MSP

This volume is a pro­ceed­ings of the sixth Ban­ff In­ter­na­tion­al Re­search Sta­tion work­shop on “In­ter­ac­tions of gauge the­ory with con­tact and sym­plect­ic to­po­logy in di­men­sions 3 and 4”, held in 2020 over the in­ter­net. The volume con­tains six­teen ref­er­eed pa­pers, with an em­phas­is on new de­vel­op­ments and in­ter­con­nec­tions of gauge the­ory, low-di­men­sion­al to­po­logy, con­tact and sym­plect­ic to­po­logy. It is rep­res­ent­at­ive of the high level of talks and res­ults presen­ted at the In­ter­ac­tions work­shops over the years since its in­cep­tion in 2007.

The Al­gorithmic Num­ber The­ory Sym­posi­um (ANTS), held bi­en­ni­ally since 1994, is the premi­er in­terna­tion­al for­um for re­search in com­pu­ta­tion­al and al­gorithmic num­ber the­ory. ANTS is de­voted to al­gorithmic as­pects of num­ber the­ory, in­clud­ing ele­ment­ary, al­geb­ra­ic, and ana­lyt­ic num­ber the­ory, the geo­metry of num­bers, arith­met­ic al­geb­ra­ic geo­metry, the the­ory of fi­nite fields, and cryp­to­graphy.

This volume is the pro­ceed­ings of the four­teenth ANTS meet­ing, which took place 29 June to 4 Ju­ly 2020 via video con­fer­ence, the plans for hold­ing it at the Uni­versity of Auck­land, New Zea­l­and, hav­ing been dis­rup­ted by the COV­ID-19 pan­dem­ic. The volume con­tains re­vised and ed­ited ver­sions of 24 ref­er­eed pa­pers and one in­vited pa­per presen­ted at the con­fer­ence.

Poin­caré du­al­ity is cent­ral to the un­der­stand­ing of man­i­fold to­po­logy. Di­men­sion 3 is crit­ic­al in vari­ous re­spects, be­ing between the known ter­rit­ory of sur­faces and the wil­der­ness mani­fest in di­men­sions ≥ 4. The main thrust of 3-man­i­fold to­po­logy for the past half cen­tury has been to show that as­pher­ic­al closed 3-man­i­folds are de­term­ined by their fun­da­ment­al groups. Re­l­at­ively little at­ten­tion has been giv­en to the ques­tion of which groups arise. This book is the first com­pre­hens­ive ac­count of what is known about PD₃-com­plexes, which mod­el the ho­mo­topy types of closed 3-man­i­folds, and PD₃-groups, which cor­res­pond to as­pher­ic­al 3-man­i­folds. In the first half we show that every P²-ir­re­du­cible PD₃-com­plex is a con­nec­ted sum of in­decom­pos­ables, which are either as­pher­ic­al or have vir­tu­ally free fun­da­ment­al group, and largely de­term­ine the lat­ter class. The pic­ture is much less com­plete in the as­pher­ic­al case. We sketch sev­er­al pos­sible aproaches for tack­ling the cent­ral ques­tion, wheth­er every PD₃-group is a 3-man­i­fold group, and then ex­plore prop­er­ties of sub­groups of PD₃-groups, uni­fy­ing many res­ults of 3-man­i­fold to­po­logy. We con­clude with an ap­pendix list­ing over 60 ques­tions. Our gen­er­al ap­proach is to prove most as­ser­tions which are spe­cific­ally about Poin­caré du­al­ity in di­men­sion 3, but oth­er­wise to cite stand­ard ref­er­ences for the ma­jor sup­port­ing res­ults.

Tar­get read­er­ship: Gradu­ate stu­dents and math­em­aticians with an in­terest in low-di­men­sion­al to­po­logy.

The Al­gorithmic Num­ber The­ory Sym­posi­um (ANTS), held bi­en­ni­ally since 1994, is the premi­er inter- na­tion­al for­um for re­search in com­pu­ta­tion­al num­ber the­ory. ANTS is de­voted to al­gorithmic as­pects of num­ber the­ory, in­clud­ing ele­ment­ary, al­geb­ra­ic, and ana­lyt­ic num­ber the­ory, the geo­metry of num­bers, arith­met­ic al­geb­ra­ic geo­metry, the the­ory of fi­nite fields, and cryp­to­graphy.

This volume is the pro­ceed­ings of the thir­teenth ANTS meet­ing, held Ju­ly 16–20, 2018, at the Uni­versity of Wis­con­sin-Madis­on. It in­cludes re­vised and ed­ited ver­sions of 28 ref­er­eed pa­pers presen­ted at the con­fer­ence.

The Al­gorithmic Num­ber The­ory Sym­posi­um (ANTS), held bi­en­ni­ally since 1994, is the premi­er in­ter­na­tion­al for­um for re­search in com­pu­ta­tion­al num­ber the­ory. ANTS is de­voted to al­gorithmic as­pects of num­ber the­ory, in­clud­ing ele­ment­ary, al­geb­ra­ic, and ana­lyt­ic num­ber the­ory, the geo­metry of num­bers, arith­met­ic al­geb­ra­ic geo­metry, the the­ory of fi­nite fields, and cryp­to­graphy.

