Victor Enolski (1945–2019) Emma Previato

Victor Enolski (1945–2019) Emma Previato Our friend and coauthor Victor Enolski passed away in Ukraine, on April 26, 1945. His father, Eugene (Zelik) 0 June 2019, after a brief but valiant struggle with pancre- Ènol ski˘ı, was a major in the Soviet Army during WWII atic cancer. Victor’s profound and multipronged contribu- and afterwards Professor of History in Kiev State Univer- tion to the mathematical sciences flew under the radar, to sity; his mother, Elizabeth Enolskaya (n´eeVainrub), served some extent; “Victor’s modesty is legend” says one of the in hospitals during the war as a senior lieutenant; after the pieces below, and indeed he was self-effacing to a fault, war she was a medical doctor in child psychiatry. Victor promoting instead others’ work, creating interdisciplinary earned an MSc from Kiev State University under the super- collaborations among scientists of diverse backgrounds, re- vision of Professor D. Ya. Petrina, National Academy of discovering classical work with childlike enthusiasm and Sciences of Ukraine (NASU), with a Master’s thesis Differ- breathing new life into it. We write this homage to his ential Equations for Feynman Integrals. Another part of his life with the hope of bringing his contribution to the at- MSc research, on the Mandelstam hypothesis in elemen- tention of the larger mathematical community, particu- tary particle physics, is the topic of his earliest MathSciNet larly because the subjects of his main focus, the special entry [15]. At NASU, Victor earned a PhD in Theoretical functions of classical geometry, mathematical physics, an- Physics and Applied Mathematics (1977) with his thesis alytic number theory and wide-ranging applications (for Analytical Properties of Feynman Integrals, as well as a Doc- cosmology, e.g., see Kunz and Lämmerzahl’s piece below), tor of Science degree in Theoretical Physics and Applied are gaining more power, partly on the strength of com- Mathematics (1996) with the thesis Method of Reduction putational differential algebra and geometry (see Eilbeck’s to Elliptic Functions in the Theory of Solitons. Victor was a piece below). Buchstaber’s piece offers a broad overview junior research fellow at NASU from 1975 to 1982, and of the foundational aspects of these specific special func- a senior research fellow thereafter. In 2016, he became tions, a view from the top that allows you to glimpse their a Professor in the Physics and Mathematics Department significance in topology, combinatorics, algebraic geom- of the National University of Kiev-Mohyla Academy. Vic- etry, deformation theory, and their crowning application tor conducted extended visits as an invited scholar at over to integrability of non-linear PDEs, one main reason for 20 universities or research institutes in Canada, Denmark, the resurgence they enjoyed the world over starting in the France, Germany, Italy, Japan, Norway, Spain, Sweden, the 1970s (see Braden’s and Matveev’s pieces). We tried to pro- UK and the US. vide a somewhat technical (although, we hope, accessible) Victor moved to Germany in the late 1990s, with his guide to enable you to follow your specific interest and wife Rena and their daughter, but he still spent most of read more about Victor’s profound ideas. There is so much his working time in Ukraine and untiringly applied for farther to go, for future researchers. workshop grants so that he could bring an international 0 Victor Zelikovich Ènol ski˘ı, as first transliterated in community to Kiev; the last workshop he organized, “Alge- Mathematical Reviews (most recently he chose the sim- braic Curves, Integrable Systems, and Cryptography,” (Au- pler version Enolski) was born in Odessa (then USSR), gust 24–25, 2018) gave rise to several new collaborative projects and publications. The announcement of the work- Emma Previato is a professor of mathematics in the Department of Mathematics shop on the university’s webpage allows us to preview the and Statistics at Boston University. Her email address is [email protected]. arc of Victor’s interests we’ll follow below, as part of it Communicated by Notices Associate Editor Steven Sam. reads (verbatim): “different aspects of the theory of alge- For permission to reprint this article, please contact: braic curves (hyperelliptic and nonhyperelliptic, and some [email protected]. particular cases): fields of Abelian functions and addition DOI: https://doi.org/10.1090/noti2190 DECEMBER 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1755 laws in these fields, multivariative entire functions which The goal was motivated, among other things, by the fol- generate the Abelian functions.” lowing problem. Let a differential equation with param- He spent the last few months of his life at home in Mu- eters be given. Its solution depends on the parameters nich, where he died on June 20, 2019. and initial data. The problem is to build a new differen- In this short article all we can do is try to give you a tial equation