Functions on $\R^n$
, directional derivatives, total derivative, higher order derivatives and Taylor series.The inverse and implicit function theorem,
Integration on $\R^n$
, differential forms on $\R^n$
, closed and exact forms. Green’s theorem, Stokes’ theorem and the Divergence theorem.
This course introduces various aspects of Computer Proofs, both interactive and fully automated. We will consider proofs of mathematical results as well as of correctness of programs. We will primarily use the Lean Theorem Prover 4
, which is a formal proof system as well as a programming language. The foundations on which the Lean prover is based, Dependent Type Theory, allow a seamless integration of mathematical objects, theorems, proofs and algorithms.
Topics covered will be among the following.
This course is an introduction to standard material in logic, based on classical first-order logic, after which it ventures into modern treatments of some non-classical logics. Although other proof methods will be discussed, the emphasis will be on proofs using tableaus.
Topics:
This course is an introduction to logic and foundations from both a modern point of view (based on type theory and its relations to topology) as well as in the traditional formulation based on first-order logic.
Topics:
Vector spaces, Bases and dimension, Direct ums, linear transformations, Matrix algebra, Eigenvalues and eigenvectors, Cayley Hamilton Theorem, Jordan canonical form., Orthogonal matrices and rotations, Polar decomposition., Bilinear forms.
The modular group and its subgroups, the fundamental domain. Modular forms, examples, Eisenstein series, cusp forms. Valence (dimension) formula, Petersson inner product. Hecke operators. L-functios: definition, analytic continution and functional equation.
Graphs, subgraphs, Eulerian tours, trees, matrix tree theory and Cayley’s formula, connectedness and Menger’s theorem, planarity and Kuratowski’s theorem, chromatic number and chromatic polynomial, Tutte polynomial, the five-colour theorem, matchings, Hall’s theorem, Tutte’s theorem, perfect matchings and Kasteleyn’s theorem, the probabilistic method, basics of algebraic graph theory
No prerequisites are expected, but we will assume a familiarity with linear algebra.
Combinatorics: Basic counting techniques. Principle of inclusion and exclusion. Recurrence relations and generating functions. Pigeon-hole principle, Ramsey theory. Standard counting numbers, Polya enumeration theorem.
Graph Theory: Elementary notions, Shortest path problems. Eulerian and Hamiltonian graphs, The Chinese postman problem. Matchings, the personal assignment prolem. Colouring or Graphs.
Number Theory: Divisibility Arithmetic functions. Congruences. Diophantine equations. Fermat’s big theorem, Quadratic reciprocity laws. Primitive roots.
Algebraic Number Theory: Algebraic numbers and algebraic integers, Class groups, Groups of units, Quadratic fields, Quadratic reciprocity law, Class number formula.
Analytic Number Theory: Fundamental theorem of arithmetic, Arithmetical functions, Some elementary theorems on the distribution of prime numbers, Congruences, Finite Abelian groups and their characters, Dirichlet theorem on primes in arithmetic progression.
Vector spaces: Definition, Basis and dimension, Direct sums. Linear transformations: Definition, Rank-nullity theorem, Algebra of linear transformations, Dual spaces, Matrices.
Systems of linear equations: Elementary theory of determinants, Cramer’s rule. Eigenvalues and eigenvectors, the characteristic polynomial, the Cayley- Hamilton Theorem, the minimal polynomial, algebraic and geometric multiplicities, Diagonalization, The Jordan canonical form. Symmetry: Group of motions of the plane, Discrete groups of motion, Finite groups of SO(3). Bilinear forms: Symmetric, skew symmetric and Hermitian forms, Sylvester’s law of inertia, Spectral theorem for the Hermitian and normal operators on finite dimensional vector spaces.
Representation of finite groups, irreducible representations, complete reducibility, Schur’s lemma, characters, orthogonality, class functions, regular representations and induced representation, the group algebra.
Linear groups: Representation of the group $SU(2)$
Construction of the field of real numbers and the least upper-bound property. Review of sets, countable & uncountable sets. Metric Spaces: topological properties, the topology of Euclidean space. Sequences and series. Continuity: definition and basic theorems, uniform continuity, the Intermediate Value Theorem. Differentiability on the real line: definition, the Mean Value Theorem. The Riemann-Stieltjes integral: definition and examples, the Fundamental Theorem of Calculus. Sequences and series of functions, uniform convergence, the Weierstrass Approximation Theorem. Differentiability in higher dimensions: motivations, the total derivative, and basic theorems. Partial derivatives, characterization of continuously-differentiable functions. The Inverse and Implicit Function Theorems. Higher-order derivatives.
Construction of Lebesgue measure, Measurable functions, Lebesgue integration, Abstract measure and abstract integration, Monotone convergence theorem, Dominated convergence theorem, Fatou’s lemma, Comparison of Riemann integration and Lebesgue integration, Product sigma algebras, Product measures, Sections of measurable functions, Fubini’s theorem, Signed measures and Radon-Nikodym theorem, Lp-spaces, Characterization of continuous linear functionals on Lp - spaces, Change of variables, Complex measures, Riesz representation theorem.
Basic topological concepts, Metric spaces, Normed linear spaces, Banach spaces, Bounded linear functionals and dual spaces, Hahn-Banach theorem. Bounded linear operators, open-mapping theorem, closed graph theorem. The Banach- Steinhaus theorem. Hilbert spaces, Riesz representation theorem, orthogonal complements, bounded operators on a Hilbert space up to (and including) the spectral theorem for compact, self-adjoint operators.
Complex numbers, holomorphic and analytic functions, Cauchy-Riemann equations, Cauchy’s integral formula, Liouville’s theorem and proof of fundamental theorem of algebra, the maximum-modulus principle. Isolated singularities, residue theorem, Argument Principle. Mobius transformations, conformal mappings, Schwarz lemma, automorphisms of the disc and complex plane. Normal families and Montel’s theorem. The Riemann mapping theorem. If time permits - analytic continuation and/or Picard’s theorem.
Harmonic and subharmonic functions, Green’s function, and the Dirichlet problem for the Laplacian; the Riemann mapping theorem (revisited) and characterizing simple connectedness in the plane; Picard’s theorem; the inhomogeneous Cauchy–Riemann equations and applications; covering spaces and the monodromy theorem.
