# Profile: Steve Evans

**David Steinsaltz, University of Oxford, UK, profiles Steve Evans, one of the five IMS Fellows elected in 2016 to the US National Academy of Sciences [profiles of the other four have appeared in previous issues].**

In May 2016, Steven Neil Evans was elected to the US National Academy of Sciences, in recognition of his contributions to probability theory, statistics and mathematical biology. His works have been dotted across diverse mathematical topics and multiple scientific problems—superprocesses, coalescence processes, phylogenetics, random matrices, genomics, population dynamics, linguistics, and theoretical statistics. Seminal, each one sprouting its individual world of research, they remain connected to the rest by his rambling curiosity and idiosyncratic mathematical imagination.

When I asked some of his colleagues and collaborators for their impressions, I heard numerous variants of “a remarkable listener”. “It’s relaxing to speak with him, because to Steve nothing is irrelevant.” He was praised for his “encyclopedic knowledge”, and his capacity for refining and simplifying an inchoate problem into mathematically tractable form. He is an indefatigable calculator who sifts the examples for insights, and translates his mathematical rigor into the language and mind-frame of a scientific collaborator.

In print he has been described (in a popular mathematics book) as “a burly Australian in denims who looked as if he could have stepped off a building site”. Indeed, his accent, literary tastes, and ruthless personal modesty all show the deep traces of his childhood in rural Australia. After completing his degree at the University of Sydney and a stint in the Commonwealth Banking Corporation of Australia, the wool bankers’ suits and Australian summer combined to push him toward the cool fens of Cambridge, England. He completed his doctorate in 1987, and then, following a two-year postdoctoral appointment at the University of Virginia, joined the department of statistics at U.C. Berkeley, where he has remained ever since. He was awarded the Rollo Davidson prize in 1990 and a Sloan Fellowship for 1993–4. He is a fellow of IMS and the American Mathematical Society.

In the late 1980s and into the 1990s he published a series of papers, many of them in collaboration with Edwin Perkins, developing a set of flexible and intuitive tools that opened up the study of superprocesses. These are diffusions on measure-valued state spaces that arise as limits of stochastic processes with branching particles, often used to represent populations evolving in space, whether concretely geographic or abstract spaces of biological traits. One of the highlights of this period was the construction now known as the Evans immortal particle, which first appeared in 1993 in [1], a forerunner of the now-fundamental technique of spine decomposition. A superprocess conditioned on long-term survival, he showed, may be decomposed into clumps of mass (behaving like the unconditioned process) thrown off at random intervals by a single particle that is immune to the killing. While his research quickly moved on to other topics, he returned repeatedly to extend the standard superprocess to new questions of interest in mathematical biology, such as the behaviour of competing spatially-structured populations [3], and the role of damage-accumulation in the evolution of aging [7].

Mathematical biology has been a major theme throughout his research career, ranging over population dynamics, population genetics, theoretical evolution and phylogenetics. His work on David Aldous’s continuum random tree model includes a foundational set of lecture notes, and the papers [8, 4] (with Anita Winter and Jim Pitman) that provided a rigorous intuitive picture of how the continuum tree arises as the stationary distribution of a process of pruning and re-grafting subtrees. His collaboration with Montgomery Slatkin yielded new ways of thinking about the evolution of allele frequency spectra [6], and methods of calculating from SDEs that move beyond the standard asymptotic formulas. One remarkable paper [10] with Frederick Matsen upended the common “intuition that genetic and genealogical ancestry are equivalent”** ^{1}**, clarifying the discrepancy between the log

*N*time scale on which a population of

*N*individuals has a common ancestor and the order

*N*time to a common ancestor for a fixed genetic locus. More recently, his expertise extracting interpretable conclusions from coupled stochastic differential equations has shed light on the population growth rates of migrating populations. This work [5] showed how spatiotemporal environmental randomness can interact with migration rates in surprising ways to make the difference between global survival and extinction.

Together with many and disparate collaborators he has staked out territory on the frontiers between probability theory and far-flung regions of mathematics’ empire. Random algebraic objects have been a repeated object of study, such as his work calculating the expected number of zeros of random *p*-adic polynomials [2], novel stochastic processes on unconventional spaces (Markov processes on “vermiculated” spaces, Brownian motion on ℝ-trees, processes indexed by trees or local fields), and dynamical systems on spaces of probability measures (the core of his work on mutation–selection dynamics for age-structured populations that culminated in the monograph [9]).

While recognising his published works, it would be wrong to conclude without mentioning his more informal contributions to mathematical sciences as a teacher, colleague, and collaborator. The collected works that mention “thanks to Steve Evans”, “explained to us by Steve Evans”, or “assisted by Steve Evans” comprise by themselves a significant literature in probability, statistics, and mathematical biology. As an audience member at conference talks he throws off penetrating insights and links to far-flung mathematical concepts. His tone typically suggests that the speaker, and perhaps all the rest of the audience, surely understood all this long ago.

__

**1** M. Slatkin, private communication

**References**

[1] Steven N Evans. Two representations of a conditioned superprocess. *Proceedings of the Royal Society of Edinburgh: Section A Mathematics*, 123(05):959–971, 1993.

[2] Steven N. Evans. The expected number of zeros of a random *p*-adic polynomial. *Electronic Communications in Probability*, 11:278–290, 2006.

[3] Steven N Evans and Edwin A Perkins. Collision local times, historical stochastic calculus, and competing superprocesses. *Electronic Journal of Probability*, 3, 1998.

[4] Steven N Evans, Jim Pitman, and Anita Winter. Rayleigh processes, real trees, and root growth with re-grafting. *Probability Theory and Related Fields*, 134(1):81–126, 2006.

[5] Steven N. Evans, Peter L. Ralph, Sebastian J. Schreiber, and Arnab Sen. Stochastic population growth in spatially heterogeneous environments. *Journal of Mathematical Biology, *66(3):423–76, February 2013.

[6] Steven N Evans, Yelena Shvets, and Montgomery Slatkin. Non-equilibrium theory of the allele frequency spectrum. *Theoretical Population Biology*, 71(1):109–119, 2007.

[7] Steven N. Evans and David Steinsaltz. Damage segregation at fissioning may increase growth rates: A superprocess model. *Theoretical Population Biology*, 71(4):473–90, 2007.

[8] Steven N Evans and Anita Winter. Subtree prune and regraft: a reversible real tree-valued markov process. *The Annals of Probability*, pp.918–961, 2006.

[9] Steven Neil Evans, David Steinsaltz, and Kenneth W Wachter. A mutation-selection model with recombination for general genotypes, volume 222 of *Memoirs of the American Mathematical Society*. AMS, 2013.

[10] Frederick A Matsen and Steven N Evans. To what extent does genealogical ancestry imply genetic ancestry? *Theoretical Population Biology*, 74(2):182–190, 2008.

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