Credit Hours - 3

This course introduces undergraduate students to the qualitative theory of Ordinary Differential Equations. We will use the Picard-Lindelöf Theorem to analyze whether an ODE or a system of ODEs has a solution and the behavior of the solution as the parameter is varied (bifurcation). We will especially consider autonomous linear and nonlinear systems and investigate the stability of the solutions that result. We will also introduce the concept of a Lyapunov function. Other topics might include partial differential equations, the method of characteristics and classification.

References

• Argawal, R., & O'Regan, D. (2008). An Introduction to Ordinary Differential Equations. Universitext.
• Hirsch, S., & Devaney, (2004). Differential Equations, Dynamical Systems and An Introduction to Chaos. Elsevier.
• Kelley, W., & Peterson, A. (2014). The theory of differential equations: classical and qualitative. Pearson New Jersey
• O'Neil, P. V. (2008). Beginning partial differential equations. Wiley New York.
• Schroers, B. (2011). Ordinary Differential Equations. AIMS Library Series, Cambridge University Press.

Credit Hours - 3

The objective of this course is to introduce the theory of Fourier series and Fourier transforms on the real line.  Topics include: convolutions, summability kernels, convergence of Cesaro means. Mean-square convergence, pointwise convergence.

Fourier transform on the real line, inversion formula, Plancherel formula, Weierstrass approximation theorem. Applications to partial differential equations, Poisson summation formula. The Heisenberg uncertainty principle.

Reading List:

• Katznelson, Y. (2002). An introduction to harmonic analysis, available at www.mat.uniroma2.it/~picard/SMC/..../Katznelson/Katznelson.pdf
• Pinkus, A. & Zafrany, S. (1997). Fourier Series and Integral Transforms. Cambridge.
• Sneddon I. N. (2010). Fourier Transforms. Dover Books on Mathematics
• Stein, E.M., & Shakarchi, R. (2013). Fourier analysis, an introduction. Princeton.
• Weaver, H. J. (1989). Theory of discrete and continuous Fourier analysis.   JohnWiley and Sons, NewYork.

Credit Hours - 3

The detail of this course may be informed by the student choice(s) of project topic and could include: (i) modelling of biological systems using partial differential equations. Derivation of conservation equations. Different models for movement. Connection between diffusion and probability.

(ii) Linear reaction-diffusion equations and fundamental solutions. Speed of a wave of invasion. Non-linear reaction-diffusion equations. Travelling wave solutions for monostable equations and bistable equations.

(iii) Systems of reaction-diffusion equations and travelling wave solutions. Pattern formations. Pattern formations in chemotaxis equations. (iv) Mathematical modelling of infectious diseases

Derivation of a simple SIR model and travelling wave solutions.

Reading List:

• Britton, N. F. (2003). Essential Mathematical Biology. Springer.
• De Vries,  G.,  Hillen, T.,   Lewis, M.,  Muller, J., & Schonfisch, B. (2006). A Course in Mathematical Biology. Quantitative Modeling SIAM.
• Edelstein-Keshet, L. (1987). Mathematical Biology I, An introduction. Springer.
• Segel, L.A., & Edelstein-Keshet, (2013).  A primer on mathematical models in biology. SIAM.
• Murray, J.D. (2007). Mathematical Biology I, An introduction. Springer

Credit Hours - 3

This course will cover the basic theory of Lie groups and Lie algebras. Topics may include: topological groups and Haar measure, vector fields and groups of linear transformations. The exponential map. Linear groups and their Lie algebras. Structure of semi-simple Lie algebras, Cartan and Iwasawa decompositions. Connectedness. Closed subgroups. The classical groups. Manifolds, homogeneous spaces and Lie groups. Integration and representations.
 

Reading List:

• Carter, R., Segal, G., & Macdonald, I. (1995). Lectures on Lie Groups and Lie Algebras. Cambridge
• Hall, B. (2009). Lie groups, Lie algebras and Representations-an elementary introduction. Springer
• Krillov, A. (2006 ). An introduction to Lie groups and Lie algebras Cambridge. University Press
• Sagel, A. A., & Walde, R. E. (2013). An introduction to Lie groups and Lie algebras.  Academic Press Inc.
• Serre, J. P. (2005). Lie Algebras and Lie Groups: 1964 Lectures given at Harvard University. Springer

Credit Hours - 3

By employing the mathematics of sets, mappings and relations, we aim to develop an ability to think relativistically, exploring the relationship between space and time. Topics include: Galilean relativity, postulates of special relativity; Lorentz transformations. Lorentz-Fitzgerald contraction, time dilation. 4-vectors, relativistic mechanics, kinematics and force, conservation laws; decay of particles; collision problems, covariant formulation of electrodynamics.

Reading List:

• Dray, T. (2012).  The geometry of special relativity. CRC Press
• Frankel, T. (2013). The Geometry of Physics (3rd Edition). Cambridge
• Matsko, V. J., & Noll, W. (1993).  Mathematical Structures of Special Relativity. available at http://repository.cmu.edu/cgi/viewcontent.cgi Article=1014&content=math
• Susskind, L., & Friedman, A. (2017). Special Relativity and Classical Field Theory: The Theoretical Minimum. Basic Books.
• Woodhouse, N. M. J. (2003). Special Relativity. Springer Undergraduate Mathematics Series

Credit Hours - 3

In this course we shall study the mathematical objects called modules. The use of modules was pioneered by one of the most prominent mathematicians, Emmy Noether (a German), who led the way in demonstrating the power and elegant of this structure. Topics include: modules, submodules, homomorphism of modules. Quotient modules, free (finitely generated) modules. Exact sequences of modules. Direct sum and product of modules.Chain conditions, Noetherian and Artinian modules. Projective and injective modules. Tensor product, categories and functors. Hom and duality of modules. 

Reading List:

• Blyth, T. S. (2015).  Module theory; an approach to linear algebra. Oxford University Press
• Dummit, D. S., & Foote, R. M. (2003). Abstract Algebra (3rd Edition). Wiley.
• Lam, T.Y. (1998). