Celebratio Mathematica

I really got to know Yvonne in 1982, while we were strolling along Lake Tai in the city of Wuxi in China. Wuxi was one of the stops on the tour which ac­com­pan­ied the 1982 Mar­cel Gross­man Con­fer­ence in Shang­hai. There were about 40 people on that six-day tour of three cit­ies, and along with the won­der­ful scenery, the tour gave us lots of op­por­tun­it­ies to get to know each oth­er. I had ac­tu­ally met Yvonne very briefly al­most 10 years earli­er, when she came to give a lec­ture at Prin­ceton Uni­versity, where I was a ju­ni­or un­der­gradu­ate. I re­mem­ber the lec­ture very well, be­cause I was work­ing on the con­form­al meth­od for con­struct­ing and para­met­er­iz­ing solu­tions of the Ein­stein ini­tial value con­straint equa­tions, and Yvonne was one of the pi­on­eers in de­vel­op­ing this meth­od. After her lec­ture, my un­der­gradu­ate ad­visor Jimmy York in­tro­duced me to Yvonne, but as a young stu­dent meet­ing one of the icon­ic fig­ures in math­em­at­ic­al re­lativ­ity, I didn’t think that I had much to say to her that would be in­ter­est­ing to her. I do re­mem­ber try­ing to talk to Yvonne with my high school French, but I think she found it pretty in­ad­equate, so we quickly re­ver­ted to Eng­lish. Our con­ver­sa­tion las­ted only a couple of minutes.

I don’t think that Yvonne had re­membered our 1973 con­ver­sa­tion when we met again in China in 1982, but dur­ing that six-day tour, with all 40 of us in an ex­cit­ing but re­l­at­ively un­fa­mil­i­ar set­ting, there was quite a bit of time to get to know each oth­er, and my op­por­tun­ity to chat with Yvonne happened on that stroll around Lake Tai. I told Yvonne about how I had vis­ited Par­is on a high school ex­cur­sion in 1967 and all the won­der­ful things that I had seen dur­ing that ex­cur­sion — in­clud­ing see­ing per­form­ances by Johnny Hally­day and Charles Aznavour — but I don’t think that Yvonne was that im­pressed, so our con­ver­sa­tion moved on from there to our com­mon in­terest at the time: the newly de­veloped the­ory of su­per­grav­ity. I knew that Yvonne was cel­eb­rated for her epic early 1950s proof that the Cauchy prob­lem for Ein­stein’s grav­it­a­tion­al field the­ory was well-posed, and I men­tioned to her that my work with Jim Nester on su­per­grav­ity had led me to won­der if the Cauchy prob­lem for su­per­grav­ity was also well-posed. That ques­tion really got our con­ver­sa­tion go­ing, be­cause Yvonne had also been in­ter­ested in su­per­grav­ity, and we spent much of our stroll talk­ing about that. One cru­cial fea­ture of su­per­grav­ity is that it in­cludes Fer­mi­on­ic fields — spin 3/2 — which ne­ces­sar­ily anti-com­mute. This fea­ture res­ults in the stand­ard tools for prov­ing the well-posed­ness of the Cauchy prob­lem for a hy­per­bol­ic PDE sys­tem re­quir­ing sig­ni­fic­ant modi­fic­a­tion. Yvonne and I dis­cussed some of the modi­fic­a­tions which would be ne­ces­sary dur­ing that stroll, and our sub­sequent com­mu­nic­a­tions — which back then had to be car­ried out us­ing reg­u­lar mail — led to our first joint pa­per along with our col­lab­or­at­ors Dav­id Bao and Phil Yasskin. I re­mem­ber be­ing ex­tremely proud to pub­lish a pa­per with Yvonne.

I be­lieve that our stroll in Wuxi led to a long peri­od of friend­ship and col­lab­or­a­tion. It was not long after our joint pub­lic­a­tion of the pa­per on su­per­grav­ity that Yvonne in­vited me along with Vince Mon­crief to spend nine months work­ing with her at Par­is VI (Jussieu). That was a won­der­ful time both sci­en­tific­ally as well as so­cially. The three of us met reg­u­larly, both in pairs and al­to­geth­er. A lot of our dis­cus­sions in­volved vari­ous spe­cial ver­sions of the con­form­al meth­od for solv­ing the Ein­stein con­straints, which was the top­ic of that first lec­ture I heard Yvonne present in Prin­ceton in 1973.

