Credit Hours - 3

There are diverse specialist applied areas in mathematical physics, for example, spin foam models,  which are of current interest and are not yet covered by the above courses. This course aims to provide the opportunity for a student to study such specialist areas subject to availability of expertise.

Credit Hours - 3

Introduction to scalar field theory and its canonical quantization in flat and curved spacetimes. The flat space effects of Casimir and Unruh. Quantum fluctuations of scalar fields and of the metric on curved space-times and application to inflationary cosmology. Hawking radiation. Creation of fermions by rotating charged black holes .

Reading List

• Birrell,N.D. and Davies,P.C.W. (1984), Quantum Fields in Curved Space, Cambridge Uuniversity Press
• Fulling,S.A. (1989), Aspects of Quantum Field Theory in Curve Space-Time, Cambridge University Press
• Liddle,A.R. and Lyth,D.H. (2000), Cosmological Inflation and Large-Scale Structure, Cambridge University Press
• Mukhanov,V. and Winitzki,S. (2007), Introduction to Quantum Effects in Gravity, Cambridge  University Press
• Smoller, J.  and Temple, B. (2012), General Relativistic Self-Similar Waves that Induce an Anomalous Acceleration into the Standard Model of Cosmology, AMS.

Credit Hours - 3

Review of relativistic quantum mechanics and classical field theory. Quantization of free quantum fields (the particle interpretation of field quanta). Canonical quantization of interacting fields (Feynman rules). Application of the formalism of interacting quantum fields to lowest-order quantum electrodynamic processes. The varied landscape of operator theory. Radiative corrections and methods of renormalization. 

 Reading List

• Banks,T. (2008), Modern Quantum Field Theory, A Concise Introduction, Cambridge University Press.
• Dimock,J. (2011), Quantum Mechanics and Quantum Field Theory: A Mathematical Primer, Cambridge University Press.
• (Eds) Gaspar, D. Timotin, D. Vasilescu, F-H. and Zsidó, L. (2015), The Varied Landscape of Operator Theory: Conference Proceedings, Timişoara, July 2-7, 2012, A publication of the Theta Foundation. 
• Osborn, H (2014), Advanced quantum field theory, Lecture notes, Cambridge
• Shifman,M. (2012), Advanced Topics in Quantum Field Theory, Cambridge University Press

Credit Hours - 3

Introduction to the differential geometry of Lorentzian manifolds. The priniciples of general relativity. Einstein metrics and their conformal boundaries.  Causal structure and cosmological singularities. Cosmological space-times with Killing vector fields. Friedmann-Lemaitre cosmologies, scalar, vector and tensor perburbations in the linear and nonlinear regimes. De Sitter space-times and inflationary models.

Reading List

• Dodelson,S. (2003), Modern Cosmology, Academic Press, San Diego.
• Hawking,S. and Ellis,G.F.R. (1973), The Large Scale Structure of Space-Time Cambridge.
• Liddle, A.R. and Lyth,D.H.(2000), Cosmological Inflation and Large-Scale Structure, Cambridge University Press.
• Stewart, J. (1991), Advanced General Relativity, Cambridge.
• Straumann,N. (2004), General Relativity with Applications to Astrophysics, Springer.

Credit Hours - 3

Review of elementary general relativity. Timelike and null geodesic congruences. Hypersurfaces and junction conditions. Lagrangian and Hamiltonian formulations of general relativity. Mass and angular momentum of a gravitating body. The laws of black-hole mechanics and perturbation theory. Advances in string theory. Advances in Lorentzian geometry and global nonlinear stability of Minkowski spacetime. Possibly, assume less decay in the power of $r$ and one less derivative than in the Christodoulou-Klainerman result.

Reading List

• Bieri, L. and Zipser, N. (2009), Extensions of the Stability Theorem of the Minkowski Space in General Relativity, A co-publication of the AMS and International Press of Boston, Inc.
• (Eds) Braham, S.P. Gegenberg, J.D. and McKellar, R.J. (1997), The Sixth Canadian Conference on General Relativity and Relativistic Astrophysics, AMS/FI.
• (Eds) Ghomi, M. Li, J. et al (2013), Geometric Analysis, Mathematical Relativity, and Nonlinear Partial Differential Equations, AMS.
• Pani, P. (2013) Advanced methods in black-hole perturbation theory, International Journal of Modern Physics A, World Scientific Publ. Co.
• Sharpe, E. and Greenspoon, A. (2008), Advances in String Theory: The First Sowers Workshop in Theoretical Physics, AMS/ INP.
• Stewart, J. (1991), Advanced General Relativity, Cambridge.

Credit Hours - 3

There are diverse specialist applied areas in computational mathematics, for example signal and image processing, which are of current interest and are not yet covered by the above courses. This course aims to provide the opportunity for a student to study such specialist areas subject to availability of expertise.

Credit Hours - 3

Discretization methods for partial differential equations, including finite difference, finite volume and finite element methods. Application to elliptic, hyperbolic and parabolic equations. Convergence and stability issues, properties of discrete equations, and treatment of non-linearities. Stiffness matrix assembly and use of sparse matric software. Spectral methods evolution to complex geometries.

Reading List

• Canuto, C.S. Hussaini,M.Y. Quarteroni,A. and  Zang, T.A.(2007), Spectral Methods Evolution to Complex Geometries and Applications to Fluid Dynamics, Springer-Verlag New York.
• Denzler, J. Koch, H. and McCann, R.J. (2014), Higher-Order Time Asymptotics of Fast Diffusion in Euclidean Space: A Dynamical Systems Approach, AMS.
• Evans L.C. (1998), Partial differential equations, Graduate studies in mathematics, vol. 19, American Mathematical Society.
• LeVeque, R.J. (2002), Finite volume methods for hyperbolic problems, Cambridge University Press.
• Tsuzuki, M. (2014), Spectral Means of Central Values of Automorphic L-Functions for GL(2), AMS.

Credit Hours - 3

Topics could include direct and iterative methods for linear systems, eigenvalue decompositions and orthonormal with upper triangular (QR)/singular value decomposition (SVD) factorizations, stability and accuracy of numerical algorithms, sparse and structured matrices, preconditioning, linear algebra software. Hybridization and randomization. Convergence diagnostics and output analysis of the implementing software. 

Reading List

• Avron, H., Ng, E., and Toledo, S. (2009), Using perturbed QR factorizations to solve linear least squares problems, SIAM Journal on Matrix Analysis and Applications, 31, 674-693.
• Bernstein, D. S. (2005), Matrix Mathematics: Theory, Facts, and Formulas with Applications to Linear Systems Theory, Princeton University Press.
• Börm S. (2010), Efficient Numerical Methods for Non-local Operators: H2-Matrix Compression, Algorithms and Analysis, EMS tracts in mathematics, Vol 14, EMS.
• Stewart, G. W. (2001), Matrix algorithms, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA.
• Trefethen L. N. and Bau D. (1997), Numerical Linear Algebra, SIAM.

Credit Hours - 3

Asymptotic methods are usually most powerful precisely when numerical approaches encounter their most serious difficulties, such as in cases of small parameters or phenomena on vastly different scales. Perturbation/asymptotic analysis can then provide accurate information in analytic forms which are very well suited for both understanding and for further analysis. Elements of asymptotic analysis, asymptotic analysis for 2nd order ordinary differential equations. Techniques of perturbation theory such as Poincare-Lindstedt, matched asymptotic expansions and multiple scales. Applications to various areas form an essential aspect of the course.

