Graduation 2014. Monday 21 July 2014 the University of Sheffield
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Stability Conditions and the A2 Quiver 10
STABILITY CONDITIONS AND THE A2 QUIVER TOM BRIDGELAND, YU QIU AND TOM SUTHERLAND Abstract. For each integer n > 2 we describe the space of stability conditions on the derived category of the n-dimensional Ginzburg algebra associated to the A2 quiver. The form of our results points to a close relationship between these spaces and the Frobenius- Saito structure on the unfolding space of the A2 singularity. 1. Introduction In this paper we study spaces of stability conditions [2] on the sequence of CYn trian- gulated categories Dn associated to the A2 quiver. Our main result is Theorem 1.1 below. There are several striking features. Firstly, we obtain uniform results for all n > 2: the space of stability conditions quotiented by the action of the spherical twists is independent of n, although the identification maps are highly non-trivial. Secondly, there is a close link between our spaces of stability conditions and the Frobenius-Saito structure on the unfolding space of the A2 singularity: in fact this structure is precisely what encodes the identifications between our stability spaces for various n. A third interesting feature is that the space of stability conditions on the derived category of the path algebra of the A2 quiver arises as a kind of limit of the spaces for the categories Dn as n ! 1. 1.1. Statement of results. For each integer n > 2 we let Dn = DCYn (A2) denote the bounded derived category of the CYn complex Ginzburg algebra associated to the A2 quiver. It is a triangulated category of finite type over C, and is characterised by the following two properties: (a) It is CYn, i.e. -
June 2014 Society Meetings Society and Events SHEPHARD PRIZE: NEW PRIZE Meetings for MATHEMATICS 2014 and Events Following a Very Generous Tions Open in Late 2014
LONDONLONDON MATHEMATICALMATHEMATICAL SOCIETYSOCIETY NEWSLETTER No. 437 June 2014 Society Meetings Society and Events SHEPHARD PRIZE: NEW PRIZE Meetings FOR MATHEMATICS 2014 and Events Following a very generous tions open in late 2014. The prize Monday 16 June donation made by Professor may be awarded to either a single Midlands Regional Meeting, Loughborough Geoffrey Shephard, the London winner or jointly to collaborators. page 11 Mathematical Society will, in 2015, The mathematical contribution Friday 4 July introduce a new prize. The prize, to which an award will be made Graduate Student to be known as the Shephard must be published, though there Meeting, Prize will be awarded bienni- is no requirement that the pub- London ally. The award will be made to lication be in an LMS-published page 8 a mathematician (or mathemati- journal. Friday 4 July cians) based in the UK in recog- Professor Shephard himself is 1 Society Meeting nition of a specific contribution Professor of Mathematics at the Hardy Lecture to mathematics with a strong University of East Anglia whose London intuitive component which can be main fields of interest are in page 9 explained to those with little or convex geometry and tessella- Wednesday 9 July no knowledge of university math- tions. Professor Shephard is one LMS Popular Lectures ematics, though the work itself of the longest-standing members London may involve more advanced ideas. of the LMS, having given more page 17 The Society now actively en- than sixty years of membership. Tuesday 19 August courages members to consider The Society wishes to place on LMS Meeting and Reception nominees who could be put record its thanks for his support ICM 2014, Seoul forward for the award of a in the establishment of the new page 11 Shephard Prize when nomina- prize. -
September 2014
LONDONLONDON MATHEMATICALMATHEMATICAL SOCIETYSOCIETY NEWSLETTER No. 439 September 2014 Society Meetings HIGHEST HONOUR FOR UK and Events MATHEMATICAN Professor Martin Hairer, FRS, 2014 University of Warwick, has become the ninth UK based Saturday mathematician to win the 6 September prestigious Fields Medal over Mathematics and the its 80 year history. The medal First World War recipients were announced Meeting, London on Wednesday 13 August in page 15 a ceremony at the four-year- ly International Congress for 1 Wednesday Mathematicians, which on this 24 September occasion was held in Seoul, South Korea. LMS Popular Lectures See page 4 for the full report. Birmingham page 12 Friday LMS ANNOUNCES SIMON TAVARÉ 14 November AS PRESIDENT-DESIGNATE LMS AGM © The University of Cambridge take over from the London current President, Professor Terry Wednesday Lyons, FRS, in 17 December November 2015. SW & South Wales Professor Tavaré is Meeting a versatile math- Plymouth ematician who has established a distinguished in- ternational career culminating in his current role as The London Mathematical Director of the Cancer Research Society is pleased to announce UK Cambridge Institute and Professor Simon Tavaré, Professor in DAMTP, where he NEWSLETTER FRS, FMedSci, University of brings his understanding of sto- ONLINE: Cambridge, as President-Des- chastic processes and expertise newsletter.lms.ac.uk ignate. Professor Tavaré will in the data science of DNA se- (Cont'd on page 3) LMS NEWSLETTER http://newsletter.lms.ac.uk Contents No. 439 September 2014 15 44 Awards Partial Differential Equations..........................37 Collingwood Memorial Prize..........................11 Valediction to Jeremy Gray..............................33 Calendar of Events.......................................50 News LMS Items European News.................................................16 HEA STEM Strategic Project........................... -
