Celebratio Mathematica

Joel Hass has made ground­break­ing con­tri­bu­tions in sev­er­al areas. Firstly, Hass, to­geth­er with out­stand­ing col­lab­or­at­ors Agol, Thur­ston, Lagari­as, and Pip­penger, has made ma­jor ad­vances in our un­der­stand­ing of the com­pu­ta­tion­al com­plex­ity of al­gorithms in 3-man­i­fold to­po­logy and knot the­ory. For ex­ample, they gave a beau­ti­ful meth­od for bound­ing the num­ber of Re­idemeister moves needed to con­vert any dia­gram of the un­knot in­to stand­ard form. Pre­vi­ously no ex­pli­cit bounds were known to solve this prob­lem. Haken, in 1960, had giv­en the first solu­tion to the un­knot re­cog­ni­tion prob­lem, but Hass and his col­lab­or­at­ors fi­nally es­tab­lished the com­plex­ity of this im­port­ant ques­tion. It is still open wheth­er or not re­cog­nising the un­knot has a poly­no­mi­al-time al­gorithm. This is a sub­ject of con­tinu­ing in­tense in­terest. Hass also re­vis­ited this prob­lem in in­ter­est­ing work with Thur­ston, Lagari­as and Snoeyink, look­ing at ex­pli­cit classes of ex­amples where bounds can be giv­en on the area and com­plex­ity of disks, de­pend­ing on the de­scrip­tion of the bound­ary un­knot.

In the the­ory and ap­plic­a­tions of min­im­al sur­faces, Hass has made cru­cial dis­cov­er­ies. In a series of pa­pers on in­ter­sec­tions and self-in­ter­sec­tions of least area sur­faces in 3-di­men­sion­al man­i­folds with Scott and the key ini­tial pa­per also with Freed­man, Hass de­veloped tech­niques used by many oth­er au­thors. The “man­tra” of their work is that least area sur­faces have least self-in­ter­sec­tions and in­ter­sec­tions. Scott used this meth­od to prove to­po­lo­gic­al ri­gid­ity of Seifert fibred man­i­folds in a ma­jor pa­per in the An­nals. (Ri­gid­ity here means that ho­mo­topy equi­val­ence im­plies homeo­morph­ism). Hass and Scott then ex­ten­ded to­po­lo­gic­al ri­gid­ity to 3-man­i­folds which have im­mersed es­sen­tial sur­faces sat­is­fy­ing cer­tain com­bin­at­or­i­al prop­er­ties (4-plane and 1-line con­di­tions). Aitchis­on and Ru­bin­stein then ap­plied this to the class of 3-man­i­folds with Cat(0) cu­bings. This was fol­lowed by the work of Sageev on pairs of groups with more than one end and ac­tions on Cat(0) cu­bic­al com­plexes, which then was fur­ther de­veloped by Wise us­ing spe­cial cu­bic­al com­plexes cul­min­at­ing in Agol’s re­cent won­der­ful solu­tion of the vir­tu­al Haken and vir­tu­al fib­ring con­jec­tures of Thur­ston.

As a byproduct, Hass and Scott wrote sev­er­al el­eg­ant pa­pers on prop­er­ties of geodesics on sur­faces, us­ing sim­il­ar meth­ods. This is a very clas­sic­al area of re­search, but their ap­proach gave in­ter­est­ing in­sights in­to the com­bin­at­or­ics of geodesics us­ing gen­er­al Rieman­ni­an met­rics.

Hass, Ru­bin­stein and Scott were able to show that any ir­re­du­cible 3-man­i­fold with an im­mersed es­sen­tial sur­face is covered by Eu­c­lidean 3-space, us­ing the Hass–Scott tech­no­logy for least area sur­faces. Hass, with Ru­bin­stein and Wang, gave a strong bound on the re­la­tion between genus and bound­ary slope of es­sen­tial sur­faces in 3-man­i­folds with hy­per­bol­ic met­rics, us­ing min­im­al sur­faces. Fi­nally Hass, Nor­bury and Ru­bin­stein gave the first ex­pli­cit con­struc­tion of em­bed­ded min­im­al spheres with ar­bit­rar­ily high Morse in­dex.

With Sch­lafly, Hass solved the fam­ous double bubble con­jec­ture, show­ing that two soap bubbles which are stuck to­geth­er and have equal volumes must form a ro­ta­tion­ally sym­met­ric shape. Their ar­gu­ment is a tour de force of del­ic­ate geo­metry and nu­mer­ic­al ana­lys­is. There have since been fur­ther ad­vances in the un­der­stand­ing of the struc­tures of com­bin­a­tions of bubbles, but their work was the first big break­through on this top­ic.

Hass has re­cently worked with en­gin­eers and com­puter sci­ent­ists on im­port­ant prob­lems in com­puter-aided design, es­pe­cially ef­fi­ciently in­ter­pol­at­ing sur­face shapes (splines).

To­geth­er with Thompson and Thur­ston, Hass gave a won­der­ful con­struc­tion, us­ing har­mon­ic maps, of two Hee­gaard split­tings which re­quire many sta­bil­isa­tions (adding trivi­al handles) to make them iso­top­ic. This was an old cent­ral prob­lem in the study of handle present­a­tions (Hee­gaard split­tings of 3-di­men­sion­al man­i­folds). The idea is that “turn­ing over” a sur­face, i.e., iso­top­ing it to re­verse the sides, should re­quire adding \( g \) handles, where \( g \) is the genus of the sur­face. To make this pre­cise, area es­tim­ates are used em­ploy­ing the the­ory of har­mon­ic maps. In par­tic­u­lar, the man­i­fold has a “long thin shape” with small cross-sec­tion giv­en by the sur­face.

More re­cently, Hass and Scott have de­veloped a the­ory of dis­crete har­mon­ic map­pings. This is a very ex­cit­ing top­ic — har­mon­ic map­pings oc­cur in many phys­ic­al prob­lems such as sigma mod­els, but ana­lys­ing their sin­gu­lar­it­ies in­volves del­ic­ate prob­lems in par­tial dif­fer­en­tial equa­tions. The dis­crete ver­sion, as de­veloped by Hass and Scott, should have many in­ter­est­ing ap­plic­a­tions but without the dif­fi­cult ana­lys­is. Moreover, do­ing com­pu­ta­tion­al ex­per­i­ment­a­tion should be much easi­er than with the clas­sic­al smooth ver­sion.

Hass, to­geth­er with col­lab­or­at­ors from the bio­molecu­lar area, has been study­ing the geo­metry of pro­teins, es­pe­cially the prob­lem of ef­fi­ciently loc­at­ing po­ten­tial dock­ing sites. With his ex­tremely broad skills in geo­met­ric ana­lys­is, Hass is an ideal col­lab­or­at­or for mo­lecu­lar bio­lo­gists.