Likelihood Geometry

We study the critical points of monomial functions over an algebraic subset of the probability simplex. The number of critical points on the Zariski closure is a topological invariant of that embedded projective variety, known as its maximum likelihood degree. We present an introduction to this theory and its statistical motivations. Many favorite objects from combinatorial algebraic geometry are featured: toric varieties, A-discriminants, hyperplane arrangements, Grassmannians, and determinantal varieties. Several new results are included, especially on the likelihood correspondence and its bidegree. These notes were written for the second author's lectures at the CIME-CIRM summer course on Combinatorial Algebraic Geometry at Levico Terme in June 2013.Comment: 45 pages; minor changes and addition

Classical R-Matrices and the Feigin-Odesskii Algebra via Hamiltonian and Poisson Reductions

We present a formula for a classical $r$-matrix of an integrable system obtained by Hamiltonian reduction of some free field theories using pure gauge symmetries. The framework of the reduction is restricted only by the assumption that the respective gauge transformations are Lie group ones. Our formula is in terms of Dirac brackets, and some new observations on these brackets are made. We apply our method to derive a classical $r$-matrix for the elliptic Calogero-Moser system with spin starting from the Higgs bundle over an elliptic curve with marked points. In the paper we also derive a classical Feigin-Odesskii algebra by a Poisson reduction of some modification of the Higgs bundle over an elliptic curve. This allows us to include integrable lattice models in a Hitchin type construction.Comment: 27 pages LaTe

МЕЗОПРОТЕРОЗОЙСКИЙ БАЗИТОВЫЙ МАГМАТИЗМ БАШКИРСКОГО МЕГАНТИКЛИНОРИЯ (ЮЖНЫЙ УРАЛ): ВОЗРАСТНЫЕ ОГРАНИЧЕНИЯ, ПЕТРОЛОГИЧЕСКИЕ И ГЕОХИМИЧЕСКИЕ ОСОБЕННОСТИ

Among the Riphean–Vendian dyke complexes of the basic composition, which intersect the Precambrian strata of the Bashkirian meganticlinorium (Southern Urals), one of the most common is the Kurgas gabbro-dolerite complex dated to the Early – Middle Riphean. This article presents the isotopic dating, petrological and geochemical features of the rocks belonging to the Kurgas complex. It gives the first description of the regional geochemical zoning that is reflected in a decrease in SiO2 contents from the north to the south along the Bashkirian meganticlinorium, and an increase in MgO contents in the same direction, which is attributed by the authors to an increase in the permeability of the crust to the south during opening of the Mashak riftogenic structure.New data on isotopic ages were obtained for the dyke that cuts the Satka formation of the Lower Riphean – 1318±10 (40Ar/39Ar), the intrusion in the exocontact zone of the Berdyaush rapakivi granite massif – 1349±11 Ma (U-Pb), and the andesite dyke among the metamorphic rocks of the Taratash complex – 1365.6±6.6 Ma (U-Pb). These ages, in combination with the previously obtained data, suggest that the complex formed during a rather long period of time (from 1385 to 1318 Ma, as a minimum), which corresponds to the beginning of the Middle Riphean. Isotopic U-Pb age of the dyke that cuts the contact zone of the Berdyaush massif, suggests that the major portion of the massif had already crystallized by that time and was exhumed into the shallow zone of brittle deformation.Среди рифейско-вендских дайковых комплексов основного состава, секущих докембрийские толщи Башкирского мегантиклинория (Южный Урал), одним из самых распространенных является кургасский габбро-долеритовый раннесреднерифейский комплекс. В работе собраны и обобщены данные по изотопным датировкам, петрологическим и геохимическим особенностям пород кургасского комплекса. Впервые показано наличие региональной геохимической зональности в породах комплекса, которая выражена в уменьшении содержаний SiO2 с севера на юг вдоль Башкирского мегантиклинория и увеличении содержаний MgO в этом направлении, что авторами связывается с увеличением проницаемости коры к югу при раскрытии Машакской рифтогенной структуры.Получены новые изотопные датировки для дайки, секущей саткинскую свиту нижнего рифея, – 1318±10 (40Ar/39Ar), тела в зоне экзоконтакта Бердяушского массива гранитов рапакиви – 1349±11 млн лет (U-Pb) и для дайки андезитов среди метаморфитов тараташского комплекса – 1365.6±6.6 млн лет (U-Pb). Эти данные, в сочетании с полученными ранее, указывают на формирование комплекса на достаточно большом протяжении времени – как минимум от 1385 до 1318 млн лет, что отвечает началу среднего рифея. Полученный возраст для дайки, секущей контактовую зону Бердяушского массива, позволяет предполагать, что к этому моменту массив уже в значительной мере кристаллизовался и вышел в условия хрупкой деформации