Reading List

• Denzler, J. Koch, H. and McCann, R.J. (2014), Higher-Order Time Asymptotics of Fast Diffusion in Euclidean Space: A Dynamical Systems Approach, AMS.
• Hunter, J.K. (2004), Asymptotic analysis and singular perturbation theory, University of California at Davis
• Kevorkian, J. and Cole,J.D. (1996), Multiple Scale and Singular Perturbation Methods, Springer-Verlag, New York.
• Paulsen, W. (2013), Asymptotic analysis and perturbation theory, Chapman and Hall/CRC Press.
• Tsuzuki, M. (2014), Spectral Means of Central Values of Automorphic L-Functions for GL(2), AMS.

Credit Hours - 3

There are diverse specialist areas of geometry and of topology  which are of current interest and are not yet covered by the above courses. This course aims to provide the opportunity for a student to study such specialist areas subject to availability of expertise.

Credit Hours - 3

This course aims to develop rigorously and comprehensively the theory of convex sets and functions, in the classical tradition of Fenchel and Rockafellar. Topics would include cutting plane method and simplicial decomposition, proximal methods, bundle methods, incremental methods and subgradient methods. Nonquadratic proximal algorithms, entropy minimization and entropic descent algorithm could be one direction the course takes. 

Reading List

• Bertsekas, D. (2003), Convex analysis and optimization, Athena Scientific, Belmont, Massachusetts. 
• Boyd, S and Vandenberghe L.(2004), Convex Optimization, Cambridge , Cambridge University Press.
• Censor, Y. and Zenios, S. A. (1997), Parallel Optimization: Theory, Algorithms and Applications, New York: Oxford University Press.
• Herman, G. T. (2009), Fundamentals of Computerized Tomography: Image Reconstruction from Projections 2nd edn, New York: Springer.
• Hiriart-Urruty J-B and Lemarechal C. (1996), Convex analysis and minimization algorithms, A series of comprehensive studies in mathematics, Vol 306, Springer-Verlag Berlin.
• Rockafellar, R.T. (1997), Convex analysis, Princeton Landmarks in Mathematics, Princeton University Press

Credit Hours - 3

Point-set topology; smooth manifolds, smooth maps and tangent vectors; the tangent bundle; vector fields, tensor fields and differential forms. Other topics may include: de Rham cohomology; Frobenius Theorem; Riemannian metrics, connections and curvature.

It could be of interest to specialize to hyperbolic manifolds or examine diffeology. Flag varieties: an interplay of geometry, combinatorics, and representation theory.

Reading List

• Bredon, G. (1993), Topology and Geometry, Graduate Text in Mathematics 139, New York, Springer-Verlag New York Inc.
• Iglesias-Zemmour, P. (2013), Diffeology, AMS.
• Hodgson, C.D., Jaco, W.H., Scharlemann, M.G. and Tillman, S (Eds) (2013), Geometry and Topology Down Under, Contemporary Mathematics, Vol. 597, AMS.
• Lakshmibai,V and Brown,J (2009), Flag Varieties: An Interplay of Geometry, Combinatorics, and Representation Theory, Hindustan Book Agency HIN/40.
• Melrose, R., Vasy A. and Wunsch J. (2013), Diffraction of Singularities for the Wave Equation on Manifolds with Corners, Ast\'erisque, Soci\'et\'e Math\'ematique de France.
• Ratcliffe, J.G. (2006), Foundations of hyperbolic manifolds, Graduate Text in Mathematics 149, New York, Springer-Verlag New York Inc.

Credit Hours - 3

In addition to the “canons” of the subject (Ext, Tor, cohomology of groups, and spectral sequences), there will be introductions to several other subjects: sheaves, lim’, local cohomology, hypercohomology, profinite groups, the classifying space of a group, Affine Lie algebras, the Dold-Kan correspondence with simplicial modules, triple cohomology, Hochschild and cyclic homology, and the derived category. The historical connections with topology, regular local rings, and semisimple Lie algebras may also be studied.

Reading List

• Hatcher, A. (2002), Algebraic Topology, Cambridge, Cambridge University Press.
• (Eds) Katzarkov, L. Lupercio,E, and  Turrubiates, F.J. (2014), The Influence of Solomon Lefschetz in Geometry and Topology: 50 Years of Mathematics at CINVESTAV, AMS.
• Lambrechts, P.  and  Volić, I. (2013), Formality of the Little $N$-disks Operad, Memoirs of the AMS, vol 230, AMS.
• May, J.P. (1993), Simplicial Objects in Algebraic Topology, Chicago Lectures in Mathematics. University of Chicago Press, Chicago.
• Sharfarevich I R (Ed) (1991), Algebraic geometry II, Encyclopaedia of Mathematical Sciences, Vol. 35, Springer-Verlag.
• Weibel, C. (1995), An Introduction to Homological Algebra, Cambridge University Press

Credit Hours - 3

This course would be a detailed study of one of the following topics, in particular to address current results and open problems: cellular cohomology of posets with local coefficients; the geometry and topology of 3-manifolds; Khovanov homology of links and 4-manifolds; topology of 4-manifolds; cobordism theory; surgery theory; hyperbolic knot theory. 

Reading List

• Bar Natan D (2005), Khovanov's homology for tangles and cobordisms, Geometry and topology, Vol 9, Issue 3, pp 1443--1499, Mathematical Science Publishers
• Bessières, L., Besson, G., Boileau,M. and Maillot, S. (2010), Geometrisation of 3-Manifolds, EMS Tracts in Mathematics, EMS.
• Everitt, B. and Turner, P. (2014), Cellular cohomology of posets with local coefficients, arXiv:math.AT/1112.3460v4.
• Everitt, B. and Turner, P. (2014), The homotopy theory of Khovanov homology, Algebraic and Geometric Topology, 14:2747-2781.
• Gompf, R.E. and Stipsicz, A.I. (1999), 4-Manifolds and Kirby Calculus, Graduate Studies in Mathematics, Vol 20, AMS.
• Morgan, J (2014), The Geometrization Conjecture, Clay Mathematics Monographs, Vol. 5, a co-publication of the AMS and the Clay Mathematics Institute. 

Credit Hours - 3

There are diverse specialist areas  of differential equations and of dynamical systems which are of current interest and are not yet covered by the above courses. This course aims to provide the opportunity for a student to study such specialist areas subject to availability of expertise. 

Credit Hours - 3

The course will examine the long-time behavior of solutions of degenerate parabolic dissipative equations arising in the study of biological, ecological, and physical problems. Examples include porous media equations, p-Laplacian and doubly nonlinear equations, as well as degenerate diffusion equations with chemotaxis and ODE-PDE coupling systems. The long-time dynamics of various classes of degenerate parabolic equations, both semilinear and quasilinear, are systematically studied in terms of their global and exponential attractors. Applications will direct specific content.

Reading List

• Ammari, H., Capdeboscq, Y. and  Kang, H. (Eds)(2012), Multi-Scale and High-Contrast PDE: From Modelling, to Mathematical Analysis, to Inversion, Contemporary Mathematics, Vol 577, AMS.
• Efendiev, M. (2013), Attractors for Degenerate Parabolic Type Equations, Mathematical Surveys and Monographs, Vol. 192, AMS.
• Giráldez, F. and Herrero, M.A. (Eds) (2009), Mathematics, Developmental Biology and Tumour Growth Contemporary Mathematics, Vol 492, AMS.
• Gumel, A.B. and Lenhart, S. (Eds) (2010), Modeling Paradigms and Analysis of Disease Transmission Models, DIMACS: Series in Discrete Mathematics and Theoretical Computer Science, a co-publication of the AMS and DIMACS.
• Sigmund, K (Ed) (2011), Evolutionary Game Dynamics, Proceedings of Symposia in Applied Mathematics, AMS.

