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{equation}\label{eq1} \|(I+\lambda A)^{-1}\|\leq 1 \end{equation} for \( \lambda > 0 \), then \begin{equation}\label{eq2} u(t)=\lim_{n\to\infty}\biggl(\mkern-2mu I+\frac{t}{n}A\mkern-2mu \biggr)^{\!-n}x \end{equation} ex­ists for \( t\geq0 \) and \( x\in X \). Moreover, the func­tion defined in \eqref{eq2} is a solu­tion of the Cauchy prob­lem \begin{equation}\label{eq3} \frac{\mathrm{d} u}{\mathrm{d} t}+Au=0,\quad u(0)=x \end{equation} if \( x\in D(A) \). Roughly speak­ing, the main res­ult of this work states that the lim­it \eqref{eq2} ex­ists even if \( A \) is non­lin­ear (and mul­ti­val­ued) provided that the non­lin­ear ana­logue of \eqref{eq1} holds for \( A \). In ad­di­tion we prove that the (mul­ti­val­ued) ana­logue of \eqref{eq3} has a solu­tion in a strong sense if and only if the lim­it in \eqref{eq2} is strongly dif­fer­en­ti­able, and in this case it is the solu­tion to \eqref{eq3}. 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 \[ \begin{cases} \frac{\mathrm{d}}{\mathrm{d}s}v(x,s)=\Delta v(x,s) &\text{for all } (x,s)\in\mathbb{R}^{m}\times(0,\infty),\\ \phantom{\frac{\mathrm{d}}{\mathrm{d}\, s}} v(x,0)=1_{C}(x) &\text{for all }x\in\mathbb{R}^{m}. \end{cases} \] We then threshold this heat equa­tion solu­tion, and define a new set \( C_{t}\subset\mathbb{R}^{m} \) by \[ C_{t}:= \bigl\{x\in\mathbb{R}^{m}: v(x,t)\geq\textstyle\frac12\bigr\}. \] 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{equation}\label{eq4} \lim_{n\to\infty}[\mathcal{H}(t/n)]^{n} C. \end{equation} The work [e3] sug­gests that \eqref{eq4} 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.