arXiv subjects

Carl McTague

Publications and source records attributed to Carl McTague.

7 recordsLinked to original sources

arXiv2018

On the Greatest Common Divisor of Binomial Coefficients ${n \choose q}, {n \choose 2q}, {n \choose 3q}, \dots$

Every binomial coefficient aficionado knows that the greatest common divisor of the binomial coefficients $\binom n1,\binom n2,\dots,\binom n{n-1}$ equals $p$ if $n=p^i$ for some $i>0$ and equals 1 otherwise. It is less well known that the greatest common divisor of the binomial coefficients $\binom{2n}2,\binom{2n}4,\dots,\binom{2n}{2n-2}$ equals (a certain power of 2 times) the product of all odd primes $p$ such that $2n=p^i+p^j$ for some $0\le i\le j$. This note gives a concise proof of a tidy generalization of these facts.

math.CO↗

arXiv2017

tmf Is Not a Ring Spectrum Quotient of String Bordism

This paper shows that $\mathrm{tmf}[1/6]$ is not a ring spectrum quotient of $\mathrm{MO}\langle8\rangle[1/6]$. In fact, for any prime $p>3$ and any sequence $X$ of homogeneous elements of $π_*\mathrm{MO}\langle8\rangle$, the $π_*\mathrm{MO}\langle8\rangle$-module $$π_*\big(\mathrm{MO}\langle8\rangle_{(p)}/X\big)$$ is not (even abstractly) isomorphic to $π_*\mathrm{tmf}_{(p)}$. It does so by showing that, for any commutative ring spectrum $R$ and any sequence $X$ of homogeneous elements of $π_*(R)$, there is an isomorphism of graded $\mathbf{Q}$-vector spaces $$π_*(R/X)\otimes\mathbf{Q} \cong \mathrm{H}_*(\mathrm{Tot}(\mathrm{K}(X)))\otimes\mathbf{Q},$$ where the right-hand side is the rational homology of the (total) Koszul complex of $X$, which is strictly bigger than $π_*(R)/(X)\otimes\mathbf{Q}$ unless $X$ is a $π_*(R)\otimes\mathbf{Q}$-quasi-regular sequence. The result then follows from the fact that the kernel of the $p$-local Witten genus cannot be generated by a $π_*\mathrm{MO}\langle8\rangle\otimes\mathbf{Q}$-quasi-regular sequence.

math.AT↗

arXiv2015

A New Approach to Euler Calculus for Continuous Integrands

Euler calculus is based on integrating simple functions with respect to the Euler characteristic. This paper makes the case for extending Euler calculus to continuous integrands by integrating with respect to (Gaussian) curvature. This requires a metric but is nevertheless defined within any O-minimal theory. It satisfies a Fubini theorem and extends to a functor. Euler calculus is the "adiabatic limit" of this "curvature calculus". All this suggests new applications of differential geometry to data analysis.

math.DG↗

arXiv2015

Shrinking the Fibers of a Submersion Splits the Riemann Tensor

This paper uses Karcher's formulation [Kar99] of the O'Neill tensors [O'N66,Gra67] to derive a concise formula for the family $Ω^ε$ of curvature forms obtained by shrinking the fibers of a submersion $π:M\to B$ of semi-Riemannian manifolds by a factor of $1-ε$. The formula clearly shows that as $ε$ approaches 1, $Ω^ε$ approaches the sum of the vertical curvature form $Ω^\mathrm{V}$ and the pullback $π^*Ω^B$ of the curvature form of $B$. The Gauss-Bonnet integrand $\mathrm{Pf}(Ω^ε)$ therefore approaches the wedge $\mathrm{Pf}(Ω^\mathrm{V})\wedgeπ^*\mathrm{Pf}(Ω^B)$. So if $π$ has compact fiber $F$, the pushforward $π_*\mathrm{Pf}(Ω^ε)$ approaches $χ(F)\cdot\mathrm{Pf}(Ω^B)$.

math.DG↗

arXiv2014

The Cayley plane and String bordism

This paper shows that, away from 6, the kernel of the Witten genus is precisely the ideal consisting of (bordism classes of) Cayley plane bundles with connected structure group, but only after restricting the Witten genus to string bordism. It does so by showing that the divisibility properties of Cayley plane bundle characteristic numbers arising in Borel-Hirzebruch Lie-group-theoretic calculations correspond precisely to the divisibility properties arising in the Hovey-Ravenel-Wilson BP-Hopf-ring-theoretic calculation of string bordism at primes >3.

math.AT↗

arXiv2011

Stiefel-Whitney Numbers for Singular Varieties

This paper determines which Stiefel-Whitney numbers can be defined for singular varieties compatibly with small resolutions. First an upper bound is found by identifying the F_2-vector space of Stiefel-Whitney numbers invariant under classical flops, equivalently by computing the quotient of the unoriented bordism ring by the total spaces of RP^3 bundles. These Stiefel-Whitney numbers are then defined for any real projective normal Gorenstein variety and shown to be compatible with small resolutions whenever they exist. In light of Totaro's result [Tot00] equating the complex elliptic genus with complex bordism modulo flops, equivalently complex bordism modulo the total spaces of twisted(CP^3) bundles, these findings can be seen as hinting at a new elliptic genus, one for unoriented manifolds.

math.AT↗

arXiv2010

The Cayley Plane and the Witten Genus

This paper defines a new genus, the Cayley plane genus. By definition it is the universal multiplicative genus for oriented Cayley plane bundles. The main result (Theorem 2) is that it factors (tensor Q) through the product of the Ochanine elliptic genus and the Witten genus---revealing a synergy between these two genera---and that its image is the homogeneous coordinate ring Q[Kum,HP^2,HP^3,CaP^2]/(CaP^3).(HP^3,CaP^2-(HP^2)^2) of the union of the curve of Ochanine elliptic genera and the surface of Witten genera meeting with multiplicity 2 at the point CaP^2=HP^3=HP^2=0 corresponding to the Â-genus. This all remains true if the word "oriented" is replaced with the word "spin" (Theorem 3). This paper also characterizes the Witten genus (tensor Q) as the universal genus vanishing on total spaces of Cayley plane bundles (Theorem 1, a result proved independently by Dessai in [Des09].)

math.AT↗