The combinatorics of plane curve singularities. How Newton polygons blossom into lotuses

This survey may be seen as an introduction to the use of toric and tropical geometry in the analysis of plane curve singularities, which are germs $(C,o)$ of complex analytic curves contained in a smooth complex analytic surface $S$. The embedded topological type of such a pair $(S, C)$ is usually defined to be that of the oriented link obtained by intersecting $C$ with a sufficiently small oriented Euclidean sphere centered at the point $o$, defined once a system of local coordinates $(x,y)$ was chosen on the germ $(S,o)$. If one works more generally over an arbitrary algebraically closed field of characteristic zero, one speaks instead of the combinatorial type of $(S, C)$. One may define it by looking either at the Newton-Puiseux series associated to $C$ relative to a generic local coordinate system $(x,y)$, or at the set of infinitely near points which have to be blown up in order to get the minimal embedded resolution of the germ $(C,o)$ or, thirdly, at the preimage of this germ by the resolution. Each point of view leads to a different encoding of the combinatorial type by a decorated tree: an Eggers-Wall tree, an Enriques diagram, or a weighted dual graph. The three trees contain the same information, which in the complex setting is equivalent to the knowledge of the embedded topological type. There are known algorithms for transforming one tree into another. In this paper we explain how a special type of two-dimensional simplicial complex called a lotus allows to think geometrically about the relations between the three types of trees. Namely, all of them embed in a natural lotus, their numerical decorations appearing as invariants of it. This lotus is constructed from the finite set of Newton polygons created during any process of resolution of $(C,o)$ by successive toric modifications.Comment: 104 pages, 58 figures. Compared to the previous version, section 2 is new. The historical information, contained before in subsection 6.2, is distributed now throughout the paper in the subsections called "Historical comments''. More details are also added at various places of the paper. To appear in the Handbook of Geometry and Topology of Singularities I, Springer, 202

Коллизионная система Западного Прибайкалья: аэрокосмическая геологическая карта Ольхонского региона (Байкал, Россия)

We announce the second edition of the Aerospace geological map of the Olkhon Region (Baikal, Russia), scale 1:40 000, which was published in 2017. The map has been considerably revised and updated, and its changes are critical for correct understanding of the regional geology, tectonics and geodynamics. Only a small number of its printed copies have been released, and therefore the map may not be available for all interested specialists. The electronic version of the map is available for studying and/or printing (see the link to its pdf file in the paper’s supplement). The pdf file is about 68 MB, i.e. small compared to the original map (more than 5 GB), but the quality is maintained. The map does not show the base layer due to the terms of the licenses owned by the companies and satellite owners.Настоящее краткое сообщение является в значительной степени анонсом второго издания Аэрокосмической геологической карты Ольхонского региона (Байкал, Россия) м-ба 1:40000, изданной в 2017 г. Изменения по сравнению с первым изданием карты весьма значительны и принципиально важны для понимания геологии, тектоники и геодинамики региона. Карта отпечатана небольшим тиражом, поэтому вряд ли будет доступна всем заинтересованным специалистам. В статье же приводится ссылка на электронный вариант карты (pdf-файл), размещенный в дополнительных материалах к статье на сайте журнала, который можно изучать или распечатывать для пользования. Размер электронного варианта файла карты (около 68 Мб) невелик по сравнению с оригиналом (более 5 Гб), однако потери качества нет, из него только удален базовый слой по условиям лицензий, полученных от компаний и владельцев спутников

Projective duality and homogeneous spaces

Projective duality is a very classical notion naturally arising in various areas of mathematics, such as algebraic and differential geometry, combinatorics, topology, analytical mechanics, and invariant theory, and the results in this field were until now scattered across the literature. Thus the appearance of a book specifically devoted to projective duality is a long-awaited and welcome event. Projective Duality and Homogeneous Spaces covers a vast and diverse range of topics in the field of dual varieties, ranging from differential geometry to Mori theory and from topology to the theory of algebras. It gives a very readable and thorough account and the presentation of the material is clear and convincing. For the most part of the book the only prerequisites are basic algebra and algebraic geometry. This book will be of great interest to graduate and postgraduate students as well as professional mathematicians working in algebra, geometry and analysis