Suzuki functor at the critical level

In this paper we define and study a critical-level generalization of the Suzuki functor, relating the affine general linear Lie algebra to the rational Cherednik algebra of type A. Our main result states that this functor induces a surjective algebra homomorphism from the centre of the completed universal enveloping algebra at the critical level to the centre of the rational Cherednik algebra at t=0. We use this homomorphism to obtain several results about the functor. We compute it on Verma modules, Weyl modules, and their restricted versions. We describe the maps between endomorphism rings induced by the functor and deduce that every simple module over the rational Cherednik algebra lies in its image. Our homomorphism between the two centres gives rise to a closed embedding of the Calogero-Moser space into the space of opers on the punctured disc. We give a partial geometric description of this embedding.Comment: Some changes in the presentation, a few minor mistakes correcte

Every countable group is the fundamental group of some compact subspace of R^4

For every countable group G we construct a compact path connected subspace K of R^4 whose fundamental group is isomorphic to G. Our construction is much simpler than the one found recently by Virk.Comment: 4 pages; figure on page 1 correcte

Homotopical localizations at a space

Our main motivation for the work presented in this paper is to construct a localization functor, in a certain sense dual to the f-localization of Bousfield and Farjoun, and to study some of its properties. We succeed in a case which is related to the Sullivan profinite completion. As a corollary we prove the existence of certain cohomological localizations.Comment: To appear in Algebraic Topology and its Application

Rational Cherednik algebras, quiver Schur algebras and cohomological Hall algebras

This thesis is devoted to three interrelated problems in representation theory. The first problem concerns the combinatorial aspects of the connection between rational Cherednik algebras at $t=0$ and Hilbert schemes. The second problem concerns the critical-level limit of the Suzuki functor, which connects the representation theory of affine Lie algebras to that of rational Cherednik algebras. The third problem concerns the properties of certain generalizations of Khovanov-Lauda-Rouquier algebras, called quiver Schur algebras, and their relationship to cohomological Hall algebras. Let us describe our results in more detail. In chapter 3, we study the combinatorial consequences of the relationship between rational Cherednik algebras of type G(l,1,n), cyclic quiver varieties and Hilbert schemes. We classify and explicitly construct C*-fixed points in cyclic quiver varieties and calculate the corresponding characters of tautological bundles. We give a combinatorial description of the bijections between C*-fixed points induced by the Etingof-Ginzburg isomorphism and Nakajima reflection functors. We apply our results to obtain a new proof as well as a generalization of a well known combinatorial identity, called the q-hook formula. We also explain the connection between our results and Bezrukavnikov and Finkelberg's, as well as Losev's, proofs of Haiman's wreath Macdonald positivity conjecture. In chapter 4, we define and study a critical-level generalization of the Suzuki functor, relating the affine general linear Lie algebra to the rational Cherednik algebra of type A. Our main result states that this functor induces a surjective algebra homomorphism from the centre of the completed universal enveloping algebra at the critical level to the centre of the rational Cherednik algebra at t=0. We use this homomorphism to obtain several results about the functor. We compute it on Verma modules, Weyl modules, and their restricted versions. We describe the maps between endomorphism rings induced by the functor and deduce that every simple module over the rational Cherednik algebra lies in its image. Our homomorphism between the two centres gives rise to a closed embedding of the Calogero-Moser space into the space of opers on the punctured disc. We give a partial geometric description of this embedding. In chapter 5, we establish a connection between a generalization of KLR algebras, called quiver Schur algebras, and the cohomological Hall algebras of Kontsevich and Soibelman. More specifically, we realize quiver Schur algebras as algebras of multiplication and comultiplication operators on the CoHA, and reinterpret the shuffle description of the CoHA in terms of Demazure operators. We introduce ``mixed quiver Schur algebras" associated to quivers with a contravariant involution, and show that they are related, in an analogous way, to the cohomological Hall modules defined by Young. Furthermore, we obtain a geometric realization of the modified quiver Schur algebra, which appeared in a version of the Brundan-Kleshchev-Rouquier isomorphism for the affine q-Schur algebra due to Miemietz and Stroppel

Generalized Schur-Weyl dualities for quantum affine symmetric pairs and orientifold KLR algebras

We define a boundary analogue of the Kang-Kashiwara-Kim-Oh generalized Schur-Weyl dualities between quantum affine algebras and Khovanov-Lauda-Rouquier (KLR) algebras. Let $\mathfrak{g}$ be a complex simple Lie algebra and $U_qL\mathfrak{g}$ the corresponding quantum affine algebra. We construct a functor ${}^{\theta}{\sf F}$ between finite-dimensional modules over a quantum symmetric pair of affine type $U_q\mathfrak{k}\subset U_qL{\mathfrak{g}}$ and an orientifold KLR ($o$KLR) algebra arising from a framed quiver with a contravariant involution, whose nodes are indexed by finite-dimensional $U_qL{\mathfrak{g}}$-modules. With respect to the Kang-Kashiwara-Kim-Oh construction, our combinatorial model is further enriched with the poles of the trigonometric K-matrices (that is trigonometric solutions of a generalized reflection equation) intertwining the action of $U_q\mathfrak{k}$ on finite-dimensional $U_qL{\mathfrak{g}}$-modules. By construction, ${}^{\theta}{\sf F}$ is naturally compatible with the Kang-Kashiwara-Kim-Oh functor in that, while the latter is a functor of monoidal categories, ${}^{\theta}{\sf F}$ is a functor of module categories. Relying on an isomorphism between suitable completions of $o$KLR algebras and affine Hecke algebras of type $\sf C$, we prove that ${}^{\theta}{\sf F}$ recovers the Schur-Weyl dualities due to Fan-Lai-Li-Luo-Wang-Watanabe in quasi-split type $\sf AIII$. Finally, we construct spectral K-matrices for orientifold KLR algebras, yielding a meromorphic braiding on its category of finite-dimensional representations. We prove that, in the case of the ${\sf A}_{\infty}$ quiver with no fixed points and no framing, the functor ${}^{\theta}{\sf F}$ is exact, factors through a suitable localization, and takes values in a boundary analogue of the Hernandez-Leclerc category.Comment: 61 page

