Centre for Symmetry and Deformation Department of Mathematical Sciences, University of Copenhagen Funded by Annual report 2013 (1 Jan - 31 Dec, 2013, “Year 4”) DNRF Centre for Symmetry and Deformation (SYM) DNRF92 Department of Mathematical Sciences University of Copenhagen Established Jan 1, 2010 1 Highlights 1 1.1 English .................................. 1 1.2 Danish .................................. 2 2 Organization 3 2.1 Scientific staff .............................. 3 2.2 Visitors .................................. 4 2.3 Administration .............................. 4 3 Research plan 4 A Homotopical group theory ........................ 4 B Symmetry of manifolds ......................... 5 C Symmetry of noncommutative manifolds ................ 5 4 Comments to the appendix 6 5 Signature 7
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Centre for Symmetry and Deformation Department of Mathematical Sciences, University of Copenhagen
Funded by
Annual report 2013(1 Jan - 31 Dec, 2013, “Year 4”)
DNRF Centre for Symmetry and Deformation (SYM)DNRF92
Department of Mathematical SciencesUniversity of Copenhagen
Centre for Symmetry and Deformation Department of Mathematical Sciences, University of Copenhagen
Funded by
CENTER HIGHLIGHTS 2013The year 2013 was an eventful one, with new research results and an exceptionalamount of workshop activity. The initial group of PhD students began to graduatein numbers, and a fresh batch of fall students and postdocs arrived in their place,marking a change for the center from a start-up venture to an established entity.
ACTIVITIES. The center hosted 5 successful workshops and 10 masterclasses in 2013, on topics ranging fromstring topology to ergodic theory and von Neumann algebras; the 214 researchvisitors to the center was an all-time high. A lot of activity was concentrated inthe summer teaching break, which many international experts once again choseto spend at the center. The fall activity was centered around symbolic dynamicalsystems, with Boyle as our distinguished visiting professor.
RESEARCH. A record-breaking 69 new preprints appeared in 2013 in our report series CPH-SYM-DNRF92on arXiv.org, with exciting results across our spectrum of research. In homotopical group theory, Henke gavea characterization of saturated fusion systems over abelian 2-groups, resolving a conjecture of Kühlshammer–Navarro–Sambale–Tiep. In symmetry of manifolds, Berglund–Madsen discovered that all unstable rational ho-mology groups of outer automorphism groups of free groups sit insidethe stable rational cohomology of homotopy automorphisms of certainhigh dimensional manifolds. And in non-commutative geometry, Monod–Kellerhals–Rørdam provided a range of new characterisations of supra-menable groups. Several of the results reported last year were meanwhileaccepted in top journals, including Invent. Math., Acta Math., and DukeMath. J. Our leadership position was again recognized internationally in that Haagerup received one of only277 honorary doctorates awarded by China the last 30 years.
EDUCATION AND OUTREACH. We held 16 graduate courses and 2 reading seminars in 2013. Our outreachactivities featured targeted talent-nurturing, such as mentoring the Georg Mohr Competition and the Danish
team in the International Math Olympiad, as well as broader outreachactivities such as a stand at the Copenhagen Culture Night. Our out-reach accomplishments did not go unnoticed, in that Eilers won the2013 UCPH Faculty of Science Dissemination Award, with a citationmentioning several of our activities from past years.
Please visit sym.math.ku.dk for more information.
Centre for Symmetry and Deformation Department of Mathematical Sciences, University of Copenhagen
Finansieret af
CENTERHØJDEPUNKTER 2013Året 2013 var rigt på begivenheder, med nye forskningsresultater og exceptioneltmange workshops. Et større antal af centrets første ph.d.-studerende forsvaredederes grad, og i efteråret ankom et nyt kuld for at overtage deres pladser, som enmarkering af centrets overgang til en etableret forskningsinstitution.
AKTIVITETER. Centret var vært for 5 succesfulde workshops og 10 masterclasser i 2013, med emnespandfra strengtopologi til ergodisk teori og von Neumann-algebraer. Antal forsk-ningsgæster var rekordhøjt: 214. En stor del af aktiviteten var koncentreret isommerens undervisningsferie idet mange internationale eksperter igen i år valg-te at tilbringe perioden ved centret. Efterårets aktiviteter var centreret omkringsymbolske dynamiske systemer foranlediget af Boyles længerevarende besøg.
