Fall 2019 – Spring 2020
Fall 2017 – Spring 2018
MATHEMATICAL COURSES 2017–2018
Algebraic Geometry: a start-up course (geometric introduction to Algebraic Geometry)
This course is oﬀered jointly with the Math in Moscow program.
Instructor: Alexey Gorodentsev
Course description: Algebraic geometry studies geometric loci deﬁned by polynomial equations, for example the complex plane curve f(x; y) = 0. It plays an important role at both elementary and sophisticated levels in many areas of mathematics and theoretical physics, and provides the most visual and elegant tools to express all aspects of the interaction between diﬀerent branches of mathematical knowledge. The course gives the ﬂavor of the subject by presenting examples and applications of the ideas of algebraic geometry, as well as a ﬁrst discussion of its technical apparatus.
Prerequisites: basic linear and multilinear algebra (tensor products, multilinear maps), basic ideas of commutative algebra (polynomial rings and their ideals). Some experience in geometry and topology (projective spaces, metric and topological spaces, simplicial complexes and homology groups) is desirable but not essential.
- Projective spaces. Geometry of projective quadrics. Spaces of quadrics.
- Projective conics and PGL(2). Rational curves and the Veronese curve. Cubic curves.
- The grassmannians, Veronese, and Segre varieties. Examples of projective maps.
- Commutative algebra drought: Integer ring extensions, polynomial ideals, aﬃne algebraic geometry, and Hilbert’s theorems.
- Plane projective algebraic curves: point multiplicities, intersection numbers, Bezout’s theorem, duality, the Pl¨ucker formulas.
- Algebraic varieties, Zarisky topology, schemes, geometry of ring homomorphisms.
- Irreducible varieties. Dimension. Dimensions of projective varieties.
- Working example: curves on surfaces, the 27 lines on a smooth cubic surface.
- Vector bundles and their sheaves of sections. Vector bundles on the projective line.
- Linear systems, invertible sheaves, and divisors. The Picard group, line bundles on aﬃne and projective spaces.
- If the time allows: Tangent, cotangent, normal and conormal bundles. The Euler exact sequence on projective spaces and grassmannians. Smothness, tangent cone, blow up.
- A. L. Gorodentsev, Algebraic Geometry Start Up Course, MCCME.
- J. Harris, Algebraic Geometry. A First Course, Springer.
- M. Reid, Undergraduate algebraic geometry, CUP. Univ. Press, 1988.
This course is oﬀered jointly with the Math in Moscow program.
Instructor: Alexey Gorinov
Course description: Topology studies properties of geometric objects that are unchanged under continuous deformations.
- The language of topology. Continuity, homeomorphism, compactness for subsets of Rn(from the epsilon-delta language to the language of neighborhoods and coverings).
- The objects of topology: topological and metric spaces, simplectic and cell spaces, manifolds. Topological constructions (product, disjoint union, wedge, cone, suspension, quotient spaces, cell spaces, examples of ﬁber bundles).
- Examples of surfaces(2-manifolds), orientability, Euler characteristic. Classiﬁcation of surfaces (geometric proof for triangulated surfaces).
- Homotopy and homotopy equivalence, the homotopy groups and their main properties.
- Vector ﬁelds on the plane. Generic singular points. The index of a plane vector ﬁeld. Vector ﬁelds on surfaces. The Poincar´e index theorem.
- Inﬁnite constructions (counterexamples to "obvious"statements): the Cantor set, the Peano curve, the Brouwer continuum , Antoine’s necklace, and Alexander’s horned sphere.
- Curves in the plane, degree of a point with respect to a curve, Whitney index (winding number) of a curve, classiﬁcation of immersions, the “fundamental theorem of algebra”. Degree of a map of a circle into itself. Brouwer ﬁxed point theorem.
- Fundamental group (main properties, simplest computations), covering spaces. Algebraic classiﬁcation of covering spaces (via subgroups of the fundamental group of the base). Branched coverings,Riemann-Hurwitz theorem.
- Knots and links in 3-space. Reidemeister moves. The Alexander-Conway polynomial.
- W.Massey, A basic course in algebraic topology, Graduate Texts in Mathematics,Springer-Verlag, New York, 1991.
- J.Munkrs, Topology (2nd Edition), Pearson, 2000.
Aﬃne Kac-Moody Lie algebras.
Instructor: Evgeny Feigin
Course description: The theory of Lie groups and Lie algebras is one of the central areas of modern mathematics. It has various interrelations with algebraic geometry, combinatorics, theory of symmetric functions, integrable systems, classical and quantum ﬁeld theories. Lie groups and Lie algebras usually show up as the sets of symmetries of objects of a theory. For example, aﬃne Kac-Moody algebras turn out to be very important for the description of many quantum ﬁeld theories showing up as symmetries of the spaces of states. These Lie algebras also play an important role in the theory of integrable systems and in algebraic geometry. Our goal is to give an introduction to the theory of aﬃne Kac-Moody algebras. We will describe main deﬁnitions, constructions and applications of the theory. The course is aimed at PhD students and master students.
Prerequisites: basic Lie theory.
- Finite-dimensional simple Lie algebras: Cartan subalgebras, root vectors: Weyl group.
- Finite-dimensional simple Lie algebras: Verma modules, irreducible representations, characters.
- Generalized Cartan matrices,Kac-Moody Lie algebras - deﬁnitions and ﬁrst properties..
- Finite, Aﬃne and IndeﬁniteKac-Moody Lie algebras.
- Classiﬁcation of aﬃneKac-Moody Lie algebras.
- Weyl group and invariant bilinear form.
- Real and imaginary roots.
- Canonical central element and imaginary roots for aﬃne Lie algebras.
- Center and the Weyl group for aﬃne Lie algebras.
- Weyl group for aﬃne Lie algebras and alcoves.
- Aﬃne walls and alcoves.
- Loop algebras and central extensions.
- V. Kac, Inﬁnite dimensional Lie algebras, Cambridge University Press.
- R.Carter, Lie Algebras of Finite and Aﬃne Type, Cambridge Studies in Advanced Mathematics .
Geometric Representation Theory
Instructor: Michael Finkelberg
Course description: Geometric representation theory applies algebraic geometry to the problems of representation theory. Some of the most famous problems of representation theory were solved on this way during the last 40 years. We will study representations of the aﬃne Hecke algebras using the geometry of aﬃne Grassmannians (Satake isomorphism) and Steinberg varieties of triples (Deligne-Langlands conjecture). This is a course for master students knowing the basics of algebraic geometry, sheaf theory, homology and K-theory.
