ALBERT

Description: 〈p〉Publication date: 7 November 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Advances in Mathematics, Volume 356〈/p〉 〈p〉Author(s): Florentin Münch, Radosław K. Wojciechowski〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Discrete time random walks on a finite set naturally translate via a one-to-one correspondence to discrete Laplace operators. Typically, Ollivier curvature has been investigated via random walks. We first extend the definition of Ollivier curvature to general weighted graphs and then give a strikingly simple representation of Ollivier curvature using the graph Laplacian. Using the Laplacian as a generator of a continuous time Markov chain, we connect Ollivier curvature with the heat equation which is strongly related to continuous time random walks. In particular, we prove that a lower bound on the Ollivier curvature is equivalent to a certain Lipschitz decay of solutions to the heat equation. This is a discrete analogue to a celebrated Ricci curvature lower bound characterization by Renesse and Sturm. Our representation of Ollivier curvature via the Laplacian allows us to deduce a Laplacian comparison principle by which we prove non-explosion and improved diameter bounds.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 October 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Advances in Mathematics, Volume 355〈/p〉 〈p〉Author(s): Brian C. Hall, Todd Kemp〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉The free multiplicative Brownian motion 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉b〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈/math〉 is the large-〈em〉N〈/em〉 limit of Brownian motion 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈msubsup〉〈mrow〉〈mi〉B〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msubsup〉〈/math〉 on the general linear group 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈mrow〉〈mi mathvariant="normal"〉GL〈/mi〉〈/mrow〉〈mo stretchy="false"〉(〈/mo〉〈mi〉N〈/mi〉〈mo〉;〈/mo〉〈mi mathvariant="double-struck"〉C〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉. We prove that the Brown measure for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉b〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈/math〉—which is an analog of the empirical eigenvalue distribution for matrices—is supported on the closure of a certain domain 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si130.svg"〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉Σ〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈/math〉 in the plane. The domain 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si130.svg"〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉Σ〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈/math〉 was introduced by Biane in the context of the large-〈em〉N〈/em〉 limit of the Segal–Bargmann transform associated to 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈mrow〉〈mi mathvariant="normal"〉GL〈/mi〉〈/mrow〉〈mo stretchy="false"〉(〈/mo〉〈mi〉N〈/mi〉〈mo〉;〈/mo〉〈mi mathvariant="double-struck"〉C〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉.〈/p〉 〈p〉We also consider a two-parameter version, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si364.svg"〉〈msub〉〈mrow〉〈mi〉b〈/mi〉〈/mrow〉〈mrow〉〈mi〉s〈/mi〉〈mo〉,〈/mo〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈/math〉: the large-〈em〉N〈/em〉 limit of a related family of diffusion processes on 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈mrow〉〈mi mathvariant="normal"〉GL〈/mi〉〈/mrow〉〈mo stretchy="false"〉(〈/mo〉〈mi〉N〈/mi〉〈mo〉;〈/mo〉〈mi mathvariant="double-struck"〉C〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 introduced by the second author. We show that the Brown measure of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si364.svg"〉〈msub〉〈mrow〉〈mi〉b〈/mi〉〈/mrow〉〈mrow〉〈mi〉s〈/mi〉〈mo〉,〈/mo〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈/math〉 is supported on the closure of a certain planar domain 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si357.svg"〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉Σ〈/mi〉〈/mrow〉〈mrow〉〈mi〉s〈/mi〉〈mo〉,〈/mo〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈/math〉, generalizing 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si130.svg"〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉Σ〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈/math〉, introduced by Ho.〈/p〉 〈p〉In the process, we introduce a new family of spectral domains related to any operator in a tracial von Neumann algebra: the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si398.svg"〉〈msub〉〈mrow〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msub〉〈/math〉〈em〉-spectrum〈/em〉 for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"〉〈mi〉n〈/mi〉〈mo〉∈〈/mo〉〈mi mathvariant="double-struck"〉N〈/mi〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"〉〈mi〉p〈/mi〉〈mo〉≥〈/mo〉〈mn〉1〈/mn〉〈/math〉, a subset of the ordinary spectrum defined relative to potentially-unbounded inverses. We show that, in general, the support of the Brown measure of an operator is contained in its 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"〉〈msubsup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msubsup〉〈/math〉-spectrum.〈/p〉 〈/div〉