This volume is the pro­ceed­ings of the tenth ANTS meet­ing, held Ju­ly 9–13, 2012, at the Uni­versity of Cali­for­nia, San Diego. It in­cludes re­vised and ed­ited ver­sions of the 25 ref­er­eed pa­pers presen­ted at the con­fer­ence, to­geth­er with ex­ten­ded ab­stracts of two of the five in­vited talks.

There has been a long his­tory of rich and subtle con­nec­tions between low-di­men­sion­al to­po­logy, map­ping class groups and geo­met­ric group the­ory. From 1 to 5 Ju­ly 2013, the con­fer­ence “In­ter­ac­tions between low di­men­sion­al to­po­logy and map­ping class groups,” held at the Max Planck In­sti­tute for Math­em­at­ics in Bonn, high­lighted these di­verse con­nec­tions and fostered new and un­ex­pec­ted col­lab­or­a­tions between re­search­ers in these areas.

The pro­ceed­ings for this con­fer­ence aim to fur­ther draw at­ten­tion to the beau­ti­ful math­em­at­ics emer­ging from di­verse in­ter­ac­tions between these areas. The art­icles col­lec­ted in this volume, in ad­di­tion to gath­er­ing new res­ults, also con­tain ex­pos­i­tions and sur­veys of the latest de­vel­op­ments in vari­ous act­ive areas of re­search at the in­ter­face of map­ping class groups of sur­faces and the to­po­logy and geo­metry of 3- and 4-di­men­sion­al man­i­folds. Many open prob­lems and new dir­ec­tions for re­search are dis­cussed.

This volume is ded­ic­ated to Mike Freed­man on the oc­ca­sion of his 60th birth­day. It con­tains pa­pers on a wide vari­ety of top­ics in To­po­logy and Phys­ics and is based on two con­fer­ences that were held in 2011. The Con­fer­ence on Low-Di­men­sion­al Man­i­folds and High-Di­men­sion­al Cat­egor­ies, in hon­or of Mi­chael Freed­man, was held at UC Berke­ley on June 6–10, 2011. It fea­tured talks by Yakov Eli­ash­berg, Ron Fin­tushel, Dav­id Gay, Camer­on Gor­don, Sergei Gukov, Ko Honda, Mi­chael Hutch­ings, Slava Krushkal, Yanki Lekili, Scott Mor­ris­on, Frank Quinn, Mar­tin Schar­le­mann, Rob Schnei­der­m­an, Cath­ar­ina Strop­pel, Clif­ford Taubes, Peter Teich­ner, Ben Web­ster, Kat­rin Wehrheim, and Ed­ward Wit­ten. The or­gan­iz­ing com­mit­tee con­sisted of Ian Agol, Ro­bi­on Kirby, Slava Krushkal, Peter Teich­ner, Abi­gail Thompson, and Kev­in Walk­er.

The Freed­man Sym­posi­um was held at the Kavli In­sti­tute for The­or­et­ic­al Phys­ics in Santa Bar­bara on April 15–17, 2011. In­vited lec­tures were giv­en by Scott Aaron­son, Sank­ar Das Sarma, Ed­die Far­hi, Mat­thew Fish­er, Dan Freed, Matt Hast­ings, Charlie Kane, Alexei Kit­aev, Steve Kiv­el­son, Laci Lovasz, An­dreas Lud­wig, John Man­fer­del­li, Greg Moore, Chet­an Nayak, John Preskill, Xiao­l­i­ang Qi, Peter Shor, Kir­ill Shten­gel, Steve Si­mon, and Xiao-Gang Wen. The con­fer­ence was or­gan­ized by Chet­an Nayak and Zhenghan Wang.

The pub­lic­a­tion of this volume and the con­fer­ences were sup­por­ted in part by Zheng-Xu He, the Na­tion­al Sci­ence Found­a­tion, and Mi­crosoft Sta­tion Q.

Pois­son geo­metry is a rap­idly grow­ing sub­ject, with many in­ter­ac­tions and ap­plic­a­tions in areas of math­em­at­ics and phys­ics, such as clas­sic­al dif­fer­en­tial geo­metry, Lie the­ory, non­com­mut­at­ive geo­metry, in­teg­rable sys­tems, flu­id dy­nam­ics, quantum mech­an­ics, and quantum field the­ory. Re­cog­niz­ing the role played by Pois­son geo­metry and the sig­ni­fic­ant re­search it has gen­er­ated, the Ab­dus Salam In­ter­na­tion­al Centre for The­or­et­ic­al Phys­ics in Trieste, Italy, sponsored a 3-week sum­mer activ­ity on this sub­ject (Ju­ly 4–22 2005) in or­der to bring it to the at­ten­tion of sci­ent­ists and stu­dents from de­vel­op­ing coun­tries. There was an over­whelm­ing re­sponse to this pro­gram, which brought to­geth­er more than 150 par­ti­cipants from all over the world with var­ied back­grounds, from gradu­ate stu­dents to ex­perts.