Prerequisites: Basic algebraic geometry (projective varieties, coherent and constructible sheaves). Basic representation theory (of compact Lie groups). Some experience in equivariant cohomology and K-theory is desirable.
- Aﬃne Grassmannians.
- Stratiﬁcations with ﬁnite and coﬁnite strata.
- Example for the general linear group.
- Semiinﬁnite orbits.
- Hyperbolic localization.
- Equivariant perverse sheaves.
- Mirkovi´c-Vilonen ﬁber functor.
- Exactness of convolution.
- Drinfeld fusion.
- Commutativity constraint.
- Tannakian formalism and the Langlands dual group.
- Geometric Satake equivalence.
- Aﬃne Hecke algebras.
- Steinberg variety of triples.
- Polynomial representation.
- Kazhdan-Lusztig-Ginzburg isomorphism.
- Standard representations
- Shapovalov form
- Deligne-Langlands-Lusztig classiﬁcation.
N.Chriss and V.Ginzburg, Representation theory and complex geometry, Birkh¨auser, Boston, 1997.
Instructor: Michael Finkelberg
Course description: Symmetric functions is a lively subject developing rapidly for the last 200 years, at the crossroads of combinatorics and representation theory. We will start from scratch (elementary symmetric polynomials) and hopefully get to Macdonald polynomials. Everybody starting from the 2nd year students is welcome, but some basic representation theory will be helpful.
Prerequisites: basic linear and multilinear algebra (tensor products, multilinear maps). Basic group theory and basic representation theory.
- Symmetric functions ring.
- Schur functions.
- Skew Schur functions.
- Transition matrices.
- Characters of symmetric groups.
- Littlewood-Richardson rule
- Polynomial functors
- Hall algebra
- Hall polynomials
- Hall-Littlewood functions.
- Green functions.
- Parabolic induction.
- Characters of general linear groups over ﬁnite ﬁelds.
- Macdonald polynomials.
I.G.Macdonald, Symmetric Functions and Hall Polynomials, second edition. Oxford University Press, 1995.
Introduction to Number Theory.
Instructor: Vadim Vologodsky
Course description: The beginnings of number theory can be traced to Diophantine equations: polynomial equations such that only the integer solutions are sought or studied. Surprisingly this is a highly structured part of mathematics: there are general results and conjectures which have many concrete nontrivial corollaries. Number theory uses tools from algebra, analysis, and topology. The course covers some of the most important results obtained by the beginning of the 20th century.
Prerequisites: basic abstract algebra (linear algebra, rings, groups, the Galois theory) and elementary complex analysis.
- Finite ﬁelds
- Integers represented by binary quadratic forms
- Quadratic reciprocity law
- Division rings over number ﬁelds
- The ideal class group
- Dirichlet’s theorem on units in number ﬁelds
- Local ﬁelds
- Hasse-Minkowski theorem
- Dirichlet’s theorem on primes in arithmetic progressions
- Analytic class number formula
- J.P. Serre, A course in Arithmetic. Springer (1973)
- Z.I. Borevich and I. R. Shafarevich, Number Theory. Academic Press Inc (1966)
- Algebraic Number Theory, Proceedings of an Instructional Conference Organized by the London Mathematical Society, Edited by Cassels and Frohlich (1967)
Introduction to Ergodic Theory
Instructor: Michael Blank
Course description: Is it possible to distinguish deterministic chaotic dynamics from a purely random and whether this question makes sense? Does irreversibility inﬂuence qualitative characteristics of the process? Ergodic theory studies these and other statistical properties of dynamical systems. Interest in this subject stems from the fact that “typical” deterministic dynamical systems (eg, diﬀerential equations) exhibit chaotic behavior: their trajectories look similar to the implementation of random processes. We begin with the classical results by Poincare, Birkhoﬀ, Khinchin, Kolmogorov, and get to modern productions (including yet unresolved) problems. This is an introductory course designed for 2-4 bachelors and graduate students.
Prerequisites: Prior knowledge except for the course in mathematical analysis is not required (although it is desirable).
- Dynamical systems: trajectories, invariant sets, simple and strange attractors and their classiﬁcation, chaos.
- Action in the space of measures, concept of the transfer operator, invariant measures. Comparison with Markov chains.
- Ergodicity, Birkhoﬀ ergodic theorem, mixing, CLT.Sinai-Bowen-Ruelle measures and natural / observable measures.
- Basic ergodic constructions: direct and skew products, Poincare and integral maps, natural extension and irreversibility problem.
- Ergodic approach to number theoretical problems.
- Hyperbolic dynamical systems and Lyapunov exponents.
- Entropy: metric and topological approaches.
- Operator formalism. Spectral theory of dynamical systems. Banach spaces of measures, random perturbations.
- Multicomponent systems: synchronization and phase transitions.
- Mathematical foundation of numerical simulations of chaotic dynamics.
- M. Blank. Stability and localization in chaotic dynamics’, MCCME, Moscow, 2001.
- I.P. Kornfeld, Ya.G. Sinai, S.V. Fomin. Ergodic theory, Nauka, Moscow, 1980.
- A. Katok, B. Hasselblatt. Introduction to the modern theory of dynamical systems, Cambridge Univ. Press, 1995.
Problems in Commutative Algebra
Instructor: Chris Brav
Course description: The seminar will illustrate the general theory of algebraic geometry and commutative algebra by focusing on the special case of varieties over a ﬁeld, using concrete techniques, particularly Groebner bases, resultants, and elimination theory.
Prerequisites: Some familiarity with rings, ideals, and modules.
- Division algorithm for polynomials in many variables.
- Groebner bases.
- Hilbert’s Basissatz
- Ideal membership problem.
- Computing solutions of polynomial systems of equations.
- Computing the closure of the image of a map of aﬃne varieties (elimination theory).
- Resultants and elimination theory.
- Hilbert’s Nullstellensatz
- Irreducible components. Dimension.
- Projective elimination theory.
- Cox, Little, O’Shea. Ideals, Varieties, and Algorithms. Springer.
- Hassett. Introduction to Algebraic Geometry. Cambridge Univ. Press.
Complex-Analytic Algebraic Geometry
Instructor: Misha Verbitsky
Course description: Complex algebraic geometry was developed by W. V. D. Hodge in 1930-es and was given its modern form by A. Weil and S.-S. Chern in 1940-ies and 1950-ies. This is a discipline allowing one to get results of classical algebraic geometry using basic observations of analysis, diﬀerential geometry and topology instead of complicated algebraic computations. As an additional bonus, the methods of Hodge theory can be used to study non-algebraic objects, such as general K¨ahler manifolds and more complicated geometric objects.
