On finitely generated profinite groups, II: products in quasisimple groups

We prove two results. (1) There is an absolute constant $D$ such that for any finite quasisimple group $S$, given $2D$ arbitrary automorphisms of $S$, every element of $S$ is equal to a product of $D$ ‘twisted commutators’ defined by the given automorphisms.

(2) Given a natural number $q$, there exist $C=C(q)$ and $M=M(q)$ such that: if $S$ is a finite quasisimple group with $\left| S/\mathrm{Z}(S)\right| > C$, $\beta_{j}$ $ (j=1,\ldots,M)$ are any automorphisms of $S$, and $q_{j}$ $ (j=1,\ldots,M)$ are any divisors of $q$, then there exist inner automorphisms $\alpha_{j}$ of $S$ such that $S=\prod_{1}^{M}[S,(\alpha _{j}\beta_{j})^{q_{j}}]$.

These results, which rely on the classification of finite simple groups, are needed to complete the proofs of the main theorems of Part I.

Pages 239-273 by Nikolay Nikolov, Dan Segal | From volume 165-1

On finitely generated profinite groups, I: strong completeness and uniform bounds

We prove that in every finitely generated profinite group, every subgroup of finite index is open; this implies that the topology on such groups is determined by the algebraic structure. This is deduced from the main result about finite groups: let $w$ be a `locally finite’ group word and $d\in\mathbb{N}$. Then there exists $f=f(w,d)$ such that in every $d$-generator finite group $G$, every element of the verbal subgroup $w(G)$ is equal to a product of $f$ $w$-values.

An analogous theorem is proved for commutators; this implies that in every finitely generated profinite group, each term of the lower central series is closed.

The proofs rely on some properties of the finite simple groups, to be established in Part II.

Pages 171-238 by Nikolay Nikolov, Dan Segal | From volume 165-1

Quantum Riemann–Roch, Lefschetz and Serre

Given a holomorphic vector bundle $E$ over a compact Kähler manifold $X$, one defines twisted Gromov-Witten invariants of $X$ to be intersection numbers in moduli spaces of stable maps $f:\Sigma \to X$ with the cap product of the virtual fundamental class and a chosen multiplicative invertible characteristic class of the virtual vector bundle $H^0(\Sigma,f^* E) \ominus H^1(\Sigma,f^* E)$. Using the formalism of quantized quadratic Hamiltonians [25], we express the descendant potential for the twisted theory in terms of that for $X$. This result (Theorem $1$) is a consequence of Mumford’s Grothendieck-Riemann-Roch theorem applied to the universal family over the moduli space of stable maps. It determines all twisted Gromov-Witten invariants, of all genera, in terms of untwisted invariants.

When $E$ is concave and the $\mathbb{C}^\times$-equivariant inverse Euler class is chosen as the characteristic class, the twisted invariants of $X$ give Gromov-Witten invariants of the total space of $E$. “Nonlinear Serre duality” [21], [23] expresses Gromov-Witten invariants of $E$ in terms of those of the super-manifold $\Pi E$: it relates Gromov-Witten invariants of $X$ twisted by the inverse Euler class and $E$ to Gromov-Witten invariants of $X$ twisted by the Euler class and $E^*$. We derive from Theorem 1 nonlinear Serre duality in a very general form (Corollary 2).

When the bundle $E$ is convex and a submanifold $Y\subset X$ is defined by a global section of $E$, the genus-zero Gromov-Witten invariants of $\Pi E$ coincide with those of $Y$. We establish a “quantum Lefschetz hyperplane section principle” (Theorem $2$) expressing genus-zero Gromov-Witten invariants of a complete intersection $Y$ in terms of those of $X$. This extends earlier results [4], [9], [18], [29], [33] and yields most of the known mirror formulas for toric complete intersections.

Pages 15-53 by Thomas Coates, Alexander Givental | From volume 165-1

Isoparametric hypersurfaces with four principal curvatures

Let $M$ be an isoparametric hypersurface in the sphere $S^n$ with four distinct principal curvatures. Münzner showed that the four principal curvatures can have at most two distinct multiplicities $m_1, m_2$, and Stolz showed that the pair $(m_1,m_2)$ must either be $(2,2)$, $(4,5)$, or be equal to the multiplicities of an isoparametric hypersurface of FKM-type, constructed by Ferus, Karcher and Münzner from orthogonal representations of Clifford algebras. In this paper, we prove that if the multiplicities satisfy $m_2 \geq 2m_1 – 1$, then the isoparametric hypersurface $M$ must be of FKM-type. Together with known results of Takagi for the case $m_1 = 1$, and Ozeki and Takeuchi for $m_1 = 2$, this handles all possible pairs of multiplicities except for four cases, for which the classification problem remains open.