Five of the pro­jects that I worked on with Yvonne — see the list be­low — fo­cused on the con­form­al meth­od. As Yvonne’s proof of the well-posed­ness of the Cauchy prob­lem for Ein­stein’s (va­cu­um) the­ory of grav­ity shows (to­geth­er with some­what later col­lab­or­at­ive work of hers with Bob Ge­roch), if one spe­cifies an ini­tial data set on a three-di­men­sion­al man­i­fold \( \Sigma \), which in­cludes a Rieman­ni­an met­ric \( h \) and a sym­met­ric tensor \( k \) which sat­is­fies the Ein­stein con­straint equa­tions, then there is a unique (up to space­time dif­feo­morph­ism) max­im­al space­time solu­tion \( (\Sigma\times I,g) \) of the full Ein­stein equa­tions, with \( g \) a Lorent­zi­an met­ric. As well, for this space­time solu­tion \( g \), \( h \) is the in­duced first fun­da­ment­al form on \( \Sigma \) and \( k \) is the in­duced second fun­da­ment­al form on \( \Sigma \). The Ein­stein va­cu­um con­straint equa­tions take the form of an un­der-de­term­ined non­lin­ear set of four par­tial dif­fer­en­tial equa­tions to be solved for \( h \) and \( k \). The Cauchy prob­lem for Ein­stein’s equa­tions is an es­pe­cially ef­fect­ive and prac­tic­al way to ob­tain space­time solu­tions, both ana­lyt­ic­ally and nu­mer­ic­ally. But it does rely on de­vel­op­ing a sys­tem­at­ic way to pro­duce solu­tions of the con­straint equa­tions. This is the goal of the con­form­al meth­od. The idea is to spe­cify “seed data” on \( \Sigma \) con­sist­ing of a con­form­al equi­val­ence class \( [h] \), a di­ver­gence-free and trace-free sym­met­ric tensor \( \mu \), a func­tion \( \tau \), as well as a func­tion \( N \), and then use these seed data to con­struct a de­term­ined par­tial dif­fer­en­tial equa­tion set — the “con­form­al con­straint equa­tions” — to be solved for a vec­tor field W and a con­form­al factor \( \phi \). Pre­sum­ing this set of con­form­al con­straint equa­tions can be solved, the seed data to­geth­er with \( W \) and \( \phi \), al­low one to con­struct \( h \) and \( k \) which sat­is­fy the con­straint equa­tions them­selves.

Work done by An­dré Lich­ner­ow­icz, Yvonne, Jimmy York, Ni­all Ó Murchadha, Vince Mon­crief, Dav­id Max­well, Daniel Pol­lack and me as well as by oth­ers shows that so long as the func­tion \( \tau \) — which cor­res­ponds to the mean curvature of the ini­tial data set — is con­stant, the con­form­al meth­od works very well. All of these names of con­trib­ut­ing re­search­ers re­flects the fact that the con­form­al meth­od has been ap­plied not just to the Ein­stein va­cu­um con­straint equa­tions on com­pact man­i­folds \( S \), but also to the Ein­stein con­straint equa­tions coupled to vari­ous “mat­ter fields” in­clud­ing Max­well (elec­tro­mag­net­ic) fields, Yang–Mills fields, vari­ous scal­ar fields, neut­rino fields, as well as vari­ous flu­id fields. The con­form­al meth­od has also been ap­plied for seed data which is asymp­tot­ic­ally Eu­c­lidean, asymp­tot­ic­ally hy­per­bol­ic as well as with oth­er asymp­tot­ic con­di­tions. Yvonne has played a sig­ni­fic­ant role in de­vel­op­ing the con­form­al meth­od for many of these vari­ous cases, and I am very priv­ileged to have worked with her on this re­search.

While some of my col­lab­or­a­tion with Yvonne in­volved email, al­most all of our work to­geth­er was done face-to-face. I am very happy that this was the case, be­cause we had the op­por­tun­ity to meet in Par­is, in Italy, in Ore­gon (at my farm), as well as a num­ber of oth­er places. The fo­cus was usu­ally on work­ing to­geth­er on re­search, but we also had oc­ca­sions to en­joy each oth­er’s com­pany dur­ing “road trips”, and I learned what a warm and caring and in­ter­est­ing per­son Yvonne was. I have won­der­ful memor­ies of time that we spent to­geth­er in many loc­a­tions.

