#### by Steven Heilman

#### Introduction to Professor Liggett

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.

#### Some other recollections

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.

#### Conclusion

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.*