Credit Hours - 3

 In recent years, many new exciting connections have been established between the spectral theory of elliptic operators and the theory of dynamical systems. The subject features a diversity of topics, dynamical systems and their applications in biology, quantum chaos, spectral geometry, semiclassical analysis, number theory and ergodic theory any of which could be developed in this course. 

Reading List

• Jakobson, D., Nonnenmacher, S. and  Polterovich, I (Eds) (2010), Spectrum and Dynamics:  Proceedings of the Workshop Held in Montréal, QC, April 7-11, 2008, CRM Proceedings and Lecture Notes, a co-publication of the Centre de Recherches Mathématiques (CRM) and the     AMS.
• Jordan, D.W. and Smith, P. (2007), Nonlinear Ordinary Differential Equations, An Introduction to Dynamical Systems (4th Edition), Oxford, Oxford University Press. 
• Kolyada, S., Manin, Y. and Ward, T. (Eds) (2005), Algebraic and Topological Dynamics, Contemporary Mathematics, Vol. 385, AMS.
• (Eds) Ruan, S. Wolkowicz, G.S.K. and Wu, J (2003), Dynamical Systems and Their Applications in Biology, AMS/FI.
• Zehnder, E.(2010), Lectures on Dynamical Systems: Hamiltonian Vector Fields and Symplectic Capacities, Berlin, European Mathematical Society.

Credit Hours - 3

There are several directions this course could take. We could study optimal control problems for (time-)delayed stochastic differential equations with jumps. Sufficient and necessary stochastic maximum principles for an optimal control of such systems. The associated adjoint processes satisfy a (time-) advanced backward stochastic differential equation (ABSDE). Or develop a systematic theory of Taylor expansions of evolutionary-type stochastic partial differential equations (SPDEs). Then show how Taylor expansions can be used to derive higher order numerical methods for SPDEs, with a focus on pathwise and strong convergence.

 Reading List 

• Baudoin, F. (2014), Diffusion Processes and Stochastic Calculus, EMS Textbooks in Mathematics.
• Jentzen, A and Kloeden,P.E. (2011), Taylor Approximations of Stochastic Partial Differential Equations, CBMS Lecture Series, SIAM, Philadelphia. 
• Latala, R. Ruci\'nski, Strzelecki, P et al. (Eds) (2013), European Congress of Mathematics: Krak\'ow, July 2-7, 2012, EMS. 
• Lian,Z and  Lu,K. (2010), Lyapunov Exponents and Invariant Manifolds for Random Dynamical Systems in a Banach Space, Memoirs of the AMS, vol. 206, AMS.
• Oksendal, B. Sulem, A. and Zhang, T. (2011), Optimal control of stochastic delay equations and time-advanced backward stochastic differential equations, Stochastic methods and models, No 7518, INRIA

Credit Hours - 3

The main themes are well-posedness of problems, Hilbert space methods, variational principles and integral equation methods. Topics include: first-order nonlinear partial differential equations, quasilinear hyperbolic systems, potential theory, eigenfunctions and eigenvalues, semi-groups, and power series solutions. Or nonlinear elliptic equations. Applications from various areas will be used to motivate and illustrate the theory.

Reading List

• Apushkinskaya D and Nazarov A I (Eds) (2014), Proceedings of the St. Petersburg Mathematical Society, Volume XV: Advances in Mathematical Analysis of Partial Differential Equations, AMS.
• Evans L.C. (1998), Partial differential equations, Graduate studies in mathematics, vol. 19, American Mathematical Society.
• Jost J.(2007), Partial Differential Equations (2nd Edition), New York, Springer. 
• (Eds) Latala, R. Ruciński, A. et al. (2013), European Congress of Mathematics: Kraków, July 2-7, 2012, A publication of the European Mathematical Society.
• Nadirashvili, N., Tkachev,V. and Vlăduţ S. (2014), Nonlinear Elliptic Equations and Nonassociative Algebras, Mathematical Surveys and Monographs, AMS.
• Serrin, J.B.  Mitidieri, E.L and  Rădulescu, V.D. (2013), Recent Trends in Nonlinear Partial Differential Equations I: Evolution Problems, AMS.

Credit Hours - 3

Qualitative theory of systems of ODEs. Topics can include: existence/uniqueness of solutions, comparison principle, iterative techniques, stability and boundedness, Lyapunov method, periodic solutions, Floquet theory and Poincare maps, hyperbolicity, stable, unstable and center manifolds, structural stability and bifurcation. Shape, smoothness and invariant stratification of an attracting set for delayed monotone positive feedback. Applications from various areas will be used to motivate and illustrate the theory.

Reading List

• Ammari,H. Kang, H. and Lee, H. (2009), Layer Potential Techniques in Spectral Analysis AMS.
• Dzhamay A, Maruno K and Pierce V.U. (2013), Algebraic and Geometric Aspects of Integrable Systems and Random Matrices, AMS.
• Hale, J. (2009), Ordinary differential equations, Dover.
• Hartman, P. (2002), Ordinary differential equations, Classics in Applied Mathematics (Book 38), Society for Industrial and Applied Mathematics
• Krisztin, T. Walther, H-O. and Wu, J. (1999), Shape, Smoothness and Invariant Stratification of an Attracting Set for Delayed Monotone Positive Feedback York University - AMS/FI.
• Perko, L. (2001), Differential equations and dynamical systems, Springer-Verlag New York.

Credit Hours - 3

There are diverse specialist areas  of modern algebra and of number theory which are of current interest and are not yet covered by the above courses. This course aims to provide the opportunity for a student to study such specialist areas subject to availability of expertise. 

Credit Hours - 3

Banach algebras, functional calculus, Gelfand transform, Jacobson radical, Banach space and Hilbert space operators, Fredholm alternative, spectral theorem for compact normal operators, ideals in C* -algebras, linear functionals and states, GNS construction, von Neumann algebras, strong/weak operator topologies, Double Commutant theorem, Kaplansky's Density Theorem, spectral theorem for normal operators.

 Reading List 

• Douglas, R. G. (1998), Banach algebra techniques and operator theory , Graduate texts in mathematics 179, Springer-Verlag New York, Inc.
• Gaspar, D. Timotin, D. et al. (Eds) (2015), The Varied Landscape of Operator Theory, Conference Proceedings, Timişoara, July 2-7, 2012  A publication of the Theta Foundation.
• Stratila, S. and  Zsido,L. (1975), Lectures on von Neumann algebras, Editura Academiei, Bucharest.
• Sunder, V.S. (1986), An invitation to von Neumann algebras, Springer, New York.
• Zhu,K. (2007), Operator Theory in Function Spaces, 2nd edition, Mathematical Surveys and Monographs, 138, American Mathematical Society, Providence, RI.   

Credit Hours - 3

Summation methods; analytic theory of the Riemann zeta function; Prime Number Theorem (PNT), applications of the PNT, relations equivalent to the PNT, Weiner-Ikechara Tauberian theorem; primitive roots; quadratic reciprocity; Dirichlet characters and infinitude of primes in arithmetic progressions; specialised topics, for example, the lectures of Iwaniec, H on the Riemann zeta function. 