Functions of several variables, Directional derivatives and continuity, total derivative, mean value theorem for differentiable functions, Taylor’s formula. The inverse function and implicit function theorems, extreme of functions of several variables and Lagrange multipliers. Sard’s theorem. Manifolds: Definitions and examples, vector fields and differential forms on manifolds, Stokes theorem.
Point-set topology: Open and closed sets, continuous functions, Metric topology, Product topology, Connectedness and path-connectedness, Compactness, Countability axioms, Separation axioms, Complete metric spaces, Quotient topology, Topological groups, Orbit spaces.
The fundamental group: Homotopic maps, Construction of the fundamental group, Fundamental group of the circle, Homotopy type, Brouwer’s fixed-point theorem, Separation of the plane.
The fundamental group: Homotopy of maps, multiplication of paths, the fundamental group, induced homomorphisms, the fundamental group of the circle, covering spaces, lifting theorems, the universal covering space, Seifert-van Kampen theorem, applications.
Simplicial Homology: Simplicial complexes, chain complexes, definitions of the simplicial homology groups, properties of homology groups, applications.
Suggested books :
Curves in Euclidean space: Curves in R3, Tangent vectors, Differential derivations, Principal normal and binomial vectors, Curvature and torsion, Formulae of Frenet.
Surfaces in R3: Surfaces, Charts, Smooth functions, Tangent space, Vector fields, Differential forms, Regular Surfaces, The second fundamental form, Geodesies, Parellel transport, Weingarten map, Curvatures of surfaces, Rules surfaces, Minimal surfaces, Orientation of surfaces.
Metric geometry is the study of geometric properties such as curvature and dimensions in terms of distances, especially in contexts where the methods of calculus are unavailable, An important instance of this is the study of groups viewed as geometric objects, which constitutes the field of geometric group theory. This course will introduce concepts, examples and basic results of Metric Geometry and Geometric Group theory.
A review of continuity and differentiability in more than one variable. The inverse, implicit, and constant rank theorems. Definitions and examples of manifolds, maps between manifolds, regular and critical values, partition of unity, Sard’s theorem and applications. Tangent spaces and the tangent/cotangent bundles, definition of general vector bundles, vector fields and flows, Frobenius’ theorem. Tensors, differential forms, Lie derivative and the exterior derivative, integration on manifolds, Stokes’ theorem. Introduction to de Rham cohomology.
Basics concepts:Introduction and examples through physical models, First and second order equations, general and particular solutions, linear and nonlinear systems, linear independence, solution techniques. Existence and Uniqueness Theorems :Peano’s and Picard’s theorems, Grownwall’s inequality, Dependence on initial conditions and associated flows. Linear system:The fundamental matrix, stability of equilibrium points, Phase- plane analysis, Sturm-Liouvile theory . Nonlinear system and their stability:Lyapunov’s method, Non-linear Perturbation of linear systems, Periodic solutions and Poincare- Bendixson theorem.
First order partial differential equation and Hamilton-Jacobi equations; Cauchy problem and classification of second order equations, Holmgren’s uniqueness theorem; Laplace equation; Diffusion equation; Wave equation; Some methods of solutions, Variable separable method.
Matrix Algebra: Systems of linear equations, Nullspace, Range, Nullity, Rank, Similarity, Eigenvalues, Eigenvectors, Diagonalization, Jordan Canonical form. Ordinary Differential Equations: Singular points, Series solution Sturm Liouville problem, Linear Systems, Critical points, Fundamental matrix, Classification of critical points, Stability.
Complex Variables: Analytic functions, Cauchy’s integral theorem and integral formula, Taylor and Laurent series, isolated singularities, Residue and Cauchy’s residue theorem chwarz lemma.
Numerical solution of algebraic and transcendental equations, Iterative algorithms, Convergence, Newton Raphson procedure, Solutions of polynomial and simultaneous linear equations, Gauss method, Relaxation procedure, Error estimates, Numerical integration, Euler-Maclaurin formula. Newton-Cotes formulae, Error estimates, Gaussian quadratures, Extensions to multiple integrals.
Numerical integration of ordinary differential equations: Methods of Euler, Adams, Runge-Kutta and predictor - corrector procedures, Stability of solution. Solution of stiff equations.
Solution of boundary value problems: Shooting method with least square convergence criterion, Quasilinearization method, Parametric differentiation technique and invariant imbedding technique.
Solution of partial differential equations: Finite-difference techniques, Stability and convergence of the solution, Method of characteristics. Finite element and boundary element methods.
Finite difference methods for two point boundary value problems, Laplace equation on the square, heat equation and symmetric hyperbolic systems in 1 D. Lax equivalence theorem for abstract initial value problems. Introduction to variational formulation and the Lax-Milgram lemma. Finite element methods for elliptic and parabolic equations.
Introduction: Floating point representation of numbers and roundoff errors, Interpolation Numerical integration.
Linear systems and matrix theory: Various factorizations of inversion of matrices, Condition number and error analysis.
Non-linear systems: Fixed point iteration, Newton-Rapson and other methods, Convergence acceleration.
Numerical methods for ODE: Introduction and analysis of Taylor, Runge-kutta and other methods.
Numerical methods for PDE: Finite difference method for Laplace, Heat and wave equations.
Sample spaces, events, probability, discrete and continuous random variables, Conditioning and independence, Bayes’ formula, moments and moment generating function, characteristic function, laws of large numbers, central limit theorem, Markov chains, Poisson processes.
Financial market. Financial instruments: bonds, stocks, derivatives. Binomial no-arbitrage pricing model: single period and multi-period models. Martingale methods for pricing. American options: the Snell envelope. Interest rate dependent assets: binomial models for interest rates, fixed income derivatives, forward measure and future. Investment portfolio: Markovitz’s diversification. Capital asset pricing model (CAPM). Utility theory.
Conservative Systems: Hamiltonians, canonical transformations, nonlinear pendulum, perturbative methods, standard map, Lyapunov exponents, chaos, KAM theorem, Chirikov criterion.
