Combinatorial Adventures in Analysis, Algebra, and Topology Stephen Melczer, Marni Mishna, and Robin Pemantle The authors of this piece are organizers of the AMS 2020 Mathematics Research Communities summer conference Combinatorial Applications of Computational Geometry and Algebraic Topology, one of five topical research conferences offered this year that are focused on collaborative research and professional development for early-career mathematicians. Researchers with experience in one or more of the areas of combinatorics, computational algebra, analysis, algebraic topology, and singularity Figure 1. ACSV has been used to determine the shape of large random quantum random walks [3] (left) and cube groves [5] theory are welcomed. Additional information can be (right). found at https://www.ams.org/programs /research-communities/2020MRC-CompGeom. Applications are open until February 15, 2020. Perhaps surprisingly, complex analysis, algebraic ge- ometry, topology, and computer algebra are important Capturing large-scale behavior of combinatorial objects ingredients in such results, combining to form the rela- can be difficult but very revealing. Consider Figure 1, for tively new field of analytic combinatorics in several variables instance, which illustrates the behavior of random objects (ACSV). ACSV builds on classic generating function tech- of large size in two combinatorial classes inspired by quan- niques in combinatorics to provide new tools for the study tum computing and statistical mechanics. of sequences and discrete objects. Applications previously studied in the literature include queuing systems, bioinfor- matics, data structures, random tilings, special functions, and integrable systems. Ongoing research efforts from the Stephen Melczer is a CRM-ISM postdoctoral fellow at the Universit´edu Qu´ebec combinatorial side concern an evolving study of lattice à Montr´eal.His email address is [email protected]. paths [8] and objects from representation theory and al- Marni Mishna is a professor of mathematics at Simon Fraser University. Her email address is [email protected]. gebraic combinatorics [14]. Robin Pemantle is the Merriam Term Professor of Mathematics at the University The purpose of this note is to give a brief overview of of Pennsylvania. His email address is [email protected]. ACSV and its history to drive interest among those in any For permission to reprint this article, please contact: of the diverse areas touched by the topic. A large collection [email protected]. of worked examples can be found in [15] and a detailed DOI: https://doi.org/10.1090/noti2031 treatment of ACSV in [16]. 262 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 2 Analytic Combinatorics expansion The use of analytic techniques in combinatorics and prob- 퐢 푖1 푖푑 퐹(퐳) = ∑ 푓퐢퐳 = ∑ 푓푖1,…,푖푑 푧1 ⋯ 푧푑 . ability theory has a long history, dating back to the 퐢∈ℕ푑 퐢∈ℕ푑 eighteenth-century work of Laplace [12] and contempo- Given 퐫 ∈ ℕ푑, the 퐫-diagonal of 퐹(퐳) is the sequence (푓 ). raries such as Stirling [18] and de Moivre [9]. In mod- 푛퐫 Many mathematical sequences arise naturally as diagonals ern times, the use of real and complex analysis to derive of explicit rational functions. Furthermore, coefficient se- asymptotic behavior is the domain of analytic combina- ries for all algebraic functions may be embedded as gen- torics [11], a field which finds application in many areas of eralized diagonals of rational functions [17]. It is conjec- mathematics, computer science, and the natural sciences. tured that the same is true of all globally bounded D-finite The key idea behind such results is that important prop- functions [7, Conjecture 4]. erties of a sequence (푓푛) = (푓0, 푓1,…) of real or complex numbers are easily deduced from the generating function Example 1. A key step in Ap´ery’s proof [19] of the ir- rationality of 휁(3) is determining asymptotics of the se- 푛 퐹(푧) = ∑ 푓푛푧 = 푓0 + 푓1푧 + ⋯ . quence 푛 2 2 푛≥0 푛 푛 + 푘 푏 = ∑ ( ) ( ) . 푛 푘 푘 Although much can be deduced treating 퐹 only as a for- 푘=0 mal series [20], when 푓푛 grows at most exponentially as 푛 Because it is a sum of nonnegative terms, saddle point ap- goes to infinity then 퐹 defines a complex-analytic function proximations will produce the desired asymptotics. at the origin. The powerful methods of complex analysis Such sums with mixed signs are notoriously difficult to can then be used to study properties of 푓푛, including its estimate. However, any binomial sum sequence regardless asymptotic behavior. of signs can automatically be written as the ퟏ-diagonal of Indeed, the Cauchy integral formula implies that for ev- a rational function [6]. For instance, Ap´ery’s sum is the ery natural number 푛 the coefficient 푓푛 can be represented ퟏ-diagonal of the four-variable rational function by a contour integral 1 . 