Prerequisites: One would need a good understanding of analysis on manifolds (vector bundles, diﬀerential forms, de Rham cohomology, sheaf cohomology, Riemannian manifolds) and diﬀerential geometry (connections, parallel transoport along the connections, principal bundles). We assume working knowledge of linear algebra (tensor spaces, Grassmann algebra, Hermitian structures), topology (de Rham cohomology, fundamental groups, coverings), one-dimensional complex analysis (Cauchy formula) and representation theory (Lie groups and Lie algebras).
- Kahler manifolds and algebraic manifolds
- Hodge theory on Riemannian and Kahler manifolds
- Poincare-Dolbeault-Grothendieck lemma and its applications
- Line bundles, Chern connection and its curvature,ddc-lemma and its applicattions
- Kodaira-Nakano vanishing theorem
- Kodaira embedding theorem
- Andrei Moroianu, Lectures on Kahler geometry,http://www.math.polytechnique.fr/~moroianu/tex/kg.pdf
- J.-P. Demailly, Complex analytic and diﬀerential geometry,http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf
- W. Ballmann, Lectures on K¨ahler manifoldshttp://people.mpim-bonn.mpg.de/hwbllmnn/notes.html
- C. Voisin, “Hodge theory”.
- D. Huybrechts, “Complex Geometry �An Introduction”
- A. Besse, “Einstein manifolds”.
Instructor: Takashi Takebe
Course description: An elliptic function is deﬁned as a doubly periodic meromorphic function on the complex plane. The theory of elliptic functions was born in the eighteenth century. At ﬁrst Fagnano, Euler, Legendre, Gau�and others studied elliptic integrals. Then in the nineteenth century Abel and Jacobi changed the viewpoint and studied the inverse functions of elliptic integrals, namely, elliptic functions, which was a great advance. Riemann and Weierstrass developed the theory further. The theory of elliptic functions thus founded is a prototype of today’s algebraic geometry.
On the other hand, elliptic functions appear in various problems in mathematics as well as in physics. In fact the name “elliptic integral” comes from calculation of length of an arc of an ellipse.
Other examples are:
- The arithmetic-geometric mean is closely related to the elliptic integral, as Gau�observed.
- Solutions of various problems in physics: motion of a pendulum, form of skipping rope, motion of a top (= a rigid body), KdV equation (typical integrable non-linear diﬀerential equation),... are expressed in terms of elliptic functions and elliptic integrals.
- One can solve a general ﬁfth order equation, using an elliptic function.
In this course we shall put emphasis on analytic aspects and applications of elliptic functions.
Prerequisites: calculus and complex analysis.
- Real elliptic integrals as arclength of curves.
- Classiﬁcation of elliptic integrals.
- Applications of elliptic integrals.
- Jacobi’s elliptic functions (real case).
- Riemann surfaces of algebraic functions.
- Elliptic curves.
- Complex elliptic integrals.
- Abel-Jacobi theorem.
- Elliptic funtions; general theory.
- Weierstrass function.
- Theta functions.
- Jacobi’s elliptic functions (complex case).
Textbooks: (They are helpful, but not indispensable for the lecture)
- N. I. Akhiezer, Elements of the Theory of Elliptic Functions, Moscow, 1970; translated into English as AMS Translations of Mathematical Monographs Volume 79, AMS, Rhode Island, 1990.
- E. T. Whittaker and G. N. Watson. A course of modern analysis, Cambridge University Press, 1952.
- A. Hurwitz, R. Courant: Vorlesungen uber¨ allgemeine Funktionentheorie und elliptische Funktionen. 4. Auﬂage. Springer, Berlin, 1964. (Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Bd. 3.) 5. Auﬂage. Springer, Berlin / Heidelberg 2000.
- M. Toda: Introduction to Elliptic Functions,Nihon-Hyoron-sha, Tokyo, 2001.
- H. Umemura: Theory of Elliptic Functions. analysis on elliptic curves, University of Tokyo Press, Tokyo, 2000.
Convex and algebraic geometry
Semester: Fall, Spring
Seminar organizers: Alexander Esterov, Valentina Kiritchenko, Evgeny Smirnov
Our research seminar is devoted to the many connections between convex and algebraic geometry. This interaction has many important applications in various areas of mathematics: combinatorics, representation theory, mathematical physics to name a few. A classical and one of the most well-known examples is the combinatorial description of an important class of algebraic varieties – the so-called toric varieties – in terms of polytopes and fans (collections of cones). Yet another recent andup-to-date application is the theory of Newton-Okounkov bodies. Participants will tell about recent papers that they ﬁnd important, providing extensive background material for those less familiar with the subject.
Geometrically oriented 2nd year students and higher are welcome.
Prerequisites: There are no prerequisites beyond the compulsory curriculum of the ﬁrst 1 + " years at our bachelor program. Most of the talks will be accessible to second year students.
- Newton polytopes and Kushnirenko theorem
- Multidimensional Pick’s formula
- Toric varieties and fans
- Newton-Okounkov convex bodies
- Convex polytopes in representation theory
- Grassmannians, ﬂag varieties andGelfand-Zetlin polytopes
Textbooks and research papers:
- M.Beck, S.Robins, Computing the Continuous Discretely, Undergraduate Texts in Mathematics, Springer, 2007
- D. Cox, J.B. Little, H.K. Schenck, Toric varieties, Graduate Studies in Mathematics, vol. 124, American Mathematical Society, Providence, RI, 2011
- K. Kaveh, A. Khovanskii, Newton convex bodies, semigroups of integral points, graded algebras and intersection theory, Ann. of Math.(2), vol. 176 (2012), no.2,925-978
Introduction to diﬀerential geometry
Instructor: Ossip Schwarzman
Course description: The course will serve as an introductory guide to basic topics of Diﬀerential and Riemann geometry
Prerequisites: linear algebra, Euclidean geometry, Calculus on Manifolds,
Basic Topology Curriculum:
- Diﬀerential and Riemann geometry of smooth hypersurfaces in the Euclidean space: Parallel transport. The Gauss Map. Shape operator. Curvature. Geodesics.
- Riemannian manifolds: The metric connection. Covariant derivatives. Parallel transport Completeness and geodesics. The Exponential Map. The Hopf- - Rinow theorem. Geometry of a compact classical Lie group.