Dissipative Systems: logistics map, period doubling, chaos, strange attractors, fractal dimensions, Smale horseshoe, coupled maps, synchronization, control of chaos.
Linear stability analysis, attractors, limit cycles, Poincare-Bendixson theorem, relaxation oscillations, elements of bifurcation theory: saddle-node, transcritical, pitchfork, Hopf bifurcations, integrability, Hamiltonian systems, Lotka-Volterra equations, Lyapunov function & direct method for stability, dissipative systems, Lorenz system, chaos & its measures, Lyapunov exponents, strange attractors, simple maps, period-doubling bifurcations, Feigenbaum constants, fractals. Both flows (continuous time systems) & discrete time systems (simple maps) will be discussed. Assignments will include numerical simulations.
Prerequisites, if any: familiarity with linear algebra - matrices, and ordinary differential equations
Desirable: ability to write codes for solving simple problems.
This is an introductory course on the foundations of mechanics, focusing mainly on classical mechanics. The laws of classical mechanics are most simply expressed and studied in the language of symplectic geometry. This course can also be viewed as an introduction to symplectic geometry. The role of symmetry in studying mechanical systems will be emphasized.
The core syllabus will consist of Lagrangian mechanics, Hamiltonian mechanics, Hamilton-Jacobi theory, moment maps and symplectic reduction. Additional topics will be drawn from integrable systems, quantum mechanics, hydrodynamics and classical field theory.
$C^*$-algebras, Calkin algebra, Compact and Fredholm operators, Index spectral theorem, the Weyl-von Neumann-Berg Theorem and the Brown-Douglas-Fillmore Theorem.
Transversality, Morse functions, stable and unstable manifolds, Morse-Smale moduli spaces, the space of gradient flows, compactification of the moduli spaces of flows, Morse homology, applications.
Riemann surfaces are one-dimensional complex manifolds, obtained by gluing together pieces of the complex plane by holomorphic maps. This course will be an introduction to the theory of Riemann surfaces, with an emphasis on analytical and topological aspects. After describing examples and constructions of Riemann surfaces, the topics covered would include branched coverings and the Riemann-Hurwitz formula, holomorphic 1-forms and periods, the Weyl’s Lemma and existence theorems, the Hodge decomposition theorem, Riemann’s bilinear relations, Divisors, the Riemann-Roch theorem, theorems of Abel and Jacobi, the Uniformization theorem, Fuchsian groups and hyperbolic surfaces.
The material to be covered will include:
Refresher on Commutative Algebra: localization, local rings, integral closure, Krull dimension.
Zariski topology, Hochster’s characterization of Zariski topology, spectral spaces.
Zariski spectrum as a frame
Refresher on categories : Categories, functors, Yoneda Lemma, equivalence of categories, adjoints.
Grothendieck sites : Zariski, '{e}tale and Nisnevich sites.
Presheaves and Sheaves
Locally ringed spaces and schemes
Separated schemes, proper schemes, irreducible schemes, reduced schemes, integral schemes, noetherian schemes.
Morphisms : separated, proper, finite morphisms, finite type morphisms, affine morphisms
Sheaves of algebras : affine morphisms as sheaves of algebras
Sheaves of modules over a scheme, Quasi-coherent and coherent sheaves
Divisors and Line Bundles, Weil divisors, Cartier divisors, Line bundles on Projective spaces, Serre sheaves.
Projective morphisms, ample and very ample line bundles
Formal schemes
Sheaves of differentials.
Background on homological algebra : resolutions, derived functors, $\delta$-categories.
Triangulated categories, Derived categories of abelian categories.
Injective and flasque resolutions.
Cohomology of sheaves of abelian groups
Vanishing theorems for cohomology
Serre’s criterion for affineness
Čech cohomology
Cohomology of projective space, twisting by Serre sheaves
$Ext$ and $Tor$ for sheaves
Serre duality theorem
Schemes as functors of points, the idea of stacks
Noetherian rings and Modules, Localisations, Exact Sequences, Hom, Tensor Products, Hilbert’s Null-stellensatz, Integral dependence, Going-up and Going down theorems, Noether’s normalization lemma , Discrete valuation rings and Dedekind domains.
Number fields and rings of integers, Dedekind domains; prime factorization, ideal class group, finiteness of class number, Dirichlet’s unit theorem, cyclotomic fields, theory of valuations, local fields.
Abstract relations and Dickson’s Lemma; Hilbert Basis theorem, Buchberger Criterion for Grobner Bases and Elimination Theorem; Field Extensions and the Hilbert Nullstellensatz; Decomposition, Radical, and Zeroes of Ideals; Syzygies, Grobner Bases for Modules, Computation of Hom, Free Resolutions; Universal Grobner Bases and Toric Ideals.
LIE ALGEBRAS AND THEIR REPRESENTATIONS
Finite dimensional Lie algebras, Ideals, Homomorphisms, Solvable and Nilpotent Lie algebras, Semisimple Lie algebras, Jordan decomposition, Kiling form, root space decomposition, root systems, classification of complex semisimple Lie algebras Representations Complete reducibility, weight spaces, Weyl character formula, Kostant, steinberg and Freudenthal formulas
Polynomial ring, Projective modules, injective modules, flat modules, additive category, abelian category, exact functor, adjoint functors, (co)limits, category of complexes, snake lemma, derived functor, resolutions, Tor and Ext, dimension, local cohomology,group (co)homology, sheaf cohomology, Cech cohomology, Grothendieck spectral sequence, Leray spectral sequence.
Review of arithmetical functions, averages of arithmetical functions, elementary results on the distribution of prime numbers, Dirichlet characters, Dirichlet’s theorem on primes in arithmetic functions, Dirichlet series and Euler products, the Riemann zeta function and related objects, the prime number theorem. (Time permitting: advanced topics like sieves, bounds on exponential sums, zeros of functions. the circle method.)
Counting problems in sets, multisets, permutations, partitions, trees, tableaux; ordinary and exponential generating functions; posets and principle of inclusion-exclusion, the transfer matrix method; the exponential formula, Polya theory; bijections, combinatorial identities and the WZ method.