The Space of Stability Conditions for $ a 3 $-Quiver
THE SPACE OF STABILITY CONDITIONS FOR A3-QUIVER TAKAHISA SHIINA Abstract. The author studied in [Shi11] the covering map property of the local homeomorphism associating to the space of stability conditions over the n-Kronecker quiver. In this paper, we discuss the covering map property for stability conditions over the Dynkin quiver of type A3. The local homeo- morphism from a connected component of stability conditions over A3 to 3- dimensional complex vector space becomes a covering map when we restrict it to the complement of six codimension one subspaces. 1. Introduction T. Bridgeland introduced the notion of stability conditions on triangulated cate- gories ([Bri07]). The idea comes from Douglas’s work on π-stability for D-branes in string theory ([Dou02]). Bridgeland defined a topology (generalized metric) on the set of all (locally finite) stability conditions, Stab(T ), on a triangulated category T . Each connected component Σ ⊂ Stab(T ) is equipped with a local homeo- morphism Z to a certain topological vector space V (Σ) ([Bri07, Theorem 1.2]). Moreover Bridgeland showed that Σ is a (possibly infinite-dimensional) complex manifold. It is an important problem to show simply connectivity of Σ, connectivity of Stab(T ), or (universal) covering map property of the local homeomorphism Z : Σ → V (Σ). It have been studied intensively, for example, stability conditions over K3 surfaces, over curves, and over Kleinian singularities are studied [Bri09, Bri08, BT11, IUU10, Mac07, ST01, Tho06]. In [Shi11], the author studied the space of stability conditions Stab(Pn) con- b structed on the bounded derived category D (Pn) of the finite dimensional repre- sentations over n-Kronecker quiver Pn – the quiver with two vertices and n-parallel arXiv:1309.0922v1 [math.CT] 4 Sep 2013 arrows. -
Scattering Diagrams, Hall Algebras and Stability Conditions
SCATTERING DIAGRAMS, HALL ALGEBRAS AND STABILITY CONDITIONS TOM BRIDGELAND In memory of Kentaro Nagao Abstract. To any quiver with relations we associate a consistent scattering diagram taking values in the motivic Hall algebra of its category of representations. We show that the chamber structure of this scattering diagram coincides with the natural chamber structure in an open subset of the space of stability conditions on the associated triangulated category. In the three- dimensional Calabi-Yau situation, when the relations arise from a potential, we can apply an integration map to give a consistent scattering diagram taking values in a tropical vertex group. 1. Introduction The concept of a scattering diagram has emerged from the work of Kontsevich and Soibelman [15], and Gross and Siebert [9], on the Strominger-Yau-Zaslow approach to mirror symmetry. Scattering diagrams, and the associated combinatorics of broken lines, have been used recently by Gross, Hacking, Keel and Kontsevich [7] to settle several important conjectures about cluster algebras. The aim of this paper is to explain some very general connections between spaces of stability conditions and scattering diagrams. In particular, we explain the relationship between the scattering diagram associated to an acyclic quiver Q, and the wall-and-chamber structure of the space of stability conditions on the associated CY3 triangulated category. Our longer-term goal is to better understand the geometrical relationship between cluster varieties and spaces of stability conditions. It turns out that to give a categorical description of the scattering diagram one does not need to use triangulated categories. In fact, for the most part, we work in the abelian category of representations of a fixed quiver with relations (Q; I), and use King's notion of θ{stability. -
FBU Firefighter Magazine • October/November 2916
THE MAGAZINE OF THE FIRE BRIGADES UNION | WWW.FBU.ORG.UK OCTOBER/NOVEMBER 2016 SAFETY FOR SALE TIMES HAVE CHANGED THE RISKS HAVE NOT SEE P14 Plus FBU AT THE LABOUR PARTY CONFERENCE SEE P10 FIREFIGHTER FATALITIES: MORE THAN JUST GUIDANCE NEEDED SEE P12 GENERAL SECRETARY’S COMMENT MATT WRACK CUTS, AND THOSE WHO FORCE THEM THROUGH, MUST BE RESISTED During September our BOX/REPORTDIGITAL.CO.UK PAUL current state of affairs where members in Greater response times are increasing, Manchester were facing a putting lives at risk. bleak future. We have campaigned More than 1,250 relentlessly over the past six firefighters faced the sack – years, warning politicians only those who agreed to sign that the scale of cuts to the up to worse conditions of fire and rescue service is service would be re-engaged. such that public safety is Greater Manchester seriously compromised. Fire and Rescue Service We now have fire authority has to find savings of more Rapid response – but continuing cuts will take effect members also publicly than £14m, but this is no expressing this fear, as the justification for threatening negotiating process was of mental ill health than the link between an increased such action. stopped, but our officials in rest of the population. response time and rising fire FBU officials mounted Greater Manchester are well We are proud that our deaths is inextricably linked. a vigorous campaign that aware that further battles lottery will see a higher But this government attracted a lot of public and lie ahead. They will get our percentage of the monies remains deaf to the political support. -