Reading List

• Cojoaru, A.C and Murty, M.R. (2006), An introduction to sieve methods and their applications, London Mathematical Society.
• Davenport, H. (2000), Multiplicative number theory, Graduate texts in mathematics vol. 74, Springer-Verlag New York.
• Iwaniec, H. (2014), Lectures on the Riemann Zeta Function, Rutgers University - AMS.
• Iwaniec, H. and Kowalski, E. (2004), Analytic number theory, Colloquium Publication 53,  American Mathematical Society.
• Kowalski, E. (2013), Gaps between prime numbers and primes in arithmetic progressions, S\'eminaire BOURBAKI 66\`eme ann\'ee, 2013-14, no.1084.
• Montgomery, H.L. and Vaughan, R.C. (2006), Multiplicative number theory 1: classical theory, Cambridge studies in advanced mathematics 97, Cambridge University Press.

Credit Hours - 3

The framework of  ∞-categories is developed. The collection of all spectra can be organized into an ∞-category, Sp: it is an ∞-categorical counterpart of the ordinary category of abelian groups. The smash product functor on spectra. Symmetric monoidal ∞-category:  the E ∞ -ring of the ∞-category of spectra. Diverse applications abound along with many excellent books and the choice would be guided by the research interests.

Reading List

• Joyal, A. and Tierney, M. (2007), Quasi-categories vs. Segal Spaces. Categories in algebra, geometry and mathematical physics, 277326, Contemp. Math., 431, Amer. Math. Soc., Providence, RI.
• Lurie, J. (2009), On the Classification of Topological Field Theories, Current Developments in Mathematics, International Press. 
• Moerdijk, I (1995), Classifying spaces and classifying topoi, Lecture notes in mathematics 1616, Springer.
• Weibel, C.(1995), An Introduction to Homological Algebra, Cambridge University Press.

Credit Hours - 3

Basic definitions and examples: subrepresentations and irreducible representations, tensor products of representations. Character theory. Representations as modules over the group ring, Artin-Wedderburn structure theorem for semisimple rings. Induced representations, Frobenius reciprocity, Mackey's irreducibility criterion.  Representation theory of matrix Lie groups. Quantum groups and algebraic geometry.

Reading List

• Achar, P.N., Jakelić, D., et al. (Eds), (2010), Recent Advances in Representation Theory, Quantum Groups, Algebraic Geometry, and Related Topics, AMS.
• Adams J.F., (2009), Lectures on Exceptional Lie Groups, (Eds) Z. Mahmoud and M. Mimira, University of Chicago Press.
• Donaldson S.K. (2011), Lectures on Lie Groups and Geometry, Cambridge.
• Fulton, W.  and Harris, J. (1999), Representation Theory, Graduate Texts in Mathematics, Springer New York.
• Ziller, W. (2010), Lie Groups, Representation Theory and Symmetric Spaces, University of Pennslyvania.

Credit Hours - 3

This course introduces some of the more advanced tools for solving combinatorial and graph theoretic problems. The topics covered will depend on the research interest and available expertise but significant emphasis will be on the techniques used as well as the results proved. It could introduce hypergraphs and examine Baranyai theorem, normal hypergraphs and perfect graphs, constructive super polynomial lower bound for Ramsey numbers, Bursuk conjecture. Or it could develop expertise in enumerative combinatorics.

Reading List

• Stanley R.P. (2011), Enumerative Combinatorics, vol. I and II, Cambridge Studies in Advanced Mathematics, Cambridge University Press.
• Wilf, H.S. (2005), Generating functionology, Academic Press Inc.
• Comtet L. (1974), Advanced Combinatorics: The Art of Finite and Infinite Expansions, Reidel Publishing Company, Dordrecht–Holland/Boston–U.S.A.
• Flajolet Ph. and Sedgewick R. (2009), Analytic Combinatorics, Cambridge University Press.
• Riordan J. (1968), Combinatorial Identities, Wiley.

Credit Hours - 3

Numerous specialist areas of analysis of current interest are not yet included in the above, leave alone those areas which may come into existence in the near future. This course aims to provide the opportunity for a student to study such specialist areas subject to availability of expertise in the area.

Credit Hours - 3

This course assumes knowledge of the central ideas of variational problems and seeks to consider an analysis of the second variation and of discontinuous solutions;  a discussion of the general properties of the solutions of Sturm-Liouville systems.

Other typical topics might include: Sobolev functions, distributions, Sobolev spaces, functions of bounded variation, Besov spaces and their use to find the trace of Sobolev spaces and symmetrization. 

Reading List

• Ambrosio, L. et al. (2000), Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs.
• Leoni, G. (2009), A first course in Sobolev Spaces, GTM vol. 105, American Mathematical Society.
• Lieb,E.H. and  Loss, M. (2001), Analysis (2nd Edition), Graduate Studies in Mathematics, Volume 14, American Mathematical Society. 
• Martin, J. (2014), Fractional Sobolev Inequalities: Symmetrization, Isoperimetry and Interpolation, A publication of the Société Mathématique de France.
• Ziemer, W.P. (1989), Weakly differentiable functions: Sobolev spaces and functions of bounded variation, GTM Book 120, Springer.

Credit Hours - 3

This course may take any of several directions from geometric function theory which could include Courant's sewing theorem, uniformization, Riemann mapping theorem, to various analytic topics or to several complex variables which could include, Siegel's modular group, Koecher's principle and Igusa's congruence subgroups.Toeplitz approach to the problems of the uncertainty principle. \newline

Reading List 

• Frietag, E.and Busam, R. (2009), Complex analysis, Universitext, Springer.
• Frietag, E. (2012), Complex analysis 2, Universitext, Springer.
• Grauert, H. et al. (1991), Several complex variables VII: sheaf-theoretical methods in complex analysis, Encylopedia of mathematical sciences Vol 74, Springer. 
• Poltoratski, A. (2015), Toeplitz Approach to Problems of the Uncertainty Principle, Texas A and M University - AMS/CBMS.
• Scheidemann, V. (2005), Introduction to complex analysis in several variables, Birkh\"auser Basel 
• Schlag, W. (2014), A Course in Complex Analysis and Riemann Surfaces, Graduate Studies in Mathematics vol. 154, AMS.
• Taylor, J.L. (2000), Several complex variables with connections to algebraic geometry and Lie groups, GTM, vol. 46, Springer.

Credit Hours - 3

We have at least two broad fields of harmonic analysis: real-variable and abstract harmonic analysis related to representation theory and functional analysis. In the first field topics could include Calder\'on-Zygmund theory, Carleson measures and bounded mean oscillation, Littlewood-Paley theory, the theory of Sobolev spaces and theorems for singular integrals. Commutative and non-commutative harmonic analysis.

Reading List

• Mayeli,A, Iosevich, A., et al (2013), Commutative and Noncommutative Harmonic Analysis and Applications,AMS. 
• Folland, G.B. (1995), A course in abstract harmonic analysis, Studies in advanced mathematics, Chapman and Hall/CRC Press LLC. 
• Stein, E.M. (1993), Harmonic analysis: real-variable methods, orthogonality and oscillatory integrals, Princeton University Press.

Credit Hours - 3

It is possible to find two research mathematicians both of whom call themselves functional analysts who have difficulty understanding the work of the other. There is however a common thread, a linear space with a topology (or several topologies). This course aims to follow the thread through a few of the many possible tours of the subject. Likely topics might include: weak topologies, Banach algebras and spectral theory for operators on a Banach space, $C^*$-algebras, Fredholm theory.