The algebra of symmetric functions, Schur functions, RSK algorithm, Murnaghan- Nakayama Rule, Hillman-Grassl correspondence, Knuth equivalence, jeu de taquim, promotion and evacuation, Littlewood-Richardson rules.
No prior knowledge of combinatorics is expected, but a familiarity with linear algebra and finite groubs will be assumed.
Lie groups, Lie algebras, matrix groups , representations, Schur’s orthogonality relations, Peter-Weyl theorem, structure of compact semisimple Lie groups, maximal tori, roots and rootspaces, classification of fundamental systems Weyl group, Highest weight theorem, Weyl integration formula, Weyl’s character formula.
Theory of Distributions: Introduction, Topology of test functions, Convolutions, Schwartz Space, Tempered distributions.
Fourier transform and Sobolev-spaces:Definitions, Extension operators, Continuum and Compact imbeddings, Trace results.
Elliptic boundary value problems: Variational formulation, Weak solutions, Maximum Principle, Regularity results.
Harmonic Analysis on the Poincare disc-Fourier transform, Spherical functions, Jacobi transform, Paley-Wiener theorem, Heat kernels, Hardy’s theorem etc.,
Review of basic notions from Banach and Hilbert space theory.
Bounded linear operators: Spectral theory of compact, self adjoint, and normal operators. Sturm-Liouville problems, Green’s function, Fredholm integral operators.
Unbound linear operators on Hilbert spaces: Symmetric and self adjoint operators, Spectral theory, Banach algebras, Gelfand representation theorem, $C^*$-algebras, Gelfand-Naimark-Segal construction.
The general theory of holomorphic mappings between bounded domains, automorphisms of bounded domains, discussions on the non-existence of a classical Riemann Mapping Theorem in several variables, discussion of the various forms of the one-variable Riemann Mapping Theorem, the Rosay-Wong Theorem, other Riemann-Rosay-Wong-type results (e.g., the work of Pinchuk) to the extent that time permits.
Sz.-Nagy Foias theory: Dilation of contractions on a Hilbert space, minimal isometric dilation, unitary dilation. Von Neumann’s inequality.
Ando’s theorem: simultaneous dilation of a pair of commuting contractions. Parrott’s example of a triple of contractions which cannot be dilated simultaneously. Creation operators on the full Fock space and the symmetric Fock space.
Operators spaces. Completely positive and completely bounded maps. Endomorphisms. Towards dilation of completely positive maps. Unbounded operators: Basic theory of unbounded self-adjoint operators.
Introduction to Fourier Series; Plancherel theorem, basis approximation theorems, Dini’s Condition etc. Introduction to Fourier transform; Plancherel theorem, Wiener-Tauberian theorems, Interpolation of operators, Maximal functions, Lebesgue differentiation theorem, Poisson representation of harmonic functions, introduction to singular integral operators.
In this course we begin by stating many wonderful theorems in analysis and proceed to prove them one by one. In contrast to usual courses (where we learn techniques and see results as “applications of those techniques). We take a somewhat experimental approach in stating the results and then exploring the techniques to prove them. The theorems themselves have the common feature that the statements are easy to understand but the proofs are non-trivial and instructive. And the techniques involve analysis.
We intend to cover a subset of the following theoremes: Isoperimetric inequality, infinitude of primes in arithmetic progressions, Weyl’s equidistribution theorem on the circle, Shannon’s source coding theorem, uncertainty, principles including Heisenberg’s Wigner’s law for eigenvalue of a random matrix, Picard’s theorem on the range of an entire function, principal component analysis to reduce dimensionality of data.
Preliminaries: Holomorphic functions in $C^n$ : definition , the generalized Cauchy integral formula, holomorphic functions: power series development(s), circular and Reinhardt domains, analytic continuation : basic theory and comparisons with the one- variable theory.
Convexity theory: Analytic continuation: the role of convexity, holomorphic convexity, plurisub-harmonic functions, the Levi problem and the role of the d-bar equation.
The d- bar equation: Review of distribution theory, Hormander’s solution and estimates for the d-bar operator.
This topics course is being run as an experiment in approaching the properties of holomorphic maps in several complex variables (SCV) in a self-contained manner (i.e., without requiring any prior exposure to SCV).
The course will begin with a complete and rigorous introduction to holomorphic functions in several variables and their basic properties. This will pave the way to motivating and studying some objects that are, perhaps, entirely indigenous to SCV: e.g., plurisubharmonic functions and invariant metrics. This will allow us to discuss the inequivalence of the (Euclidean) ball and the polydisc in higher dimensions, and to discuss appropriate analogues of the one-variable Riemann Mapping Theorem in higher dimensions.
Next, we shall study the properties of the Kobayashi metric (which is one of the invariant metrics mentioned above) and the Kobayashi distance. This will be used to study the behaviour of automorphisms of bounded domains and refinements of some of the results hinted at above – to the extent that time permits.
TOPOLOGY - II
Point Set Topology: Continuous functions, metric topology, connectedness, path connectedness, compactness, countability axioms, separation axioms, complete metric spaces, function spaces, quotient topology, topological groups, orbit
The fundamental group: Homotopy of maps, multiplication of paths, the fundamental group, induced homomorphisms, the fundamental group of the circle, covering spaces, lifting theorems, the universal covering space, Seifert-Van Kampen theorem, applications.
Manifolds: Differentiable manifolds, differentiable maps and tangent spaces, regular values and Sard’s theorem, vector fields, submersions and immersions, Lie groups, the Lie algebra of a Lie group.
Fundamental Groups: Homotopy of maps, multiplication of paths, the fundamental group, induced homomorphisms, the fundamental group of the circle, covering spaces, lifting theorems, the universal covering space, Seifert-Van Kampen theorem, applications.
Homology : Singular homology, excision, Mayer-Vietoris theorem, acyclic models, CW-complexes, simplicial and cellular homology, homology with coefficients.
Cohomology : Comology groups, relative cohomology,cup products, Kunneth formula, cap product, orientation on manifolds, Poincare duality.