The pro­gram con­sisted of a two-week in­tens­ive school com­pris­ing 10 minicourses, fol­lowed by a week-long in­ter­na­tion­al re­search con­fer­ence. The lec­tur­ers at the school were asked to turn their notes in­to sec­tions of a book that could serve as a quick in­tro­duc­tion to the cur­rent state of re­search in Pois­son geo­metry. We hope that the present volume will be use­ful to people who want to learn about Pois­son geo­metry and its ap­plic­a­tions.

In the late twen­ti­eth cen­tury, stable ho­mo­topy the­ory ex­pan­ded rap­idly and be­came in­creas­ingly soph­ist­ic­ated in de­fin­ing ho­mo­top­ic­ally in­vari­ant al­geb­ra­ic ma­chinery as­so­ci­ated with mul­ti­plic­at­ive co­homo­logy the­or­ies and their in­tern­al op­er­a­tions. In­puts to these de­vel­op­ments have in­cluded es­tab­lished math­em­at­ic­al ideas from sub­jects such as al­geb­ra­ic geo­metry and num­ber the­ory. The work­shop “New To­po­lo­gic­al Con­texts for Galois The­ory and Al­geb­ra­ic Geo­metry” brought to­geth­er to­po­lo­gists in­volved in de­vel­op­ing or us­ing these new tech­niques and al­lowed for the in­ter­ac­tions with oth­er sub­ject areas by in­clud­ing non-to­po­lo­gist par­ti­cipants who would con­trib­ute to this.

This book is de­voted to the study of mod­uli spaces of Seiberg–Wit­ten mono­poles over spin^c Rieman­ni­an 4-man­i­folds with long necks and/or tu­bu­lar ends. The ori­gin­al pur­pose of this work was to provide ana­lyt­ic­al found­a­tions for a cer­tain con­struc­tion of Flo­er ho­mo­logy of ra­tion­al ho­mo­logy 3-spheres; this is car­ried out in “Mono­pole Flo­er ho­mo­logy for ra­tion­al ho­mo­logy 3-spheres” [arX­iv 0809.4842]. However, along the way the pro­ject grew, and, ex­cept for some of the trans­vers­al­ity res­ults, most of the the­ory is de­veloped more gen­er­ally than is needed for that con­struc­tion. Flo­er ho­mo­logy it­self is hardly touched upon in this book, and, to com­pensate for that, I have in­cluded an­oth­er ap­plic­a­tion of the ana­lyt­ic­al ma­chinery, namely a proof of a “gen­er­al­ized blow-up for­mula” which is an im­port­ant tool for com­put­ing Seiberg–Wit­ten in­vari­ants.

The book is di­vided in­to three parts. Part 1 is al­most identic­al to my pa­per “Mono­poles over 4-man­i­folds con­tain­ing long necks, I” [Geom. To­pol. 9 (2005) 1–93]. The oth­er two parts con­sist of pre­vi­ously un­pub­lished ma­ter­i­al. Part 2 is an ex­pos­it­ory ac­count of glu­ing the­ory in­clud­ing ori­ent­a­tions. The main nov­el­ties here may be the for­mu­la­tion of the glu­ing the­or­em, and the ap­proach to ori­ent­a­tions. In Part 3 the ana­lyt­ic­al res­ults are brought to­geth­er to prove the gen­er­al­ized blow-up for­mula.

This volume is ded­ic­ated to Hein­er Zi­eschang, who has been a teach­er, ment­or and friend to us and to most of those that have con­trib­uted their work con­tained in this volume.

Con­tri­bu­tions by: Jochen Abhau, Carl-Friedrich Bödigheimer, Anne Bauval, Se­meon Bogatyi, Oleg Bogo­pol­ski, Michel Boileau, Steve Boy­er, Daryl Cooper, Ralf Ehren­fried, Jan Fricke, Olga Frolk­ina, Claude Hay­at, C. Hog-An­geloni, Don­ald J. Collins, An­dré Jäger, Mi­chael Ka­povich, Akio Kawau­chi, Ul­rich Kos­chorke, Elena Kudryavt­seva, Da­ciberg L. Gonçalves, Maria Her­mínia de Paula Leite Mello, Mar­tin Lust­ig, S. Matveev, Mat­tia Mec­chia, To­motada Oht­suki, Lu­isa Paoluzzi, Joan Porti, Le­onid Potya­gailo, Mar­tin R. Brid­son, Robert Ri­ley, Makoto Sak­uma, Mar­tin Schar­le­mann, Wil­helm Sing­hof, Juan Souto, Kon­stantin Svi­ridov, Satoshi To­moda, Alina Vdov­ina, Ern­est Vin­berg, El­mar Vo­gt, Shicheng Wang, Richard Weidmann, Hein­er Zi­eschang, Bruno Zi­m­mer­mann, Peter Zven­growski.

