Journal of the London Mathematical Society-Second Series最新文献

Many articles have recently been devoted to Mahler equations, partly because of their links with other branches of mathematics such as automata theory. Hahn series (a generalization of the Puiseux series allowing arbitrary exponents of the indeterminate as long as the set that supports them is well ordered) play a central role in the theory of Mahler equations. In this paper, we address the following fundamental question: is there an algorithm to calculate the Hahn series solutions of a given linear Mahler equation? What makes this question interesting is the fact that the Hahn series appearing in this context can have complicated supports with infinitely many accumulation points. Our (positive) answer to the above question involves among other things the construction of a computable well-ordered receptacle for the supports of the potential Hahn series solutions.

This is a corrigendum to the paper Transcendental Brauer groups of products of CM elliptic curves, published in (J. Lond. Math. Soc. 93 (2016), 397–419). In this note, we point out a mistake affecting the statements of Theorems 1.3, 5.2, 5.3 and Proposition 5.1 in op. cit., and provide a correction.

We show that two semi-infinite positive temperature polymers coalesce on the scale predicted by KPZ (Kardar–Parisi–Zhang) universality. The two polymer paths have the same asymptotic direction and evolve in the same environment, independently until coalescence. If they start at distance $�k$ apart, their coalescence occurs on the scale $�k^{�3�/�2�}$. It follows that the total variation distance of two semi-infinite polymer measures decays on this same scale. Our results are upper and lower bounds on probabilities and expectations that match, up to constant factors and occasional logarithmic corrections. Our proofs are done in the context of the solvable inverse-gamma polymer model, but without appeal to integrable probability. With minor modifications, our proofs give also bounds on transversal fluctuations of the polymer path. As the free energy of a directed polymer is a discretization of a stochastically forced viscous Hamilton–Jacobi equation, our results suggest that the hyperbolicity phenomenon of such equations obeys the KPZ exponent.

This is the second in a series of two papers that develops a theory of relatively Anosov representations using the original “contracting flow on a bundle” definition of Anosov representations introduced by Labourie and Guichard–Wienhard. In this paper, we focus on building families of examples.

We study the $�n$-point differentials corresponding to Kadomtsev–Petviashvili (KP) tau functions of hypergeometric type (also known as Orlov–Scherbin partition functions), with an emphasis on their $�{\hslash }^2$-deformations and expansions. Under the naturally required analytic assumptions, we prove certain higher loop equations that, in particular, contain the standard linear and quadratic loop equations, and thus imply the blobbed topological recursion. We also distinguish two large families of the Orlov–Scherbin partition functions that do satisfy the natural analytic assumptions, and for these families, we prove in addition the so-called projection property and thus the full statement of the Chekhov–Eynard–Orantin topological recursion. A particular feature of our argument is that it clarifies completely the role of $�{\hslash }^2$-deformations of the Orlov–Scherbin parameters for the partition functions, whose necessity was known from a variety of earlier obtained results in this direction but never properly understood in the context of topological recursion. As special cases of the results of this paper, one recovers new and uniform proofs of the topological recursion to all previously studied cases of enumerative problems related to weighted double Hurwitz numbers. By virtue of topological recursion and the Grothendieck–Riemann–Roch formula, this, in turn, gives new and uniform proofs of almost all Ekedahl–Lando–Shapiro–Vainshtein (ELSV)-type formulas discussed in the literature.

We prove that all rational slopes are characterizing for the knot $�5_2$, except possibly for positive integers. Along the way, we classify the Dehn surgeries on knots in $�S^3$ that produce the Brieskorn sphere $��\Sigma �(�2�,�3�,�11�)�$, and we study knots on which large integral surgeries are almost L-spaces.

In this paper, we show the existence of uniform rational lc polytopes for foliations with functional boundaries in dimension $��⩽�3�$. As an application, we prove the global ACC for foliated threefolds with arbitrary DCC coefficients. We also provide applications on the accumulation points of lc thresholds of foliations in dimension $��⩽�3�$.

We consider the mixed local and nonlocal functionals with nonstandard growth

We consider the generalized parabolic Cahn–Hilliard equation