CASA is a special-purpose system for computational algebra and constructive algebraic geometry. The system has been developed since The system is built on the kernel of the widely used computer algebra system Maple. It is able to perform simple and sophisticated operations on multivaraiate polynomial s and on various data related to them ideals, modules, matrices, rational functions.

For example, it can readily compute Grobner bases, syzygies and minimal free resolution, intersection, division, the radical of an ideal, the ideal of zero-dimensional schemes, Poincare' series and Hilbert functions, factorization of polynomial s, toric ideals. The capabilities of CoCoA and the flexibility of its use are further enhanced by the dedicated high-level programming language.

For convenience, the system offers a textual interface, an Emacs mode, and a graphical user interface common to most platforms. Fermat is a super calculator - computer algebra system, in which the basic items being computed can be rational numbers, modular numbers, elements of finite fields, multivariable polynomial s, multivariable rational functions, or multivariable polynomial s modulo other polynomial s.

It is shareware. Further, polynomial s p, q,.. It is possible to allow Laurent polynomial s. Once the computational ring is established in this way, all computations are of elements of this ring. GAP is a system for computational discrete algebra, with particular emphasis on Computational Group Theory. GAP provides a programming language, a library of thousands of functions implementing algebraic algorithms written in the GAP language as well as large data libraries of algebraic objects.

GAP is used in research and teaching for studying groups and their representations, rings, vector spaces, algebras, combinatorial structures, and more. GAP is developed by international cooperation.

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You can study and easily modify or extend GAP for your special use. It contains different program modules and is fully compatible with the LinBox linear algebra library and the Athapascan environment, which permits parallel programming. It has been developed under the project leadership of Prof. LiE is the name of a software package that enables mathematicians and physicists to perform computations of a Lie group theoretic nature. It focuses on the representation theory of complex semisimple reductive Lie groups and algebras, and on the structure of their Weyl groups and root systems.

LiE does not compute directly with elements of the Lie groups and algebras themselves; it rather computes with weights, roots, characters and similar objects. LinBox has the following top-level functions: solve linear system, matrix rank, determinant, minimal polynomial , characteristic polynomial , Smith normal form and trace. A good collection of finite field and ring implementations is provided, for use with numerous black box matrix storage schemes. Magma is a large, well-supported software package designed to solve computationally hard problems in algebra, number theory, geometry and combinatorics.

It provides a mathematically rigorous environment for computing with algebraic, number-theoretic, combinatoric and geometric objects. Maple is an environment for scientific and engineering problem-solving, mathematical exploration, data visualization and technical authoring. Mathematica seamlessly integrates a numeric and symbolic computational engine, graphics system, programming language, documentation system, and advanced connectivity to other applications. Mathomatic is a free, portable, general-purpose CAS Computer Algebra System and calculator software that can symbolically solve, simplify, combine, and compare equations, perform complex number and polynomial arithmetic , etc.

It does some calculus and is very easy to use. MuPAD is a mathematical expert system for doing symbolic and exact algebraic computations as well as numerical calculations with almost arbitrary accuracy. For example, the number of significant digits can be chosen freely. Apart from a vast variety of mathematical libraries the system provides tools for high quality visualization of 2- and 3-dimensional objects. On Microsoft Windows, Apple Macintosh and Linux systems, MuPAD offers a flexible notebook concept for creating mathematical documents combining texts, graphics, formulas, computations and mathematical visualizations and animations.

Thus it offers a natural integration in Office applications like Word or PowerPoint as well as others. Normaliz is a tool for computations in affine monoids, vector configurations, lattice polytopes, and rational cones. Its input data can be specified in terms of a system of generators or vertices or a system of linear homogeneous Diophantine equations, inequalities and congruences or a binomial ideal. Normaliz computes the dual cone of a rational cone in other words, given generators, Normaliz computes the defining hyperplanes, and vice versa , a placing or lexicographic triangulation of a vector configuration resulting in a triangulation of the cone generated by it , the Hilbert basis of a rational cone, the lattice points of a lattice polytope, the normalization of an affine monoid, the Hilbert or Ehrhart series and the Hilbert or Ehrhart quasi polynomial under a Z-grading for example, for rational polytopes , NEW: generalized or weighted Ehrhart series and Lebesgue integrals of polynomial s over rational polytopes via NmzIntegrate, a description of the cone and lattice under consideration by a system of inequalities, equations and congruences.