describing this dependence. Such a problem glimpse into Victor’s legacy to mathematics and physics. naturally arises in theoretical and mathematical physics. There are no words that could describe his legacy to his In memory of Victor Enolski, I prepared, addressed to a collaborators, of whom MathSciNet lists 51, and to his wide mathematical audience, a review of the theory whose friends: since Victor could find humor under the most creation was largely due to Enolski. trying circumstances and used the English language with Victor Enolski attained deep results on the relationship gusto, and in view of his expansive professional pilgrim- between theta and sigma functions of algebraic curves, ages, we will remember him with the humorous words of which gave answers to long-standing questions (see [11]). the Head of the School of Mathematical and Computer In this review, multidimensional theta functions are not Sciences at Heriot-Watt University, who had been chasing explicitly involved. However, it should be noted that now him for days with a travel-reimbursement form to sign. there are many representations of multidimensional sigma Victor, Chris Eilbeck and I were working in Chris’ office functions in the form of modified multidimensional theta when he walked in and said “You are peripatetic and mis- functions. We can say that in essence we are talking about chievous.” one class of functions, but in different “coordinates.” In the multidimensional case, as in the elliptic, there are problems in which either theta or sigma functions can 1. Multidimensional Sigma be successfully used. But in well-known problems, to obtain an effective description of solutions of equations, it is Functions and Applications important to use one and not the other. For example, if you want to investigate the dependence on the variations V. M. Buchstaber of individual branch points of a curve, theta functions are preferred. But, if, for example, it is necessary to obtain 1.1. Introduction. We met with Victor Enolski in 1995. results that require solutions of differential equations, in- It was a time when the theory of multidimensional theta cluding differentiation with respect to the parameters of functions and methods of algebraic geometry were the fo- the curve, or require limit transitions to the variety of de- cus of attention of a large mathematical community, in- generate curves, then sigma functions are preferable. At spired by fundamental results in a number of relevant ar- the same time, there are remarkable results that use sigma eas of theoretical and mathematical physics. Our close col- functions as modified theta functions and vice versa. laboration and friendship continued until his last days. He 1.2. Theory of sigma functions. In the classical sense, an was a brilliant scientist and a very good friend. Abelian function is a meromorphic function on a complex Almost immediately after the start of our collaboration, Abelian torus 푇푔 = ℂ푔/Γ, where Γ ⊂ ℂ푔 is a rank-2푔 lattice we put forward an ambitious program: to develop the the- satisfying the Riemann conditions. That is, 푓 is an Abelian ory of multidimensional sigma functions based on mod- function if and only if 푓 is a meromorphic function and ern achievements of differential topology and algebraic ge- 푔 푓(퐮) = 푓(퐮 + 휔) for all 퐮 = (푢1, … , 푢푔) ∈ ℂ and 휔 ∈ Γ. ometry. We decided to follow Weierstrass’ approach to Abelian functions form a differential field 퐹. Basic facts the theory of elliptic functions, which essentially uses the about Abelian functions: remarkable model of elliptic curves. Our theory was de- 퐴 푓 ∈ 퐹 휕 푓 ∈ 퐹 푖 = 1, … , 푔 signed to ensure the construction of the fundamental equa- 1 If , then ᵆ푖 , . tions of mathematical physics based on the uniformiza- 퐴2 For any nonconstant 푓1, … , 푓푔+1 from 퐹 there exists tion of the Jacobians of a certain class of algebraic curves. 푃 ∈ ℂ[푧1, … , 푧푔+1] such that 푃(푓1, … , 푓푔+1) = 0. The key goal was to obtain solutions of these equations, in- 퐴3 If 푓 ∈ 퐹 is any nonconstant function, then any ℎ ∈ 퐹 (푓, 휕 푓, … , 휕 푓) dependent of the choice of a basis in the lattice spanned by is a rational function of ᵆ1 ᵆ푔 . 푔 the periods of holomorphic differentials on these curves. 퐴4 There exists an entire function 휃 ∶ ℂ → ℂ such that 휕ᵆ푖 휕ᵆ푗 log 휃 ∈ 퐹, 푖, 푗 = 1, … , 푔. Victor Buchstaber is a professor at the Faculty of Mathematics and Mechanics, The theory of Abelian functions was a central topic of Moscow State University, and an emeritus professor at the School of Mathemat- 19th century mathematics. In the mid-seventies of the last ics, University of Manchester. His email address is [email protected]. century, a new wave of investigation in this field arose in response to the discovery that Abelian functions provide 1756 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 11 solutions to a number of challenging problems of modern The work in [9] and subsequent papers is devoted to theoretical and mathematical physics.

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