Review of differentiable manifolds and tensors, Riemannian metrics, Levi-Civita connection, geodesics, exponential map, curvature tensor, first and second variation formulas, Jacobi fields, conjugate points and cut locus, Cartan-Hadamard and Bonnet Myers theorems. Special topics - Comparison geometry (theorems of Rauch, Toponogov, Bishop-Gromov), and Bochner techniques.
This course introduces homotopy type theory, which provides alternative foundations for mathematics based on deep connections between type theory, from logic and computer science, and homotopy theory, from topology. This connection is based on interpreting types as spaces, terms as points and equalities as paths. Many homotopical notions have type-theoretic counterparts which are very useful for foundations.
Such foundations are far closer to actual mathematics than the traditional ones based on set theory and logic, and are very well suited for use in computer-based proof systems, especially formal verification systems.
The course will also include background material in Algebraic Topology (beyond a second course in Algebraic Topology).
This is an introduction to hyperbolic surfaces and 3-manifolds, which played a key role in the development of geometric topology in the preceding few decades. Topics that shall be discussed will be from the following list: Basic notions of Riemannian geometry, Models of hyperbolic space, Fuchsian groups, Thick-thin decomposition, Teichmüller space, The Nielsen Realisation problem, Kleinian groups, The boundary at infinity, Mostow rigidity theorem, 3-manifold topology and the JSJ-decomposition, Statement of Thurston’s Geometrization Conjecture (proved by Perelman)
Bochner formula, Laplace comparison, Volume comparison, Heat kernel estimates, Cheng-Yau gradient estimates, Cheeger-Gromoll splitting theorem, Gromov-Haudorff convergence, epsilon regularity, almost rigidity, quantitative structure theory of Riemannian manifolds with Ricci curvature bounds. If time permits, we will discuss the proof of the co-dimension four conjecture due to Cheeger and Naber.
The goal of this course is to use computers to address various questions in Topology and Geometry, with an emphasis on arriving at rigorous proofs. The course will consist primarily of projects which will be contributions to open source software written in the scala programming language.
Differentiable manifolds, differentiable maps, regular values and Sard’s theorem, submersions and immersions, tangent and cotangent bundles as examples of vector bundles, vector fields and flows, exponential map, Frobenius theorem, Lie groups and Lie algebras, exponential map , tensors and differential forms, exterior algebra, Lie derivative, Orientable manifolds, integration on manifolds and Stokes Theorem . Covariant differentiation, Riemannian metrics, Levi-Civita connection, Curvature and parallel transport, spaces of constant curvature.
Basics of Riemannian geometry (Metrics, Levi-Civita connection, curvature, Geodesics, Normal coordinates, Riemannian Volume form), The Laplace equation on compact manifolds (Existence, Uniqueness, Sobolev spaces, Schauder estimates), Hodge theory, more general elliptic equations (Fredholmness etc), Uniformization theorem.
Banach algebras, Gelfand theory, $C^{*}$-algebras the GNS construction, spectral theorem for normal operators, Fredholm operators. The L-infinity functional calculus for normal operators.
This course explores matrix positivity and operations that preserve it. These involve fundamental questions that have been extensively studied over the past century, and are still being studied in the mathematics literature, including with additional motivation from modern applications to high-dimensional covariance estimation. The course will bring together techniques from different areas: analysis, linear algebra, combinatorics, and symmetric functions.
List of topics (time permitting):
1. The cone of positive semidefinite matrices. Totally positive/non-negative matrices. Examples of PSD and TP/TN matrices (Gram, Hankel, Toeplitz, Vandermonde, $\mathbb{P}_G$). Matrix identities (Cauchy-Binet, Andreief). Generalized Rayleigh quotients and spectral radius. Schur complements.
2. Positivity preservers. Schur product theorem. Polya-Szego observation. Schoenberg’s theorem. Positive definite functions to correlation matrices. Rudin’s (stronger) theorem. Herz, Christensen-Ressel.
3. Fixed-dimension problem. Introduction and modern motivations. H.L. Vasudeva’s theorem and simplifications. Roger Horn’s theorem and simplifications.
4. Proof of Schoenberg’s theorem. Characterization of (Hankel total) positivity preservers in the dimension-free setting.
5. Analytic/polynomial preservers – I. Which coefficients can be negative? Bounded and unbounded domains: Horn-type necessary conditions.
6. Schur polynomials. Two definitions and properties. Specialization over fields and for real powers. First-order approximation.
7. Analytic/polynomial preservers – II. Sign patterns: The Horn-type necessary conditions are best possible. Sharp quantitative bound. Extension principle I: dimension increase.
8. Entrywise maps preserving total positivity. Extension principle II: Hankel TN matrices. Variants for all TP matrices and for symmetric TP matrices. Matrix completion problems.
9. Entrywise powers preserving positivity. Application of Extension principle I. Low-rank counterexamples. Tanvi Jain’s result.
10. Characterizations for functions preserving $\mathbb{P}_G$. Extension principle III: pendant edges. The case of trees. Chordal graphs and their properties. Functions and powers preserving $\mathbb{P}_G$ for $G$ chordal. Non-chordal graphs.
Introduction to distribution theory and Sobolev spaces, Fundamental solutions for Laplace, heat and wave operations.
Second order elliptic equations: Boundary value problems, Regularity of weak solutions, Maximum principle, Eigenvalues.
Semi group theory:Hille-Yosida theorem, Applications to heat, Schroedinger and wave equations.
System of first order hyperbolic equations: Bicharacteristics, Shocks, Ray theory, symmetric hyperbolic systems.
Ando dilation of a commuting pair of contractions, Distinguished varieties of the bidisc, Description of all distinguished varieties, Construction of a distinguished variety corresponding to a pair of commuting matrices, Sharpening of Ando’s inequality, Extending the sharpened Ando inequality to operators with finite dimensional defect spaces, The extension property, Holomorphic retracts.
Review of Distributions, Sobolev spaces and Variational formulation. Introduction to Homogenization. Homogenization of elliptic PDEs. Specific Cases: Periodic structures and layered materials. Convergence Results: Energy method, Two-scale multi-scale methods, H-Convergence, Bloch wave method. General Variational convergence: G -convergence and G- convergence, Compensated compactness. Study of specific examples and applications
Distribution Theory - Introduction, Topology of Test functions, Convolutions, Schwartz Space, Tempered Distributions, Fourier Transform;
Sobolev Spaces - Definitions, Extension Operators, Continuous and Compact Imbeddings, Trace results; Weak Solutions - Variational formulation of Elliptic Boundary Value Problems, Weak solutions, Maximum Principle, Regularity results;
Finite Element Method (FEM) - Introduction to FEM, Finite element solution of Elliptic boundary value problems.