Description: 〈p〉Publication date: 15 October 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Advances in Mathematics, Volume 355〈/p〉 〈p〉Author(s): François Greer〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The Gromov-Witten theory of threefolds admitting a smooth K3 fibration can be solved in terms of the Noether-Lefschetz intersection numbers of the fibration and the reduced invariants of a K3 surface. Toward a generalization of this result to families with singular fibers, we introduce completed Noether-Lefschetz numbers using toroidal compactifications of the period space of elliptic K3 surfaces. As an application, we prove quasi-modularity for some genus 0 partition functions of Weierstrass fibrations over ruled surfaces, and show that they satisfy a holomorphic anomaly equation.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 October 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Advances in Mathematics, Volume 355〈/p〉 〈p〉Author(s): Radu Saghin, Jiagang Yang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉In this paper we obtain local rigidity results for linear Anosov diffeomorphisms in terms of Lyapunov exponents. More specifically, we show that given an irreducible linear hyperbolic automorphism 〈em〉L〈/em〉 with simple real eigenvalues with distinct absolute values, any small perturbation preserving the volume and with the same Lyapunov exponents is smoothly conjugate to 〈em〉L〈/em〉.〈/p〉 〈p〉We also obtain rigidity results for skew products over Anosov diffeomorphisms. Given a volume preserving partially hyperbolic skew product diffeomorphism 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉f〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈/math〉 over an Anosov automorphism of the 2-torus, we show that for any volume preserving perturbation 〈em〉f〈/em〉 of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉f〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈/math〉 with the same average stable and unstable Lyapunov exponents, the center foliation is smooth.〈/p〉 〈/div〉

Description: 〈p〉Publication date: 7 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Advances in Mathematics, Volume 353〈/p〉 〈p〉Author(s): Alan D. Logan〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Fix an equilateral triangle group 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉T〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mo stretchy="false"〉〈〈/mo〉〈mi〉a〈/mi〉〈mo〉,〈/mo〉〈mi〉b〈/mi〉〈mo〉;〈/mo〉〈msup〉〈mrow〉〈mi〉a〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msup〉〈mo〉,〈/mo〉〈msup〉〈mrow〉〈mi〉b〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msup〉〈mo〉,〈/mo〉〈msup〉〈mrow〉〈mo stretchy="false"〉(〈/mo〉〈mi〉a〈/mi〉〈mi〉b〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉〉〈/mo〉〈/math〉 with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mi〉i〈/mi〉〈mo〉≥〈/mo〉〈mn〉6〈/mn〉〈/math〉 arbitrary. Our main result is: for every presentation 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si20.svg"〉〈mi mathvariant="script"〉P〈/mi〉〈/math〉 of every countable group 〈em〉Q〈/em〉 there exists an HNN-extension 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si28.svg"〉〈msub〉〈mrow〉〈mi〉T〈/mi〉〈/mrow〉〈mrow〉〈mi mathvariant="script"〉P〈/mi〉〈/mrow〉〈/msub〉〈/math〉 of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si25.svg"〉〈msub〉〈mrow〉〈mi〉T〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈/math〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.svg"〉〈mi mathvariant="normal"〉Out〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈msub〉〈mrow〉〈mi〉T〈/mi〉〈/mrow〉〈mrow〉〈mi mathvariant="script"〉P〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈mo〉≅〈/mo〉〈mi〉Q〈/mi〉〈/math〉. We construct the HNN-extensions explicitly, and examples are given. The class of groups constructed have nice categorical and residual properties. In order to prove our main result we give a method for recognising malnormal subgroups of small cancellation groups, and we introduce the concept of “malcharacteristic” subgroups.〈/p〉〈/div〉

