We review the mathematical analysis of some uniaxial, liquid crystal phases. Firstly, we state the models for the two different studied phases: nematic and smectic-A liquid crystals. The spatial and temporal profiles of the liquid crystal configurations will be described by means of strongly nonlinear parabolic partial differential systems, which are presented at the same time. Then we will state some results about existence, regularity, time-periodicity and stability of solutions at infinite time for both models. It is our aim to show that, although nematic and smectic-A phases have different physical properties and are modelled by different nonlinear parabolic problems, there exists a common mathematical machinery to rewrite the models and obtain analytical results.

Let G be a string graph (an intersection graph of continuous arcs in the plane) with m edges. Fox and Pach proved that G has a separator consisting of $O(m^{3/4}\sqrt{\log m})$ vertices, and they conjectured that the bound of $O(\sqrt m)$ actually holds. We obtain separators with $O(\sqrt m \,\log m)$ vertices.

We prove uniform Sobolev estimates $\Vert u\Vert _{{L}^{p\prime } } \leq C\Vert (\Delta - \alpha )u\Vert _{{L}^{p} } $ for $\alpha \in \mathbb{C} $ and $p= 2n/ (n+ 2),{p}^{\prime } = 2n/ (n- 2)$ on non-trapping asymptotically conic manifolds of dimension $n\geq 3$ , generalizing to non-constant coefficient Laplacians a result of Kenig, Ruiz and Sogge [13].

A brief survey of some aspects of noetherian Hopf algebras is given, concentrating on structure, homology, and classification, and accompanied by a panoply of open problems.

We study the automorphism group of the algebra $\co_q(M_n)$ of n × n generic quantum matrices. We provide evidence for our conjecture that this group is generated by the transposition and the subgroup of those automorphisms acting on the canonical generators of $\co_q(M_n)$ by multiplication by scalars. Moreover, we prove this conjecture in the case when n = 3.

We present a unifying framework for the key concepts and results of higher Koszul duality theory for N-homogeneous algebras: the Koszul complex, the candidate for the space of syzygies and the higher operations on the Yoneda algebra. We give a universal description of the Koszul dual algebra under a new algebraic structure. For that we introduce a general notion: Gröbner bases for algebras over non-symmetric operads.

Let k be an algebraically closed field of characteristic zero. I. M. Musson and M. Van den Bergh (Mem. Amer. Math. Soc., vol. 136, 1998, p. 650) classify primitive ideals for rings of torus invariant differential operators. This classification applies in particular to subquotients of localized extended Weyl algebras $\mathcal{A}_{r,n-r}=k[x_1,\ldots,x_r,x_{r+1}^{\pm1}, \ldots, x_{n}^{\pm1},\partial_1,\ldots,\partial_n],$ where it can be made explicit in terms of convex geometry. We recall these results and then turn to the corresponding primitive quotients and study their Goldie ranks. We prove that the primitive quotients fall into finitely many families whose Goldie ranks are given by a common quasi-polynomial and then realize these quasi-polynomials as Ehrhart quasi-polynomials arising from convex geometry.

We prove that special ample line bundles on toric varieties arising from root systems are projectively normal. Here the maximal cones of the fans correspond to the Weyl chambers, and special means that the bundle is torus-equivariant such that the character of the line bundle that corresponds to a maximal Weyl chamber is dominant with respect to that chamber. Moreover, we prove that the associated semi-group rings are quadratic.

Using Beilinson–Bernstein localisation, we give another proof of Levasseur's theorem on the Krull dimension of the enveloping algebra of a complex semisimple Lie algebra. The proof also extends to the case of affinoid enveloping algebras.

Let R be an affine PI-algebra over an algebraically closed field $\mathbb{k}$ and let G be an affine algebraic $\mathbb{k}$-group that acts rationally by algebra automorphisms on R. For R prime and G a torus, we show that R has only finitely many G-prime ideals if and only if the action of G on the centre of R is multiplicity free. This extends a standard result on affine algebraic G-varieties. Under suitable hypotheses on R and G, we also prove a PI-version of a well-known result on spherical varieties and a version of Schelter's catenarity theorem for G-primes.

Erdős asked in 1962 about the value of f(n,k,l), the minimum number of k-cliques in a graph with order n and independence number less than l. The case (k,l)=(3,3) was solved by Lorden. Here we solve the problem (for all large n) for (3,l) with 4 ≤ l ≤ 7 and (k,3) with 4 ≤ k ≤ 7. Independently, Das, Huang, Ma, Naves and Sudakov resolved the cases (k,l)=(3,4) and (4,3).

We study the ring theory of the multi-parameter deformations of the quantum Schubert cell algebras obtained from 2-cocycle twists. This is a large family, which extends the Artin–Schelter–Tate algebras of twisted quantum matrices. We classify set theoretically the spectra of all such multi-parameter quantum Schubert cell algebras, construct each of their prime ideals by contracting from explicit normal localizations and prove formulas for the dimensions of their Goodearl–Letzter strata for base fields of arbitrary characteristic and all deformation parameters that are not roots of unity. Furthermore, we prove that the spectra of these algebras are normally separated and that all such algebras are catenary.

In this paper we study rational Cherednik algebras at t = 1 in positive characteristic. We study a finite-dimensional quotient of the rational Cherednik algebra called the restricted rational Cherednik algebra. When the corresponding pseudo-reflection group belongs to the infinite series G(m, d, n), we describe explicitly the block decomposition of the restricted algebra. We also classify all pseudo-reflection groups for which the centre of the corresponding rational Cherednik algebra is regular for generic values of the deformation parameter.