It will be useful to have some familiarity with programming. Topics: Basic type theory: terms and types, function types, dependent types, inductive types. Most of the material will be developed using the dependently typed language Idris. Connections with programming in functional languages will be explored. Manin, Yu. Srivastava, S. Suggested books : Artin, M.

Hoffman, K and Kunze R. Halmos, P. Greub, W. Suggested books : Artin, Algebra , M. Prentice-Hall of India, Dummit, D. Herstein, I. Lang, S. Atiyah, M. Suggested books : T. Becker and V. Adams and P. Sturmfels, Grobner bases and convex polytopes , American Mathematical Society Suggested books : Serre, J. Koblitz, N. Iwaniec, H. Diamond, F. Suggested books : Adrian Bondy and U. Springer-Verlag, Berlin, ISBN: Douglas B. West, Introduction to graph theory , Prentice Hall, Inc. Suggested books : Bondy, J. Burton, D. Clark, J. Polya G. Suggested books : Narasimham, R.

Niven, I.

## Oberwolfach References on Mathematical Software

Apostol, T. Ireland, K. Hoffman, K. American Mathematical Society. Serre, Linear representations of finite groups , Graduate Texts in Mathematics. New York-Heidelberg. Suggested books : Rudin, W. Suggested books : Royden, H. Folland, G. Hewitt, E. Suggested books : Rudin, Functional Anaysis 2nd Ed.

Yosida, K. Goffman, C. Suggested books : Ahlfors, L. Conway, J. Suggested books : Narasimhan, R. Nievergelt , Birkhauser 2nd ed. Greene, R. Suggested books : Spivak, M. Hirsh, M. Suggested books : Armstrong, M. Munkres, K. Viro, O. MA Introduction to algebraic topology Prerequisite courses: MA and MA 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.

Hatcher, A. Kosniowski, C. Press, Croom, F. Suggested books : do Carmo, M. Thorpe, J. O'Neill, B. Gray, A. MA Metric Geometry of Spaces and Groups Prerequisite courses: MA Pre-requisites : A first course in Topology can be taken concurrently 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.

MA Ordinary Differential Equations Prerequisite courses: MA 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. Birkhaeuser, Coddington, E. Perko, L.

Suggested books : Garabedian, P. Renardy, M. Suggested books : Hoffman, K.

Simmons G. Churchill, R. MA Numerical Methods 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. Suggested books : Gupta, A.

Conte, S. Hildebrand, F. Froberg, C. MA Numerical Methods for Partial Differential Equations Finite difference methods for two point boundary value problems, Laplace equation on the square, heat equation and symmetric hyperbolic systems in 1 D. Suggested books : Smith, G. Evans, G. Blackledge, J. Suggested books : Faires, J. Stoer, J. Iserlas, A. Suggested books : Ross, S. Taylor, H. Suggested books : Luenberger, D. Shiryaev, A. Shreve, S. Suggested books : Lichtenberg, A. Guckenheimer, J. Prerequisites, if any: familiarity with linear algebra - matrices, and ordinary differential equations Desirable: ability to write codes for solving simple problems.

Suggested books : S. Alligood, T. Tabor, Chaos and Integrability in Non-linear Dynamics , Rudin, W. Rudin, Functional Anaysis 2nd Ed. House, New Delhi, Munkres, J. Milnor, Morse Theory , Ann. Press, Princeton, Zariski spectrum as a frame Refresher on categories : Categories, functors, Yoneda Lemma, equivalence of categories, adjoints. 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.