Reading list

• Lin, H. (2014), Locally AH-Algebras, University of Oregon, and The Research Center for Operator Algebras, East China Normal University - American Mathematical Society (AMS).
• Kaliuzhnyi-Verbovetskyi, D.S. and Vinnikov, V. (2014), Foundations of Free Noncommutative Function Theory, Ben Gurion University of the Negev - AMS.
• Knapp, A.W. (2005), Advanced Real Analysis,   Boston, Birkh\"auser.
• Hirsch, F. and Lacombe, G. (1999), Elements of Functional Analysis, Graduate texts in Mathematics 192, New York, Springer-Verlag Inc.
• Berberian, S. (1974), Lectures in functional analysis and operator theory Graduate texts in Mathematics 15, New York, Springer-Verlag Inc. 
• Conway, J.B. (1990), A course in functional analysis,  Graduate texts in Mathematics 96, New York, Springer-Verlag Inc. 

Credit Hours - 3

This course builds on the course MATH 699, Mathematical Research, Writing and Publication as well as MATH 797, Mathematical Problem Solving. In MATH 699 the emphasis was on acquiring and using available software to produce a thesis or research paper in Latex and on the sites available for searching and storing data. It touches on the conventions in mathematical writing and referencing to avoid plagiarism. This course will develop the latter two more fully as well as how to find the “right” journal to submit a research or review paper to for publication. It will also study available ethical codes of practice for mathematicians and ethical conduct of research in mathematics.

Reading List

• Krantz, S.G.(1997), A Primer of Mathematical Writing, Rhode Island, American Mathematical Society 
• Krantz, S.G.(2005), Mathematical Publishing: A Guidebook, Rhode Island, American Mathematical Society

Credit Hours - 3

This course begins by considering a variety of (possibly) elementary but challenging problems in different branches of pure mathematics. Investigation and comparison of different methods of attack, literature searches, partial solutions, solutions and possible generalisations will arise. The objective is to illustrate and gain practical experience of different approaches to problem solving and research. The student will produce a 3 week module as part of a 13 week course at the upper undergraduate level, in mathematical problem solving and will demonstrate teaching and assessment of his/her module. 

Reading List

• Beardon, A.F. (2009), Creative mathematics: a gateway to research, Cambridge University Press.
• Beardon, A.F. (2005), Algebra and Geometry, Cambridge University Press.
• Courant, R., Robbins, H. and Stewart, I. (1996), What is mathematics?, Oxford University Press, USA.
• Mason, J.,Burton, L. and Stacey, K. (2010), Thinking Mathematically, Prentice Hall.
• Schoenfeld, A. (1985), Mathematical Problem Solving, Academic Press Inc (London) Ltd

Credit Hours - 3

In this course candidates attend seminars given by faculty and visiting scholars and present at least two seminars detailing the major findings of their research, which have been accepted for publication in an internationally respected peer reviewed journal.

Credit Hours - 3

In this course candidates give at least 1 seminar each semester to report progress in their own research programme.

Credit Hours - 3

 In this course, the student is expected to participate to an on-going research in the department or in other institutes as detailed in the research programmes for year II section. The student is expected to give two seminars the second of which would normally be towards the end of year II and should be a report on the collaborative research undertaken. It should detail  particular  contributions made by the student.

Credit Hours - 3

In this course, students should attend all seminars presented by faculty and visiting scholars. In addition they would present at least 1 seminar on a current research paper assigned by the supervisory committee.

Credit Hours - 3

During the first semester of the second year, students will be required to give a seminar on the progress of their research. In the second semester and towards the conclusion of the preparation of their thesis, students would present their research findings.

PHD MATHEMATICS & MATHEMATICAL PHYSICS

Components of the programme:

• Core course (s);  MATH 797  Mathematical Problem Solving, MATH 798 Mathematical Research, Writing and Publication II 
• Elective course (s); all other courses are elective; they are shown in the table at (h).
• Mandatory course(s); there are no mandatory courses, these are in the Bachelors programme.
• Research component; 

The major research component is to produce a thesis which exposes an original 

contribution to knowledge. This could be supported by publication(s) in a recognised reputable journal.

• Competence-Based Training(CBT) component PhD Comprehensive Examination and MATH 798    Mathematical Research, Writing and Publication II
• Problem-Based Learning(PBL) component MATH 797  Mathematical Problem Solving
• Practical training, industrial attachment, internship, clinical experience, etc.,

The Year II activities include participating in an ongoing research programme either in the department or at the research centre, AIMS-Ghana or any other institute with which faculty members are doing collaborative research. The aim of this component of the programme is to guide the students in acquiring skills and knowledge relevant to doing research in mathematics. Projects currently include the following:

AIMS-Ghana Research Centre

 The AIMS Research Centre attracts local, regional and international students and researchers.  The Research Chair is 

 held by Dr Wilfred Ndifon who is the IDRC-IMS Career Development Chair of Quantitative Biology. The research area is quantitative assessment of  T-cell diversity in health and disease. The candidate will conduct research under the supervision of Dr Ndifon.

 The AIMS-Ghana Research Centre is envisaged to grow, offering further opportunities to our PhD candidates. 

On the classification of immersed curves which extend to immersed surfaces- MmcIntyre:

In recent work in the department, some progress was made on the geometric interpretation of cancelling a negative group in the Blank word of a curve. A negative group was defined to be a pair of negative letters, yet we were able to construct an example and give a geometric interpretation of cancellation of three consecutive negative letters. This raises the possibility of redefining a negative group and exploring other possibilities for cancellation of subwords in the Blank word. A second line of research would investigate patterns in the Blank word which inhibit extension and a third line of research would be to develop a code which could generate the Blank word of a curve, up to the obvious equivalences induced by choice of starting point and choice of basepoint.

Analytic and Harmonic function spaces and associated operators- B Sehba:

The use of harmonic analysis in the study of functions of one or several complex variables is not a new phenomenon but recent developments in real harmonic analysis and the difficulty of applying complex analysis methods opened the study of connections between the two areas to find a way to deal with some unanswered questions of analytic function spaces. In particular,

a general weak factorisation theorem for Orlicz-type generalisations of Hardy spaces has been established and used to obtain results about Hankel operators. A candidate would work on a remaining open case and further weak factorisation results. A second line of research involves Carleson measures in Békollé-Bonami weights and applications for example to finding conditions that are necessary and sufficient for the Toeplitz product to be bounded.

Noncommutative geometry, quantum groups and quantum gravity- P K Osei and Bianca Dittrich: (Perimeter Institute)/Bernd Schroers Heriot-Watt University

Quantum gravity aims at unifying Einstein's vision of space-time as a dynamical object with the realization that fundamental physics and hence space-time has to be quantum. The first goal is to study analogue spin foam models with q-deformed quantum groups; in particular, to  investigate the linearisation of coarse graining relations around fixed points of quantum group spin net models and analyse stability around these fixed points. A second goal is to construct semiduals of various twistor spaces and to interpret the emerging structures in the context of 4d noncommutative spacetimes. A third direction is to review the modern gauge theory formulation of 4 dimensional gravity and to study the Lie 2-group, a new algebraic structure proposed in the context of quantum gravity. 

Diffeological spaces, category theory, Kacs-Moody algebras, cluster algebras-R A Twum :

The Borel-Weil theorem characterizes all irreducible representations of a connected Lie Group G as certain line bundles of the homogeneous space G/B. Analogues of this theorem exist for  G a Kac-Moody group (possibly infinite dimensional). Work completed includes using enriched category theory to approach the problem. A categorical version of the Borel-Weil theorem with  G a diffeological Lie group attached to a Kac-Moody algebra g and B a Lie subgroup of G is still being sought. In addition recently work has begun on cluster algebras.