Pages 1-76 by Thomas E. Cecil, Quo-Shin Chi, Gary R. Jensen | From volume 166-1

Proofs without syntax

Proofs are traditionally syntactic, inductively generated objects. This paper presents an abstract mathematical formulation of propositional calculus (propositional logic) in which proofs are combinatorial (graph-theoretic), rather than syntactic. It defines a combinatorial proof of a proposition $\phi$ as a graph homomorphism $h:C\to G(\phi)$, where $G(\phi)$ is a graph associated with $\phi$ and $C$ is a coloured graph. The main theorem is soundness and completeness: $\,\phi$ is true if and only if there exists a combinatorial proof $h:C\to G(\phi)$.

Pages 1065-1076 by Dominic J. D. Hughes | From volume 164-3

The number of extensions of a number field with fixed degree and bounded discriminant

We give an upper bound on the number of extensions of a fixed number field of prescribed degree and discriminant $\leq X$; these bounds improve on work of Schmidt. We also prove various related results, such as lower bounds for the number of extensions and upper bounds for Galois extensions.

Pages 723-741 by Jordan S. Ellenberg, Akshay Venkatesh | From volume 163-2

Schubert induction

We describe a Schubert induction theorem, a tool for analyzing intersections on a Grassmannian over an arbitrary base ring. The key ingredient in the proof is the Geometric Littlewood-Richardson rule of [V2].

As applications, we show that all Schubert problems for all Grassmannians are enumerative over the real numbers, and sufficiently large finite fields. We prove a generic smoothness theorem as a substitute for the Kleiman-Bertini theorem in positive characteristic. We compute the monodromy groups of many Schubert problems, and give some surprising examples where the monodromy group is much smaller than the full symmetric group.

Pages 489-512 by Ravi Vakil | From volume 164-2

A geometric Littlewood–Richardson rule

We describe a geometric Littlewood-Richardson rule, interpreted as deforming the intersection of two Schubert varieties into the union of Schubert varieties. There are no restrictions on the base field, and all multiplicities arising are $1$; this is important for applications. This rule should be seen as a generalization of Pieri’s rule to arbitrary Schubert classes, by way of explicit homotopies. It has straightforward bijections to other Littlewood-Richardson rules, such as tableaux, and Knutson and Tao’s puzzles. This gives the first geometric proof and interpretation of the Littlewood-Richardson rule. Geometric consequences are described here and in [V2], [KV1], [KV2], [V3]. For example, the rule also has an interpretation in $K$-theory, suggested by Buch, which gives an extension of puzzles to $K$-theory.

Pages 371-422 by Ravi Vakil | From volume 164-2

A local regularity theorem for mean curvature flow

This paper proves curvature bounds for mean curvature flows and other related flows in regions of spacetime where the Gaussian densities are close to $1$.

Pages 1487-1519 by Brian White | From volume 161-3

Universal bounds for hyperbolic Dehn surgery

This paper gives a quantitative version of Thurston’s hyperbolic Dehn surgery theorem. Applications include the first universal bounds on the number of nonhyperbolic Dehn fillings on a cusped hyperbolic 3-manifold, and estimates on the changes in volume and core geodesic length during hyperbolic Dehn filling. The proofs involve the construction of a family of hyperbolic cone-manifold structures, using infinitesimal harmonic deformations and analysis of geometric limits.

Pages 367-421 by Craig Hodgson, Steven P. Kerckhoff | From volume 162-1

The symplectic sum formula for Gromov–Witten invariants

In the symplectic category there is a ‘connect sum’ operation that glues symplectic manifolds by identifying neighborhoods of embedded codimension two submanifolds. This paper establishes a formula for the Gromov-Witten invariants of a symplectic sum $Z=X\# Y$ in terms of the relative GW invariants of $X$ and $Y$. Several applications to enumerative geometry are given.