FORSKNING. Rekordmange preprints, 69, blev lagt på arXiv.org i 2013 under vores referencenummer CPH-SYM-DNRF92, og de indeholdt spændende resultater fra hele centrets forskningsspektrum. Inden for ho-motopisk gruppeteori gav Henke en karakterisation af mættede fusionssystemer over abelske 2-grupper ogløste dermed en formodning af Kühlshammer–Navarro–Sambale–Tiep. Inden for symmetri af mangfoldighe-der opdagede Berglund–Madsen at alle ikke-stabile rationale homologi-grupper af ydre automorfigrupper til frie grupper sidder inde i den stabilerationale kohomologi af homotopi-automorfier af visse højdimensionalemangfoldigheder. Og inden for ikke-kommutativ geometri gav Monod–Kellerhals–Rørdam en række nye karakterisationer af superamenable grup-per. Adskillige af sidste års resultater blev imens optaget i toptidsskrifter,herunder Invent. Math., Acta Math., og Duke Math. J. Vores førende position vakte atter international bevå-genhed idet Haagerup modtog en af kun 277 æresdoktorgrader tildelt af Kina de sidste 30 år.
UDDANNELSE OG FORMIDLING. Vi afholdt 16 kandidatkurser og 2 læsekurser i 2013. Vores formidlingsak-tiviter indbefattede målrettet talentpleje, såsom mentorskab for Georg Mohr–konkurrencen og det danske hold
ved den Internationale Matematikolympiade, såvel som mere bredtfav-nende aktiviteter som en stand ved den københavnske Kulturnat. Voresformidlingsindsatser gik ikke upåagtet hen idet Eilers vandt KU SCIEN-CE Formidlingsprisen 2013, med en udtalelse der nævnte adskillige afvores aktiviteter fra tidligere år.
Besøg vores hjemmeside sym.math.ku.dk for mere information.
Annual report 2013 Centre for Symmetry and Deformation
2 OrganizationBelow is an updated diagram of the center’s scientific structure, based on the appendix staff list:
E. Schrohe
C
BSymmetry of Manifolds
Group leader: Jesper Grodal
of Non−Commutative Manifolds
Symmetry and Deformation
Group leader: Ryszard Nest
15 PhD students
Symmetry and Deformation
Director Jesper Grodal
Jesper Møller Søren Eilers
Uffe Haagerup
Homotopical Group Theory
A
14 postdocs Magdalena Musat
Mikael Rørdam
Erik Christensen
Erik Pedersen
Ib Madsen
Group leader: Nathalie Wahl
J. Ventura
A. Chermak
E. Yalcin
Bob Oliver
Kasper Andersen
Søren Galatius
Lars Hesselholt
Bohr Professor
M. Boyle
I. Farah
H. Matui
T. Natsume
G. Resdorff
E. Ruiz
J. Savinien
2.1 Scientific staffThe composition of the scientific staff has been adjusted in the following ways since last year:
• Musat has replaced Pedersen as core member, who continues in an associated role.
• Henke was promoted from postdoc to assistant professor per 1/9/2013.
• Ultimo 2013 the number of postdocs/assist. prof was 14 (ult. 2012: 13; contract: 6).
• Ultimo 2013 the number of PhD students was 15 (ult. 2012: 17; contract: 9).
Five PhD students (de Kleijn, Laude, Olesen, Prytula, and Ungheretti) were hired during 2013 to the3-year PhD program, with a sixth (Canlubo) to start Jan 2014; a seventh (A. Høgenhaven) is affiliatedthrough Hesselholt’s Bohr Professorship. This is a larger enrolment than last years 3, but reflects thenumber of graduations, keeping the total number steady. Of the six PhD students who graduated(Anevski, Bentmann, Danielsen, Klamt, de Laat and Nørgard-Sørensen), Bentmann went to Gottingen andde Laat to Leuven as postdocs. Anevski got a job at the Swedish Security Service, Danielsen works at ahigh school, Nørgard-Sørensen got a job at a computer company, while Klamt is trying out consulting.Hence this year saw a more diverse choice of career paths amongst PhDs than in previous years, wherethe majority continued in academia. But we notice with pride that those who continued in academiaagain this year secured jobs at top institutions, and that the people continuing in other directions havehad success in finding high-level jobs in a diverse range of fields.
Five postdocs where hired in 2013 (Degrijse, Enders, Gandini, Patchkoria, Shulman), and further fouraffiliated; three (W. Steimle, D. Clausen and C. Davis) via Hesselholt’s Bohr professorship, and one(F. Lutz) as part of an off-shoot project on experimental mathematics directed by Eilers. Three postdocsleft the center: Gimperlein got a lectureship at Heriot-Watt University Edinburgh, Lahtinen got a postdocat KTH, and Wagner got a postdoc at Helsinki.