This volume is the pro­ceed­ings of the con­fer­ence “Groups, Ho­mo­topy and Con­fig­ur­a­tion Spaces” held at the Uni­versity of Tokyo, Ju­ly 5–11, 2005, in hon­or of the 60th birth­day of Fred Co­hen. The em­phas­is of the con­fer­ence was on co­homo­logy of groups, clas­sic­al and mod­ern ho­mo­topy the­ory, geo­metry and to­po­logy of con­fig­ur­a­tion spaces and re­lated top­ics. However, the con­fer­ence was in­ten­ded to have a broad scope, with talks on a vari­ety of top­ics of cur­rent in­terests in to­po­logy. The or­gan­iz­ing com­mit­tee con­sisted of Norio Iwase, Toshi­take Kohno, Ran Levi, Dai Ta­maki and Jie Wu. The con­fer­ence was sup­por­ted by the COE pro­gram of the Gradu­ate School of Math­em­at­ic­al Sci­ences, The Uni­versity of Tokyo.

The no­tion of a Hee­gaard split­ting of a 3-man­i­fold is as old as 3-di­men­sion­al to­po­logy it­self; we may re­call, for ex­ample, that Poin­caré de­scribed his do­deca­hed­ral space by means of a Hee­gaard dia­gram. It also seems to be the case that one of the early mo­tiv­a­tions for the study of auto­morph­isms of sur­faces was the de­sire to un­der­stand 3-man­i­folds through their Hee­gaard split­tings. Nev­er­the­less, for many years know­ledge about Hee­gaard split­tings was lim­ited; the fol­low­ing is a more or less com­plete list of things that were known up to 1970: 3-man­i­folds are tri­an­gulable, and hence pos­sess Hee­gaard split­tings (Moise, 1952); any two Hee­gaard split­tings of a giv­en 3-man­i­fold be­come iso­top­ic afer some num­ber of sta­bil­iz­a­tions (Re­idemeister, Sing­er, 1933); S³ has a unique split­ting of any genus up to iso­topy (Wald­hausen, 1968); Hee­gaard genus is ad­dit­ive un­der con­nec­ted sum (Haken, 1968); the al­geb­ra­ic char­ac­ter­iz­a­tion of Hee­gaard split­tings in terms of split­ting ho­mo­morph­isms (Stallings, 1966).

Start­ing in the 1980s, pro­gress in the sub­ject began to ac­cel­er­ate and it entered more and more in­to the main­stream of 3-di­men­sion­al to­po­logy, with de­vel­op­ments com­ing from sev­er­al dif­fer­ent dir­ec­tions. There are now enough gen­er­al res­ults and tech­niques es­tab­lished to jus­ti­fy speak­ing of the the­ory of Hee­gaard split­tings. A (cer­tainly in­com­plete) list of re­cent ad­vances in the sub­ject is the fol­low­ing: the clas­si­fic­a­tion of Hee­gaard split­tings of Seifert fiber spaces; the no­tion of strong ir­re­du­cib­il­ity; the in­tro­duc­tion of the curve com­plex in­to the study of Hee­gaard split­tings; the use of nor­mal and al­most nor­mal sur­faces; res­ults ob­tained us­ing Cerf the­ory (sweep-outs); the ap­plic­a­tion of the the­ory of min­im­al sur­faces; geo­met­ric to­po­lo­gic­al meth­ods, in­clud­ing the the­ory of lam­in­a­tions; res­ults re­lat­ing Hee­gaard split­tings to hy­per­bol­ic struc­tures, for in­stance hy­per­bol­ic volume; res­ults on the tun­nel num­ber of knots; the use of Hee­gaard split­tings to define Hee­gaard–Flo­er ho­mo­logy.

It was against this back­ground that the Tech­nion Work­shop on Hee­gaard Split­tings was held in the sum­mer of 2005. The goal was to gath­er people with a spe­cif­ic in­terest in Hee­gaard split­tings in one work­shop where the state of the art could be ex­posed and dis­cussed.

It was de­cided by the par­ti­cipants to pub­lish a pro­ceed­ings of the work­shop with the hope of mak­ing avail­able to in­ter­ested people a con­cen­trated source of in­form­a­tion about the cur­rent state of re­search on Hee­gaard split­tings. Some pa­pers were so­li­cited from non-par­ti­cipants whose in­terests are close to the field. We wish to thank all those who con­trib­uted to this volume for their ef­forts.

The volume also con­tains a list of prob­lems about Hee­gaard split­tings, con­trib­uted by some of the work­shop par­ti­cipants.

The Pro­ceed­ings of the in­ter­na­tion­al School and Con­fer­ence in Al­geb­ra­ic To­po­logy, Hà Nội 2004, is a col­lec­tion of art­icles in hon­our of Huỳnh Mùi, the founder of the Vi­et­nam school in Al­geb­ra­ic To­po­logy.

