Celebratio Mathematica

Dmitry Fuchs

Introductory notes

by Boris Khesin, Fedor Malikov, Valentin Ovsienko and Sergei Tabachnikov

A group photo from the ArnoldFest in 1997. Top row (left to right): E. Bierstone, T. Ratiu, Ya. Eliashberg, A. Givental, A. Neishtadt. Bottom row: V. Vassiliev, Yu. Ilyashenko, D. Fuchs.

This volume of Cel­eb­ra­tio Math­em­at­ica is de­voted to Dmitry Bor­iso­vich Fuchs, an out­stand­ing math­em­atician and our teach­er. It com­prises our in­ter­view with Fuchs of June 2020, as well as con­tri­bu­tions from Yakov Eli­ash­berg, Bor­is Fei­gin, Daniel Ruther­ford, and Vic­tor Vassiliev, who re­flec­ted vari­ously on their in­ter­ac­tions with D.B. and his math­em­at­ic­al in­terests. There are many (semi-le­gendary) stor­ies about Fuchs and, by way of in­tro­duc­tion, we present a few spe­ci­men of this lore, bor­rowed from [e2].

We can­not res­ist start­ing with D.B. Fuchs’ own re­col­lec­tions of Arnold’s sem­in­ar in Mo­scow (see Chapter 12 of [e1]):

My role there was well es­tab­lished: I had to re­solve any to­po­logy-re­lated dif­fi­culty. Some of my friends said that at Arnold’s sem­in­ar I was a “cold to­po­lo­gist”. Cer­tainly, a non-Rus­si­an speak­er can­not un­der­stand this, so let me ex­plain. In many Rus­si­an cit­ies there were “cold shoe­makers” in the streets who could provide an ur­gent re­pair to your foot­wear. They sat in their booths, usu­ally with no heat­ing (this is why they were “cold”), and shouted, “Heels!…Soles!….” So I ap­peared as if sit­ting in a cold booth and yelling, “Co­homo­logy rings!…Ho­mo­topy groups!…Char­ac­ter­ist­ic classes!…”

The family: Dmitry, his wife, Ira, and the daughters, Katya and Lyalya.

Fuchs’ to­po­lo­gic­al repu­ta­tion ex­ten­ded far bey­ond Arnold’s sem­in­ar. There was a fam­ous story where a group of math­em­aticians, “all the best people” as Fuchs says, helped one of them, A. A. Kir­illov, to move to his new apart­ment on the 14th floor of a 16-story build­ing. It turned out that a huge cus­tom-made book­case, which would oc­cupy the en­tire wall of a room, did not fit in­to the el­ev­at­or. It had to be car­ried to the 14th floor via the stair­case but, even more im­port­antly, it was so large that there was a unique way of mov­ing it through each flight of stairs, with only one way pos­sible of turn­ing the book­case with­in the con­fines of the stair­well. (One of the par­ti­cipants was Ya. G. Sinai, who first for­mu­lated the prob­lem as “keep this damned book­case con­nec­ted” and later “keep the num­ber of its com­pon­ents not great­er than two.”) Even­tu­ally, when they got to the apart­ment, there was a unique way to bring the book­case in­to the room, with no way to turn it around or oth­er­wise change its ori­ent­a­tion.

Left to right: Sergei Tabachnikov, Alexander Kirillov, and Dmitry Fuchs.

An­oth­er memory: Fuchs was giv­ing a talk at USC. At some point, he said that, for a suf­fi­ciently small ep­si­lon, such and such held. One of the at­tendees, who was drows­ing, sud­denly woke up and asked how small this ep­si­lon must be. Fuchs replied im­me­di­ately: “one tenth of an inch.”

And once in­side, it turned out that the book­case fit the room in the only pos­sible po­s­i­tion, with the shelves fa­cing the wall and the back­side fa­cing the room! The only way to fix this was to carry the case back down all 14 flights of stairs and get it back out onto the street. Then Fuchs, as the lead­ing to­po­lo­gist, was giv­en the task of find­ing the cor­rect “ini­tial con­di­tions” in or­der to start the pro­cess anew. He thought about it, gave the in­struc­tions how to turn it, they did all the lift­ing via 14 flights of stairs up — and the book­case in­deed fit in the room per­fectly!

Left to right: Dmitry Fuchs, Katerina Malikov, Louisa Kirillova, Vadim Schechtman, Alexander Kirillov, and Fedor Malikov.

And here is one more story told by Fuchs him­self. He re­called that some­time in the 70s or 80s he was at one of A. Fo­men­ko’s pub­lic lec­tures on “The New Chro­no­logy,” which was a “hot” sub­ject at the time. At some point, when com­par­ing time pat­terns re­lated to two se­quences of the Ro­man em­per­ors, Fo­men­ko said that the prob­ab­il­ity that such pat­terns were in­de­pend­ent and not copied from one an­oth­er was one out of a mil­lion. At this mo­ment Fo­men­ko tried to cla­ri­fy his point: “In or­der to see how neg­li­gible this prob­ab­il­ity of one mil­lionth is, just ima­gine that you put a kettle on a hot stove, but in­stead of boil­ing the wa­ter freezes up! This is how rare such events are!” Fuchs said that once he heard that ex­plan­a­tion, he thought: “Hmmm…Mo­scow’s pop­u­la­tion is 8 mil­lion…So every morn­ing 8 people in this city put their kettles on the stove and have the wa­ter in the kettle frozen?!”

As they write in book re­views, “if the read­er liked those stor­ies, (s)he will cer­tainly en­joy many more an­ec­dotes” which Fuchs shared with us in his in­ter­view for this volume.