- Curvature and The Ricci tensor: Calculations with curvature tensor. The Gauss curvature. The Ricci tensor. Spaces of constant curvature . . .
- Variational theory of geodesics: First and second variation of arc length. Jacobi’s equation and conjugate points. The Gauss lemma and polar coordinates.
- Connections: Parallel transport and Covariant derivatives in vector bundles.
- (Gallot, Hulin, Lafontaine, Riemannian Geometry.
- J. Milnor, Morse Theory.
- Taubes, Diﬀerential Geometry. Bundles, connections, metrics and curvature.
- I. Chavel, Riemannian Geometry: a modern introduction.
Introduction to complex dynamics and analytic theory of ordinary diﬀerential equations
Instructor: Alexey Glutsyuk
Complex dynamics and analytic theory of ordinary diﬀerential equations are situated on crossing of many domains of contemporary mathematics. The analytic theory of ordinary diﬀerential equations and its global extension, the theory of holomorphic foliations were born in the ﬁrst half of XX-th century, in studying the Painlev´ equations and the second part of Hilbert 16-th Problem about limit cycles of realplanar polynomial vector ﬁelds. Studying the 16-th Hilbert Problem led to a lot of important results in local dynamics, normal forms and global properties of holomorphic foliations that became classical. Now holomorphic foliations and complex dynamics are quickly developing areas on the crossing of several domains in mathematics, including dynamical systems, analysis, complex and Riemannian geometry, ergodic theory. For example, both foliations and holomorphic dynamics arise in classiﬁcation problems in complex geometry and in some problems of mathematical physics. The course will present selected classical results on local dynamics, with an accent on moduli of analytic classiﬁcation, Stokes phenomena and also applications of Stokes phenomena and holomorphic foliations in other domains of mathematics.
Prerequisites: basic calculus, analysis of one complex variable, linear algebra, basic theory of ordinary diﬀerential equations.
- Holomorphic vector ﬁelds, singular points.
- Resonances.Non-resonant formal normal forms
- Analytic normal forms of singularities with linear parts in the Poincar´e domain.
- Linear equations, Fuchsian singularities, normal forms.
- TheRiemann–Hilbert problem.
- Irregular singularities. Stokes phenomena.
- Applications of the Stokes phenomena to real dynamics: model of Josephson eﬀect.
- Resolution of singularities oftwo-dimensional holomorphic vector ﬁelds.
- Saddle node singularities and their monodromy; parabolic germs of conformal mappings.
- Analytic classiﬁcation of parabolic germs.Ecalle–Voronin moduli.
- Analytic classiﬁcation ofsaddle-node singularities of holomorphic vector ﬁelds: Martinet–Ramis moduli.
- Applications to CR geometry.
- One-dimensional holomorphic foliations. Main conjectures on minimal sets and topology of leaves.
- Density of leaves of a generic foliation with an invariant line.
- Blue sky catastrophe.Deroin–Guillot–Frenkel example of a Riccati-like foliation on CP2with non-extendable holonomy.
- Simultaneous uniformization of leaves.
- Relation of local dynamics to classiﬁcation problems in complex geometry.
- Ilyashenko, Yulij; Yakovenko, Sergei. Lectures on analytic diﬀerential equations. Graduate Studies in Mathematics, 86. American Mathematical Society, Providence, RI, 2008. xiv+625.
- Arnold, V.I. Geometrical methods in the theory of ordinary diﬀerential equations. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 250.Springer-Verlag, New York, 1988.
Functional analysis 1
Instructor: Andrei Pogrebkov
Course description: Functional analysis is analysis in inﬁnite-dimensional vector spaces endowed with a norm and of operators in such spaces. Its methods and results have numerous applications in diﬀerent ﬁelds of mathematics and mathematical physics.
Prerequisites: analysis of one and several real variables, linear algebra.
- Banach spaces and operators on them;
- Hilbert spaces, orthonormal bases, Parseval’s identity and theRiesz-Fischer theorem;
- bounded and unbounded operators in the Hilbert space closed operators, orthogonal projectors;
- Hermitian andself-adjoint operators, resolvent, spectrum of an operator;
- spectral theorem for unboundedself-adjoint operators;
- nested Hilbert space, distributions.
- A.N. Kolmogorov and S.V. Fomin,Elements of the Theory of Functions and Functional Analysis Vol. 1:Metric and Normed Spaces, Graylock Press, Rochester, N.Y., 1957
- M. Reed and B. Simon,Methods of Modern Mathematical Physics Vol. 1: Frunctional analysis, Academic Press, N.y., London, 1972
Instructor: Christopher Brav
Course description: We give an introduction to commutative algebra and its applications to linear algebra, algebraic geometry, and algebraic number theory, focusing on basic notions and explicit examples.
Prerequisites: basic abstract algebra, linear algebra.
Examples from elementary mathematics: rings of polynomials k[x]over a ﬁeld kand the ring of integers Z, principal ideal domains, unique factorisation, modules over a principal ideal domain, application to normal forms of matrices.
Examples from algebraic geometry and algebraic number theory: polynomial functions on aﬃne n-space An, varieties in Anas solutions of systems of polynomial equations, rings of integers in a number ﬁeld.
(2) Basic notions and results
Prime ideals and integral domains, nilpotents, radicals of ideals, the spectrum Spec(A)of a commutative ring with its Zariski topology as the basic object of algebraic geometry, localisation/inverting elements, Noetherian rings and ﬁnite generation of ideals, Hilbert’s Basissatz stating that ideals in polynomial rings are ﬁnitely generated. Hilbert’s Nullstellensatz on existence of solutions of systems of polynomial equations with coeﬃcients in an algebraically closed ﬁeld. Irreducible components of varieties.
(3) More on modules
Submodules and quotient modules, exact sequences, localisation of modules, modules as geometric objects spread over Spec(A), tensor products, projective modules, locally free modules, exterior powers and determinants, symmetric powers, Nakayama’s lemma and its uses, module of K¨ahler diﬀerentials, smooth and singular points of algebraic varieties, the de Rham complex.
(4) Finite ring extensions
Integral and ﬁnite extensions of rings, normal/integral closure, rings of integers in an algebraic number ﬁeld as ﬁnite extensions of Z, Noether’s normalisation lemma stating that ﬁnitely generated algebras over a ﬁeld are ﬁnite extensions of polynomial rings. Branching and ramiﬁcation in number theory and algebraic geometry.
- Bosch, Algebraic Geometry and Commutative Algebra.
- Reid, Undergraduate Commutative Algebra.