Banach algebras – Gelfand theory, L-infinity functional calculus for bounded normal operators, Pick - Nevanlinna and Caratheodory Interpolation problems, Distinguished varieties in the bidisc.
Arithmetical functions, Primes in Arithmetic Progressions, Prime number theorem for arithmetic progressions and zeros of Dirichlet L-functions, Bombieri-Vinogradov theorem, Equidistribution, circle method and applications (ternary Goldbach in mind), the Large Sieve and applications, Brun’s theorem on twin primes.
(Further topics if time permits: more on sieves, automorphic forms and L-functions, Hecke’s L-functions for number fields, bounds on exponential sums etc.)
Semigroup Theory: Introduction, Continuous and Contraction Semigroups, Generators, Hille-Yoshida and Lumer-Philips Theorems
Evolution Equations: Semigroup Approach to Heat, Wave and Schrodinger Equations
Review of arithmetical functions, Averages of arithmetical functions, Elementary results on the distribution of prime numbers, Dirichlet characters, Dirichlet’s theorem on primes in arithmetic functions, Dirichlet series and Euler products, Riemann zeta function and related objects, The prime number theorem.
(Time permitting: More advanced topics like Sieves, bounds on exponential sums, zeros of zeta functions, circle method etc.)
Elliptic curves are smooth projective curves of genus 1 with a marked point. Over a field of characteristic zero they are given by an equation of the form $y^2 = x^3+ax+b$
. They are at the boundary of our (conjectural) understanding of rational points on varieties and are subject of many famous conjectures as well as celebrated results. They play an important role in number theory.
The course will begin with an introduction to algebraic curves. We will then study elliptic curves over complex number, over finite fields, over local fields of characteristic zero and finally over number fields. Our goal will be to prove the Mordell-Weil theorem.
The goal is to give an introduction to adeles and some of their uses in modern number theory, discussing also some topics which are not too common in textbooks.
Topics to be covered: absolute values and Ostrowski’s Theorem; classification of locally compact fields; definition of adeles and some applications (finiteness of class number and of the generators of the group of S-units; structure of modules over Dedekind domains; applications to the geometry of curves); an introduction to the Strong Approximation Theorem; adelic points of varieties and schemes; possibly other topics (depending on time left and interests of the audience; for example Tate’s thesis, quasi-characters of the idele class group and p-adic L-functions).
This course would be a survey of fundamental results as well as current research. Topics will be related to the following areas: geometric structures on surfaces, hyperbolic 3-manifolds, Riemann surfaces and Teichmüller theory, and will focus on the various interactions between these fields. Students will be encouraged to explore open-ended questions and/or write related computer programs. The following is the course plan:
Part I
Part II
Part III
Part IV
The topic covered will be the control of discrete-time infinite state-space Markovian systems. These techniques appear frequently in the analysis and optimization of stochastic systems e.g. control of queues, resource allocation problems in networks, machine learning, reinforcement learning, operations research, etc. The course is aimed at students who work in applied probability, stochastic control, machine learning, networking. Course is divided into the following three parts:
Wigner’s semicircle law: (a) combinatorial method, (b) Stieltjes’ transform method, (c) Chatterjee’s invariance principle method.
Gaussian unitary and orthogonal ensembles: (a) Exact density of eigenvalues. (b) Orthogonal polynomials and determinantal formulas leading to another proof of Wigner’s semicircle law.
Tridiagonal reduction for GUE and GOE: (a) Another derivation of eigenvalue density. (b) Another proof of Wigner’s semicircle law. (c) Matrix models for Beta ensembles. (d) Selberg’s integral.
Other models of random matrices - Wishart and Jacobi ensembles.
Free probability: (a) Noncommutative probability space and free independence. (b) Combinatorial approach to freeness. (c) Limiting spectra of sums of random matrices.
Non-hemitian random matrices: (a) Ginibre ensemble. (b) Circular law for matrices with i.i.d entries.
Fluctuation behaviour of eigenvalues (if time permits).
Probability measures and randown variables, pi and lambda systems, expectation, the moment generating function, the characteristic function, laws of large numbers, limit theorems, conditional contribution and expectation, martingales, infinitely divisible laws and stable laws.
First Construction of Brownian Motion, convergence in $C[0,\infty)$, $D[0,\infty)$, Donsker’s invariance principle, Properties of the Brownian motion, continuous-time martingales, optional sampling theorem, Doob-Meyer decomposition, stochastic integration, Ito’s formula, martingale representation theorem, Girsanov’s theorem, Brownian motion and the heat equation, Feynman- Kac formula, diffusion processes and stochastic differential equations, strong and weak solutions, martingale problem.
This course will be aimed at understanding the behavior of random geometric objects in high dimensional spaces such as random vectors, random graphs, random matrices, and random subspaces, as well. Topics will include the concentration of measure phenomenon, non-asymptotic random matrix theory, chaining and Gaussian processes, empirical processes, and some related topics from geometric functional analysis and convex geometry. Towards the latter half of the course, a few applications of the topics covered in the first half will be considered such as community detection, covariance estimation, randomized dimension reduction, and sparse recovery problems.
Linear time series analysis - modelling time series using stochastic processes, stationarity, autocovariance, auto correlation, multivariate analysis - AR, MA, ARMA, AIC criterion for order selection;
Spectral analysis - deterministic processes, concentration problem, stochastic spectral analysis, nonparametric spectral estimation (periodogram, tapering, windowing), multitaper spectral estimation; parametric spectral estimation (Yule-Walker equations, Levinson Durbin)(recursions);
Multivariate analysis - coherence, causality relations; bootstrap techniques for estimation of parameters;
Nonlinear time series analysis - Lyapunov exponents, correlation dimension, embedding methods, surrogate data analysis.