Not long ago, Hà Nội, the cap­it­al city, was known as the cent­ral icon of the long and ter­rible Vi­et­nam War. Nowadays, Hà Nội is proud to be known as a young centre of math­em­at­ics. The in­ter­na­tion­al School and Con­fer­ence in Al­geb­ra­ic To­po­logy, Hà Nội 2004, was the first not­able meet­ing on Al­geb­ra­ic To­po­logy in Vi­et­nam with the par­ti­cip­a­tion of an im­press­ive num­ber of both young and in­ter­na­tion­ally es­tab­lished Al­geb­ra­ic To­po­lo­gists.

The Hà Nội 2004 Pro­ceed­ings’ main top­ics are the Steen­rod al­gebra, in­vari­ant the­ory, clas­si­fy­ing spaces, and group co­homo­logy. It con­tains tran­scripts of some of the school courses and the con­fer­ence talks as well as re­lated art­icles sub­mit­ted spe­cific­ally for the Pro­ceed­ings. Most of the art­icles in the Pro­ceed­ings present ori­gin­al re­search with proofs. Oth­ers are sur­vey art­icles writ­ten by lead­ing ex­perts.

A ma­jor in­ter­na­tion­al meet­ing on ho­mo­topy the­ory took place in Kino­saki, Ja­pan, from Ju­ly 28–Au­gust 1 2003, fol­lowed on Au­gust 4–8 by an in­tense satel­lite con­fer­ence at the Nagoya In­sti­tute of Tech­no­logy. This volume con­tains the Pro­ceed­ings of those con­fer­ences. They, and this volume, are ded­ic­ated to Pro­fess­or Gôrô Nishida on the oc­ca­sion of his 60th birth­day.

Nishida’s earli­est work grew out of the study of in­fin­ite loopspaces. His first pa­per (in 1968, on what came even­tu­ally to be known as the Nishida re­la­tions) ac­counts for in­ter­ac­tions between Steen­rod and Dyer–Lashof (Kudo–Araki) op­er­a­tions. This was fol­lowed by early work with H. Toda on the ex­ten­ded power con­struc­tion, which led in 1973 to his mile­stone proof of the nil­po­tence of pos­it­ive-de­gree ele­ments in the stable ho­mo­topy ring of spheres.

This res­ult, whose echoes con­tin­ue to re­ver­ber­ate today in work of Dev­in­atz, Hop­kins, Smith, and oth­ers on the chro­mat­ic pic­ture, and in work on motives in al­geb­ra­ic geo­metry, stood at the time as an isol­ated beacon of hope in the (then very mys­ter­i­ous) world of stable ho­mo­topy the­ory. It, to­geth­er with the Kahn–Priddy the­or­em, was one of the first signs that the sub­ject pos­sesses deep glob­al prop­er­ties—that it held struc­tur­al secrets well bey­ond its already for­mid­able com­pu­ta­tion­al as­pects. Nishida next turned his at­ten­tion to a circle of ideas sur­round­ing the Segal con­jec­ture, trans­fer ho­mo­morph­isms, and stable split­tings of clas­si­fy­ing spaces of groups. The ideas in this series of pa­pers have by now grown in­to a rich sub­field of ho­mo­topy the­ory, with im­port­ant con­tri­bu­tions by Ben­son, Fesh­bach, Mar­tino, Mi­n­ami, Priddy, Webb, and many oth­ers; it con­tin­ues today in (for ex­ample) the the­ory of p-com­pact groups. In re­cent years much of his work has been con­cerned with vari­ous as­pects of el­lipt­ic co­homo­logy. His deep in­sight from the early 90s, that work of Eichler and Shimura on mod­u­lar forms, high­er S¹-trans­fers, and the dif­feo­morph­ism group of the two-tor­us are all in­tim­ately con­nec­ted, is still not ad­equately un­der­stood; its ex­ploit­a­tion may de­pend on new geo­met­ric ideas from the de­vel­op­ing the­ory of el­lipt­ic ob­jects.

The Work­shop on Exot­ic Ho­mo­logy Man­i­folds took place at MFO (Math­em­at­isches Forschungsin­sti­tut Ober­wolfach) in Ger­many on June 29th – Ju­ly 5th, 2003.