Instructor: Serge Lvovski
Course description: This is an introduction to the theory of Riemann surfaces, particularly compact Riemann surfaces. The course, which is concentrated on examples rather than on a systematic exposition of general theories, can serve as an elementary introduction to algebraic geometry.
- A course of complex analysis.
- Rudiments of algebra of polynomials (discriminant, Gauss lemma).
- Rudiments of topology (topological spaces, coverings).
- Modest familiarity with diﬀerential forms (just main deﬁnitions, not Stokes theorem).
- Deﬁnition of Riemann surfaces. First examples.
- Meromorphic functions. Orders of zeroes and poles.
- Holomorphic and meromorphic forms on Riemann surfaces.
- Compact Riemann surface associated to an algebraic equation. Examples.
- Integration of1-forms. Residues. Theorem on the sum of residues.
- Holomorphic Morse lemma. Compact Riemann surfaces associated to smooth and nodal plane projective curves.
- Ramiﬁed coverings. Degree of a morphism. Genus. Sum of orders of meromorphic diﬀerentials.Riemann–Hurwitz formula.
- Poincare residue.
- Ellipgtic curves.}-function. Proof that ellip[tic curves are “algebraic”.
- Classiﬁcation of elliptic curves.
- Mittag–Leﬄer problem. Statement of Riemann–Roch theorem. Its proof for elliptic curves.
- Divisors, linear equivalence, linear sustems.
- Line bundles; interpretation of linear systems in terms of line bundles.
- Canonical mapping. Structure of hyperelliptic curves. Examples of curves of low genus.
- Abel–Jacobi theorem for elliptic curves.
- Proof of Riemann–Roch theorem for Riemann surfaces associated to algebraic equations.
- Springer, George. Introduction to Riemann surfaces. Addison-Wesley Publishing Company, Inc., Reading, Mass. 1957.
- Gunning, R. C. Lectures on Riemann surfaces. Princeton Mathematical Notes Princeton University Press, Princeton, N.J. 1966
- Lang, Serge. Introduction to algebraic and abelian functions.Addison-Wesley Publishing Co., Inc., Reading, Mass., 1972.
Real and complex analysis
Instructor: Serge Lvovski
Course description: This is an introduction into measure theory and functional analysis. The aim of the course is to introduce basic concepts and to develop abstract theory to the extent that allows one to show how the fundamental results of functional analysis can be applied to obtain results in more classical branches of real and complex analysis.
- Basic analysis (including multidimensional Riemann integral and rigirous construction of real numbers).
- Basic linear algebra, including deﬁnition of abstract linear spaces and linear operators.
- Rudiments ofpoint-set topology, including deﬁnitions of metric and topological spaces, continuous mappings, and compact spaces.
- Semi-rings and Caratheodory extension theorem.
- Urysohn’s lemma and its analogues.
- Riesz construction of measures on locally compact spaces.
- Measurable functions.
- Lebesgue integral and its properties.
- L1 and L2 spaces. Hilbert spaces.
- Fourier series. Parseval’s formula.
- Dirichlet and Fejer kernels. Pointwise and unifrom convergence of Fourier series.
- Baire theorem.Banach–Steinhaus theorem. Application to Fourier series.
- Open mapping and closed graph theorems. Application to Fourier series.
- Harmonic functions and Poisson kernel.
- Hahn–Banach theorem. Application to Poisson kernel.
- Radon–Nikodym theorem. Complex measures. Riesz representation theorem.
- Tychonoﬀ theorem.Banach–Aalaoglu theorem.
- Banach algebras. Spectrum, spectral radius.
- Commutative Banach algebras. Gelfand transform. Characterixation of commutative C*-algebras.
- Rudin, Walter. Real and complex analysis. Third edition.McGraw-Hill Book Co., New York, 1987.
- Rudin, Walter. Functional analysis.McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York-D¨usseldorf-Johannesburg, 1973.
Elementary introduction to the theory of integrable diﬀerential equations
Instructor: Andrei Pogrebkov
Course description: Creation and development of the theory of integrable equations is one of main achievements of the mathematical physics of the fall of the previous century. In our times ideas and results of this theory penetrate in many branches of the modern mathematics: from string theory to the theory of Riemann surfaces.
Prerequisites: analysis of one and several real variables, theory of complex variable, linear algebra, theory of linear partial diﬀerential equations.
- Commutator identities on associative algebras;
- dressing operators;
- Lax pairs;
- Kadomtsev–Petviashvili equation;
- Two-dimensional reduction: KdV equation;
- Soliton solutions of the KP and KdV equations, their properties.
- S. Novikov, S.V. Manakov, L.P. Pitaevskij, V.E. Zakharov, Theory of solitons. The inverse scattering methods, Contemporary Soviet Mathematics. New York – London: Plenum Publishing Corporation. Consultants Bureau, 1984
- F. Calogero, A. Degasperis, Spectral Transform and solitons, I. North-Holland, Amsterdam, N.Y., Oxford, 1982
Topological vector spaces and applications to geometry
Instructor: Alexei Yu. Pirkovskii
The traditional functional analysis deals mostly with Banach spaces and, in particular, Hilbert spaces. However, many classical vector spaces have canonical topologies that cannot be determined by a single norm. For example, many spaces of smooth functions, holomorphic functions, and distributions belong to the above class. Such spaces are the subject of the theory of topological vector spaces. Although the golden age of topological vector spaces was in the 1950ies, their theory is still evolving nowadays, contrary to a stereotyped view coming from incompetent sources. The current development of topological vector spaces is directed not so much towards general theory as towards applications in PDEs and in complex analytic geometry.
We plan to discuss the basics of the theory of topological vector spaces, including some applications to complex analytic geometry.
Prerequisites: basic functional analysis (Banach and Hilbert spaces, bounded linear operators).
- Topological vector spaces. Seminorms and locally convex spaces. Continuous linear maps. Normability and metrizability criteria. Completeness. Examples: spaces of continuous, smooth, holomorphic functions, the Schwartz space.
- Quotients, products, coproducts, inverse and direct limits, completions, topological tensor products.
- Bornological and barrelled spaces. Equicontinuity. TheBanach-Steinhaus theorem, the Open Mapping theorem, the Banach-Alaoglu-Bourbaki theorem.
- Dual pairs and weak topologies. The Bipolar theorem. TheMackey-Arens theorem. The Mackey topology, the strong topology. Reﬂexivity. Relations between properties of linear maps and their duals.