Baltic Sea coastal erosion; a case study from the Jastrzębia Góra region

The coastline in the Jastrzębia Góra area can be divided into three major zones of general importance: a beach and barrier section, a cliff section, and a section protected by a heavy hydrotechnical construction. These areas are characterised by a diverse geology and origin, and hence different vulnerability to erosion. In addition, observations have demonstrated a different pace of erosion within each zone. Based on the results obtained by remote sensing methods (analysis of aerial photographs and maps), it has been determined that the coastline in the barrier area, i.e., to the west of Jastrzębia Góra, moved landwards by about 130 m, in a period of 100 years, and 80 m over about 50 years. A smaller displacement of the shoreline could be observed within the cliff. Between the middle of the twentieth and the start of the twenty-first centuries the shore retreated by about 25 m. However, in recent years, an active landslide has led to the displacement of the uppermost part of the cliff locally up to 25 m. Another issue is, functioning since 2000, a heavy hydrotechnical construction which has been built in order to protect the most active part of the cliff. The construction is not stable and its western part, over a distance of 50 m, has moved almost 2 m vertically downwards and c. 2.5 m horizontally towards the sea in the past two years. This illustrates that the erosional factor does not comprise only marine abrasion, but also involves land-based processes determined by geology and hydrogeology. Changes in the shoreline at the beach and barrier part are constantly conditioned by rising sea levels, the slightly sloping profile of the sea floor and low elevation values of the backshore and dune areas. Cliffs are destroyed by mass wasting and repetitive storm surges that are responsible for the removal of the colluvium which protects the coast from adverse wave effects. Presumably, mass movements combined with groundwater outflow from the cliff, plus sea abrasion cause destabilisation of the cliff protection construction

The RESET project: constructing a European tephra lattice for refined synchronisation of environmental and archaeological events during the last c. 100 ka

This paper introduces the aims and scope of the RESET project (. RESponse of humans to abrupt Environmental Transitions), a programme of research funded by the Natural Environment Research Council (UK) between 2008 and 2013; it also provides the context and rationale for papers included in a special volume of Quaternary Science Reviews that report some of the project's findings. RESET examined the chronological and correlation methods employed to establish causal links between the timing of abrupt environmental transitions (AETs) on the one hand, and of human dispersal and development on the other, with a focus on the Middle and Upper Palaeolithic periods. The period of interest is the Last Glacial cycle and the early Holocene (c. 100-8 ka), during which time a number of pronounced AETs occurred. A long-running topic of debate is the degree to which human history in Europe and the Mediterranean region during the Palaeolithic was shaped by these AETs, but this has proved difficult to assess because of poor dating control. In an attempt to move the science forward, RESET examined the potential that tephra isochrons, and in particular non-visible ash layers (cryptotephras), might offer for synchronising palaeo-records with a greater degree of finesse. New tephrostratigraphical data generated by the project augment previously-established tephra frameworks for the region, and underpin a more evolved tephra 'lattice' that links palaeo-records between Greenland, the European mainland, sub-marine sequences in the Mediterranean and North Africa. The paper also outlines the significance of other contributions to this special volume: collectively, these illustrate how the lattice was constructed, how it links with cognate tephra research in Europe and elsewhere, and how the evidence of tephra isochrons is beginning to challenge long-held views about the impacts of environmental change on humans during the Palaeolithic. © 2015 Elsevier Ltd.RESET was funded through Consortium Grants awarded by the Natural Environment Research Council, UK, to a collaborating team drawn from four institutions: Royal Holloway University of London (grant reference NE/E015905/1), the Natural History Museum, London (NE/E015913/1), Oxford University (NE/E015670/1) and the University of Southampton, including the National Oceanography Centre (NE/01531X/1). The authors also wish to record their deep gratitude to four members of the scientific community who formed a consultative advisory panel during the lifetime of the RESET project: Professor Barbara Wohlfarth (Stockholm University), Professor Jørgen Peder Steffensen (Niels Bohr Institute, Copenhagen), Dr. Martin Street (Romisch-Germanisches Zentralmuseum, Neuwied) and Professor Clive Oppenheimer (Cambridge University). They provided excellent advice at key stages of the work, which we greatly valued. We also thank Jenny Kynaston (Geography Department, Royal Holloway) for construction of several of the figures in this paper, and Debbie Barrett (Elsevier) and Colin Murray Wallace (Editor-in-Chief, QSR) for their considerable assistance in the production of this special volume.Peer Reviewe