Ho­mo­logy man­i­folds were de­veloped in the first half of the 20th cen­tury to give a pre­cise set­ting for Poin­caré’s ideas on du­al­ity. Ma­jor res­ults in the second half of the cen­tury came from two dif­fer­ent areas. Meth­ods from the point-set tra­di­tion were used to study ho­mo­logy man­i­folds ob­tained by di­vid­ing genu­ine man­i­folds by fam­il­ies of con­tract­ible sub­sets. Exot­ic ho­mo­logy man­i­folds are ones that can­not be ob­tained in this way, and these have been in­vest­ig­ated us­ing al­geb­ra­ic and geo­met­ric meth­ods. The Mini-Work­shop brought to­geth­er ex­perts from both point-set and al­geb­ra­ic areas, along with new PhDs and ex­perts in re­lated areas. This was the first time this was done in a meet­ing fo­cused only on ho­mo­logy man­i­folds. The 17 par­ti­cipants had 14 form­al lec­tures and a prob­lem ses­sion. There was a par­tic­u­lar fo­cus on the proof, 10 years ago, of the ex­ist­ence of exot­ic ho­mo­logy man­i­folds. This gave ex­perts in each area an the op­por­tun­ity to learn more about de­tails com­ing from the oth­er area. There had also been con­cerns about the cor­rect­ness of one of the lem­mas, and this was dis­cussed in de­tail. One of the high points of the con­fer­ence was the dis­cov­ery of a short and beau­ti­ful new proof of this lemma. Ex­tens­ive dis­cus­sions of ex­amples and prob­lems have un­doubtedly helped pre­pare for fu­ture pro­gress in the field.

These pro­ceed­ings for the meet­ing in­clude an art­icle on the his­tory of the sub­ject and a prob­lem list.

There was also a won­der­ful in­ter­ac­tion with the Mini-Work­shop Henri Poin­caré and to­po­logy, which was held in the same week. There was a joint dis­cus­sion on the early his­tory of man­i­folds, and both groups offered even­ing lec­tures on top­ics of in­terest to the oth­er. Sev­er­al of the day­time his­tory lec­tures also drew large num­bers of ho­mo­logy man­i­fold par­ti­cipants.

In the sum­mer of 2001, we (Dave and Jim) were at the Gökova con­fer­ence in Tur­key talk­ing about BIRS, the new math­em­at­ics in­sti­tute that was go­ing to open in Ban­ff, Canada, in 2003. Al­though 2003 seemed like a long way off at the time, we wanted to pro­pose a work­shop. For­tu­it­ously, Jim had re­cently heard about some ex­cit­ing work in phys­ics by Go­pak­u­mar, Vafa and oth­ers that had found some very ex­pli­cit con­nec­tions between to­po­lo­gic­al string the­ory and Chern–Si­mons gauge the­ory—the very same phys­ic­al the­or­ies that led to the math­em­at­ic­al the­or­ies of Gro­mov–Wit­ten in­vari­ants and fi­nite type in­vari­ants. Al­though these ideas had not yet taken hold in the math com­munity, it seemed likely that with­in a few years they would be timely and war­rant a work­shop.

In­deed, by 2003, the top­ic was very timely. Phys­i­cists Aganagic, Klemm, Marino, and Vafa had de­veloped the “to­po­lo­gic­al ver­tex”, a gad­get which (con­jec­tur­ally) com­puted Gro­mov–Wit­ten of tor­ic Calabi–Yau threefolds in terms of cer­tain in­vari­ants of fi­nite type: Chern–Si­mons in­vari­ants. Math­em­aticians Li, Liu, Liu, and Zhou had be­gun to de­vel­op a math­em­at­ic­al frame­work for the to­po­lo­gic­al ver­tex. Garoufal­id­is and Le had just proven the LMOV con­jec­ture. This con­jec­ture en­coded in­teg­ral­ity prop­er­ties of the HOM­FLY(PT) poly­no­mi­al that must hold if the con­jec­tur­al large-N du­al­ity was in­deed true. In ad­di­tion new con­jec­tures re­lat­ing large N du­al­ity with Khovan­ov ho­mo­logy were form­ing.

This volume con­tains pa­pers on a wide range of top­ics in low-di­men­sion­al to­po­logy. It arose out of two events that were held in 2003. The first was the 28th Uni­versity of Arkan­sas Spring Lec­ture Series in the Math­em­at­ic­al Sci­ences, which took place April 10–12, 2003. These an­nu­al con­fer­ences fo­cus on a spe­cif­ic top­ic of cur­rent in­terest in math­em­at­ics, and fea­ture a prin­cip­al lec­turer who gives a series of five lec­tures and se­lects ad­di­tion­al in­vited speak­ers. In 2003 the prin­cip­al lec­turer was An­drew Cas­son, and the title of his lec­ture series was "The An­drews-Curtis and the Poin­care Con­jec­tures". The in­vited speak­ers were Steph­en Bi­gelow, Mar­tin Brid­son, Danny Calegari, Nath­an Dun­field, Camer­on Gor­don, Alan Re­id, Mar­tin Schar­le­mann, Zlil Sela, and Peter Shalen. A spe­cial pub­lic lec­ture was giv­en by Jeff Weeks. There were also sev­er­al con­trib­uted talks. The or­gan­izers were Chaim Good­man-Strauss and Yo'av Rieck. The con­fer­ence was sup­por­ted by NSF Grant DMS-0245047 and by the De­part­ment of Math­em­at­ic­al Sci­ences, Ful­bright Col­lege of Arts and Sci­ences and Gradu­ate School of the Uni­versity of Arkan­sas.