- Nuclear maps. Nuclear spaces and their properties. Examples of nuclear spaces. A characterization of nuclear spaces in terms of tensor products.
- Coherent analytic sheaves. TheCartan-Serre ﬁniteness theorem. The Grauert direct image theorem (if time permits).
- H. H. Schaefer. Topological vector spaces. Springer, 1971.
- A. P. Robertson, W. Robertson. Topological vector spaces. Cambridge, 1964.
- V. I. Bogachev, O. G. Smolyanov. Topological vector spaces. Springer, 2017.
- R. Meise, D. Vogt. Introduction to Functional Analysis. Clarendon Press, Oxford, 1997.
- F. Treves. Topological Vector Spaces, Distributions, and Kernels. Academic Press, NewYork–London, 1967.
- J. Horvath. Topological vector spaces and distributions.Addison-Wesley, 1966.
- A. Pietsch. Nuclear locally convex spaces. Springer, 1972.
- H. Jarchow. Locally convex spaces. Teubner, Stuttgart, 1981.
- G. K¨othe. Topological Vector Spaces. Vol I, Springer, 1969; Vol. II, Springer, 1979.
- N. Bourbaki. Topological vector spaces. Springer, 1987.
- J. L. Kelley, I. Namioka. Linear topological spaces. Springer, 1963.
- A. Grothendieck. Produits tensoriels topologiques et espaces nucl´eaires. Mem. Amer. Math. Soc. 1955, no. 16A.
- A. Grothendieck. Topological vector spaces. Gordon and Breach, 1973.
- J. L. Taylor. Notes on locally convex topological vector spaces. Univ. of Utah, 1995.
- E. Thomas. Nuclear spaces and topological tensor products. 2001.
- R. Douady. Produits tensoriels topologiques et espaces nucleaires. In: “Quelques problemes de modules” (S´em. Geom. Anal. Ecole Norm. Sup., Paris, 1971– 1972), pp. 7–32. Asterisque, No. 16, Soc. Math. France, Paris, 1974.
- P. Perez Carreras, J. Bonet. Barrelled locally convex spaces.North-Holland, 1987.
- S. M. Khaleelulla. Counterexamples in topological vector spaces. Springer, 1982.
- L. Schwartz. Th´eorie des distributions. Hermann, Paris, 1978.
- J. Eschmeier, M. Putinar. Spectral decompositions and analytic sheaves. Clarendon Press, Oxford, 1996.
Instructor: Alexander Esterov, Alexander Kolesnikov
Course description: This is an introductory course on variational methods, extremal problems, and convexity. In the ﬁrst part of the course we discuss classical variational methods with examples of applications from geometry, physics, and engineering. We start with the variational calculus of Lagrange and Euler and ﬁnish with a brief discussion of the Pontryagin maximum principle. In the second (less standard) part we try to demonstrate the power of convex analysis and explain how convexity appears in various branches of science, starting with engineering and economical applications and ﬁnishing with abstract algebraic problems.
Prerequisites: standard linear algebra and analysis, ordinary diﬀerential equations. Some experience in functional analysis is desirable.
- Classical extremal problems from geometry and physics (isoperimetric problem, brachistochrone, geodesics, and others).
- Euler-Lagrange equations.
- Pontryagin maximum principle.
- Applications of the Pontryagin maximum principle in physics and engineering.
- Convex functions, convex bodies, convex polytopes. Support functions, Minkowski summation.
- Legendre transform. Lagrangians and Hamiltonians.
- Hamilton–Jacobi equation. Bellman principle and Bellman equation.
- Mixed volumes, geometric inequalities(Brunn–Minkowski,Alexandrov–Fenchel).
- Minimax principle.Kuhn–Tucker theorem.
- Linear programming. Kantorovich duality.
- Elements of the game theory. Von Neuman theorem.
- Some applications in graph theory: bipatrite graphs, shortest path, tropical algebra.
- Giaquinta M., Hildebrandt S., "Calculus of Variations."Vol.1-2. Grundlehren der Mathematischen Wissenschaften, Berlin, Springer, 1996.
- Galeev E.M., Zelikin M.I., Konyagin S.V. “Optimal’noe upravlenije” (Russian), MCCME 2008
- Gardner R.J., TheBrunn-Minkowski inequality, Bull. Amer. Math. Soc. 39 (2002), 355–405
- Ziegler G.M., Lectures on Polytopes, Springer, Graduate Texts in Mathematics, Springer, 2007.
- Burago Yu.D., Zalgaller V.A., Geometric inequalities, Springer, 1988.
- Ferguson T.S., Linear programming: a concise intoduction.
- Ferguson T.S., Game theory. 2014.
Semester: Full Year
Instructor: Vladimir Zhgoon
Course description: The aim of this course is to give introduction to the spherical varieties i.e. such compactiﬁcations of homogeneous spaces of reductive group for which a Borel subgroup B acts with a dense open orbit. The particular case of these varieties are Grassmanians, ﬂag varieties, symmetric spaces and toric varieties. The remarkable property of spherical homogeneous spaces is that their compactiﬁcations can be described in combinatorial way (Luna-Vust theory of scherical embedding). In our course we shall study the Luna-Vust theory, Local structure of spherical varieties, G-equivariant invariants of homogeneous spaces, such as rank lattice, little Weyl group and spherical roots. We also shall discuss the equivariant geometry of cotangent bundle and its moment map and give the relations with the little Weyl group.
- Scherical varieties. General properties and equivalent deﬁnitions.
- Local structure of spherical varieties.
- Akhiezer theorem on modality and complexity. Vinberg theorem on complexity. Horospherical contraction.
- Finiteness of number ofB-orbits in spherical variety. Springer-Richardson monoid. Action of the Weyl group on the set of B-orbits.Bott–Samelson resolutions of Schubert varieties.
- Luna-Vust theory of spherical embeddings. Invariant valuations, colours.
- Demazure construction of wonderful compactiﬁcations.Bialynicki-Birula cell decomposition.
- Equivariant geometry of cotangent bundle. Structure of the moment map. Variety of generic horosheres and the little Weyl group.
- Moment polytope and Brion’s polytope.
