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 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.
Part A
Part B
Part A
Part B
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.
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.
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 S0(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 Randon-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, complex-analytic functions, Cauchy’s integral formula, power series, Liouville’s theorem. The maximum-modulus theorem. Isolated singularities, residue theorem, the Argument Principle, real integrals via contour integration. Mobius transformations, conformal mappings. The Schwarz lemma, automorphisms of the dis. Normal families and Montel’s theorem. The Riemann mapping 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.
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.
REAL ANALYSIS
The Lebesgue Integral:Riemann-Stieltjes integral, Measures and measurable sets, measurable functions, the abstract Lebesgue integral. Product measures and Fubini’s theorem. Complex measures and the lebesgue - Radon - Nikodym theorem and its applications. Function Spaces and Banach Spaces: Lpspaces, Abstract Banach Spaces. The conjugate spaces. Abstract Hilbert spaces.
Differentiation:Basic definitions and theorems, Partial derivatives, Derivatives (as linear maps), Inverse and Implicit function theorems. Integration:Basic definitions and theorems, Integrable functions, Partitions of unity, Change of variables. Manifolds:Basic definitions, forms on manifolds, Stokes theorem on manifolds, Volume element, classical theorems (Green’s and divergence).
$C^*$-algebras, Calkin algebra, Compact and Fredholm operators, Index spectral theorem, the Weyl-von Neumann-Berg Theorem and the Brown-Douglas-Fillmore Theorem.
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
\v{C}ech 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.
Algebraic preliminaries: Algebraic field extensions: Normal, separable and Galois extensions. Euclidean rings, principal ideal domains and factorial rings. Quadratic number fields. Cyclotomic number fields. Algebraic integers: Integral extensions: Algebraic number fields and algebraic integers. Norms and traces. Resultants and discriminants. Integral bases. Class numbers:Lattices and Minkowski theory. Finiteness of class number. Dirichlet’s unit theorem. Ramification Theory: Discriminants. Applications to cryptography.
Affine algebraic sets, Hilbert basis theorem, Hilbert Nullstellensatz, function field, plane curves, Bezout’s theorem, product, normality, morohisms, Noether normalisation, Graded rings, projective varieties, rational functions, tangent spaces, non- singularity, blowing up points, Riemann-Roch for curves. Schemes examples.
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 basic concepts in several complex variables with the eventual aim of studying some topics in multi-variable complex dynamics. By “complex dynamics”, we mean the the study of the dynamical system that arises in iterating a holomorphic map.
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 a concept that is, perhaps, entirely indigenous to several complex variables: the notion of plurisubharmonicity.
Next, we shall look at some of the motivations behind the study of complex dynamics in several variables. Using the tools developed, we shall undertake a crash-course in currents, which are objects central to the study of some aspects of complex dynamics. We shall then cover as much of the following topics as 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./p> Cohomology : Comology groups, relative cohomology,cup products, Kunneth formula, cap product, orientation on manifolds, Poincare duality.
Riemannian metric, Levi-Civita connection, geodesics, exponential map, Hopf-Rinow theorem, curvature tensior, first and second variational formula, jacobi fields, Myers Bonnet theorem, Bishop-Gromov volume comparison theorem, Cartan-Hadamard theorem, Synge’s theorem, de Rham cohomology and the Bochner techniques. Topological implications of positive or negative curvature.
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)
The first half of the course will focus on convergence theory of Riemannian manifolds. Gromov-Hausdorff convergence, Lipschitz convergence and collapsing theory will be discussed.
The second half will be about the Ricci flow. Existence and uniqueness, maximum principles and Hamilton’s theorem for 3-manifolds with positive Ricci curvature will be covered.
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
Introduction to Calculus of Variations and Morse Theory. Critical Point Theory for Gradient Mappings: Mountain Pass theorem, linking Theorems, Saddle Point theorem, Dual formulation etc.
Classification of integral equations and occurrence in boundary value problems for Ordinary and Partial Differential Equations, Abel’s integral equations, Integral equations of the second kind, Degenerate Kernels, The Neumann series solutions, Fredholm theorems, The eigenvalue problems, Rayleigh-Ritz method, Galerkin method, The Hilbert-Schmidt theory, Singular integral equations, Riemann-Hilbert problems, The Wiener-Hopf equations, The Wiener-Hopf technique, Numerical methods, Applications of integral equations of problems of Elasticity, Fluid Mechanics and Electromagnetic theory.
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.
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.)
Holomorphic Modular forms: motivation and introduction, Eisentein series, cusp forms, Fourier expansion of Poincare series and Petersson trace formula, Hecke operators and overview of newform theory, Kloosterman sums and bounds for Fourier coefficients, Automorphic L-functions, Dirichlet-twists and Weil’s converse theorm, Theta functions and representation by quadratic forms, Convolution: the Rankin-Selberg method. (Further topics if time permits: Non-holomorphic modular forms (overview), Siegel modular forms (introduction), Elliptic curves and cusp forms, spectral theory, analytic questions related to modular forms.)
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.)
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.
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 and Applications, Ergodic Theory, Random Walks, Branching Processes.
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.
Topological groups, locally compact groups, Haar measure, Modular function, Convolutions, homogeneous spaces, unitary representations, Gelfand-Raikov Theorem. Functions of positive type, GNS construction, Potrjagin duality, Bochner’s theorem, Induced representations, Mackey’s impritivity theorem.
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.
Semigroup Theory: Introduction, Continuous and Contraction Semigroups, Generators, Hille-Yoshida and Lumer-Philips Theorems
Evolution Equations: Semigroup Approach to Heat, Wave and Schrodinger Equations
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.
Plurisubharmonic functions, domains of holomorphy, the d-bar problem and potential theory.
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.
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.
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 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.