Pages 935-1025 by Eleny-Nicoleta Ionel, Thomas H. Parker | From volume 159-3

Corrigendum to “Classification of prime 3-manifolds with Yamabe invariant greater than $\mathbb{R}P^3$”

Pages 887-887 by Hubert L. Bray, André Neves | From volume 159-2

Cover times for Brownian motion and random walks in two dimensions

Let $\mathcal{T}(x,\varepsilon)$ denote the first hitting time of the disc of radius $\varepsilon$ centered at $x$ for Brownian motion on the two dimensional torus $\mathbb{T}^2$. We prove that $\sup_{x\in \mathbb{T}^2} \mathcal{T}(x,\varepsilon)/|\log \varepsilon|^2 \to 2/\pi$ as $\varepsilon \rightarrow 0$. The same applies to Brownian motion on any smooth, compact connected, two-dimensional, Riemannian manifold with unit area and no boundary. As a consequence, we prove a conjecture, due to Aldous (1989), that the number of steps it takes a simple random walk to cover all points of the lattice torus $\mathbb{Z}_n^2$ is asymptotic to $4n^2(\log n)^2/\pi$. Determining these asymptotics is an essential step toward analyzing the fractal structure of the set of uncovered sites before coverage is complete; so far, this structure was only studied nonrigorously in the physics literature. We also establish a conjecture, due to Kesten and Révész, that describes the asymptotics for the number of steps needed by simple random walk in $\mathbb{Z}^2$ to cover the disc of radius $n$.

Pages 433-464 by Amir Dembo, Yuval Peres, Jay Rosen, Ofer Zeitouni | From volume 160-2

The main conjecture for ${\rm CM}$ elliptic curves at supersingular primes

At a prime of ordinary reduction, the Iwasawa “main conjecture” for elliptic curves relates a Selmer group to a $p$-adic $L$-function. In the supersingular case, the statement of the main conjecture is more complicated as neither the Selmer group nor the $p$-adic $L$-function is well-behaved. Recently Kobayashi discovered an equivalent formulation of the main conjecture at supersingular primes that is similar in structure to the ordinary case. Namely, Kobayashi’s conjecture relates modified Selmer groups, which he defined, with modified $p$-adic $L$-functions defined by the first author. In this paper we prove Kobayashi’s conjecture for elliptic curves with complex multiplication.

Pages 447-464 by Robert Pollack, Karl Rubin | From volume 159-1

Classification of prime 3-manifolds with $\sigma$-invariant greater than $\Bbb{RP}^3$

In this paper we compute the $\sigma$-invariants (sometimes also called the smooth Yamabe invariants) of $\mathbb{RP}^3$ and $\mathbb{RP}^2 \times S^1$ (which are equal) and show that the only prime $3$-manifolds with larger $\sigma$-invariants are $S^3$, $S^2 \times S^1$, and $S^2 \tilde\times S^1$ (the nonorientable $S^2$ bundle over $S^1$). More generally, we show that any $3$-manifold with $\sigma$-invariant greater than $\mathbb{RP}^3$ is either $S^3$, a connect sum with an $S^2$ bundle over $S^1$, or has more than one nonorientable prime component. A corollary is the Poincaré conjecture for $3$-manifolds with $\sigma$-invariant greater than $\mathbb{RP}^3$.

Surprisingly these results follow from the same inverse mean curvature flow techniques which were used by Huisken and Ilmanen in [7] to prove the Riemannian Penrose Inequality for a black hole in a spacetime. Richard Schoen made the observation [18] that since the constant curvature metric (which is extremal for the Yamabe problem) on $\mathbb{RP}^3$ is in the same conformal class as the Schwarzschild metric (which is extremal for the Penrose inequality) on $\mathbb{RP}^3$ minus a point, there might be a connection between the two problems. The authors found a strong connection via inverse mean curvature flow.

Pages 407-424 by Hubert L. Bray, André Neves | From volume 159-1

$(\log t)^2/3$ law of the two dimensional asymmetric simple exclusion process

We prove that the diffusion coefficient for the two dimensional asymmetric simple exclusion process with nearest-neighbor-jumps diverges as $(\log t)^{2/3}$ to the leading order. The method applies to nearest and non-nearest neighbor asymmetric simple exclusion processes.

Pages 377-405 by Horng-Tzer Yau | From volume 159-1

Real polynomial diffeomorphisms with maximal entropy: Tangencies

Pages 1-26 by Eric Bedford, John Smillie | From volume 160-1

Global existence and convergence for a higher order flow in conformal geometry

Pages 323-343 by Simon Brendle | From volume 158-1

Relative Gromov-Witten invariants

We define relative Gromov-Witten invariants of a symplectic manifold relative to a codimension-two symplectic submanifold. These invariants are the key ingredients in the symplectic sum formula of [IP4]. The main step is the construction of a compact space of ‘$V$-stable’ maps. Simple special cases include the Hurwitz numbers for algebraic curves and the enumerative invariants of Caporaso and Harris.

Pages 45-96 by Eleny-Nicoleta Ionel, Thomas H. Parker | From volume 157-1