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Annual report 2013 Centre for Symmetry and Deformation
2.2 VisitorsOur total number of long and short-term visitors continued their steady climb in 2013: We had 32 long-term visitors, visiting the center for a minimum of 3 weeks, as well as 214 short-term guests (164 in 2012),some of these joint with Hesselholt’s Bohr professorship. In particular Matui was visiting during thespring and Boyle during the fall, working on different aspects of operator algebras and dynamical systemsin (C). One further highlight was the active summer program, where several teams of researchers gottogether at the center during the teaching break, mixing buzzing activity and summer feel: Related to(A), Chermak, Oliver, Ventura, and Yalcin were all visiting throughout the summer; Galatius was visiting,working on (B), and a group around Buch was working on one of our off-shoot projects.
2.3 AdministrationThe basic administrative structure within the center continued unchanged from last year, with hiringdecisions being made collectively by the permanent members as described in previous reports, and withthe interaction with the department following the guidelines set out in our business protocol createdin 2012. A new steering group put in place for the 2015-2019 research plan, already started havingplanning meetings, and played an important role in connection with the renewal process, which took upa lot of energy during the year. One big administrative success in 2013 was the successful handing-overof the administration of the annual postdoc hiring and the bi-annual PhD hiring from the center to thedepartment: As detained in earlier reports, the center had set up these procedures on the departmentlevel, with the hope that the department would eventually run it to everyones benefit, and it is a greatsatisfaction to see this come true. In general several department-wide improvements in the operation,in particular with respect to visitors and communication, helped to ease the center’s administrativeburden. We furthermore hired a student assistant Jørgensen for 10hr/week, to aid center administratorArklint with tasks like webpages and data collection.
3 Research planWe here provide an update on our research, which is structured under three headings (A) Homotopi-cal group theory, (B) Symmetry of manifolds, and (C) Symmetry and deformation of non-commutativemanifolds, with 3 research goals each. We have seen satisfactory progress on all fronts.
(A) Homotopical group theory
Core: J. Grodal, J. Møller.
Visiting: K. Andersen, B. Oliver, M. Szymik, E. Yalcin, J. Cantarero, J. Ventura, A. Chermak, A. Buch, E. Marberg.
Postdocs: E. Henke, E. Meir, J. Foley, D. Degrijse, I. Patchkoria.
PhDs: T. Nørgaard-Sørensen (JG), S. Reeh (JG), M. W. Jacobsen (JM), Laude (JG), Prytula (JMM).
(A1) p-local finite groups and their relationship to group and representation theory. Henke gave acharacterization of saturated fusion systems over abelian 2–groups, establishing a conjecture of Kuhlsham-mer–Navarro–Sambale–Tiep. Grodal’s student Reeh finished his work on transfer maps in p–local Burn-side rings, providing, amongst other things, a new explicit construction of characteristic idempotentsof fusion systems, and improving on a number of constructions in the existing litterature. Finally theactivity during the summer lead to several new projects ranging from the so-called glueing conjectureto the subgroups of definition of p–local finite groups, in addition to new results on Burnside rings. Wehope to report on these developments next year.
(A2) Group actions via homotopy theory. We have started up several new projects in this area duringthe last year. Grodal’s new PhD student Laude is examining the space of maps from the classifyingspace of a finite p–group to the classifying space of a p–local finite group, without introducing anyp–completion, and she has already obtained a description for certain classes of p–local finite groups.
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In another new direction, Møller’s new student Prytula is examining the homotopy theory of Davis–Januszkiewicz spaces, hence building a bridge to questions studied in geometric group theory. As anupdate to last year’s report, the mentioned paper by Benson–Grodal–Henke was accepted to the leadingjournal Invent. Math.
(A3) The homotopy theory of loop groups and affine p-compact groups. Postdoc Foley has beeninitiating a lot of activity in this area, and organized both a masterclass about perspectives on Kac–Moody groups and workshop on p–local compact groups. This has led to new results on in particularthe self-equivalences and Adams operations of classifying spaces of Kac–Moody groups. Furthermorethere has been a renewed push to understand the connections to string topology of classifying spaces,where we have seen much progress as described under (B1).
(B) Symmetry of manifolds
Core: I. Madsen, N. Wahl. (Associated: E. Pedersen.)
Visiting: T. Bargheer, S. Galatius.
Postdocs: A. Berglund, H. Gimperlein, A. Lahtinen, G. Gandini.
PhDs: T. Danielsen (ext), E. Dotto (IM), D. Egas (NW), C. Guldberg (NW), A. Klamt (NW), M. Lopez (EKP),K. Moi (IM), M. Grey (IM), M. Ungheretti (NW).