Springer-Verlag, New York-Heidelberg, Robin Hartshorne, Residues and duality , Lecture notes of a seminar on the work of A.

With an appendix by P. Lecture Notes in Mathematics, No. Triangulated categories, Derived categories of abelian categories. Injective and flasque resolutions. Suggested books : Artin, E. Borevich, Z. Cassels, J. Hasse, H. Hecke, E. Samuel, P. Suggested books : Shafarevich, I. Smith, K. Reid, M. Fulton, W. Holme, A.

MA Introduction to Homological Algebra 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. Suggested books : Cartan and Eilenberg, Homological Algebra. Weibel, Introduction to Homological Algebra. Rotman, Introduction to Homological Algebra.

Suggested books : Apostol, T. Davenport, H. MA Combinatorics Pre-requisites : Calculus, Linear algebra and some exposure to proofs and abstract mathematics. Programming in Sage will be a part of every lecture. Richard P. Suggested books : Stanley, R.

Sagan, B. Prasad, A. Stanley, R. Suggested books : V. Varadarajan, Lie groups, Lie algebras and their representations , Sringer Hall, Lie groups, Lie algebras and representations , Springer Knapp, Representation theory of semismiple lie groups. An overview based on examples , Princeton university press Kesavan, S. Evans, L. Schwartz, L. Hermann, Theorie des Distributions , Suggested books : Conway, J. Berberian, S. MA Topics in Complex Analysis 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.

Suggested books : Krantz, S. Suggested books : John B. Conway, A course in Functional Analysis , Springer, Suggested books : Dym, H. Stein, E. Sadosky, C. Suggested books : Korner, I. Robert Ash. Serre, J. Thangavelu, S. Rudin W. Students who have not seen any one-dimensional complex dynamics earlier but are highly interested in this course are encouraged to speak to the instructor. Suggested books : L.

## Polynomial Chaos Methods for Hyperbolic Partial Differential Equations

Amsterdam, Janich, K. Suggested books : Brickell, F. Guillemin, V. Milnor, John W. Suggested books : Hatcher, A. Press, Indian edition is available. Munkres, I. Shastri, A. MA Riemannian Geometry 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.

Springer-Verlag, New York, MA Introduction to Homotopy Type Theory Prerequisite courses: MA , MA , MA Pre-requisites : Algebraic Topology, Dependent Type Theory 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.

Tutorial for the Lean Theorem Prover. Benedetti-Petronio, Lectures on Hyperbolic Geometry. Martelli, Introduction to Geometric Topology. A first course in algebraic topology is helpful but not necessary. Real analysis in more than one variable. Linear algebra. Suggested books : Spivak M. Kumaresan S. Hindustan Book Agency, New Delhi, Warner F. Springer-Verlag, New York-Berlin, Lee J. Analysis multivariable calculus, some measure theory, function spaces. Ideally, the spectral theory of compact self-adjoint operators too, but we will recall the statement if not the proof 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.

Suggested books : Do Carmo, Riemannian Geometry. Griffiths and Harris, Principles of Algebraic Geometry. Nicolaescu, Lectures on the Geometry of Manifolds. Aubin, Some nonlinear problems in geometry. Evans, Partial differential equations. Gilbarg and Trudinger, Elliptic partial differential equations of the second order. Szekelyhidi, Extremal Kahler metrics. Douglas, R. List of topics time permitting : 1. Suggested books : Rajendra Bhatia, Matrix Analysis , vol. Roger A. Horn and Charles R. Johnson, Matrix analysis , Cambridge University Press, Johnson, Topics in matrix analysis , Cambridge University Press, Semi group theory:Hille-Yosida theorem, Applications to heat, Schroedinger and wave equations.

Suggested books : Evans, L. Pazy, A.

- Course Catalogue: [email protected];
- Georgian Mathematical Journal.
- Department of Mathematics!
- Royal Amber.
- Computing Approximate Greatest Common Right Divisors of Differential Polynomials!
- Course Catalogue.
- Surgeon at Arms.

