Celebratio Mathematica

After ar­riv­ing at UCLA in 2014, I gave a talk at the South­ern Cali­for­nia Prob­ab­il­ity Sym­posi­um on some re­cent joint work con­cern­ing Poin­caré in­equal­it­ies for the lin­ear heat semig­roup on the dis­crete hy­per­cube $\{-1,1{\}}^m.$ Af­ter­wards, Pro­fess­or Lig­gett walked over for a quick com­ment. He said that he did not ex­actly work on this top­ic, but in the 1970s he had proven a the­or­em about non­lin­ear semig­roups. In typ­ic­al fash­ion, he quickly and flaw­lessly re­called the res­ult, and humbly omit­ted its found­a­tion­al im­port­ance. The the­or­em that he al­luded to ap­peared in [1], and it turned out to be in­dir­ectly in­flu­en­tial in my own work. I be­lieve I am not unique in that as­sess­ment. (With over 300 cita­tions on Math­S­ciNet and over 1000 on Google Schol­ar, [1] is Pro­fess­or Lig­gett’s most cited re­search pa­per.)

Be­fore sum­mar­iz­ing [1], I will men­tion that Pro­fess­or Lig­gett al­ways gave pos­it­ive en­cour­age­ment to oth­er ju­ni­or mem­bers of the UCLA prob­ab­il­ity group such as my­self, even though I did not work dir­ectly with him in any way. For ex­ample, he read his daily arX­iv mail­ings and con­grat­u­lated any­one in the group for put­ting a pa­per on the arX­iv.

Here is an ex­cerpt from the in­tro­duc­tion of [1]:

In [e1], Hille proved that if $A$ is a closed and densely defined lin­ear op­er­at­or on a Banach space $X$ with the prop­er­ties that $(I+\lambda A{)}^{-1}$ is defined on $X$ and $$\begin{array}{}\text{(1)}& \parallel (I+\lambda A{)}^{-1}\parallel \le 1\end{array}$$ for $\lambda >0$, then $$\begin{array}{}\text{(2)}& u(t)=\underset{n\to \mathrm{\infty }}{lim}(I+\frac{t}{n}A{)}^{-n}x\end{array}$$ ex­ists for $t\ge 0$ and $x\in X$. Moreover, the func­tion defined in $\text{(2)}$ is a solu­tion of the Cauchy prob­lem $$\begin{array}{}\text{(3)}& \frac{\mathrm{d}u}{\mathrm{d}t}+Au=0,u(0)=x\end{array}$$ if $x\in D(A)$. Roughly speak­ing, the main res­ult of this work states that the lim­it $\text{(2)}$ ex­ists even if $A$ is non­lin­ear (and mul­ti­val­ued) provided that the non­lin­ear ana­logue of $\text{(1)}$ holds for $A$. In ad­di­tion we prove that the (mul­ti­val­ued) ana­logue of $\text{(3)}$ has a solu­tion in a strong sense if and only if the lim­it in $\text{(2)}$ is strongly dif­fer­en­ti­able, and in this case it is the solu­tion to $\text{(3)}$. These res­ults ex­tend earli­er ones which re­quire ad­di­tion­al re­strict­ive con­di­tions […].