Applied-Math-Catalog-Web.Pdf
Applied mathematics was once only associated with the natural sciences and en- gineering, yet over the last 50 years it has blossomed to include many fascinat- ing subjects outside the physical sciences. Such areas include game theory, bio- mathematics and mathematical economy demonstrating mathematics' presence throughout everyday human activity. The AMS book publication program on applied and interdisciplinary mathematics strengthens the connections between mathematics and other disciplines, highlighting the areas where mathematics is most relevant. These publications help mathematicians understand how math- ematical ideas may benefit other sciences, while offering researchers outside of mathematics important tools to advance their profession. Table of Contents Algebra and Algebraic Geometry .............................................. 3 Analysis ........................................................................................... 3 Applications ................................................................................... 4 Differential Equations .................................................................. 11 Discrete Mathematics and Combinatorics .............................. 16 General Interest ............................................................................. 17 Geometry and Topology ............................................................. 18 Mathematical Physics ................................................................... 19 Probability ..................................................................................... -
English Is a Welsh Language
ENGLISH IS A WELSH LANGUAGE Television’s crisis in Wales Edited by Geraint Talfan Davies Published in Wales by the Institute of Welsh Affairs. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form, or by any means without the prior permission of the publishers. © Institute of Welsh Affairs, 2009 ISBN: 978 1 904773 42 9 English is a Welsh language Television’s crisis in Wales Edited by Geraint Talfan Davies The Institute of Welsh Affairs exists to promote quality research and informed debate affecting the cultural, social, political and economic well-being of Wales. IWA is an independent organisation owing no allegiance to any political or economic interest group. Our only interest is in seeing Wales flourish as a country in which to work and live. We are funded by a range of organisations and individuals. For more information about the Institute, its publications, and how to join, either as an individual or corporate supporter, contact: IWA - Institute of Welsh Affairs 4 Cathedral Road Cardiff CF11 9LJ tel 029 2066 0820 fax 029 2023 3741 email [email protected] web www.iwa.org.uk Contents 1 Preface 4 1/ English is a Welsh language, Geraint Talfan Davies 22 2/ Inventing Wales, Patrick Hannan 30 3/ The long goodbye, Kevin Williams 36 4/ Normal service, Dai Smith 44 5/ Small screen, big screen, Peter Edwards 50 6/ The drama of belonging, Catrin Clarke 54 7/ Convergent realities, John Geraint 62 8/ Standing up among the cogwheels, Colin Thomas 68 9/ Once upon a time, Trevor -
Truth, Yes … but Justice?
SUMMER 2016 TRUTH, YES … BUT JUSTICE? REMEMBER THE 96 2 FOODWORKER EDITOR’S SUMMER 2016 NOTES TRUTH, YES "… Jeremy Corbyn is … BUT the leader, get over JUSTICE? it and unite for the benefit of those you purport to represent" REMEMBER THE 96 "Ronnie Draper @ronniebfawu INSIDE THIS ISSUE: As we approach Annual Conference, We don’t want a return to the disloyalty of Editor’s Notes . 2 which will be held in Southport, the time to 1981 when the gang of four Roy Jenkins, reflect on where the political landscape has David Owen, Shirley Williams and Bill National President . 3 moved over the last year is upon us. If we Rogers left the party to form the SDP Dennis Nash presentation . .4 are honest the only significant political although if that is the only way that we are FastFood Day of Action . .5 change has been the landslide election of going to rid our party of the treachery that Jeremy Corbyn as Labour Party leader and appears to abound then so be it. Trade Union Bill . 6 the massive increase in the number of Who is the common enemy? Thank You Pauls! . 6 grassroots members who have joined in the Working people and the disadvantaged in our Womens' TUC Conference . .7 hope that we will see a move to a more society are under constant attack from the socialist agenda. most heinous government in living memory. Signature Flatbreads . .8 Labour = Loyalty? We and all those that we elected to Poverty Wages make us Ill! . 8 Since that day we have seen a number of parliament should be concentrating all our efforts on defeating the common enemy We are Wakefield . -