A course in Gaussian processes. At first we shall study basic facts about Gaussian processes - isoperimetric inequality and concentration, comparison inequalities, boundedness and continuity of Gaussian processes, Gaussian series of functions, etc. Later we specialize to smooth Gaussian processes and their nodal sets , in particular expected length and number of nodal sets, persistence probability and other such results from recent papers of many authors.
Trading in continuous time : geometric Brownian motion model. Option pricing : Black-Scholes-Merton theory. Hedging in continuous time : the Greeks. American options. Exotic options. Market imperfections. Term-structure models. Vasicek, Hull-White and CIR models. HJM model. LIBOR model. Introduction to credit Rsik Models: structural and intensity models. Credit derivatives.
Discrete parameter martingales: Conditional expectation. Optional sampling theorems. Doob’s inequalities. Martingale convergence theorems. Applications.
Brownian motion. Construction. Continuity properties. Markov and strong Markov property and applications. Donsker’s invariance principle. Further sample path properties.
Origins, states, observables, interference, symmetries, uncertainty, wave and matrix mechanics, Measurement, scattering theory in 1 dimension, quantum computation and information, Prerequisites are analysis and linear algebra.
Optimal Control of PDE:Optimal control problems governed by elliptic equations and linear parabolic and hyperbolic equations with distributed and boundary controls, Computational methods. Homogenization:Examples of periodic composites and layered materials. Various methods of homogenization. Applications and Extensions:Control in coefficients of elliptic equations, Controllability and Stabilization of Infinite Dimensional Systems, Hamilton- Jacobi-Bellman equations and Riccati equations, Optimal control and stabilization of flow related models.
Introduction and examples, optimal control problems governed by elliptic and parabolic systems, adjoint systems, optimality conditions, optimal control and optimality systems for other PDEs like Stokes systems.
Course Objective To provide a gentle introduction to the direct methods in Calculus of Variations concerning minimizations problems, excluding minmax methods. The focus will be on illustrating the main methods using important prototype examples and not on proving the most general or the sharpest results.
Target audience This course is primarily intended for students of Mathematics with interests in Analysis, PDE and/or differential Geometry and geometric analysis, especially minimal surfaces. However, students of physics and different branches of engineering ( especially mechanical engineering ) and economics would still probably find a portion of the course useful for them.
Course contents and outline Our goal is to cover the following topics:
Classical Methods: Euler-Lagrange equations, Lagrangian and Hamiltonian formulations, Hamilton-Jacobi equations, constrained problems and Lagrange multipliers, An illustration of the methods: Geodesic curves.
Direct Methods: Dirichlet integral and $p$-Dirichlet Integral: Existence of minimizers: Existence theorem for convex functional with lower order terms, examples and counterexamples, weak form of the Euler-Lagrange equations, Dirichlet Principle, weak continuity of determinants. Regularity questions.
Plateau’s problem and minimal surfaces: Parametric Plateau’s problem: Douglas-Courant-Tonelli method, Regularity, uniqueness and nonuniqueness, Nonparametric minimal surfaces, Isoperimetric inequality.
Basic definitions in graph theory, line graphs, some matrices related to graphs and their spectral properties, the Perron-Frobenius theorem, Cauchy’s interlacing theorem, strongly regular graphs, the Laplacian matrix, cuts and flows.
The dynamics alluded to by the title of the course refers to dynamical systems that arise from iterating a holomorphic self-map of a complex manifold. In this course, the manifolds underlying these dynamical systems will be of complex dimension 1. The foundations of complex dynamics are best introduced in the setting of compact spaces. Iterative dynamical systems on compact Riemann surfaces other than the Riemann sphere – viewed here as the one-point compactification of the complex plane – are relatively simple. We shall study what this means. Thereafter, the focus will shift to rational functions: these are the holomorphic self-maps of the Riemann sphere. Along the way, some of the local theory of fixed points will be presented. In the case of rational maps, some ergodic-theoretic properties of the orbits under iteration will be studied. The development of the latter will be self-contained. The properties/ theory coverd will depend on the time available and on the audience’s interest.
The aim of this course is to provide an introduction to CR (Cauchy Riemann/Complex Real) geometry, which is broadly the study of the structure(s) inherited by real submanifolds in complex spaces. We will first give a parallel introduction to the fundamental objects of SCV and CR geometry. These include holomorphic functions in several variables, CR manifolds (embedded and abstract) and CR functions. Next, we will cover some examples, results, and techniques from the following range of topics.
a) embeddability of abstract CR structures;
b) holomorphic extendability of CR functions;
c) CR singularities.
Wherever possible (and time permitting), we will highlight the connections of this field to other areas of analysis and geometry. For instance, abstract CR structures will be discussed in the broader context of involutive structures on smooth manifolds.
Quick review of the theory of bounded operators on a Hilbert space compactoperators, Fredholm operators, spectral theory of compact self-adjoint operators.
Spectral theorem and functional calculus for a bounded self-adjoint operator.
Unbounded operators - examples, spectral theorem and functional calculus for an unbounded self-adjoint operator.
Schatten p-classes- interpolation.
Krein’s spectral shift function
Gauss Map, Gaussian Curvature, Mean curvature, Area functional etc.
Harmonic coordinates in isothermal parameters. Examples of minimal surfaces.
Minimal surfaces with boundary: Plateau’s problem.
The gauss map for minimal surfaces with some examples.
The Weierstrass-Enneper representation of minimal surfaces. Many more examples of minimal surfaces.
If time permits:
Surfaces that locally maximise area in Lorenztian space (maximal surfaces). A lot of examples and analogous results, as in minimal surface theory, for maximal surfaces.
The purpose of this course will be to understand (to an extent) and appreciate the symbiotic relationship that exists between mathematics and physics. Topics to be covered can vary but those in this edition include: a brisk introduction to basic notions of differential geometry (manifolds, vector fields, metrics, geodesics, curvature, Lie groups and such), classical mechanics (Hamiltonian and Lagrangian formulations, n-body problems with special emphasis on the n=3 case) and time permitting, an introduction to integrable systems.
This course will focus on the structure as well as on finite dimensional complex representations of the following classical groups: General and special Linear groups, Symplectic groups, Orthogonal and Unitary groups.