Textbooks and research papers:
- И. Н. Бернштейн, И. М. Гельфанд, С. И. Гельфанд, Клетки Шуберта и когомологии пространств G/P, УМН, 28:3(171) (1973),3–26
- M. Brion, Spherical varieties, Birkhauser, Progress in Mathematics 295, 2012,3-24
- M.Brion, Lectures on the geometry of ﬂag varieties, Topics in cohomological studies of algebraic varieties,33–85, Trends Math., Birkhauser, Basel, 2005
- M. Brion, Groupe de Picard et nombres caracteristiques des varietes spheriques, Duke Math J. 58, no.2 (1989),397-424
- C. De Concini, C. Procesi, Complete symmetric varieties I, Lect. Notes in Math. 996, Springer, 1983,1-43
- M.Demazure, Desingularization des Varietes de Schubert generalisees, Ann. Sc. Ec. Norm. Super. 7 (1974)53-88
- Friedrich Knop, TheLuna-Vust theory of spherical embeddings, Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989), 225- 249, Manoj Prakashan, Madras, 1991
- D.A. Timashev, Homogeneous spaces and equivariant embeddings, 138, Encyclopaedia of Mathematical Sciences, Springer, Berlin, 2011
Instructor: Vadim Vologodsky
Course description: The development of Galois Theory was prompted by the following question (whose answer is known as the Abel-Ruﬃni theorem): Is there a formula for the roots of a ﬁfth (or higher) degree polynomial equation in terms of the coeﬃcients of the polynomial, using only the usual algebraic operations (addition, subtraction, multiplication, division) and application of radicals (square roots, cube roots, etc)? Galois theory also gives a clear insight into “classical questions” concerning compass and straightedge constructions such as: Which regular polygons can be constructed using compass and straightedge? Galois Theory is a key tool in many areas of modern mathematics including number theory, topology, and algebraic geometry. The course oﬀers a concise exposition of the theory.
Prerequisites: A ﬁrst course in abstract algebra: group theory, basic linear algebra, and some elementary ring theory.
- Algebraic, transcendental, ﬁnite ﬁeld extensions; degree of an extension, algebraic closure, splitting ﬁeld of a polynomial.
- The Fundamental Theorem of Galois Theory.
- Applications to ﬁnding solutions of polynomial equations in radicals and to geometric constructions with ruler and compass.
- Finite ﬁleds. Chevalley’s theorem.
- Finite dimensional algebras over a ﬁeld, the Brauer group.
- E. Artin, Galois Theory. Dover Publications.
- N. Stewart, Galois Theory. CRC Press
- B.L. van der Waerden, "Algebra1-2. Springer
Probability Theory. Analytic and economic applications
Semester: Winter, Spring
Instructors: Alexander Kolesnikov, Valentine Konakov
Course description: We discuss all kind of problems related to probabilistic methods in analysis and various applications. The discussed topics cover a broad area and vary every year. The content highly depends on the interest of invited lectureres and participating students.
Prerequisites: standard linear algebra and analysis, ordinary diﬀerential equations. Some experience in functional analysis and stochastics is desirable.
List of some regularly discussed topics.
- Stochastic diﬀerential equations with applications in ﬁnance
- Random matrices
- Convex geometry and probability
- Probabilistic and economic applications of theMonge-Kantorovich problem and other extremal problems
- Martingale theory, its ﬁnancial applications
- Probability distributions on Lie groups
- Stochastic Riemannian geometry
- Inﬁnite-dimensional distributions, Gaussian measures
- Elements of the game theory
- Physical methods in economics
- Levy processes
In addition, in year 2017-2018 several lectures will be given by Prof. Stanislav Molchanov.
- Random walks and Brownian movements on fractals, similar to Sierpinski gasket and Sierpinski carpet.
- Mathematical problems of Biochemistry: phase transitions in proteins, singular diﬀusions in living tissues.
- Open problems in populational dynamics.
- Missing mass problem,inﬁnite-dimensional case.
Partial Diﬀerential Equations
Instructors: Alexey Glutsyuk and Ilya Vyugin
Course description: The course of partial diﬀerential equations is devoted to studying equations which describe basic phenomena of enviroment such as heat spreading, waves and stationary phisycal processes. Also we are going to discuss the generalized functions theory which are used for studying of equations of mathematical physics.
Prerequisites: calculus, linear algebra, ordinary diﬀerential equations.
- Physical sence of the basic equations of mathematical physics.
- Method of Characteristics and reduction of equations to the canonical form.
- The wave equation. d’Alembert’s formula.
- The Fourier method and theSturm–Liouville problem.
- The ﬁrstboundary-value problem for the heat equation.
- Boundary-value problems for wave equation and Laplace equation.
- Cauchy problem for heat equation.
- Petrovsky correctness and Schredinger equation.
- The concept of a generalized function
- Generalized and fundamental solutions.
- The Kirchhoﬀ and Poisson formulas.
Lawrence C. Evans. Partial Diﬀerential Equations. Second edition – Graduate Studies in Mathematics, Vol. 19, 2010.
Real algebraic and toric geometry
Instructor: Alexander Esterov
This is an introduction to real algebraic geometry (1st module) and toric varieties (2nd module). Although the two topics are formally independent, the ﬁrst one provides a natural context for the second.
The ﬁrst module will be devoted to Hilbert’s 16th problem for algebraic curves, which was one of the starting points for real algebraic geometry. The problem asks for the topological classiﬁcation of smooth plane real algebraic curves of a given degree. Hilbert himself solved the problem up to degree 6 modulo one elusive topological type, whose existence was proved only 70 years later. Our aim is Viro’s patchworking theorem, which allows to construct algebraic curves of a given degree with prescribed topology.
The second module will be devoted to toric varieties – certain algebraic varieties that can be assigned to integer polytopes. This correspondence between algebraic and geometric objects turns out to be proﬁtable for both ﬁelds of study. For instance, on the polyhedral side, it solves the Upper bound conjecture regarding the number of faces of a simple polytope, while, on the algebro-geometric side, it produces the theory of Newton polytopes and provides the technique behind Viro’s patchworking.
Prerequisites: Linear algebra and point-set topology. Familiarity with smooth manifolds and algebraic sets is a plus.
- Hilbert’s 16th problem and Harnack’s inequality
- Viro’s patchworking
- Real and complex projective toric varieties
- Newton polytopes and tropical compactiﬁcations
- Kouchnirenko–Bernstein formula
- V. Kharlamov and O. Viro, Easy reading on topology of real plane algebraic curves
- O. Viro, Patchworking Real Algebraic Varieties
- D. Cox, What is a Toric Variety?