The second event was the Con­fer­ence on the To­po­logy of Man­i­folds of Di­men­sions 3 and 4, held at the Uni­versity of Texas at Aus­tin, May 19–21, 2003, in hon­or of the 60th birth­day of An­drew Cas­son. In­vited lec­tures were giv­en by Danny Calegari, Bob Ed­wards, Mike Freed­man, Dave Gabai, Rob Kirby, Greg Ku­per­berg, Dar­ren Long, Peter Oz­s­vath, An­drew Ran­icki, Ron Stern, Peter Teich­ner, Kev­in Walk­er, and Terry Wall. The or­gan­iz­ing com­mit­tee con­sisted of Camer­on Gor­don, Bob Gom­pf, John Luecke and Alan Re­id. The con­fer­ence was sup­por­ted by NSF Grant DMS-0229035 and by the De­part­ment of Math­em­at­ics of the Uni­versity of Texas at Aus­tin.

The goal of this book is to char­ac­ter­ize al­geb­ra­ic­ally the closed 4-man­i­folds that fibre non­trivi­ally or ad­mit geo­met­ries in the sense of Thur­ston, or which are ob­tained by sur­gery on 2-knots, and to provide a ref­er­ence for the to­po­logy of such man­i­folds and knots. The first chapter is purely al­geb­ra­ic. The rest of the book may be di­vided in­to three parts: gen­er­al res­ults on ho­mo­topy and sur­gery (Chapters 2–6), geo­met­ries and geo­met­ric de­com­pos­i­tions (Chapters 7–13), and 2-knots (Chapters 14–18). In many cases the Euler char­ac­ter­ist­ic, fun­da­ment­al group and Stiefel–Whit­ney classes to­geth­er form a com­plete sys­tem of in­vari­ants for the ho­mo­topy type of such man­i­folds, and the pos­sible val­ues of the in­vari­ants can be de­scribed ex­pli­citly. The strongest res­ults are char­ac­ter­iz­a­tions of man­i­folds which fibre ho­mo­top­ic­ally over S¹ or an as­pher­ic­al sur­face (up to ho­mo­topy equi­val­ence) and in­fra­solv­man­i­folds (up to homeo­morph­ism). As a con­sequence 2-knots whose groups are poly–Z are de­term­ined up to Gluck re­con­struc­tion and change of ori­ent­a­tions by their groups alone.

This book arose out of two earli­er books: 2-Knots and their Groups and The Al­geb­ra­ic Char­ac­ter­iz­a­tion of Geo­met­ric 4-Man­i­folds, pub­lished by Cam­bridge Uni­versity Press for the Aus­trali­an Math­em­at­ic­al So­ci­ety and for the Lon­don Math­em­at­ic­al So­ci­ety, re­spect­ively. About a quarter of the present text has been taken from these books, and I thank Cam­bridge Uni­versity Press for their per­mis­sion to use this ma­ter­i­al. The ar­gu­ments have been im­proved and the res­ults strengthened, not­ably in us­ing Bowditch’s ho­mo­lo­gic­al cri­terion for vir­tu­al sur­face groups to stream­line the res­ults on sur­face bundles, us­ing L² meth­ods in­stead of loc­al­iz­a­tion, com­plet­ing the char­ac­ter­iz­a­tion of map­ping tori, re­lax­ing the hy­po­theses on tor­sion or on abeli­an nor­mal sub­groups in the fun­da­ment­al group and in de­riv­ing the res­ults on 2–knot groups from the work on 4–man­i­folds. The main tools used are co­homo­logy of groups, equivari­ant Poin­care du­al­ity and (to a less­er ex­tent) L²–co­homo­logy, 3–man­i­fold the­ory and sur­gery.

The book has been re­vised in March 2007.

Jonath­an Hill­man

The work­shop and sem­inars on “In­vari­ants of Knots and 3-Man­i­folds” took place at the Re­search In­sti­tute for Math­em­at­ic­al Sci­ences (RIMS), Kyoto Uni­versity, in Septem­ber 2001. The work­shop was held over the peri­od Septem­ber 17–21. Sem­inars were held on the Tues­days, Wed­nes­days and Thursdays of the oth­er weeks of Septem­ber, in­clud­ing “Gous­sarov day” on Septem­ber 25.