Reflection groups and their generalisations, Coxeter systems, permutation representations, reduced words, Bruhat order, Kazhdan-Lusztig theory, Chevalley’s theorem, Poincare series, root systems, classification of finite and affine Coxeter groups
No prior knowledge of combinatorics or algebra is expected, but we will assume a familiarity with linear algebra and basics of group theory.
Loop algebras, central extensions, untwisted affine Lie algebras, root systems, and Weyl groups of untwisted affine Lie algebras. Graph automorphisms of untwisted affine Lie algebras, twisted affine Lie algebras, root systems and Weyl groups of twisted affine Lie algebras. Representations of affine Lie algebras, weight space decomposition, the Category O, Verma modules, integrable modules in Category O. The generalized Casimir operator, Weyl-Kac Character formula, Weyl-Kac denominator identities and Macdonald identities.
In this course, our main aim is to develop abstract variational techniques which can be employed to study the existence of solutions of various Semi-linear elliptic Partial Differential Equations. The main fundamental results, that will be covered in this course, are functional analytic in nature and can be used in many other situations. A basic outline of the course is as follows:
Discrete parameter martingales, branching process, percolation on graphs, random graphs, random walks on graphs, interacting particle systems.
Part I - Applications of Spectral Algotihms: Best-Fit Subspaces, Mixture models, Probabilistic Clustering,Recursive Clustering, Optimization via low-rank approximation.
Part II - Algorithms: Matrix Approximation via Random Sampling, Adaptive Sampling Methods, Extensions of SVD to tensors.
Erdos - Renyi random graphs, graphs with power law degree distributions, Ising Potts and contact process, voter model, epidemic models.
Real trees, the Brownian continuum random tree, phase transition in random graphs, scaling limits of discrete combinatorial structures, random maps, the Brownian map and its geometry
We shall illustrate some important techniques in studying discrete random structures through a number of examples. The techniques we shall focus on will include (if time permits)
We shall discuss applications of these techniques in various fields such as Markov chains, percolation, interacting particle systems and random graphs.
Financial market. Financial instruments: bonds, stocks, derivatives. Binomial no- arbitrage pricing model: single period and multi-period models. Martingale methods for pricing. American options: the Snell envelope. Investment portfolio: Markovitz’s diversification. Capital asset pricing model(CAPM). Utility theory.
Trading in continuous time: geometric Brownian motion model. Option pricing: Black-Scholes-Merton theory. Hedging in continuous time: the Greeks. American options. Exotic options. Market imperfections. Term-Structure models: Vasicek, Hull-White and CIR models. HJM model. Forward LIBOR model.
We shall cover a selection of topics in probability theory coming from statistical physics models on the Euclidean lattice. A few possible examples of the models include: Ising model, O(N) model, Gaussian free field, contact process, voter model and exclusion processes.
Pre-requisites: This course will be aimed at Int-Ph.D. and PhD students working in probability theory and related areas. A course in graduate probability theory is useful, but not absolutely necessary. A student with a strong undergraduate background in probability (i.e., without measure theory) might also find this course accessible.
One-variable Calculus: Real and Complex numbers; Convergence of sequences and series; Continuity, intermediate value theorem, existence of maxima and minima; Differentiation, mean value theorem,Taylor series; Integration, fundamental theorem of Calculus, improper integrals. Linear Algebra: Vector spaces (over real and complex numbers), basis and dimension; Linear transformations and matrices.
Linear Algebra continued: Inner products and Orthogonality; Determinants; Eigenvalues and Eigenvectors; Diagonalisation of symmetric matrices. Multivariable calculus: Functions on $\R^n$, partial and total derivatives; Chain rule; Maxima, minima and saddles; Lagrange multipliers; Integration in $\R^n$, change of variables, Fubini’s theorem; Gradient, Divergence and Curl; Line and Surface integrals in $\R^2$ and $\R^3$; Stokes, Green’s and Divergence theorems. Introduction to Ordinary Differential Equations; Linear ODEs and Canonical forms for linear transformations.
Basic notions of probability, conditional probability and independence, Bayes’ theorem, random variables and distributions, expectation and variance, conditional expectation, moment generating functions, limit theorems. Samples and sampling distributions, estimation of parameters, testing of hypotheses, regression, correlation and analysis of variance.
Note: This course has been replaced by UM 205.
Divisibility and Euclid’s algorithm; Fundamental theorem of arithmetic; Infinitude of primes; Congruences; (Reduced) residue systems, Application to sums of squares; Chinese Remainder Theorem; Solutions of polynomial congruences, Hensel’s lemma; A few arithmetic functions (in particular, discussion of the floor function); the Mobius inversion formula; Recurrence relations; Basic combinatorial number theory (pigeonhole principle, inclusion-exclusion, etc.); Primitive roots and power residues, Quadratic residues and the quadratic reciprocity law, the Jacobi symbol; Some Diophantine equations, Pythagorean triples, Fermat’s descent, examples; Definitions of groups, rings and fields, motivations, examples and basic properties; polynomial rings over fields, factorisation of polynomials, content of a polynomial and Gauss’ lemma, Eisenstein’s irreducibility criterion; Elementary symmetric polynomials, the fundamental theorem on Symmetric polynomials; Algebraic and transcendental numbers (an introduction).
Basic notions from set theory, countable and uncountable sets. Metric spaces: definition and examples, basic topological notions. The topology of $\R^n$: topology induced by norms, the Heine-Borel theorem, connected sets. Sequences and series: essential definitions, absolute versus conditional convergence of series, some tests of convergence of series. Continuous functions: properties, the sequential and the open- set characterizations of continuity, uniform continuity. Differentiation in one variable. The Riemann integral: formal definitions and properties, continuous functions and integration, the Fundamental Theorem of Calculus. Uniform convergence: definition, motivations and examples, uniform convergence and integration, the Weierstrass Approximation Theorem.
Mandatory project for undergraduate mathematics majors in their fourth year, second semester.
Optional project for undergraduate mathematics majors in their fifth year, first semester.
Optional project for undergraduate mathematics majors in their fifth year, second semester.