Arxiv:1603.00416V4 [Math.AG] 15 May 2020 Ouetinuae Aeois Nfc,Frtems at W Part, ( Most Relations the with for Quiver fixed Fact, a in of Representations Categories
SCATTERING DIAGRAMS, HALL ALGEBRAS AND STABILITY CONDITIONS TOM BRIDGELAND In memory of Kentaro Nagao Abstract. To any quiver with relations we associate a consistent scattering diagram taking values in the motivic Hall algebra of its category of representations. We show that the chamber structure of this scattering diagram coincides with the natural chamber structure in an open subset of the space of stability conditions on the associated triangulated category. In the three- dimensional Calabi-Yau situation, when the relations arise from a potential, we can apply an integration map to give a consistent scattering diagram taking values in a tropical vertex group. 1. Introduction The concept of a scattering diagram has emerged from the work of Kontsevich and Soibelman [15], and Gross and Siebert [9], on the Strominger-Yau-Zaslow approach to mirror symmetry. Scattering diagrams, and the associated combinatorics of broken lines, have been used recently by Gross, Hacking, Keel and Kontsevich [7] to settle several important conjectures about cluster algebras. The aim of this paper is to explain some very general connections between spaces of stability conditions and scattering diagrams. In particular, we explain the relationship between the scattering diagram associated to an acyclic quiver Q, and the wall-and-chamber structure of the space of stability conditions on the associated CY3 triangulated category. Our longer-term goal is to better understand the geometrical relationship between cluster varieties and spaces of stability conditions. It turns out that to give a categorical description of the scattering diagram one does not need to use triangulated categories. In fact, for the most part, we work in the abelian category of representations of a fixed quiver with relations (Q,I), and use King’s notion of θ–stability. -
We Must Unite Behind Progressive Policies and Become the Force That the Party’S Talent Deserves to Be
AUTUMN 2016 We must unite behind progressive policies and become the force that the party’s talent deserves to be THIS IS A DECISION FOR ALL MEMBERS – NOT JUST THE PLP! 2 FOODWORKER EDITOR’S AUTUMN 2016 We must we NOTES unite behind progressive policies and become the force that the party’s "Hammering terms talent deserves to be and conditions is not the way to get a productive, proactive workforce" THIS IS A DECISION FOR ALL MEMBERS – NOT JUST THE PLP! "Ronnie Draper @ronniebfawu INSIDE THIS ISSUE: If anyone was ever in doubt about the from Eastern Europe for the dramatic Editor’s Notes . 2 negative effect a Tory government would reduction in wage levels, housing shortages have on those at the bottom of the wage and lengthening waiting times within the National President . 3 pile, you need look no further than the so NHS. Again this is massively wide of the Kumaran Bose . 5 called 'living wage', brought in by former mark. Migrant labour does not reduce Conference Report . 6 Chancellor, George Osborne. wage rates but opportunist employers do, After years which have seen pay freezes, employers who exploit financial situations Bill Morris' Conference Diary .10 the bedroom tax, cuts to in-work benefits, abroad to create a myth that is propagated by the right wing press. HEALTH & SAFETY hikes in VAT and the repeal of protective legislation, we now have a joke living wage We are the 5th richest economy in the National Safety that has been a green light for unscrupulous world run by a government who create Committee Update . -
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Editorial Standards Findings Appeals to the Trust and other editorial issues considered by the Editorial Standards Committee September 2010 issued October 2010 Getting the best out of the BBC for licence fee payers Editorial Standards Findings/Appeals to the Trust and other editorial issues considered byContents the Editorial Standards Committee Remit of the Editorial Standards Committee 1 Summary of findings 3 Findings 8 Newsnight, BBC Two, 19 January 2010 8 Newsnight, BBC Two, 19 January 2010 20 The World at One, BBC Radio 4, 8 December 2009 38 To Let, BBC Two, 20 December 2009 – Complaints Handling 44 The Culture Show, BBC Two, 4 February 2010 49 Rejected appeals 65 Fly Me To The Reverend Moon, BBC Radio 4, 19 April 2010 65 To Let, BBC Two, 29 December 2009 68 The Big Questions, BBC One (General complaint – bias against Muslims) 70 The Review Show, BBC Two, 26 February 2010 72 Scapegoat, BBC Northern Ireland, 26 October 2009 74 BBC News website reporting on Gaza conflict (2009) 77 Today, BBC Radio 4, 27 January 2010 80 Systemic bias on climate change 82 September 2010 issued October 2010 Editorial Standards Findings/Appeals to the Trust and other editorial issues considered by the Editorial Standards Committee Remit of the Editorial Standards Committee The Editorial Standards Committee (ESC) is responsible for assisting the Trust in securing editorial standards. It has a number of responsibilities, set out in its Terms of Reference at bbc.co.uk/bbctrust/about/meetings_and_minutes/bbc_trust_committees.html. The Committee comprises six Trustees: Richard Tait (Chairman), Chitra Bharucha, Mehmuda Mian, David Liddiment, Alison Hastings and Anthony Fry.