ZnZ^n-free groups are CAT(0)

We show that every group with free $\mathbb{Z}^n$-length function is CAT(0).Comment: To be published in the Journal of the London Mathematical Society. This version is very close to the accepted version. The exposition greatly improved due to the referee's comment

Global rough solutions to the cubic nonlinear Boussinesq equation

We prove that the initial value problem (IVP) for the cubic defocusing nonlinear Boussinesq equation $u_{tt}-u_{xx}+u_{xxxx}-(|u|^2u)_{xx}=0$ on the real line is globally well-posed in $H^{s}(\R)$ provided $2/3<s<1$.Comment: 21 pages, to appear, Journal of the London Mathematical Society. Many suggestions of the referee implemente

Oversampling of wavelet frames for real dilations

We generalize the Second Oversampling Theorem for wavelet frames and dual wavelet frames from the setting of integer dilations to real dilations. We also study the relationship between dilation matrix oversampling of semi-orthogonal Parseval wavelet frames and the additional shift invariance gain of the core subspace.Comment: Journal of London Mathematical Society, published online March 13, 2012 (to appear in print

Auslander algebras as quasi-hereditary algebras

Dlab V, Ringel CM. Auslander algebras as quasi-hereditary algebras. Journal of the London Mathematical Society : Ser. 2. 1989;39(3):457-466

The Kodaira dimension of some moduli spaces of elliptic K3 surfaces

We study the moduli spaces of elliptic K3 surfaces of Picard number at least 3, that is, (Formula presented.) -polarized K3 surfaces. Such moduli spaces are proved to be of general type for (Formula presented.). The proof relies on the low-weight cusp form trick developed by Gritsenko, Hulek and Sankaran. Furthermore, explicit geometric constructions of some elliptic K3 surfaces lead to the unirationality of these moduli spaces for (Formula presented.) and for 19 other isolated values up to (Formula presented.). © 2021 The Authors. Journal of the London Mathematical Society is copyright © London Mathematical Society

Csoportok és reprezentációik = Groups and their representations

Változatos kérdéseket vizsgáltunk a csoportelméletben, a csoportok reprezentációelméletében és más kapcsolódó absztrakt algebrai területeken. 39 tudományos dolgozatot publikáltunk, ezek nagy részét vezető nemzetközi folyóiratokban (pl. Bulletin of the London Mathematical Society, Duke Mathematical Journal, European Journal of Combinatorics, Journal of Algebra, Journal of Group Theory, Proceedings of the American Mathematical Society). Legfontosabb eredményeink a következők: Meghatároztuk a pozitiv karakterisztikájú globális testek feletti aritmetikai csoportok kongruenciarészcsoport-növekedését. Új példákat találtunk olyan csoportokra, amelyeknek izomorf a pro-véges lezárásuk. Teljes leirását adtuk azoknak a moduláris csoportalgebráknak, melyek Lie nilpotencia-indexe maximális. Csoportelméleti módszereket alkalmazva a loopok elméletében olyan (128 elemű) loopot konstruáltunk, amelynél a belső permutációk csoportja kommutativ és a loop nilpotnecia osztálya 3, ezzel Bruck egy 60 éves kérdésére adtunk választ. Az univerzális algebrában a véges moduláris hálók egy széles osztályára konstruáltunk véges kongruencia-reprezentációkat, mégpedig operátorcsoportok felhasználásával. A bonyolultságelméletben több algebrai problémát tanulmányoztunk. Például megmutattuk, hogy nem feloldható csoportokban az azonosságok ellenőrzése NP-teljes probléma. | We studied various questions in group theory, in representation theory of groups, and in related areas of abstract algebra. We published 39 research papers, many of them in leading international journals (for example, Bulletin of the London Mathematical Society, Duke Mathematical Journal, European Journal of Combinatorics, Journal of Algebra, Journal of Group Theory, Proceedings of the American Mathematical Society). The most important results are the following: We determined the congruence subgroup growth of arithmetic groups over global fields of positive characteristic. We found new examples of groups with isomorphic pro-finite closure. We gave a complete description of modular group algebras with maximal Lie nilpotency index. Applying group theoretic methods in loop theory, we constructed an example of a loop (of order 128) with an Abelian inner permutation group and of nilpotency class 3, thereby answering a 60-year old question of Bruck. In universal algebra we constructed finite congruence lattice representations for a large class of finite modular lattices, namely by using operator groups. In complexity theory we studied several algebraic problems. For example, we showed that for nonsolvable groups the checking of identities is an NP-complete problem

From triangulated categories to module categories via localisation II: Calculus of fractions

We show that the quotient of a Hom-finite triangulated category C by the kernel of the functor Hom(T, -), where T is a rigid object, is preabelian. We further show that the class of regular morphisms in the quotient admit a calculus of left and right fractions. It follows that the Gabriel-Zisman localisation of the quotient at the class of regular morphisms is abelian. We show that it is equivalent to the category of finite dimensional modules over the endomorphism algebra of T in C.Comment: 21 pages; no separate figures. Minor changes. To appear in Journal of the London Mathematical Society (published version is different

External definability and groups in NIP theories

We prove that many properties and invariants of definable groups in NIP theories, such as definable amenability, G/G^{00}, etc., are preserved when passing to the theory of the Shelah expansion by externally definable sets, M^{ext}, of a model M. In the light of these results we continue the study of the "definable topological dynamics" of groups in NIP theories. In particular we prove the Ellis group conjecture relating the Ellis group to G/G^{00} in some new cases, including definably amenable groups in o-minimal structures.Comment: 28 pages. Introduction was expanded and some minor mistakes were corrected. Journal of the London Mathematical Society, accepte

A local-global principle for isogenies of prime degree over number fields

We give a description of the set of exceptional pairs for a number field $K$, that is the set of pairs $(\ell, j(E))$, where $\ell$ is a prime and $j(E)$ is the $j$-invariant of an elliptic curve $E$ over $K$ which admits an $\ell$-isogeny locally almost everywhere but not globally. We obtain an upper bound for $\ell$ in such pairs in terms of the degree and the discriminant of $K$. Moreover, we prove finiteness results about the number of exceptional pairs.Comment: 22 pages, presentation improved as suggested by the referees. To appear in Journal of London Mathematical Society. arXiv admin note: text overlap with arXiv:1006.1782 by other author