Description: 〈p〉Publication date: 7 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Advances in Mathematics, Volume 353〈/p〉 〈p〉Author(s): Anton Bernshteyn〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we investigate the extent to which the Lovász Local Lemma (an important tool in probabilistic combinatorics) can be adapted for the measurable setting. In most applications, the Lovász Local Lemma is used to produce a function 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi〉f〈/mi〉〈mo〉:〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉→〈/mo〉〈mi〉Y〈/mi〉〈/math〉 with certain properties, where 〈em〉X〈/em〉 is some underlying combinatorial structure and 〈em〉Y〈/em〉 is a (typically finite) set. Can this function 〈em〉f〈/em〉 be chosen to be Borel or 〈em〉μ〈/em〉-measurable for some probability Borel measure 〈em〉μ〈/em〉 on 〈em〉X〈/em〉 (assuming that 〈em〉X〈/em〉 is a standard Borel space)? In the positive direction, we prove that if the set of constraints put on 〈em〉f〈/em〉 is, in a certain sense, “locally finite,” then there is always a Borel choice for 〈em〉f〈/em〉 that is “〈em〉ε〈/em〉-close” to satisfying these constraints, for any 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mi〉ε〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/math〉. Moreover, if the combinatorial structure on 〈em〉X〈/em〉 is “induced” by the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈mo stretchy="false"〉[〈/mo〉〈mn〉0〈/mn〉〈mo〉;〈/mo〉〈mn〉1〈/mn〉〈mo stretchy="false"〉]〈/mo〉〈/math〉-shift action of a countable group Γ, then, even without any local finiteness assumptions, there is a Borel choice for 〈em〉f〈/em〉 which satisfies the constraints on an invariant conull set (i.e., with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈mi〉ε〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉0〈/mn〉〈/math〉). A direct corollary of our results is an upper bound on the measurable chromatic number of the graph 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1206.svg"〉〈msub〉〈mrow〉〈mi〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msub〉〈/math〉 generated by the shift action of the free group 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"〉〈msub〉〈mrow〉〈mi mathvariant="double-struck"〉F〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msub〉〈/math〉 that is asymptotically tight up to a factor of at most 2 (which answers a question of Lyons and Nazarov). On the other hand, our result for structures induced by measure-preserving group actions is, at least for amenable groups, sharp in the following sense: a probability measure-preserving action of a countably infinite amenable group satisfies the measurable version of the Lovász Local Lemma if and only if it admits a factor map to the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈mo stretchy="false"〉[〈/mo〉〈mn〉0〈/mn〉〈mo〉;〈/mo〉〈mn〉1〈/mn〉〈mo stretchy="false"〉]〈/mo〉〈/math〉-shift action. To prove this, we combine the tools of the Ornstein–Weiss theory of entropy for actions of amenable groups with concepts from computability theory, specifically, Kolmogorov complexity.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 October 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Advances in Mathematics, Volume 354〈/p〉 〈p〉Author(s): Nero Budur, Marcel Rubió〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Over a field of characteristic zero, every deformation problem with cohomology constraints is controlled by a pair consisting of a differential graded Lie algebra together with a module. Unfortunately, these pairs are usually infinite-dimensional. We show that every deformation problem with cohomology constraints is controlled by a typically finite-dimensional 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mo〉∞〈/mo〉〈/mrow〉〈/msub〉〈/math〉 pair. As a first application, we show that for complex algebraic varieties with no weight-zero 1-cohomology classes, the components of the cohomology jump loci of rank one local systems containing the constant sheaf are tori. This imposes restrictions on the fundamental groups. The same holds for links and Milnor fibers.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 October 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Advances in Mathematics, Volume 355〈/p〉 〈p〉Author(s): Gandalf Lechner, Ulrich Pennig, Simon Wood〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉Every unitary involutive solution of the quantum Yang-Baxter equation (R-matrix) defines an extremal character and a representation of the infinite symmetric group 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉S〈/mi〉〈/mrow〉〈mrow〉〈mo〉∞〈/mo〉〈/mrow〉〈/msub〉〈/math〉. We give a complete classification of all such Yang-Baxter characters and determine which extremal characters of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉S〈/mi〉〈/mrow〉〈mrow〉〈mo〉∞〈/mo〉〈/mrow〉〈/msub〉〈/math〉 are of Yang-Baxter form.〈/p〉 〈p〉Calling two involutive R-matrices equivalent if they have the same character and the same dimension, we show that equivalence classes can be parameterized by pairs of Young diagrams, and construct an explicit normal form R-matrix for each class. Using operator-algebraic techniques (subfactors), we prove that two R-matrices are equivalent if and only if they have similar partial traces.〈/p〉 〈p〉Furthermore, we describe the algebraic structure of the set of equivalence classes of all involutive R-matrices, and discuss several families of examples. These include unitary Yang-Baxter representations of the Temperley-Lieb algebra at loop parameter 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mi〉δ〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉2〈/mn〉〈/math〉, which can be completely classified in terms of their trace and dimension.〈/p〉 〈/div〉

Description: 〈p〉Publication date: 15 October 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Advances in Mathematics, Volume 355〈/p〉 〈p〉Author(s): Kateryna Tatarko, Elisabeth M. Werner〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉We prove an analogue of the classical Steiner formula for the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈/math〉 affine surface area of a Minkowski outer parallel body for any real parameter 〈em〉p〈/em〉. We show that the classical Steiner formula and the Steiner formula of Lutwak's dual Brunn Minkowski theory are special cases of this new Steiner formula. This new Steiner formula and its localized versions lead to new curvature measures that have not appeared before in the literature. They have the intrinsic volumes of the classical Brunn Minkowski theory as well as the dual quermassintegrals of the dual Brunn Minkowski theory as special cases.〈/p〉 〈p〉Properties of these new quantities are investigated, a connection to information theory among them. A Steiner formula for the 〈em〉s〈/em〉-th mixed 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈/math〉 affine surface area of a Minkowski outer parallel body for any real parameters 〈em〉p〈/em〉 and 〈em〉s〈/em〉 is also given.〈/p〉 〈/div〉

Description: 〈p〉Publication date: 15 October 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Advances in Mathematics, Volume 355〈/p〉 〈p〉Author(s): Anton Freund〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉A dilator is a particularly uniform transformation 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mi〉X〈/mi〉〈mo stretchy="false"〉↦〈/mo〉〈msub〉〈mrow〉〈mi〉T〈/mi〉〈/mrow〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈/msub〉〈/math〉 of linear orders that preserves well-foundedness. We say that 〈em〉X〈/em〉 is a Bachmann-Howard fixed point of 〈em〉T〈/em〉 if there is an almost order preserving collapsing function 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈mi〉ϑ〈/mi〉〈mo〉:〈/mo〉〈msub〉〈mrow〉〈mi〉T〈/mi〉〈/mrow〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉→〈/mo〉〈mi〉X〈/mi〉〈/math〉 (precise definition to follow). In the present paper we show that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msubsup〉〈mrow〉〈mi mathvariant="normal"〉Π〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msubsup〉〈/math〉-comprehension is equivalent to the assertion that every dilator has a well-founded Bachmann-Howard fixed point. This proves a conjecture of M. Rathjen and A. Montalbán.〈/p〉〈/div〉