Local conditions for global convergence of gradient flows and proximal point sequences in metric spaces - Dello Schiavo, Lorenzo; Maas, Jan; Pedrotti, Francesco (Transactions of the American Mathematical Society)
This paper deals with local criteria for the convergence to a global minimiser for gradient flow trajectories and their discretisations. To obtain quantitative estimates on the speed of convergence, we consider variations on the classical Kurdyka–Łojasiewicz inequality for a large class of parameter functions. Our assumptions are given in terms of the initial data, without any reference to an equilibrium point. The main results are convergence statements for gradient flow curves and proximal point sequences to a global minimiser, together with sharp quantitative estimates on the speed of convergence. These convergence results apply in the general setting of lower semicontinuous functionals on complete metric spaces, generalising recent results for smooth functionals on Rn. While the non-smooth setting covers very general spaces, it is also useful for (non)-smooth functionals on Rn. .

Extension complexity of low-dimensional polytopes - Kwan, Matthew Alan; Sauermann, Lisa; Zhao, Yufei (Transactions of the American Mathematical Society)
Sometimes, it is possible to represent a complicated polytope as a projection of a much simpler polytope. To quantify this phenomenon, the extension complexity of a polytope P is defined to be the minimum number of facets of a (possibly higher-dimensional) polytope from which P can be obtained as a (linear) projection. This notion is motivated by its relevance to combinatorial optimisation, and has been studied intensively for various specific polytopes associated with important optimisation problems. In this paper we study extension complexity as a parameter of general polytopes, more specifically considering various families of low-dimensional polytopes. First, we prove that for a fixed dimension d, the extension complexity of a random d-dimensional polytope (obtained as the convex hull of random points in a ball or on a sphere) is typically on the order of the square root of its number of vertices. Second, we prove that any cyclic n-vertex polygon (whose vertices lie on a circle) has extension complexity at most 24√n. This bound is tight up to the constant factor 24. Finally, we show that there exists an no(1)-dimensional polytope with at most n vertices and extension complexity n1−o(1). Our theorems are proved with a range of different techniques, which we hope will be of further interest.

Integral points on the congruent number curve - Chan, Yik Tung (Transactions of the American Mathematical Society)
Abstract. We study integral points on the quadratic twists ED : y2 = x3 − D2x of the congruent number curve. We give upper bounds on the number of integral points in each coset of 2ED(Q) in ED(Q) and show that their total is (3.8)rank ED(Q). We further show that the average number of non-torsion integral points in this family is bounded above by 2. As an application we also deduce from our upper bounds that the system of simultaneous Pell equations aX2 − bY 2 = d, bY 2 − cZ2 = d for pairwise coprime positive integers a, b, c, d, has at most (3.6)ω(abcd) integer solutions.

Isometric study of Wasserstein spaces - the real line - Geher, Gyorgy Pal; Titkos, Tamas; Virosztek, Daniel (Transactions of the American Mathematical Society)
Recently Kloeckner described the structure of the isometry group of the quadratic Wasserstein space W_2(R^n). It turned out that the case of the real line is exceptional in the sense that there exists an exotic isometry flow. Following this line of investigation, we compute Isom(W_p(R)), the isometry group of the Wasserstein space W_p(R) for all p \in [1,\infty) \setminus {2}. We show that W_2(R) is also exceptional regarding the parameter p: W_p(R) is isometrically rigid if and only if p is not equal to 2. Regarding the underlying space, we prove that the exceptionality of p = 2 disappears if we replace R by the compact interval [0,1]. Surprisingly, in that case, W_p([0,1]) is isometrically rigid if and only if p is not equal to 1. Moreover, W_1([0,1]) admits isometries that split mass, and Isom(W_1([0,1])) cannot be embedded into Isom(W_1(R)).
Keywords: Wasserstein space; isometric embeddings; isometric rigidity; exotic isometry flow

Proof of a conjecture on induced subgraphs of Ramsey graphs - Kwan, Matthew Alan; Sudakov, Benny (Transactions of the American Mathematical Society)
An n-vertex graph is called C-Ramsey if it has no clique or independent set of size C log n. All known constructions of Ramsey graphs involve randomness in an essential way, and there is an ongoing line of research towards showing that in fact all Ramsey graphs must obey certain “richness” properties characteristic of random graphs. More than 25 years ago, Erdős, Faudree and Sós conjectured that in any C-Ramsey graph there are Ω(n^5/2) induced subgraphs, no pair of which have the same numbers of vertices and edges. Improving on earlier results of Alon, Balogh, Kostochka and Samotij, in this paper we prove this conjecture.

Sieving rational points on varieties - Browning, Timothy D; Loughran, Daniel (Transactions of the American Mathematical Society)
An upper bound sieve for rational points on suitable varieties isdeveloped, together with applications tocounting rational points in thin sets,to local solubility in families, and to the notion of “friable” rational pointswith respect to divisors. In the special case of quadrics, sharper estimates areobtained by developing a version of the Selberg sieve for rational points.

Scattering for the cubic Klein-Gordon equation in two space dimensions - Killip, Rowan; Stovall, Betsy; Visan, Monica (Transactions of the American Mathematical Society)
We consider both the defocusing and focusing cubic nonlinear Klein–Gordon equations utt − Δu + u ± u3 =0 in two space dimensions for real-valued initial data u(0) ∈ H1/x and ut(0) ∈ L2/x. We show that in the defocusing case, solutions are global and have finite global L4/t,x spacetime bounds. In the focusing case, we characterize the dichotomy between this behaviour and blowup for initial data with energy less than that of the ground state. These results rely on analogous statements for the two-dimensional cubic nonlinear Schr¨odinger equation, which are known in the defocusing case and for spherically-symmetric initial data in the focusing case. Thus, our results are mostly unconditional. It was previously shown by Nakanishi that spacetime bounds for Klein– Gordon equations imply the same for nonlinear Schr¨odinger equations.

The defocusing energy-supercritical nonlinear wave equation in three space dimensions - Killip, Rowan; Visan, Monica (Transactions of the American Mathematical Society)
We consider the defocusing nonlinear wave equation (mathematical formular) in the energy-supercritical regime p > 4. For even values of the power p, we show that blowup (or failure to scatter) must be accompanied by blowup of the critical Sobolev norm. An equivalent formulation is that solutions with bounded critical Sobolev norm are global and scatter. The impetus to consider this problem comes from recent work of Kenig and Merle who treated the case of spherically-symmetric solutions.