An Invitation to Nonstandard Analysis and its Recent Applications Isaac Goldbring and Sean Walsh Introduction approximate groups. However, its roots go back to Robin- Nonstandard analysis has been used recently in major re- son’s formalization of the infinitesimal approach to calcu- sults, such as Jin’s sumset theorem in additive combina- lus. After first illustrating its very basic uses in calculus torics and Breuillard–Green–Tao’s work on the structure of in “Calculus with Infinitesimals,” we go on to highlight a selection of its more serious achievements in “Selected Classical and Recent Applications,” including the afore- Isaac Goldbring is an associate professor in the Department of Mathematics at mentioned work of Jin and Breuillard–Green–Tao. After UC Irvine. His email address is [email protected]. presenting a simple axiomatic approach to nonstandard Sean Walsh is an associate professor in the Department of Philosophy at UCLA. analysis in “Axioms for Nonstandard Extensions” we ex- His email address is [email protected]. amine Jin’s theorem in more detail in “The Axioms in Ac- For permission to reprint this article, please contact: tion: Jin’s Theorem.” Finally, in “The Ultraproduct Con- [email protected]. struction” we discuss how these axioms can be justified DOI: https://doi.org/10.1090/noti1895 842 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 6 with a particular concrete construction (akin to the verifica- To illustrate this method, let us say that two elements tion of the axioms for the real field using Dedekind cuts or 푎 and 푏 of 퐑∗ are infinitely close to one another, denoted Cauchy sequences), and in “Other Approaches” we com- 푎 ≈ 푏, if their difference is an infinitesimal. Then one has: pare our axiomatic approach to other approaches. While Theorem 1. Suppose that 푓 ∶ 퐑 → 퐑 is a function and 푎 ∈ brief, our hope is that this survey can quickly give the read- 퐑. Then 푓 is continuous at 푎 if and only if: whenever 푏 ∈ 퐑∗ er a sense of both how nonstandard methods are being and 푎 ≈ 푏, then 푓(푎) ≈ 푓(푏). Likewise, 푓 is differentiable used today and how these methods can be rigorously pre- at 푎 with derivative 퐿 if and only if: whenever 푏 is a nonzero sented and justified. ∗ 푓(푎+푏)−푓(푎) infinitesimal in 퐑 , one has that 푏 ≈ 퐿. Calculus with Infinitesimals Proof. We only discuss the continuity statement, since the Every mathematician is familiar with the fact that the differentiability statement is entirely analogous. Suppose founders of calculus such as Newton and Leibniz made that the assumption of the “if” direction holds and fix 휖 > free use of infinitesimal and infinite quantities, e.g., in ex- 0 in 퐑. In 퐑∗, by choosing an infinitesimal 훿 > 0, the ′ pressing the derivative 푓 (푎) of a differentiable function following elementary statement about 휖 is true: “there is 푓 at a point 푎 as being the real number that is infinitely 훿 > 0 such that, for all 푏, if |푎 − 푏| < 훿, then |푓(푎) − 푓(푎+훿)−푓(푎) close to 훿 for every nonzero infinitesimal “quan- 푓(푏)| < 휖.” The logical similarity mentioned above then tity” 훿. Here, when we say that 훿 is infinitesimal, we mean implies that this elementary statement about 휖 is also true that |훿| is less than 푟 for every positive real number 푟. in 퐑. But this is precisely what is needed to prove that 푓 is Of course, the mathematical status of such quantities continuous at 푎. The other direction is similar. □ was viewed as suspect and the entirety of calculus was put ∗ on firm foundations in the nineteenth century by the likes Every finite element 푎 of ℝ is infinitely close to a unique of Cauchy and Weierstrass, to name a few of the more real number, called its standard part, denoted st(푎). Thus, significant figures in this well-studied part of the history the first part of the above theorem reads: 푓 is continuous of mathematics. The innovations of their “휖-훿 method” at 푎 if and only if: whenever st(푏) = 푎, then st(푓(푏)) = (much to the chagrin of many real analysis students today) 푓(푎). In these circumstances, one can then write 푏 = 푎 + allowed one to give rigor to the naïve arguments of their 훿푥 and 푓(푏) = 푓(푎)+훿푦 where 훿푥, 훿푦 are infinitesimals. predecessors. In his elementary calculus textbook based on nonstandard In the 1960s, Abraham Robinson realized that the ideas methods, Keisler pictured this as a “microscope” by which and tools present in the area of logic known as model the- one can zoom in and study the local behavior of a function ory could be used to give precise mathematical meanings to at a point. infinitesimal quantities. Indeed, one of Robinson’s stated Besides extensionally characterizing familiar concepts