(B1) Symmetry of Riemann surfaces. Hepworth and Lahtinen’s construction of a field theory associatedto free loop spaces on classifying spaces came out as a 93 pages preprint. They construct what they calla homological h-graph field theory for such spaces, which extends a more classical homological conformalfield theory, build from surfaces, earlier constructed by Chataur–Menichi. Egas, in three upcomingpapers, gives a direct comparison of different graph models for the moduli space of Riemann surfaces.The different models were each constructed in view of particular applications, and a direct comparisonallows to transport good properties of one model to another. In particular, she gives the first proofthat a combinatorial graph gluing used by Konstevich–Soibelman while studying Hochschild homologyof certain algebras is, as was hoped, a combinatorial model of the gluing of Riemann surfaces along theirboundary.
(B2) Symmetry of 3-manifolds. Berglund and Madsen continued their study of diffeomorphisms and ho-motopy automorphisms of high dimensional manifolds. They obtained the striking result that the stablerational cohomology of the homotopy automorphisms of a connected sum of Sd ×Sd, d large, comprisesall unstable rational homology groups of all outer automorphism groups of free groups, otherwise knownto relate to the mapping class group of connected sums of S1 × S1. Galatius and Randal-Williams havelikewise made continuous progress on the study of diffeomorphisms of high dimensional manifolds. Oneof their new results is a proof of the non-triviality of certain characteristic classes of bundles of smoothclosed manifolds.
(B3) Algebraic K-theory and classification of manifolds. Dotto and Moi continued the study ofHesselholt and Madsen’s real K–theory, and its relation to equivariant Goodwillie calculus and theequivariant completion theorem. Gandini–Meinert–Rueping showed that the Farrell–Jones Conjectureholds for fundamental groups of graphs of groups with abelian vertex groups—as a special case thisshows that the Farrall–Jones conjecture holds for generalized Baumslag–Solitar groups.
(C) Symmetry of noncommutative manifolds
Core: S. Eilers, U. Haagerup, R. Nest, M. Rørdam, M. Musat. (Associated: E. Christensen.)
Visiting: H. Ando, M. Boyle, I. Farah, H. Matui, T. Natsume, G. Resdorff, E. Ruiz, J. Savinien, E. Schrohe.
Postdocs: S. Deprez, T. Crisp, R. Johansen. S. Arklint, S. Wagner, D. Enders.
PhDs: S. Anevski (RN), R. Bentmann (SE/RN), J. Gabe (SE), S. Knudby (UH), T. de Laat (MM/UH), K. Li(RN/UH), N. de Kleijn (RN), K. Olesen (MM/UH).
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(C1) Baum–Connes conjecture and KK-theory as a triangulated category. Li and Deprez extendedthe notions of Property A of discrete metric spaces to locally compact groups and related it to amenabilityat infinity and to uniform embeddability into a Hilbert space. As one of the corollaries of this workthey proved that, for a locally compact group that embeds uniformly into a Hilbert space, the Baum–Connes assembly map with coefficients is split-injective and that the continuous analogue of the Novikovconjecture holds for such groups.
(C2) Deformation quantization and index theorems. Nest with Kaad studied the torsion for com-muting families of bounded operators and, in particular, computed the determinant of the associatedKoszul complex. This is a ‘higher index’ on algebraic K2, and in particular recovers the Tate tame sym-bol for singular complex varieties. Nest with Schrohe constructed a symbol calculus for Fourier integraloperators on manifolds with boundary. As a byproduct they obtained invariants of a certain class ofsymplectomorphisms of manifolds with boundary.
(C3) Classification of C∗-algebras. Monod, Kellerhals, and Rørdam have given new geometric, ana-lytic, and dynamical characterisations of supramenable groups, and they have shown how these groupsgive rise to classifiable Kirchberg algebras. Eilers and his group have done extensive work on the clas-sification of non-simple purely infinite C∗–algebras by sophisticated K–theoretical invariants, inspiredby the groundbreaking work of Kirchberg, and the later work by Meyer–Nest. Eilers, Restorff and Ruizhave obtained new strong classification results for extensions of classifiable C∗–algebras. Eilers’ studentBentmann together with Dadarlat have classified continuous fields over the interval of stable Kirchbergalgebras; and Arklint, Bentmann and Katsura have determined the range of the invariant for Cuntz–Krieger algebras and for graph algebras, giving an abstract characterisation of these algebras.
4 Comments to the appendixOverall we see healthy benchmark numbers in both research, activities, education, and outreach. Belowwe give a discussion of the individual parts and a comparison to previous years.
Appendix—A: External relations. The table lists 79 external collaborators on journal articles (2012:66), the vast majority international. We also mention that we in 2013 hosted 214 scientific guests (2012:164) at the center, some of these joint with Hesselholt’s Bohr professorship. (Here a guest is someonewho gets a building key and office space during their visit.)
