Page 1. THE UNIVERSALITY THEOREMS ON THE CLASSIFICATION PROBLEM OF CONFIGURATION VARIETIES AND CO... more Page 1. THE UNIVERSALITY THEOREMS ON THE CLASSIFICATION PROBLEM OF CONFIGURATION VARIETIES AND CONVEX POLYTOPES VARIETIES NE .Mnev Institute for Social and Economic Problems USSR Academy of Sciences The results which we present here form the part of guiding by AM Vershic topological investigations of combinatorially defined confi- guration spaces (see his article in this volume). In this paper we shall outline the proof of coincidence of the two classes of variety: first -the spaces of point configurations in ...
On 2 October, 2020, just over two months before his centenary, the outstanding representative of ... more On 2 October, 2020, just over two months before his centenary, the outstanding representative of the St Petersburg School of Geometry, honourary member of the St Petersburg Mathematical Society, Professor Viktor Abramovich Zalgaller passed away. He was born on 25 December 1920 in the village of Parfino, Novgorod Province, in the family of engineer Abram Leont’evich Zalgaller and attorney Tat’yana Markovna Shabad-Zalgaller. In 1922 the family moved to Petrograd. In 1931 his father was convicted under Article 581 and spent 16 years in Ukhtpechlag2, in prison, and then in exile. After graduating from School no. 103 in the Smol’ninskii district of Leningrad in 1937, Viktor enrolled in the Faculty of Mathematics and Mechanics of Leningrad University. His abilities were noticed by L. V. Kantorovich, his lecturer on mathematical analysis, who asked the third-year student to prepare a textbook based on his notes. In 1940, as part of a mobilisation announced by the Young Communist League, Zalgaller was transferred to the Aviation Institute. There he received an engineering education, which played a significant role in his further research career. In the first days of the war he volunteered for the People’s Militia (Second Division) and was sent to the front almost immediately. During almost the entire war he was a signaller. The combat path of Viktor Zalgaller was not easy: the defence of Leningrad, the Oranienbaum Bridgehead, an injury during the lifting of the Leningrad blockade, the storming of Vyborg, battles in the Baltic states, the storming of Danzig, reaching the Elbe. . . Zalgaller was awarded the Order of the Red Star, the medal “For Courage”, three medals “For Battle Merit”, the medal “For the Defence of Leningrad”, and others. He met the end of the war as
Discrete and Computational Geometry: Papers from the DIMACS Special Year, 1991
The problem of describing the class of realization spaces of oriented matroids was posed by AM Ve... more The problem of describing the class of realization spaces of oriented matroids was posed by AM Vershik in the 1970s. This problem was solved in 1984 [2, 3]. The solution was unexpected. By definition, the set of all matrix realizations of rank three oriented matroids is a semialgebraic subset of the space of all 3 xm real matrices. The following phenomenon was established: up to trivial stabilization, any elementary semialgebraic variety can be represented as the realization space of an oriented matroid of rank three.
Here we are fixing an output of a trivial calculation based on Konsevich's differential 2-for... more Here we are fixing an output of a trivial calculation based on Konsevich's differential 2-form for the Chern class of polygon bundle. As a result an interesting combinatorics and arithmetics jumps right out of a jukebox. The calculation gives very simple rational combinatorial characteristics (we call it "curvature") of a triangulated $S^1$ bundle over a 2-simplex, which is a local combinatorial formula for the first Chern class. The curvature is expressed in terms of cyclic word in 3-character alphabet associated to the bundle. From the point of view of simplicial combinatorics the word is a canonical shelling of the total complex. If you know a triangulation of a bundle - you can really easily compute the Chern class.
We present a local combinatorial formula for the Euler class of a n-dimensional PL spherical ber ... more We present a local combinatorial formula for the Euler class of a n-dimensional PL spherical ber bundle as a rational number eCH associated to a chain of n + 1 abstract subdivisions of abstract n-spherical PL cell complexes. The number eCH is a combinatorial (or matrix) Hodge theory twisting cochain in Guy Hirsch's homology model of the bundle associated with PL combinatorics of the bundle.