aims was to rescue the vision of Newton and Leibniz ([20]). such as continuity and differentiability, the use of infinites- While there are complicated historical and philosophical imals can help to abbreviate proofs. To illustrate, let us questions about whether Robinson succeeded entirely in prove the following, noting how we can avoid the usual 휖 this (cf. [5]), our goal is to discuss some recent applica- “ 2 ” argument and instead appeal to the simple fact that tions of the methods Robinson invented. finite sums of infinitesimals are again infinitesimal: To this end, let us turn now to the basics of Robinson’s Corollary 2. Suppose that 푓 ∶ 퐑 → 퐑 and 푔 ∶ 퐑 → 퐑 are approach. On this approach, an infinitesimal 훿 is merely 푎 ∈ 퐑 푓 + 푔 ∗ both continuous at . Then is also continuous at an element of an ordered field 퐑 properly containing 푎. the ordered field 퐑 of real numbers. While such nonar- chimedean fields were already present in, for example, al- Proof. Consider 푏 ≈ 푎. By assumption, we have 푓(푏) ≈ gebraic number theory, the new idea that allowed one to 푓(푎) and 푔(푏) ≈ 푔(푎). Then since the sums of two in- correctly apply the heuristics from the early days of calcu- finitesimals is infinitesimal, we have ∗ lus was that 퐑 is logically similar to 퐑, in that any ele- (푓+푔)(푏) = 푓(푏)+푔(푏) ≈ 푓(푎)+푔(푎) = (푓+푔)(푎). mentary statement true in the one is also true in the other. 푏 푓 + 푔 (Formally, elementary statements are defined using first- Since was arbitrary, we see that is continuous at 푎 □ order logic: see “Axioms for Nonstandard Extensions” for . a self-contained presentation.) A particular feature of this Selected Classical and Recent Applications approach is that one functorially associates to every func- The primary reason for the contemporary interest in non- tion 푓 ∶ 퐑 → 퐑 an extension 푓 ∶ 퐑∗ → 퐑∗ (and similarly standard analysis lies in its capacity for proving new re- for relations). In this light, 퐑 is referred to as the standard sults. In this section, we briefly describe a handful of the field of real numbers whilst 퐑∗ is referred to as a nonstandard more striking results that were first proven using nonstan- field of real numbers or a hyperreal field. dard methods. We remark that nonstandard methods have JUNE/JULY 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 843 proven useful in nearly every area of mathematics, includ- only the large-scale geometry of the space. The prime ex- ing algebra, measure theory, functional analysis, stochas- ample of this phenomenon is that an asymptotic cone of tic analysis, and mathematical economics to name a few. the discrete space 퐙 is the continuum 퐑. Here, we content ourselves with a small subset of these ap- In [8], van den Dries and Wilkie used the nonstandard plication areas. perspective to give a much cleaner account of the asymp- Bernstein-Robinson theorem on invariant subspaces. totic cone construction. Indeed, given a metric space (푋, 푑), The famous Invariant Subspace Problem asks whether every a fixed point 푥0 ∈ 푋, and an infinite hyperreal number ∗ bounded operator 푇 on a separable Hilbert space 퐻 has 푅 ∈ 퐑 , one can look at the subspace a 푇-invariant closed subspace besides {0} and 퐻. In the ∗ 푑(푥, 푥0) 1930s, von Neumann proved that compact operators have 푋푅 ∶= {푥 ∈ 푋 ∶ is finite}. nontrivial closed invariant subspaces. (Recall an operator 푅 is compact if it is the norm-limit of finite-rank operators.) One can then place the metric 푑푅 on 푋푅 given by 푑푅(푥, 푦) Little progress was made on the Invariant Subspace Prob- 푑(푥,푦) ∶= st( ). When 푋 is the Cayley graph associated to a lem until: 푅 group, then the polynomial growth condition on Γ allows 푅 푋 Theorem 3 (Bernstein-Robinson [3]). If 푇 is a polynomi- one to find so that 푅 is locally compact; the verifica- 푋 ally compact operator on 퐻 (meaning that there is a nonzero tion of this and other important properties of 푅 are very polynomial 푝(푍) ∈ ℂ[푍] such that 푝(푇) is a compact oper- clear from the nonstandard perspective. While Gromov’s ator), then 푇 has a nontrivial closed invariant subspace. original proof did not use nonstandard methods, the non- standard perspective on asymptotic cones has now in fact One of the main ideas of the proof is to use the nonstan- become the one that is presented in courses and textbooks dard version of the basic fact that operators on finite- on the subject. See, for instance, [18]. dimensional spaces have upper-triangular representations Jin’s Sumset Theorem. In additive combinatorics, the fo- to find many hyperfinite-dimensional subspaces of 퐻∗ cus is often on densities and structural properties of sub- which are 푇∗-invariant.