Since the in­ter­ac­tion between geo­metry and math­em­at­ic­al phys­ics in the 1980s, many in­vari­ants of knots and 3-man­i­folds have been dis­covered and stud­ied: poly­no­mi­al in­vari­ants such as the Jones poly­no­mi­al, Vassiliev in­vari­ants, the Kont­sevich in­vari­ant of knots, quantum and per­turb­at­ive in­vari­ants, the LMO in­vari­ant and fi­nite type in­vari­ants of 3-man­i­folds. The dis­cov­ery and ana­lys­is of the enorm­ous num­ber of these in­vari­ants yiel­ded a new area: the study of in­vari­ants of knots and 3-man­i­folds (from an­oth­er view­point, the study of the sets of knots and 3-man­i­folds). There are also de­vel­op­ing top­ics re­lated to oth­er areas such as hy­per­bol­ic geo­metry via the volume con­jec­ture and the the­ory of op­er­at­or al­geb­ras via in­vari­ants arising from 6j-sym­bols. On the oth­er hand, re­cent works have al­most com­pleted the to­po­lo­gic­al re­con­struc­tion of the in­vari­ants de­rived from the Chern-Si­mons field the­ory.

An aim of the work­shop and sem­inars was to dis­cuss fu­ture dir­ec­tions for this area. To dis­cuss these mat­ters fully, we planned one month of activ­it­ies, re­l­at­ively longer than usu­al. Fur­ther, to en­cour­age dis­cus­sions among the par­ti­cipants, we ar­ranged a short prob­lem ses­sion after each talk, and re­ques­ted the speak­er to give his/her open prob­lems there. Many in­ter­est­ing prob­lems were presen­ted in these prob­lem ses­sions and, based on them, we had valu­able dis­cus­sions in and between sem­inars and the work­shop. Open prob­lems dis­cussed there were ed­ited and formed in­to a prob­lem list, which, I hope, will cla­ri­fy the present fron­ti­er of this area and as­sist read­ers when con­sid­er­ing fu­ture dir­ec­tions.

T. Oht­suki

This mono­graph is the res­ult of the con­fer­ence on high­er loc­al fields held in Mün­ster, Au­gust 29 to Septem­ber 5, 1999. The aim is to provide an in­tro­duc­tion to high­er loc­al fields (more gen­er­ally com­plete dis­crete valu­ation fields with ar­bit­rary residue field) and render the main ideas of this the­ory (Part I), as well as to dis­cuss sev­er­al ap­plic­a­tions and con­nec­tions to oth­er areas (Part II). The volume grew as an ex­ten­ded ver­sion of talks giv­en at the con­fer­ence. The two parts are sep­ar­ated by a pa­per of K. Kato, an IHES pre­print from 1980 which has nev­er been pub­lished.

An n-di­men­sion­al loc­al field is a com­plete dis­crete valu­ation field whose residue field is an (n–1)-di­men­sion­al loc­al field; 0-di­men­sion­al loc­al fields are just per­fect (e.g., fi­nite) fields of pos­it­ive char­ac­ter­ist­ic. Giv­en an arith­met­ic scheme, there is a high­er loc­al field as­so­ci­ated to a flag of sub­s­chemes on it. One of cent­ral res­ults on high­er loc­al fields, class field the­ory, de­scribes abeli­an ex­ten­sions of an n-di­men­sion­al loc­al field via (all in the case of fi­nite 0-di­men­sion­al residue field; some in the case of in­fin­ite 0-di­men­sion­al residue field) closed sub­groups of the n-th Mil­nor K-group of F.

We hope that the volume will be a use­ful in­tro­duc­tion and guide to the sub­ject. The con­tri­bu­tions to this volume were re­ceived over the peri­od Novem­ber 1999 to Au­gust 2000 and the elec­tron­ic pub­lic­a­tion date is 10 Decem­ber 2000.

Ivan Fesen­ko and Masato Kur­i­hara

Rob Kirby’s re­search spans a broad spec­trum of top­ics, all with this strong visu­al fla­vor: to­po­lo­gic­al man­i­folds of high di­men­sion; the struc­ture of smooth 4-man­i­folds and their re­la­tion­ship to com­plex sur­faces; and the emer­ging new in­vari­ants for both 3- and 4-di­men­sion­al man­i­folds. In both di­men­sions three and four, the “Kirby Cal­cu­lus” has be­come a stand­ard ana­lyt­ic­al tool. He has helped to or­gan­ize and to de­vel­op prob­lem lists which have be­come stand­ard ref­er­ence points for pro­gress in geo­met­ric to­po­logy.

The Kirby­fest, held at the Math­em­at­ic­al Sci­ences Re­search In­sti­tute on 22–26 June 1998, at­trac­ted over 100 math­em­aticians from around the globe. Many of the par­ti­cipants were col­lab­or­at­ors, or former stu­dents; oth­ers were just fans of Kirby and his work. There were 27 plen­ary talks, cov­er­ing a wide vari­ety of to­po­lo­gic­ally re­lated sub­jects, in­clud­ing sev­er­al his­tor­ic­al sur­veys. Fields Medal­ists gave five of the talks. Sev­en present­a­tions were spe­cific­ally or­gan­ized to be eas­ily ac­cess­ible to gradu­ate stu­dents.

We hope these pro­ceed­ings con­vey some of the math­em­at­ic­al ex­cite­ment of the Kirby­fest week, and we are honored to ded­ic­ate it to Rob.

In hon­or of Dav­id Ep­stein.