Page 1. THE UNIVERSALITY THEOREMS ON THE CLASSIFICATION PROBLEM OF CONFIGURATION VARIETIES AND CO... more Page 1. THE UNIVERSALITY THEOREMS ON THE CLASSIFICATION PROBLEM OF CONFIGURATION VARIETIES AND CONVEX POLYTOPES VARIETIES NE .Mnev Institute for Social and Economic Problems USSR Academy of Sciences The results which we present here form the part of guiding by AM Vershic topological investigations of combinatorially defined confi- guration spaces (see his article in this volume). In this paper we shall outline the proof of coincidence of the two classes of variety: first -the spaces of point configurations in ...
On 2 October, 2020, just over two months before his centenary, the outstanding representative of ... more On 2 October, 2020, just over two months before his centenary, the outstanding representative of the St Petersburg School of Geometry, honourary member of the St Petersburg Mathematical Society, Professor Viktor Abramovich Zalgaller passed away. He was born on 25 December 1920 in the village of Parfino, Novgorod Province, in the family of engineer Abram Leont’evich Zalgaller and attorney Tat’yana Markovna Shabad-Zalgaller. In 1922 the family moved to Petrograd. In 1931 his father was convicted under Article 581 and spent 16 years in Ukhtpechlag2, in prison, and then in exile. After graduating from School no. 103 in the Smol’ninskii district of Leningrad in 1937, Viktor enrolled in the Faculty of Mathematics and Mechanics of Leningrad University. His abilities were noticed by L. V. Kantorovich, his lecturer on mathematical analysis, who asked the third-year student to prepare a textbook based on his notes. In 1940, as part of a mobilisation announced by the Young Communist League, Zalgaller was transferred to the Aviation Institute. There he received an engineering education, which played a significant role in his further research career. In the first days of the war he volunteered for the People’s Militia (Second Division) and was sent to the front almost immediately. During almost the entire war he was a signaller. The combat path of Viktor Zalgaller was not easy: the defence of Leningrad, the Oranienbaum Bridgehead, an injury during the lifting of the Leningrad blockade, the storming of Vyborg, battles in the Baltic states, the storming of Danzig, reaching the Elbe. . . Zalgaller was awarded the Order of the Red Star, the medal “For Courage”, three medals “For Battle Merit”, the medal “For the Defence of Leningrad”, and others. He met the end of the war as
Discrete and Computational Geometry: Papers from the DIMACS Special Year, 1991
The problem of describing the class of realization spaces of oriented matroids was posed by AM Ve... more The problem of describing the class of realization spaces of oriented matroids was posed by AM Vershik in the 1970s. This problem was solved in 1984 [2, 3]. The solution was unexpected. By definition, the set of all matrix realizations of rank three oriented matroids is a semialgebraic subset of the space of all 3 xm real matrices. The following phenomenon was established: up to trivial stabilization, any elementary semialgebraic variety can be represented as the realization space of an oriented matroid of rank three.
Here we are fixing an output of a trivial calculation based on Konsevich's differential 2-for... more Here we are fixing an output of a trivial calculation based on Konsevich's differential 2-form for the Chern class of polygon bundle. As a result an interesting combinatorics and arithmetics jumps right out of a jukebox. The calculation gives very simple rational combinatorial characteristics (we call it "curvature") of a triangulated $S^1$ bundle over a 2-simplex, which is a local combinatorial formula for the first Chern class. The curvature is expressed in terms of cyclic word in 3-character alphabet associated to the bundle. From the point of view of simplicial combinatorics the word is a canonical shelling of the total complex. If you know a triangulation of a bundle - you can really easily compute the Chern class.
We present a local combinatorial formula for the Euler class of a n-dimensional PL spherical ber ... more We present a local combinatorial formula for the Euler class of a n-dimensional PL spherical ber bundle as a rational number eCH associated to a chain of n + 1 abstract subdivisions of abstract n-spherical PL cell complexes. The number eCH is a combinatorial (or matrix) Hodge theory twisting cochain in Guy Hirsch's homology model of the bundle associated with PL combinatorics of the bundle.
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