ALBERT

Description: Fix positive integers $p$ and $q$ with $2 \leq q \leq {p \choose 2}$ . An edge coloring of the complete graph $K_n$ is said to be a $(p, q)$ -coloring if every $K_p$ receives at least $q$ different colors. The function $f(n, p, q)$ is the minimum number of colors that are needed for $K_n$ to have a $(p,q)$ -coloring. This function was introduced about 40 years ago, but Erdős and Gyárfás were the first to study the function in a systematic way. They proved that $f(n, p, p)$ is polynomial in $n$ and asked to determine the maximum $q$ , depending on $p$ , for which $f(n,p,q)$ is subpolynomial in $n$ . We prove that the answer is $p-1$ .

Description: In this paper, we study the large-scale structure of dense regular graphs. This involves the notion of robust expansion, a recent concept which has already been used successfully to settle several longstanding problems. Roughly speaking, a graph is robustly expanding if it still expands after the deletion of a small fraction of its vertices and edges. Our main result allows us to harness the useful consequences of robust expansion even if the graph itself is not a robust expander. It states that every dense regular graph can be partitioned into ‘robust components’, each of which is a robust expander or a bipartite robust expander. We apply our result to obtain (amongst others) the following. We prove that whenever ${\varepsilon } 〉 0$ , every sufficiently large $3$ -connected $D$ -regular graph on $n$ vertices with $D \geq (\tfrac {1}{4} + {\varepsilon })n$ is Hamiltonian. This asymptotically confirms the only remaining case of a conjecture raised independently by Bollobás and Häggkvist in the 1970s. We prove an asymptotically best possible result on the circumference of dense regular graphs of given connectivity. The $2$ -connected case of this was conjectured by Bondy and proved by Wei.

Description: We discuss the representation theory of the bialgebra ${\underline {\hbox {end}}}(A)$ introduced by Manin. As a side result, we give a new proof that Koszul algebras are distributive and furthermore we show that some well-known $N$ -Koszul algebras are also distributive.

Description: It is well known due to Jarník that the set ${\bf{Bad}}_{\mathbb R} ^1$ of badly approximable numbers is of Hausdorff dimension 1. If ${\bf{Bad}}_{\mathbb R} ^1(c)$ denotes the subset of $x\in {\bf{Bad}}_{\mathbb R} ^1$ for which the approximation constant $c(x) \geq c$ , then Jarník was in fact more precise and gave non-trivial lower and upper bounds on the Hausdorff dimension of ${\bf{Bad}}_{\mathbb R} ^1(c)$ in terms of the parameter $c 〉 0$ . Our aim is to determine simple conditions on a framework which allow one to extend ’Jarník's inequality’ to further examples. For many dynamical examples, these extensions are related to the Hausdorff dimension of the set of orbits that avoid a suitable given neighborhood of an obstacle. Among the applications, we discuss the set ${\bf{Bad}}_{{\mathbb R} ^n}^{\bar r}$ of badly approximable vectors in ${\mathbb R} ^n$ with weights $\bar r$ , the set of orbits in the Bernoulli shift that avoid a neighborhood of a periodic orbit, the set of geodesics in the hyperbolic space ${\mathbb H} ^n$ that avoid a suitable collection of convex sets, and the set of orbits of a toral endomorphism that avoid neighborhoods of a separated set.

Description: Let $X= \mathbb {P}^1 \setminus {\{ }0,1,\infty {\} }$ , and let $S$ denote a finite set of prime numbers. In an article of 2005, Kim gave a new proof of Siegel's theorem for $X$ : the set $X(\mathbb {Z}[S^{-1}])$ of $S$ -integral points of $X$ is finite. The proof relies on a ‘nonabelian’ version of the classical Chabauty method. At its heart is a modular interpretation of unipotent $p$ -adic Hodge theory, given by a tower of morphisms $h_n$ between certain $\mathbb {Q}_p$ -varieties. We set out to obtain a better understanding of $h_2$ . Its mysterious piece is a polynomial in $2|S|$ variables. Our main theorem states that this polynomial is quadratic, and gives a procedure for writing its coefficients in terms of $p$ -adic logarithms and dilogarithms.