Prasad, P. Treves, J. Ando, On a pair of commutative contractions , Acta Sci. Szeged 24 88— Das, B. Krishna and Sarkar, Jaydeb, Ando dilations, von Neumann inequality, and distinguished varieties. Bensoussan, J. Jikov, S. Kozlov, and O. Suggested books : Rabinowitz, Minimax methods in critical point theory with applications to differential equations , C. Ghoussoub, N. Struwe, M. Suggested books : Porter, D. Gakhov, F.

Muskhelishvilli, N. Nobe, B. Schwartz, Theorie des Distributions , Hermann, MA Topics around the Grothendieck inequality Top. Suggested books : H. Iwaniec and E. Suggested books : J.

## Introduction to Numerical Methods for Variational Problems

Diamond and J. Other models of random matrices - Wishart and Jacobi ensembles. Fluctuation behaviour of eigenvalues if time permits. Suggested books : Durrett, R. Billingsley, P. Kallenberg, O. Walsh, J. Suggested books : P. Billingsley, Convergence of probability measures. Karatzas and Shreve, Brownian motion and stochastic calculus. Revuz and Yor, Continuous martingales and Brownian motion. Oksendal, Introduction to stochastic differential equations.

Suggested books : Box, G. Jenkins, G. Efron, B. Parker, T. Isoperimetry and processes , Springer-Verlag, Berlin, Michel Ledoux, Isoperimetry and Gaussian analysis , St. Flour lecture notes Suggested books : Amman, M. Brigo, D and Mercurio, F. Hausdorff and Minkowski dimensions. Dimension computation of certain random fractals derived from Brownian motion range, graph and zero set. Random walks and discrete harmonic functions. Brownian motion and harmonic functions. Recurrence and transience.

What sets does Brownian motion hit? Polar sets and Capacity. Brownian motion in the plane : Conformal invariance, Winding number. Gaussian free field : Definition and basic properties. Suggested books : I. Karatzas and S. Suggested books : Rick Durrett, Probability: theory and examples. Olav Kallenberg, Foundations of modern Probability.

Second Edition , Springer-Verlag, MA Quantum Mechanics 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. Suggested books : Srinivas, M. Press Parthasarathy, K. MA Hermitian Analysis Top. MA Control and Homogenization Pre-requisites : Sobolev spaces Elliptic boundary value problems Heat and wave equations Variational formulation and semigroup theory Optimal Control of PDE:Optimal control problems governed by elliptic equations and linear parabolic and hyperbolic equations with distributed and boundary controls, Computational methods.

Suggested books : B. Lee and L. Lions, Controlabilite exact et Stabilisation des systemes distribues, Vol. Bardi, I. Suggested books : Folland, G. Hewitt, E and Ross, K. Gaal, S. MA Optimal Control of Distributed Systems Pre-requisites : Functional analysis PDE and advanced PDE 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.

Lions, Optimal control of systems governed by partial differential equations , Springer Verlag, Lions, Controlabilite exacte, perturbations et stabilisation de systems distribues, Tome 1 and 2 , Research in applied mathematics, Vol. Barbu, Mathematical methods in optimization of differential systems , Mathematics and its applications, Vol. MA Introduction to Complex Dynamics 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.

Suggested books : Krantz, Steven G. Hormander, Lars, An introduction to complex analysis in several variables 3rd ed. Surfaces that locally minimise area in Euclidean space minimal surfaces. Harmonic coordinates in isothermal parameters. Examples of minimal surfaces. The gauss map for minimal surfaces with some examples. Conjugate minimal surfaces. One parameter family of isometric minimal surfaces. If time permits: Surfaces that locally maximise area in Lorenztian space maximal surfaces.

Connection betwen minimal and maximal surfaces and Born Infeld solitions. Manfredo Do Carmo, Differential Geometry of curves and surfaces. Robert Osserman, A survey of minimal surfaces.