group laws, by the exterior algebras on the normal bundle to the orbits of the group laws in the space of lifts.

math.AT↗
arXiv2018

Toward the group completion of the Burau representation

Following Boardman-Vogt, McDuff, Segal, and others, we construct a monoidal topological groupoid or space of finite subsets of the plane, and interpret the Burau representation of knot theory as a topological quantum field theory defined on it. Its determinant or {\bf writhe} is an invertible braided monoidal TQFT which group completes to define a Hopkins-Mahowald model for integral homology as an $E_2$ Thom spectrum. We use these ideas to construct an infinite cyclic (Alexander) cover for the space of finite subsets of $\C$, and we argue that the TQFT defined by Burau is closely related to the SU(2)-valued Wess-Zumino-Witten model for string theory on $\R^3_+$.

math.AT↗