Description: Generalizing the super duality formalism for finite-dimensional Lie superalgebras of type $ABCD$ , we establish an equivalence between parabolic Bernstein-Gelfand-Gelfand (BGG) categories of a Kac–Moody Lie superalgebra and a Kac–Moody Lie algebra. The characters for a large family of irreducible highest weight modules over a symmetrizable Kac–Moody Lie superalgebra are then given in terms of Kazhdan–Lusztig polynomials for the first time. We formulate a notion of integrable modules over a symmetrizable Kac–Moody Lie superalgebra via super duality, and show that these integrable modules form a semisimple tensor subcategory, whose Littlewood–Richardson tensor product multiplicities coincide with those in the Kac–Moody Lie algebra setting.

Description: We give necessary and sufficient conditions for a closed connected co-orientable contact $3$ -manifold $(M,\xi )$ to be a standard lens space based on assumptions on the Reeb flow associated to a defining contact form. Our methods also provide rational global surfaces of section for non-degenerate Reeb flows on $(L(p,q),\xi _{\rm std})$ with prescribed binding orbits.

Description: The goal of this paper is to provide mathematically rigorous tools for modelling the evolution of a community of interacting individuals. We model the population by a measure space $(\Omega ,\mathcal {F},\nu )$ where $\nu$ determines the abundance of individual preferences. The preferences of an individual $\omega \in \Omega$ are described by a measurable choice $X(\omega )$ of a rough path. We aim to identify, for each individual, a choice for the forward evolution $Y_{t}(\omega )$ for an individual in the community. These choices $Y_{t}(\omega )$ must be consistent so that $Y_{t}(\omega )$ correctly accounts for the individual's preference and correctly models their interaction with the aggregate behaviour of the community. In general, solutions are continuum of interacting threads analogous to the huge number of individual atomic trajectories that together make up the motion of a fluid. The evolution of the population need not be governed by any over-arching partial differential equation (PDE). Although one can match the standard non-linear parabolic PDEs of McKean–Vlasov type with specific examples of communities in this case. The bulk behaviour of the evolving population provides a solution to the PDE. We focus on the case of weakly interacting systems, where we are able to exhibit the existence and uniqueness of consistent solutions. An important technical result is continuity of the behaviour of the system with respect to changes in the measure $\nu$ assigning weight to individuals. Replacing the deterministic $\nu$ with the empirical distribution of an independent and identically distributed sample from $\nu$ leads to many standard models, and applying the continuity result allows easy proofs for propagation of chaos. The rigorous underpinning presented here leads to uncomplicated models which have wide applicability in both the physical and social sciences. We make no presumption that the macroscopic dynamics are modelled by a PDE. This work builds on the fine probability literature considering the limit behaviour for systems where a large number of particles are interacting with independent preferences; there is also work on continuum models with preferences described by a semi-martingale measure. We mention some of the key papers.

Description: We give a bordism-theoretic characterization of those closed almost contact $(2q{+ }1)$ -manifolds (with $q\geq 2$ ) that admit a Stein fillable contact structure. Our method is to apply Eliashberg's $h$ -principle for Stein manifolds in the setting of Kreck's modified surgery. As an application, we show that any simply connected almost contact 7-manifold with torsion-free second homotopy group is Stein fillable. We also discuss the Stein fillability of exotic spheres and examine subcritical Stein fillability.

Description: Casson-type invariants emerging from Donaldson theory over certain negative-definite four-manifolds were recently suggested by Teleman. These are defined by an algebraic count of points in a zero-dimensional moduli space of flat instantons. Motivated by the cobordism programme of proving Witten's conjecture, we use a moduli space of ${\rm PU}(2)$ Seiberg–Witten monopoles to exhibit an oriented one-dimensional cobordism of the instanton moduli space to the empty space. The Casson-type invariant must therefore vanish.