Computational methods with applications to real life problems-E K A Schwinger and  Antonella Zanna (Bergen University):

Special areas of application include: biomedical data ( MRI, CT, X-ray, ECG), remote sensing datasets (radar), scheduling problems using computational graph theory, ranking problems using Perron-Frobenius theorem and computational methods for partial differential equations.\newline

The project has four possible directions. A candidate could learn and use graph representation methods in image processing which convert pixels to graph nodes and use similarity for edge representation. A second direction is image segmentation methods, an integral part of image understanding algorithms. A third direction is to use image registration methods to align multiple versions of an area of interest, for example matching different images of an organ taken at different time periods to analyse progress in treatment. A fourth direction is in image denoising methods. Such methods are under continuous development as the method must be specific to the particular problem being considered. It should also be possible for a candidate to work on interdepartmental projects where computational mathematics is required.

Other possibilities in the near future include:

• Biomathematics Research Initiative BRI-UG with Noguchi Memorial Institute for Medical Research and University of Vermont
• Combinatorics with Prof L Shapiro (Howard University)
• Several complex variables with Prof C Lutterodt (Howard University)
• Evolutionary game theory with Prof J Apaloo (St Josephs, Canada. 
• Theory of reductive Lie groups and Lie algebras with Prof Kinvi Kangni (Felix Houphouet-Boigny, Abidjan)
• Constrained convex optimization with Prof A R De Pierro (San Carlos,Brazil.)
• Mathematical modelling of epidemics with Dr Farai Nyabadza (Stellenbosch)
• Modelling of physical phenomena with Prof. Anthony Aidoo (East Connecticut, State)

There is a memorandum of understanding between the University of Ghana and AIMS-Ghana in respect of the AIMS-Ghana MSc in mathematical sciences, which will be extended to arrangements for PhD students in the near future. The research collaborations mentioned above developed through PhD supervision and the proposed collaborations have been initiated through the University of Ghana, diasporan linkage programme which brought the  above mathematicians to the department. The University of Ghana has a memorandum of understanding with Stellenbosch University, which enabled the visit of Farai Nyabadza to the department.

1. Structure of the programme (Semester-by-semester schedule/structure of course, showing the credit value of each course).

Credit Hours - 3

A weekly departmental seminar series is ongoing and graduate studentsare required to attend. They would present a written account of one seminar given during either the first or second semester. In addition, students would prepare and present the relevant introductory theory and motivation for their research proposals.

Credit Hours - 3

The course will provide a bridge between undergraduate abstract algebra and issues addressed in graduate courses in group theory. Sylow’s theorems for finite groups; the theory of groups acting on sets; the theory of group representations. Topological groups and the theory of their representation will provide the main application.

References

• Fraleigh, J.B. (2007), A first course in Abstract Algebra, New York, Addison Wesley.
• Rotman, J.J. (2010), Advanced Modern Algebra, Rhode Island,American Mathematical Society.
• Grillet, P.A. (2007), AbstractAlgebra, New York, Springer-Verlag N.Y. Inc.

Credit Hours - 3

The course willintroduce the essential features of mathematical writing and publication and it will addressresearch techniques. Onlinestorage options, mathematical websites to access research articles, e-books; well-posed questions, aspects of writing mathematics: structure of a thesis,a research paper. Current word-processing for mathematics, bibliography and table of content, diagram packages.

References

• Krantz, S.G.(1997), A Primer of Mathematical Writing, Rhode Island, American Mathematical Society
• Krantz, S.G.(2005), Mathematical Publishing: A Guidebook, Rhode Island, American Mathematical Society

Credit Hours - 3

Students would present their research proposals and preliminary results. After completion of theresearch work and before submission of the writtendissertation, candidates will  be required to give a seminar to present their research findings.

Requirements for graduation:

Provide information on the following requirements for graduation:         

• Course Requirements; 
• The coursework for the MSc programme comprises 3 core courses and 2 to 3 elective coursesin the first semester, 4 to 6 elective coursesin the second semester 
• (ii) The research component begins in the first semesterwith the corecourse MATH 699 Mathematical research, writing and publication, has a core seminar requirement and it is concluded with the submission of a dissertation within 12 months of initial registration.

Credit Hours - 3

This course is an introduction to modern numerical techniques in the study and practice of the numerical solution of differential equations. Numerical solutions for ordinary and partial differential equations, convergence, stability and consistence of schemes. Classification of PDEs, finite difference methods, finite volume methods for conservation laws, finite element methods, implicitschemes and iterative solvers and spectralmethods.

References

• Thomas, J.W. (1995), Numerical Partial Differential Equations: Finite Difference Methods, New York, Springer.
• Morton, K.W. (2005), Numerical Solution of Partial Differential Equations, 2nd Ed, Cambridge, Cambridge University Press
• LeVeque, R.J. (2002), Finite Volume Methods for Hyperbolic Problems, Cambridge, Cambridge University Press.
• William, F.A. (1992), Numerical Solution for Partial Differential Equations, 3rd Ed, Academic Press, San Diego, USA
• Braess, D. (2001), Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, Cambridge, Cambridge UniversityPress.
• Reddy, J.N. (1993), An Introduction to the FiniteElement Method, New York,McGraw- Hill.

Credit Hours - 3

This course gives students a background in the theory and methods of probability and stochastic processes. Probability as a measure, limit theorems, chain rule and change of variables. Expectation, spaces of random variables and conditional expectation. An introduction to the general theory of stochastic processes and applications to financial mathematics.

References

• Billingsley, P.(1995), Probability and Measure (3rd edition), New York, Wiley Series 1995
• Knapp, A.W.(2005), Advanced Real Analysis, Berlin, Birkh¨auser 2005
• Nikeghbali, A. (2005), An essay on the general theory of stochastic processes, Probability Surveys, Vol.3, pp345-412.

Credit Hours - 3

This course is a general introduction to statistical mechanics. Probability calculus and central limit theorem, statistical entropy and information theory, microcanonical, canonical and grandcanonical ensembles, Bose-Einstein and Fermi-Dirac statistics. Free Fermi gas, stability of white dwarfs. Free Bose gas, Bose-Einstein condensation.

References

• Halley, J.W. (2006), Foundations and Uses of Statistical Mechanics, Cambridge, Cambridge University Press
• Roy, B.N. (2002), Fundamentals of Classical and Statistical Thermodynamics, New York, John Wiley and Sons.

Credit Hours - 3

This course is a general introduction to statistical mechanics. Probability calculus and central limit theorem, statistical entropy and information theory, microcanonical, canonical and grandcanonical ensembles, Bose-Einstein and Fermi-Dirac statistics. Free Fermi gas, stability of white dwarfs. Free Bose gas, Bose-Einstein condensation.

References

• Halley, J.W. (2006), Foundations and Uses of Statistical Mechanics, Cambridge, Cambridge University Press
• Roy, B.N. (2002), Fundamentals of Classical and Statistical Thermodynamics, New York, John Wiley and Sons.

Credit Hours - 3

This course will explore the nature of mathematical models in particular as applied to natural systems. The different roles of analytic and numerical models.Methods to study space-time variability, nonlinear behaviour and a multiplicity of interacting processes. Dimensionless governing parameters in ODE/PDE models of natural systems.

References

• Howison, S.D. (2005), Practical Applied Mathematics: Modelling, Analysis, Approximation, Cambridge, Cambridge University Press.
• Jordan, D.W. and Smith, P. (2007), Nonlinear Ordinary Differential Equations, An Introduction to Dynamical Systems (4th Edition), Oxford, Oxford University Press

Credit Hours - 3

This course is to introducevarious methods in the solutionof boundary value problems. Fourier analysis, Bessel’sinequality, Parseval’s relation, Hilbert spaces,compact operators, eigenfunction expansions and Sturm-Liouvilleproblems.

