We construct a measure (of finite total mass) on the space of metrics of a fixed 1-dimensional complex manifold (of genus greater than or equal to 2) which corresponds in the physics literature to a path integral based on a Gibbs type measure with energy given by the sum of the Liouville action and the Mabuchi K-energy. To the best of our knowledge, this is the first rigorous construction of such an object and this is done by means of probabilistic tools. Both functionals (the Liouville functional and the K-energy functional) play an important role, respectively, in Riemannian geometry (in the case of surfaces) and Kähler geometry. As an output, we obtain a measure whose Weyl anomaly displays the standard Liouville anomaly plus an additional K-energy. Motivations come from theoretical physics where these type of path integrals arise as models for fluctuating metrics on surfaces when coupling certain nonconformal matter fields (mathematically noncritical statistical physics models) to quantum gravity as advocated by A. Bilal, F. Ferrari, S. Klevtsov, and S. Zelditch. On the discrete side, our measure is expected to describe the scaling limit of large planar maps decorated by some noncritical statistical physics models. Interestingly, our computations show that quantum corrections perturb the classical Mabuchi K-energy and produce a quantum Mabuchi K-energy: these type of corrections are reminiscent of the quantum Liouville theory. Our probabilistic construction relies on a variant of the theory of Gaussian multiplicative chaos (GMC) and derivative GMC (DGMC for short). The technical backbone of our construction consists in two estimates on (derivative and standard) GMC which are of independent interest in probability theory. First, we show that these DGMC random variables possess negative exponential moments, and second, we derive optimal small deviations estimates for the GMC associated to a recentered Gaussian free field (GFF).

We prove a freeness theorem for low-rank subgroups of one-relator groups. Let F be a free group, and let $w\in F$ be a nonprimitive element. The primitivity rank of w, $\mathrm{\pi }(w)$, is the smallest rank of a subgroup of F containing w as an imprimitive element. Then any subgroup of the one-relator group $G=F∕\mathrm{⟨}\mathrm{⟨}w\mathrm{⟩}\mathrm{⟩}$ generated by fewer than $\mathrm{\pi }(w)$ elements is free. In particular, if $\mathrm{\pi }(w)>2$, then G does not contain any Baumslag–Solitar groups.

The hypothesis that $\mathrm{\pi }(w)>2$ implies that the presentation complex X of the one-relator group G has negative immersions: if a compact, connected complex Y immerses into X and $\mathit{\chi }(Y)\ge 0$, then Y Nielsen reduces to a graph.

The freeness theorem is a consequence of a dependence theorem for free groups, which implies several classical facts about free and one-relator groups, including Magnus’ Freiheitssatz and theorems of Lyndon, Baumslag, Stallings, and Duncan–Howie.

The dependence theorem strengthens Wise’s w-cycles conjecture, proved independently by the authors and Helfer–Wise, which implies that the one-relator complex X has nonpositive immersions when $\mathrm{\pi }(w)>1$.

We prove a new version of isoperimetric inequality: Given a positive real m, a Banach space B, a closed subset Y of metric space X, and a continuous map $f:Y\to B$ with $f(Y)$ compact

$$\underset{F}{inf}{\mathrm{HC}}_{m+1}(F(X))\le c(m){\mathrm{HC}}_m{(f(Y))}^{\frac{m+1}{m}},$$

where ${\mathrm{HC}}_m$ denotes the m-dimensional Hausdorff content, the infimum is taken over the set of all continuous maps $F:X⟶B$ such that $F(y)=f(y)$ for all $y\in Y$, and $c(m)$ depends only on m. Moreover, one can find F with a nearly minimal ${\mathrm{HC}}_{m+1}$ such that its image lies in the $C(m){\mathrm{HC}}_m{(f(Y))}^{\frac{1}{m}}$-neighborhood of $f(Y)$ with the exception of a subset with zero $(m+1)$-dimensional Hausdorff measure.

The paper also contains a very general coarea inequality for Hausdorff content and its modifications.

As an application we demonstrate an inequality conjectured by Larry Guth that relates the m-dimensional Hausdorff content of a compact metric space with its $(m-1)$-dimensional Urysohn width. We show that this result implies new systolic inequalities that both strengthen the classical Gromov’s systolic inequality for essential Riemannian manifolds and extend this inequality to a wider class of non-simply-connected manifolds.

This paper is concerned with the problem of classifying tuples of natural numbers $a_1,\dots ,a_K$ and $b_1,\dots ,b_L$ such that the ratio of factorials ${\prod }_{i=1}^K(a_{i}n)!∕{\prod }_{j=1}^L(b_{j}n)!$ is an integer for all natural numbers n. A complete solution to this problem is only known in the case $L-K=1$, due to the work of Bober building on an observation of Rodriguez Villegas, which relies on the Beukers–Heckman classification of algebraic hypergeometric functions. We provide an alternative proof of this result, which also makes progress on the general problem when $L-K>1$.

The Benson–Solomon systems comprise an infinite ascending family of simple exotic fusion systems at the prime 2. The results we prove give significant additional evidence that these are the only simple exotic 2-fusion systems, as conjectured by Solomon. We consider a saturated fusion system $\mathcal{F}$ having an involution centralizer with a component $\mathcal{C}$ isomorphic to a Benson–Solomon fusion system, and we show under rather general hypotheses that $\mathcal{F}$ cannot be simple. Furthermore, we prove that if $\mathcal{F}$ is almost simple with these properties, then $\mathcal{F}$ is isomorphic to the next larger Benson–Solomon system extended by a group of field automorphisms. Our results are situated within Aschbacher’s program to provide a new proof of a major part of the classification of finite simple groups via fusion systems. One of the most important steps in this program is a proof of Walter’s Theorem for fusion systems, and our first result is specifically tailored for use in the proof of that step. We then apply Walter’s Theorem to treat the general Benson–Solomon component problem under the assumption that each component of an involution centralizer in $\mathcal{F}$ is on the list of currently known quasisimple 2-fusion systems.

We prove that the Floer complex associated with a convex Hamiltonian function on ${\mathbb{R}}^{2n}$ is isomorphic to the Morse complex of Clarke’s dual action functional associated with the Fenchel-dual Hamiltonian. This isomorphism preserves the action filtrations. As a corollary, we obtain that the symplectic capacity from the symplectic homology of a convex domain with smooth boundary coincides with the minimal action of closed characteristics on its boundary.