Vladislav Goldberg, PhD, a distinguished professor in the department of mathematical sciences at New Jersey Institute of Technology (NJIT), and an expert in web geometry, was honored last month for a lifetime of scholarship. The International Geometry in Odessa Conference in Ukraine lauded the 70-year-old mathematician during a multi-day conference. Goldberg, born and schooled in Moscow, emigrated to the US in 1979 during the immigration wave of the 1970s that brought into the US a number of highly educated Jewish scientists. Today, Goldberg has retained a network of scholarly friends in Russia, the US, Israel and many other countries.
Goldberg is renowned for his understanding of a little-known branch of geometry: web geometry. Only a small number of scholars study this field, although their expertise is frequently tapped by economists and physicists, especially those scientists studying thermodynamics. The late S.S. Chern, PhD, of the University of California, Berkeley, numbered among the past century’s noted mathematicians who worked in web geometry.
Web geometry focuses on the non-changing, or invariant, properties of a series of curved lines laid over a grid of horizontal and vertical lines.
Goldberg and a small international group of other web mathematicians create the rules for studying or understanding web geometry. If there is only one family of curves to overlay the grid, a curvilinear three-web is produced,” said Goldberg. “Two families of curves produce a curvilinear four-web. We can use an infinite number of curves in each web family, but only a finite number of families produce a web. The grid, which is the foundation, or starting point, always counts as the first two families.”
Economists and physicists use web geometry as a tool to prove their theories. Through the years, Goldberg has served many colleagues in other disciplines. His most memorable collaboration was in 2003 with Nobel Prize economists Paul A. Samuelson of MIT, Thomas Russell of University of Santa Clara and James B. Cooper of Johannes Kepler Universität Linz, Austria. The trio challenged him to answer an unsolved problem. “They asked to me find the conditions under which a curvilinear web can be mapped into a web with all transversal families being families of straight lines,” said Goldberg.
Goldberg accepted the challenge and in 2004 and again in 2006, he co-authored two breakthrough works refuting a 1938 assertion by Wilhelm Blaschke, a noted German mathematician credited with founding the field of web geometry. Blaschke (1862-1955), a colorful character in his own right, ended up snubbed by nations and academics later in life for his public Nazi sympathies during World War II.
In Einfuehrung in die Geometrie der Waben (1955) Blaschke wrote that it was “hopeless” for mathematicians to find the conditions under which a curvilinear web can be mapped into a new web with transversal, nonintersecting straight lines.
“By hopeless, he meant that the problem had impossible calculations to carry out by hand,” explained Goldberg. Of course, Blaschke’s words were penned prior to the arrival of personal computing. Some 60 years later, Goldberg and coauthors, with the aid of advanced computer software programs, proved Blaschke wrong.
In 2004, Maks A. Akivis, of the Jerusalem Institute of Technology, Goldberg and Valentin V. Lychagin, of the University of Tromso, Norway, each a renowned mathematical author and scholar, solved the problem for all webs, except three-webs. Selecta Mathematica published “Linearizability of d-webs, d> 4, on two-dimensional manifolds” in December of 2004. Then, in 2005, Goldberg and Lychagin solved the more difficult variant of the problem, finding the conditions under which a curvilinear three-web can be mapped into a linear three-web with transversal straight lines. The Journal of Geometric Analysis published “On the Blaschke conjecture for 3-webs” in March of 2006. Comptes Rendus Mathematique published a six-page version of the same work in August of 2005.
Since 1958, Goldberg has published four monographs, eight textbooks, three book chapters and more than 120 scientific papers. Goldberg received his master’s and doctoral degrees in mathematics from Moscow State University. From 1964 to 1978, Goldberg was a professor in the department of mathematics at Moscow Institute of Steel and Alloys.
In 1979, after immigrating to the US, Goldberg worked at Lehigh University for two years as a visiting professor. In 1981, he joined NJIT as a full professor. In 1985, NJIT made Goldberg one of the university’s earliest distinguished professors.
At the ceremony this past May in Ukraine, Goldberg’s colleagues from different countries warmly lauded his contribution to science and education. They presented him with a formal bound document which was read aloud to him at the ceremony. The document said: “We wish that you continuously move ahead and successfully complete all your plans and intentions. Let the welfare and success and health of all your family members assure your peace of mind and good mood. Let your loyal friends and your highly professional colleagues, united by joint goals, support your professional success which will bring progress and success to our common cause.”