References

• Attonah, H. et al (2005), A Variational Analysis in Sobolev and BV spaces. Appli- cations to PDEs and Optimization, Philadelphia, Society for Industrial and Applied Mathematics.
• Leoni, G. (2009), A first course in Sobolev Spaces, Rhode Island, American Mathematical Society.
• Knapp, A.W. (2005), Advanced Real Analysis, Berlin, Birkh¨auser

Credit Hours - 3

This course is to study Einstein’s theory of gravitation as the curvature of spacetime. Equivalence and general covariance, Einstein gravitational field equations. Solutions with special symmetries. Gravitational collapse, singularity, black holes. Cosmology, isotropic and homogeneous spaces, Maxwell’s equations in curved space-time.

References

• Fecko, M. (2006), Differential Geometry and Lie Groups for Physicists, Cambridge, Cambridge University Press.
• Mukhanov, V. and Winitzki, S. (2009), Introduction to Quantum Effects in Gravity, Cambridge, Cambridge University Press.
• Misner,C.W., Thorne, K.S. and Wheeler, J.A. (1973), Gravitation, London, W.H. Freeman and Co.

Credit Hours - 3

This course is to present modern developments in fundamental quantum mechanics. The action functional, the Lorentz and Poincare groups; local field transformations, the action for scalar, spinor and vector fields. Methods of quantisation, Fock space calculations and the path integral. Renormalisation. Global and local gauge symmetries. the Yang-Mills theory. Gauge theory of graviton.

References

• Ramond,   P. (1990),   Field  Theory,  A Modern  Primer (2nd edition), New York, Addison-Wesley.
• Mukhanov, V. and Winitzki, S.(2009), Introduction to Quantum Effects in Gravity, Cambridge, Cambridge University Press.
• Dunajski, M.(2009), Solitons, Instantons and Twistors, Oxford Graduate Texts in Mathematics 19, OUP, Oxford.

Credit Hours - 3

This course is to study classical electromagnetism in microscopic and macroscopic forms. Electromagnetic fields of and forces between charged particles. Covariant Maxwell’s equations. The electromagnetic field tensor, energy-momentum tensor, Maxwell’s stress tensor. Multi-pole moments, spectral and Fourier resolutions of electromagnetic waves, diffraction and other selected topics.

References

• Greiner, W. (1998), Classical Electrodynamics, New York, Springer.
• Jackson, J.D. (1999), Classical Electrodynamics (3rd edition), New York, J. Wiley and Sons Inc.

Credit Hours - 3

This course is to review time series analysisand signal processing methods used in biomedical applications, withspecial emphasis on magnetic resonance imaging. The processing methods encompass, signal detection, noisereduction and imagetransformation. Due to increasing digitalization, applications other than biomedical could also be addressed.

References

• Gonzalez,R.C. and Woods,R.E. (2009), Digital Image Processing, New York, Pear- son.
• Liang, Z.P. and Lauterbur, P.C. (2000), Principles of MRI: A Signal Processing Per- spective, California, SPIE Press.

Credit Hours - 3

First and secondorder equations, formulation of some of the classical problems of math- ematical physics, Applications of functional analysisto differential equations including distributions, generalized functions, semigroups of operators, the variational method and the Riesz-Schauder theorem.

References

• Evans, C, (2010), Partial Differential Equations (2nd Edition), Rhode Island, American Mathematical Society.
• Jost J. (2007), Partial Differential Equations(2nd Edition), New York, Springer.

Credit Hours - 3

This course is to introduce some methods for computational mathematics. Parallel nu- merical algorithms for: linear algebra (vector and matrix operations), the fast Fourier transform, Poisson solvers,domain decomposition, multigridand particle methods.

References

• Threfethen, L.N. and Bau, D. (1997), Numerical Linear Algebra,Philadelphia, Soci- ety for Industrial and Applied Mathematics.
• Chun, W.J. (2007), Core Python Programming, New York, PearsonEducation Inc. 
• Quarteroni, A. and Saleri, F. (2006), Scientific Computing with Matlab and Octave, New York, Springer.

Credit Hours - 3

Differential equations and dynamical systems, closed orbit structural stability and 2- dimensional flow. Introduction to Chaos, local bifurcations and periodic orbits, averaging and perturbation. Global bifurcations, Lorentzand Hopf bifurcations, Chaos in discrete dynamical systems.

References

• Zehnder, E. (2010), Lectures on Dynamical Systems: Hamiltonian Vector Fields and Symplectic Capacities, Berlin, European Mathematical Society.
• Williamson, R.E. (2001), Differential equations and dynamical systems, New York, McGraw-Hill.

Credit Hours - 3

Diophantine equations, arithmetic functions, quadratic residues and continued fractions, irrationalityand transcendence. Classicalresults include: the Euclidean algorithm, Wil- son’s theorem, the Euler-Fermat theorem, the law of quadratic reciprocity, Dirichlet’s theorem and the prime number theorem.

References

• Pollack,P. (2009), Not always buried deep: A second course in Elementary Number Theory, Providence, American Mathematical Society.
• Kraeft, U. (2000), Diophantine Equations, Berlin , Shaker Verlag 2000
• Kraeft, U. (2003), Arithmetic NumberTheory, Berlin, Shaker Verlag.

Credit Hours - 3

This course is to introducethe idea of group presentations and the fundamental notion of a free group. Free groups, presentations, Schreier’s method, Nielsen’s method, free products, amalgamated free products and HNN extensions. Decision problems and/or other selected topics.

References

• Magnus,W., Karass, A. and Solitar,D. (2005),Combinatorial Group Theory: Presen- tations of Groups in terms of Generators and Relations, New York ,Dover Publica- tions Inc.
• Cohen, D.E. (1989), Combinatorial Group Theory: A Topological Approach, Cam- bridge, Cambridge University Press.
• Rotman, J.J. (2010), Advanced Modern Algebra,Rhode Island, American Mathemat- ical Society.
• Grillet, P.A. (2007), Abstract Algebra, New York, Springer-Verlag N.Y.Inc.

Credit Hours - 3

This course is to introduce the fundamental theoryof Lie groups and Lie algebras. It be- gins with the matrix approachto Lie groups. Finite-dimensional Lie groups and algebras, adjoint mappings, Ado’s theorem, reducibility, semisimple Lie algebras, finite-dimensional irreducible representations of semisimple Lie groups.

References

• Kirillov jr. A. (2008), An introduction to Lie groups and Lie algebras, Cambridge, Cambridge University Press.
• Hall, B.C. (2003), Lie Groups, Lie Algebras and Representations, Graduate texts in Mathematics 222, New York, Springer-Verlag Inc.
• Fecko, M. (2006), Differential Geometry and Lie Groups for Physicists, Cambridge, Cambridge University Press.
• Deng, B. and Du, J. et al (2008), Finite Dimensional Algebras and Quantum Groups, Rhode Island, American Mathematical Society

Credit Hours - 3

This course is to introduce the geometry of convexsets, the analysisof convex functions and their associated inequalities. Convex sets, convex functions, Helly’s theorem, log-convexity, mixed volumes and extremum problems. Selected applications to geometry and to optimization.