At the time of our con­ver­sa­tion, I had heard of this type of res­ult be­ing ap­plied in a work of Evans [e5] con­cern­ing the ana­lys­is of a “level set meth­od” al­gorithm for mo­tion by mean curvature in­tro­duced in [e3]. In these works, we are giv­en a meas­ur­able set $C\subset {\mathbb{R}}^m$ and a time $t>0$. We then run the heat equa­tion on $1_C$ up to time $t$ by solv­ing $$\{\begin{array}{ll}\frac{\mathrm{d}}{\mathrm{d}s}v(x,s)=\mathrm{\Delta }v(x,s)& \text{for all }(x,s)\in {\mathbb{R}}^m\times (0,\mathrm{\infty }),\\ \phantom{\frac{\mathrm{d}}{\mathrm{d}s}}v(x,0)=1_C(x)& \text{for all }x\in {\mathbb{R}}^m.\end{array}$$ We then threshold this heat equa­tion solu­tion, and define a new set $C_t\subset {\mathbb{R}}^m$ by $$C_t:=\{x\in {\mathbb{R}}^m:v(x,t)\ge \frac{1}{2}\}.$$ We de­note $\mathcal{H}(t)C:=C_t$ to de­note one “it­er­a­tion” of this threshold­ing al­gorithm ap­plied to the set $C$. This al­gorithm can be it­er­ated many times for small time scales, lead­ing one to try to make sense of the fol­low­ing “lim­it”: $$\begin{array}{}\text{(4)}& \underset{n\to \mathrm{\infty }}{lim}[\mathcal{H}(t/n){]}^{n}C.\end{array}$$ The work [e3] sug­gests that $\text{(4)}$ cor­res­ponds to a mo­tion of the set $C$ un­der (gen­er­al­ized) mean curvature flow up to time $t$, and this is proven in [e5] (see also [e6], [e4], us­ing the fun­da­ment­al res­ult of [1], to­geth­er with a sub­sequent work [e2]. The res­ult of [e5] rig­or­ously jus­ti­fies the al­gorithm of [e3], which has been quite in­flu­en­tial in the field of nu­mer­ic­al ana­lys­is.

Though I was aware of this res­ult of [e5] us­ing [1], I (rather em­bar­rass­ingly) did not real­ize that I was speak­ing to an au­thor from [1]! In fact, the res­ult of [e5] has been in­dir­ectly in­flu­en­tial in my own work.

In sum­mary, the single pa­per [1] has been quite in­flu­en­tial to many people, in­clud­ing my­self. And des­pite my ig­nor­ance of this fact, Pro­fess­or Lig­gett wel­comed me to the South­ern Cali­for­nia prob­ab­il­ity com­munity with sin­cere in­terest and hu­mil­ity.

Pro­fess­or Lig­gett al­ways had a great, of­ten pithy, sense of hu­mor, though this hu­mor is dif­fi­cult to trans­fer to writ­ten words. Nev­er­the­less, I’ll at­tempt to cap­ture it here, in my paraphas­ing of his ra­tionale for be­com­ing an Emer­it­us Pro­fess­or: I worked at UCLA for 40 years, at which point my re­tire­ment pay­out be­came equal to my salary, so I re­tired.

I should em­phas­ize that this quote is not meant to boast, but rather to in­dic­ate a sens­ible way to eval­u­ate an in­creas­ing quant­ity.

Pro­fess­or Lig­gett was un­usu­ally kind to early ca­reer math­em­aticians. For ex­ample, he held a yearly prob­ab­il­ity pool party (it­self an ex­cel­lent use of al­lit­er­a­tion) in which “go­ing in­to the pool is NOT re­quired.” In Cali­for­nia, not every­one can eas­ily ac­cess a private pool, let alone own a home, so I think this a great ex­ample of someone in a for­tu­nate situ­ation shar­ing with the less for­tu­nate and bring­ing the com­munity to­geth­er.

The last time I saw Pro­fess­or Lig­gett was again at the South­ern Cali­for­nia Prob­ab­il­ity Sym­posi­um, but this time in 2018. We chat­ted briefly as we walked to our re­spect­ive cars, after the con­clu­sion of the con­fer­ence. Though it was the last time our paths would in­ter­sect, we con­tin­ue build­ing upon the found­a­tion he cre­ated. For his ever friendly per­son­al­ity, sharp mind, and for many oth­er reas­ons, he will be missed.

Steven Heil­man is As­sist­ant Pro­fess­or RT­PC of Math­em­at­ics at the Uni­versity of South­ern Cali­for­nia in Los Angeles.