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Тиморин Владлен Анатольевич
декан

 

Артамкин Игорь Вадимович
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Кузнецова Вера Витальевна
заместитель декана

 

Фейгин Евгений Борисович
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Эстеров Александр Исаакович
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119048, Москва,
ул. Усачёва, 6
тел. (495) 772-95-90 *12725 (секретарь)
тел. (495) 772-95-90 *12726 (декан)
тел. (495) 624-26-16
e-mail: math@hse.ru

Учебный офис:
mathstudyoffice@hse.ru

Редакторы сайта факультета:
Коршунов Дмитрий Олегович
Кузнецова Вера Витальевна

Low-dimensional Topology and Algebraic Geometry

Thursdays 12:00-13:30 Room 413 (Usacheva str. 6)

Syllabus   

List of suggested student topics

Organizers:
  •   Chris Brav          chris.i.brav@gmail.com       Room 333
  •   Ash Lightfoot      alightfoot@hse.ru                  Room 309

Current schedule of talks
Date
Speaker
Topic
Abstract
References
October 5, 2017 Chris Brav, Ash Lightfoot Organization   
October 12, 2017 Ash Lightfoot Heegaard splittings via Morse theory A 3-dimensional handlebody is an orientable 3-manifold obtained from the 3-ball by attaching handles. In this talk we show that any closed, orientable 3-manifold can be obtained by gluing together two handlebodies along their common boundary. Moreover, we show that such a decomposition of a 3-manifold, called a Heegaard splitting, is unique up to homeomorphism and a certain stabilization operation. (These results are classical, originally due to Reidemeister and Singer.) To do so, we introduce the basics of Morse theory.Morse Theory, John Milnor; An Invitation to Morse Theory, Liviu Nicolaescu
October 19, 2017 Ash LightfootHeegaard splittings via Morse theory (Part II)Continued. 
November 2, 2017 Balaram UsovThe mapping class groups of surfacesIn this talk I will define the mapping class group of a surface. We will compute this group for the torus and prove an important theorem that this group is finitely generated by Dehn twists. In doing so I will define a complex of curves and mention some of its properties.Knots and Links, Dale Rolfsen; A primer on mapping class groups, Benson Farb, Dan Margalit
November 9, 2017 Roman KrytovskyDehn's lemma and the loop theoremI will prove Dehn's lemma, which roughly says that if you have an embedding of the boundary of a 2-disk into the boundary of a 3-manifold, then you can extend the mapping to an embedding of the whole disk into this 3-manifold. A central part of the proof will be a tower construction due to Papakyriakopoulos.Knots and Links, Dale Rolfsen; Notes on 3-manifolds, Danny Calegari
November 16, 2017 Roman KrytovskyDehn's lemma and the loop theorem (Part II)Continued. 
November 16, 2017 Balaram UsovThe mapping class groups of surfaces (Part II)Continued. 
November 23, 2017 Ash LightfootSurgery on a link and Kirby calculus A fundamental result in low-dimensional topology is the Lickorish-Wallace theorem, which states that any closed, orientable 3-manifold can be obtained by integral Dehn surgery on a link in the 3-sphere. Such a 3-manifold may thus be described by a "Kirby" diagram: a diagram of a link where each component is equipped with an integer. Further, Kirby's theorem states that two such 3-manifolds are homeomorphic if and only if their Kirby diagrams are related by certain so-called Kirby moves. In this talk we discuss these results and their relation to Heegaard decompositions. 
November 30, 2017 Ash LightfootSurgery on a link and Kirby calculus (Part II)Continued from the previous talk. 
December 7, 2017 Tatiana OvchinnikovaThe Jones Polynomial and a conjecture of TaitThere were many difficulties in tabulating knots at the dawn of knot theory. In this talk we will discuss one of the Tait Conjectures, which were used to add many knots to the tables and assume they had not been seen earlier. The conjecture states that any reduced, alternating diagram of a knot has the minimal possible number of crossings. It was made in the 19th century and was finally proved by Morwen Thistlethwaite, Louis Kauffman and Kunio Murasugi in 1987, using the Jones polynomial.An Introduction to Knot Theory, W.B.Raymond Lickorish
December 14, 2017 Yurii RudkoKirby diagrams of 4-manifoldsI will talk about a way of understanding and studying 4-manifolds called a Kirby diagram. Any (closed, connected, smooth) 4-manifold can be decomposed into the union of 4-dimensional handles (each diffeomorphic to a 4-dimensional disk) with a single 0-handle. The boundary of the 0-handle is a 3-sphere, so can be considered as R^3 and a point. I will show how this allows us to construct the whole manifold by drawing a framed link in R^3, that is, a link for which each component is equipped with an integer. The Topology of 4-manifolds, Robion Kirby; 4-manifolds and Kirby calculus, Andras Stipsicz and Robert Gompf; Knots and Links, Dale Rolfsen; Lectures on the Topology of 3-Manifolds: An Introduction to the Casson Invariant (2nd revised edition), Nikolai Saveliev
TBA Semyon AbramyanAlgebraic topology of lens spacesTBAKnots and Links, Dale Rolfsen; Lectures on the Topology of 3-Manifolds: An Introduction to the Casson Invariant (2nd revised edition), Nikolai Saveliev; Introduction to Combinatorial Torsions, Vladimir Tureuv; Lecture Notes in Algebraic Topology, James Davis and Paul Kirk
TBA Anton Shlyapugin, Semyon AbramyanWhitehead's theorem (homotopy classification of oriented, simply connected 4-manifolds)TBA The Topology of 4-manifolds, Robion Kirby
TBA Alexander DunaykinTrisections of 4-manifoldsTBA Trisecting 4-manifolds, David Gay and Robion Kirby
TBA Denis TereshkinMorse homologyTBA An Invitation to Morse Theory, Liviu Nicolaescu