Appendix—B: Conferences. B-a lists no less than 15 events at the center and 8 external events. Ofthe 15 held at SYM, 5 were workshops and 10 were masterclasses. This is more than a doubling overlast year, which had 7 events at the center. The year was unusually busy since a lot of center postdocsdecided to organize masterclasses, which we actively encourage—the steady stream of workshop visitorsto the center has definitely helped raise awareness of our research around the world.
B-b lists 86 invited center talks distributed on 65 venues. The number is just below last years 91, andsignificantly above the numbers from the two first years which were 67 and 55. In particular we are happyto see a continuation of the positive trend from last year, that our junior center members frequently getinvited to seminar and conference talks around the world. We also note the wide geographic diversity ofthe talks. In terms of highlights we, e.g., mention the plenary talk by Haagerup at the EMS-DMF JointMath Weekend, and the plenary talk by Grodal at the 26th Nordic Conference of Mathematicians.
Appendix—C: Educational activities. The table lists 16 graduate courses and 2 reading seminars, andstates that we in 2013 produced 5 MS graduates and 11 BS graduates. Of the 5 MS students, 4 have beenadmitted to PhD programs, of which 2 at the center, and an additional student is currently applyingabroad. The number of MS theses is on the lower side this year (2011: 10, 2012: 7). We ascribe thisin part to statistical fluctuations, but also remark that, perhaps bizarrely, the high number of graduatecourses in the center’s area may in fact make it harder to reach the starting point of a thesis, and hence
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Annual report 2013 Centre for Symmetry and Deformation
might discourage weaker students from writing theses in the center’s area—this is consistent with thatmost of our MS students continue to do PhDs.
Appendix—D: External funding. Also in 2013 we received significantly more in external funding thanstated in the contract, again mainly due to the 3 ERC grants and the KU “Programme of Excellence”grant, totalling ca. 9 MDKK plus overhead. We are aware that several of these grants are set to expire,which makes it imperative to find new grants to replace them.
Appendix—E: Awards. In this category, Eilers won the UCPH Faculty of Science Dissemination Award,for his many successes in communicating the power and beauty of mathematics to a lay audience (LEGOtilings, guessing numbers, etc.). We also have Haagerup’s prestigious honorary doctorate from East ChinaNormal University, one of only 277 honorary doctorates awarded by China the last 30 years.
Appendix—F: Public outreach. The numbers and types of events in this category are virtually un-changed from 2012, with 14 events. Our collaboration with artist Bugge was rounded off with a publicinauguration and evening debate entitled “Words of Mathematics”. We also notice that we did get astand at the Copenhagen Culture Night, an old ambition, which was a success. We are also happy thatwe kept our engagement in UNF, Georg Mohr, and IMO, which are important recruiting tools for topstudents, though we will need a replacement organizer, since Reeh is graduating and will go to MITas a postdoc—we wish to broaden the group of PhD students and postdocs that take part in outreachactivities, but have not yet found a good way of encouraging and incentivizing this.
Appendix—H: Publications. As noted also in previous years, the usual 2-4 year difference between theyear in which a result is found and the year it is printed makes short-term statistics difficult. One of theleast lagging indicators for research volume is the number of preprints, where we had 69 new preprintsadded to our CPH-SYM preprint series in 2013, compared to 48 in 2012 and 31 in 2011, a strong increase.22/8 papers appeared in peer reviewed journals/proceedings in 2013, compared to 29/0 in 2012 and 26/3in 2011. We have another 14/2 papers in the accepted-in-2013-but-not-yet-appeared category (many ofwhich are already available from the journal as “Online First”). In terms of top journal publications in2013, we had one paper by Benson–Grodal–Henke accepted to Invent. Math. (arXiv:1210.1564) one paperby Galatius–Randal-Williams accepted to Acta Math. (arXiv:1201.3527) and one paper by Christensenet al. accepted to Duke Math. J. (arXiv:1209.4116). The papers arXiv:1203.6479 and arXiv:1201.1250 byOliver and Haagerup–de Laat accepted at the time of the last report have now appeared in Acta Math.and Duke Math. J. respectively. We hence conclude that the center’s research output is doing very well,both in terms of quantity and quality.
5 SignatureI hereby confirm the correctness of the information concerning annual accounts, including itemizations.Also, I confirm that the compiled annual reporting, including the appendices, is correct, i.e. it is free ofmaterial misstatement or omissions, and that the administration of the funds has been secure and sound,and in accordance with the conditions of the center agreement.