- G. Ewald, Combinatorial Convexity and Algebraic Geometry
Coverings and Galois theory
Instructor: Petr Dunin-Barkowski
Course description: This course is an introduction to the theory of coverings of Riemann surfaces and to the surprising analogy between the classiﬁcation of coverings and the main theorem of the Galois theory. We will consider both unramiﬁed and ramiﬁed coverings. It is remarkable that the connection between the theory of coverings and the algebraic Galois theory works in both ways: Galois theory allows us to understand the geometry of coverings better, and, at the same time geometric considerations lead to some purely algebraic results. In particular, we will see how to obtain the complete classiﬁcation of all ﬁnite algebraic extensions of the ﬁeld of rational functions of a single variable geometrically in a natural way.
Prerequisites: Basic abstract algebra: linear algebra, groups, rings, ideals. Basic topology and geometry: topological spaces, surfaces, fundamental groups. Basic complex analysis. Attending the fall 2017/2018 course “Introduction to Galois theory” by Vadim Vologodsky would be nice, though not required.
- Basic facts from algebraic Galois theory.
- Coverings of topological spaces. Classiﬁcation of coverings with marked points (through fundamental groups).
- Riemann surfaces/algebraic curves. Riemann’s existence theorem (without proof).
- Coverings and ramiﬁed coverings of Riemann surfaces.
- Analogy between the classiﬁcation of intermediate subcoverings of a given normal covering and the classiﬁcation of subﬁelds of a given Galois extension.
- Fields of meromorphic functions on Riemann surfaces and their algebraic extensions, ﬁelds of germs.
- Riemann surface of an algebraic equation over the ﬁeld of meromorphic functions.
- Application of Galois theory to the ﬁelds of germs on Riemann surfaces.
- Geometric description of all ﬁnite algebraic extensions of the ﬁeld of rational functions of a single variable.
- Khovanskii, Askold. Galois theory, coverings, and Riemann surfaces. Springer, 2013. Chapters 2, 3.
- Khovanskii, Askold. Topological Galois Theory. Springer, 2013. Chapter 4.
Sheaves And Supplying Homological Algebra
Instructor: Alexey Gorodentsev
Course description: The theory of sheaves is a commonly used tool for handling locally deﬁned objects on a manifold, e.g., functions with a restricted domain of deﬁnition, local sections of vector bundles, locally deﬁned continuous mappings, etc. In the algebraic/diﬀerential geometry and topology it allows to produce global geometric/topological invariants of the manifold from those local data. In the non- commutative geometry it gives various geometric style invariants for categories equipped with Grothendieck’s topology. We introduce the basic notions of the sheaf theory, discuss in details the underlying homological algebra (including the cohomology of sheaves), and give some applications of this technique, such as the De Rham theorem comparing the singular and De Rham comohomologies of a smooth manifold.
Prerequisites: The ﬁrst 3 semesters (6 modules) of the standard courses in Algebra, Calculus, and Geometry/Topology. Some experience in algebraic geometry/topology (projective spaces, topological spaces, simplicial complexes, (co) homology groups) is desirable but not essential.
- Categories, functors, presheaves. Working examples: the open sets of a topology and the (semi) simplicial sets. Categories of functors, the Yoneda lemma. Adjoint functors. (Co) limits of diagrams. ([GM], [W])
- Sheaves on topological spaces. Stalks and the etal´e space of a sheaf. The sheaﬁﬁcation. Pull backs and push forwards. Abelian sheaves. ([I], [GM])
- Complexes and (co)homologies, the long exact sequence of homologies. The Koszul complexes. Cohomologies commute with ﬁltered colimits. Spectral sequences for ﬁltered complexes, bicomplexes, and exact couples. ([GM], [D], [W])
- Global sections, ﬂabby sheaves, and the Godement resolution. The sheaf cohomology and hypercohomology. Acyclic resolutions. The Mayer–Vietoris exact sequence and Cech resolution. Acyclic coverings and the Cartan criterion for acyclicity. The Cech cohomology. ([I], [D])
- Fine and soft sheaves. The sheaves of diﬀeretial forms, the Poincar´e lemma and De Rham theorem. ([GM], [GH], [D])
- Higher direct images. The Leray spectral sequence. ([I], [D], [GH])
- Coherent sheaves in algebraic geometry: examples and applications. Acyclicity of aﬃne varieties. Cohomologies of invertible sheaves on projective spaces. ([D])
- If the time allows: the Grothendieck topologies and sheaves on sites. ([GM])
- [D]: V. I. Danilov, Cohomology of Algebraic Manifolds. In: Cohomology of Algebraic Varieties. Algebraic Surfaces, Springer’s Encyclopaedia of Mathematical Sciences, Book 35 (Algebraic Geometry 2).
- [GM]: S. I. Gelfand and Yu. I. Manin, Methods of Homological Algebra, Part I.
- [GH]: P. Griﬃths and J. Harris, Principles of Algebraic Geometry, Vol. I.
- [I]: B. Iversen, Cohomology of Sheaves.
- [W]: C. A. Weibel, An Introduction to Homological Algebra
Math Methods of Science (Introduction to Differential Geometry)
Instructor: Yurii Burman
Course description: The course deals with basic notions of differential geometry filling the gap between calculus (up to implicit function theorem) and more advanced differential-geometric topics (Riemannian metric, symplectic forms, etc.). We start from the concept of a smooth manifold, and then study objects inherently connected with it (vector fields, differential forms and more). We also develop technique (vector bundles, partition of unity and so on) allowing students to work with smooth manifolds, both using coordinates and in a coordinate-free setting.
Prerequisites: multivariate calculus (including implicit function theorem), basic linear algebra, basic notions of ordinary differential equations.
Items marked * are more advanced topics to be studied if time permits.
- Smooth manifolds.
Smooth manifolds, smooth maps, induced topology.
- Vector bundles.
Vector bundles. Operations over vector bundles (direct sum, tensor product). Tangent bundles. Derivative of a smooth map.
- Vector fields.
Group of diffeomorphisms. Lie algebra of vector fields. Lie derivative.
Paracompactness. Partition of unity. *Thom's transversality theorem. *Existence of Morse functions.
- Differential forms.
Superalgebra of differential forms. Exterior derivative, d2=0. Cartan's formula.
- *Frobenius' theorem.
Frobenius' theorem. Almost complex structure. Contact structure.
- Integration of differential forms.
Integral of a differential form. Stokes' theorem.
- Raoul Bott, Loring W. Tu, Differential Forms in Algebraic Topology, Springer, NY, 1982.
- Michael Spivak, Calculus on Manifolds, Addison-Wesley, 24th printing, 1995.
- Raghavan Narasimhan, Lectures on Topics in Analysis, Tata Institute of Fundamental Research, Bombay, 1965
- Smooth manifolds.