References

• Niculescu, C. and PerssonLars-Erik (2006), Convex functions and their applications- a contemporary approach,CMC books in Mathematics, New York, Springer.
• Webster, R. (1994), Convexity, Oxford, Oxford UniversityPress.
• Boyd, S. and Vandenberghe L. (2004), Convex Optimization, Cambridge, Cambridge University Press

Credit Hours - 3

This is a course to introduce the basic ideas of homotopy and homology theory and  to indicatethe associated cohomology theory. Homotopy,  the fundamental group, deck transformations and covering spaces. Van Kampen and excision for homotopy groups. Homology theory, cohomology, products and duality.

References

• Hatcher, A. (2002), Algebraic Topology, Cambridge, Cambridge University Press.
• Bredon, G. (1993), Topology and Geometry, Graduate Text in Mathematics 139, New York, Springer-Verlag New York Inc.
• Rotman, J.J. (1988), An Introduction to Algbraic Topology, Graduate Text in Math- ematics 119, New York, Springer-Verlag New York Inc.

Credit Hours - 3

The course aimsto pursue theconditions under whicha given topological space is metrizable. It examines countability and separation properties, the Urysohn metrization theorem and Tietzeextension. Local finiteness, the Nagata-Smirnov theorem,paracompactness and further metrization theorems and conjectures.

References

• Munkres, J.R. (2000), Topology-2nd edition, Upper saddle river N.J., Prentice Hall.
• Steen, L.A. and Seebach, J.A. (1978), Counterexamples in Topology, New York, Dover

Credit Hours - 3

This course aimsto demonstrate the need for a rigorousdefinition of measurewhich will be somewhat compatible with intuitive ideasof measure. It addresses abstractmeasure, Lebesgue measure, convergence theorems, measurable functions and integration theory. Special integrals and measures. Applications to probability theory and to analysis in Rn.

References

• Bartle, R.G. (1995), The Elements of Integration and Lebesgue Measure, New York, John Wiley and Sons.
• Aliprantis, C.D. and Owen B. (1998), Measure and Integration-Principles of Real Analysis,India , Academic Press.
• Aliprantis, C.D. and Owen B. (1998), Principles of Real Analysis (3rd edition), India, Academic Press Ltd.
• Stein,E. and Shakarchi,R. (2005), Real analysis. Measure theory, integration, and Hilbert  spaces, Princeton Lectures in Analysis, III., New Jersey,Princeton University Press.

Credit Hours - 3

This course aims to study the mathematical formulation of quantum mechanics. Ket and Bra vectors;equations of motion;perturbation theory; collision problems; theory of radiation; relativistic theory of the electron; representation theory; symmetry principles; spin; angular momentum; path integral formulation.

References

• Shankar, R. (1994), Principles of Quantum Mechanics (2nd edition), New York, Plenum Press.
• Griffiths, D.J.(2004), Introduction to Quantum Mechanics, New York, Pearson.

Credit Hours - 3

This course is to introduce both the Lagrangian and the Hamiltonian approach to the study of classical mechanics. Lagrangian mechanics, principle of least action, Euler- Lagrange equation. Generalized momentum and force. Hamiltonian formalism, conservative forces. Poisson bracket, canonical transformations, symmetries, conserved quantities and their consequences.

References

• Kibble, T.W.B. (2004), Classical Mechanics, London, Imperial College Press.
• Abraham, R. and Marsden, J.E. (2008), Foundations of Mechanics, Rhode Island, American Mathematical Society

Credit Hours - 3

The course is to introduce the basic ideasof functional analysis. Basic properties of topological spaces,locally convexspaces and Banachspaces; operators, compactoperators; duality; basic theoremsof functional analysis; spectral theory in Hilbert spaces;integration of vector-valued functions; examples and applications to classical analysis.

References

1. Knapp, A.W. (2005), Advanced Real Analysis, Boston, Birkh¨auser.
2. Hirsch, F. and Lacombe, G. (1999), Elements of Functional Analysis, Graduate texts in Mathematics, New York, Springer-Verlag Inc.
3. Kreyszig, E. (1989), Introductory Functional Analysis with Applications, New York, John Wiley and Sons.

Credit Hours - 3

This course presentsthe analytical justification for methods of solution to ordinary differential equations, commonly introduced in undergraduate courses.Explicit solutions for elementary cases, existence, uniqueness and representation of solutions of ordinary differential equations. A systematic study of linear systems, geometricmethods (phase planes, Lyapunov functions) with applications to physics and biology.

References

1. Schroers, B.J. (2011), Ordinary Differential Equations-A practical guide, AIMS Library Series, Cambridge University Press.
2. Agarwal, R.P., O’Regan, D. (2008), An Introduction to Ordinary Differential Equations, Springer, New York.
3. Williamson, R.E. (2001), Differential equations and dynamical systems, New York McGraw-Hill

Credit Hours - 3

This course seeks to reexamine, with more rigorousanalysis, material which commonly falls into an undergraduate programme in complex analysis. In particular the use of residues in integration. The Riemann mapping theorem and its implications. Montel’s theorem, Picard’s theorem and applications. Hilbert spaces of holomorphic functions and the Bergman kernel.

References

1. Greene, R.E. and Krantz, S.G (2002), Function theory of one Complex Variable-3rd edition, Graduate Studies in Mathematics, Vol 40, Rhode Island, American Mathematical Society.
2. Lang, S. (1993), Complex Analysis, Graduate texts in Mathematics 103, New York, Springer-Verlag Inc.

Wunsch, D.A. (2005), Complex Variables with applications-3rd edition, New York, Addison-Wesley.

Credit Hours - 3

This course presentsthe classical theory of surfacesfollowed by a study of the geometry of manifolds. In particular: tangentspaces and normalspaces, the classical fundamental forms, curvatures andsome intrinsic geometryare examined. Thendifferentiable manifolds, Riemmanian manifolds, the curvaturetensor and spaces of constantcurvature.

References

1. Ku¨hnel, W. (2005), Differential Geometry: Curves-Surfaces-Manifolds, Rhode Island, American Mathematical Society.
2. Conlon, L. (2008),Differentiable Manifolds, Boston, Birkh¨auser.

Credit Hours - 3

This course is to introduce the main ideas in topology, hone the abilityto write mathematical proofs and to examine applications to metric spaces.Topological spaces, closed sets and limits points, connectedness, compactness, separation properties and countability  properties. Metricspaces; continuity, differentiability, convergence. Completemetric spaces and function spaces.

References

1. Davis, S.W. (2005), Topology, New York, McGraw-Hill.

Munkres, J.R. (2000), Topology, 2nd edition, Upper SaddleRiver N.J., PrenticeHall.

Credit Hours - 3

The coursewill provide a bridge betweenundergraduate abstract algebraand issues addressed in graduate courses in group theory. Sylow’s theorems for finite groups; the theory of groups acting on sets; the theory of group representations. Topological groups and the theory of their representation will provide the main application.

References

1. Fraleigh, J.B. (2007), A first course in AbstractAlgebra, New York, Addison Wesley.
2. Rotman, J.J. (2010), Advanced Modern Algebra, Rhode Island,American Mathematical Society.
3. Grillet, P.A. (2007), Abstract Algebra, New York, Springer-Verlag N.Y. Inc.

Credit Hours - 3

The course willintroduce the essential features of mathematical writing and publication and it will addressresearch techniques. Onlinestorage options, mathematical websites to access research articles, e-books; well-posed questions, aspects of writing mathematics: structure of a thesis,a research paper. Current word-processing for mathematics, bibliography and table of content, diagram packages.

References

1. Krantz, S.G.(1997), A Primer of Mathematical Writing, Rhode Island, American Mathematical Society
2. Krantz, S.G.(2005), Mathematical Publishing: A Guidebook, Rhode Island, American Mathematical Society.