1 − 푡(1 + 푥)(1 + 푦)(1 + 푧)(1 + 푦 + 푧 + 푦푧 + 푥푦푧) 1 −푛−1 푓 = ∫ 퐹(푧)푧 푑푧, The methods of ACSV determine asymptotics of 푏푛 com- 푛 2휋푖 풞 pletely automatically [13]. where 풞 is a positively oriented circle sufficiently close to Although the 퐫-diagonal of a fixed rational function is the origin. The singularities of the generating function play defined only when 퐫 has integer coordinates, ACSV shows 1 푑 a crucial role in determining sequence asymptotics: the do- that asymptotics along most directions 퐫 ∈ ℝ>0 can be main of integration 풞 can be deformed without changing well defined by a limiting procedure. the value of the Cauchy integral as long it does not cross Multivariate Analysis and Singularity Theory any singularities of 퐹. Setting some technicalities aside, each singularity of 퐹 gives a “contribution” to asymptotics As in the univariate setting, the methods of ACSV start from a Cauchy integral formula of 푓푛 depending on its location in the complex plane and the local behavior of 퐹 near the singularity. This approach 1 푓 = ∫ 퐹(퐳)퐳−푛퐫−ퟏ푑푧 ⋯ 푑푧 , 푛퐫 푑 1 푑 is powerful enough to be completely automated for large (2휋푖) 풞 classes of generating functions, such as rational and al- where 풞 is now a product of positively oriented circles suf- gebraic functions, and is flexible enough to extend as re- ficiently close to the origin. The singularities of 퐹(퐳) again quired by applications. A thorough development of this play a crucial role in asymptotics. If 퐹(퐳) = 퐺(퐳)/퐻(퐳) is a univariate theory and a survey of its applications can be rational function with coprime numerator 퐺 and denomi- found in the fantastic text of Flajolet and Sedgewick [11]. nator 퐻, then the singularities of 퐹 form the set 풱 = {퐳 ∈ 푑 ACSV ℂ ∶ 퐻(퐳) = 0}. Once the dimension is greater than 1, the singular set 풱 In contrast to the well-developed univariate theory, until is no longer discrete but is an algebraic variety of positive recently there was no general framework for the study of dimension. In the geometrically simplest case, when 풱 multivariate generating functions. To rectify this gap in the literature, over the last two decades the theory of ACSV has 1 푑 Asymptotics of the 퐫-diagonal vary smoothly with 퐫 inside open cones of ℝ>0. been developed [16]. Here we focus on a rational function The boundaries of these cones form (푑 − 1)-dimensional sets where asymptotics 퐹(퐳) = 퐹(푧1, … , 푧푑) in 푑 > 1 variables with power series can sharply change. FEBRUARY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 263 forms a smooth manifold, one searches for a (generically interacting with each other to develop this exciting area of finite) set of critical points near which the Cauchy integral mathematics. can be approximated by saddle-point integrals which are easy to estimate asymptotically. When some of these crit- References ′ ical points lie on the boundary of the generating function [1] Arnol d VI, Guse˘ın-ZadeSM, Varchenko AN. Singularities domain of convergence and other mild conditions hold, of Differentiable Maps. Vol. II, Monodromy and asymptotics of integrals, translated from the Russian by Hugh Porteous, asymptotics can be computed using explicit formulas. In translation revised by the authors and James Montaldi, the absence of critical points on the boundary of conver- Monographs in Mathematics, vol. 83, Birkhäuser Boston, gence or when the geometry of this domain is difficult to Inc., Boston, MA, 1988. MR966191 establish, topological techniques may be applied to deter- [2] Atiyah MF, Bott R, G˙arding L. Lacunas for hyperbolic mine which critical points are responsible for the coeffi- differential operators with constant coefficients. I, Acta cient asymptotics. This was carried out in two dimensions Math. (124):109–189, 1970, DOI 10.1007/BF02394570. in [10]; generalizing to higher dimensions is an open prob- MR470499 lem. [3] Baryshnikov Y, Brady W, Bressler A, Pemantle R. Without restricting the geometry of 풱, recent work [4] Two-dimensional quantum random walk, J. Stat. Phys., no. 1 (142):78–107, 2011, DOI 10.1007/s10955-010-0098- has shown how stratified Morse theory helps determine 푑 2. MR2749710 the deformations of 풞 in ℂ ⧵풱 which yield integrals of the [4] Baryshnikov Y, Melczer S, Pemantle R. Critical points type analyzed in classic works on singularity theory [1, 2]. at infinity for analytic combinatorics, Submitted, arxiv The techniques of ACSV thus rely on an interesting mix .org/abs/1905.05250, 2019. of computer algebra, singularity theory, algebra, geome- [5] Baryshnikov Y, Pemantle R. Asymptotics of multivariate try, and topology. By asking different questions, ACSV pro- sequences, part III: Quadratic points, Adv. Math., no. 6 motes new work in these areas. For example, any advances (228):3127–3206, 2011, DOI 10.1016/j.aim.2011.08.004. in the automatic computation of singular integrals can be MR2844940 applied back to the wavelike partial differential equations [6] Bostan A, Lairez P, Salvy B.