Much of our col­lab­or­a­tion in­volved the con­form­al meth­od, but there is an­oth­er branch of re­search in gen­er­al re­lativ­ity on which I have worked ex­tens­ively with Yvonne. This re­search in­volves de­term­in­ing the be­ha­vi­or of the grav­it­a­tion­al field in a neigh­bor­hood of the Big Bang in solu­tions of Ein­stein’s equa­tions. Many years ago, Bel­in­skii, Khal­at­nikov and Lif­shitz con­jec­tured that in the neigh­bor­hood of the Big Bang sin­gu­lar­ity in solu­tions of Ein­stein’s equa­tions, the grav­it­a­tion­al field would ex­hib­it “asymp­tot­ic­ally ve­lo­city term dom­in­ated” (AVTD) be­ha­vi­or which means that with re­spect to a giv­en co­ordin­ate sys­tem, the spa­tial de­riv­at­ive terms in the solu­tions would be dom­in­ated by the time de­riv­at­ive terms. The ef­fect­ive res­ult would be that ob­serv­ers fol­low­ing time-like paths to­wards the sin­gu­lar­ity would see the grav­it­a­tion­al field evolve much like spa­tially ho­mo­gen­eous solu­tions. A later con­jec­ture of these three phys­i­cists was that the be­ha­vi­or would not be of this sort, but rather that the ob­serv­ers ap­proach­ing the sin­gu­lar­ity would see an in­fin­ite se­quence of epochs of spa­tially ho­mo­gen­eous solu­tion be­ha­vi­or (which has been labeled by Charles Mis­ner as “mix­mas­ter” be­ha­vi­or). While neither of these con­jec­tures is be­lieved to hold for gen­er­al va­cu­um cos­mo­lo­gic­al solu­tions, there ex­ists nu­mer­ic­al and sta­bil­ity ana­lys­is evid­ence that AVTD be­ha­vi­or is found in solu­tions of the Ein­stein-scal­ar field and Ein­stein-stiff flu­id field equa­tions, as well as in solu­tions of the Ein­stein va­cu­um equa­tions with sig­ni­fic­ant sym­metry. Bey­ond the in­trins­ic in­terest to re­search­ers in the pres­ence of AVTD be­ha­vi­or in cos­mo­lo­gic­al solu­tions, it has been found that if AVTD does ex­ist in a fam­ily of solu­tions, then it is re­l­at­ively straight­for­ward to de­term­ine if that fam­ily of solu­tions sat­is­fies the “strong cos­mic cen­sor­ship con­jec­ture” of Pen­rose, which is one of the ma­jor out­stand­ing ques­tions in math­em­at­ic­al re­lativ­ity.

Not long after Vince Mon­crief and I spent those nine months in Par­is with Yvonne, Vince and I were able to prove that the “po­lar­ized Gowdy” fam­ily of solu­tions do ex­hib­it AVTD be­ha­vi­or, and fol­low­ing that, Vince and I and Pio­tr Chruś­ciel were able to prove that strong cos­mic cen­sor­ship does hold for this very lim­ited fam­ily. In sub­sequent years, Vince and I fairly reg­u­larly vis­ited the IHES In­sti­tute in Burres-sur-Yvette just out­side of Par­is, and con­sequently had lots and lots of con­ver­sa­tions with Yvonne about vari­ous top­ics in math­em­at­ic­al re­lativ­ity, in­clud­ing the is­sue of AVTD be­ha­vi­or in fam­il­ies of solu­tions of Ein­stein’s equa­tions. While prov­ing that this be­ha­vi­or ex­ists in en­tire fam­il­ies of solu­tions is gen­er­ally quite dif­fi­cult, a meth­od based on “Fuch­sian” ana­lys­is was de­veloped by Alan Rend­all and his col­lab­or­at­ors to show that AVTD be­ha­vi­or oc­curs at least in an in­fin­ite di­men­sion­al sub­fam­ily of solu­tions. In our dis­cus­sions at IHES, Vince and I were able to con­vince Yvonne to work with us on ap­ply­ing the Fuch­sian tech­niques to de­term­ine if AVTD can be found in sub­fam­il­ies of solu­tions with con­sid­er­ably less sym­metry than the Gowdy solu­tions. We were suc­cess­ful in show­ing that the po­lar­ized \( U(1) \) sym­met­ric solu­tions as well as those which are “half po­lar­ized” do in­clude in­fin­ite di­men­sion­al sub­fam­il­ies with AVTD be­ha­vi­or in a neigh­bor­hood of the Big Bang. When I was last able to vis­it Yvonne at IHES, we were hop­ing to ex­tend this ana­lys­is to wider sub­fam­il­ies of va­cu­um solu­tions, as well as to Ein­stein-scal­ar solu­tions with no sym­met­ries.

Whenev­er Yvonne and I got to­geth­er out­side of France, without hes­it­a­tion we spoke in Eng­lish. But when we got to­geth­er in Par­is, Yvonne really wanted me to speak with her in French. I might have guessed that she would know from our earli­est in­ter­ac­tions that my French was not very good. Vince’s French was much bet­ter. I re­mem­ber he told me that he was able to de­vel­op very good French by watch­ing French TV while we were to­geth­er in Par­is in 1986. I should have tried that. I was ac­tu­ally okay speak­ing French — al­though it was prob­ably gram­mat­ic­ally quite flawed — but I was not very good at un­der­stand­ing Parisi­an spoken French. I pre­ten­ded to un­der­stand what was be­ing said by peri­od­ic­ally say­ing “d’ac­cord”. Yvonne was al­ways po­lite enough to not let me know that I was fak­ing it.

Yvonne has been a won­der­ful re­search col­lab­or­at­or and friend. My ac­ci­dent in 2017 has pre­ven­ted me from vis­it­ing her since then. I do wish her a won­der­ful 100th birth­day and I wish I could tell her that face-to-face.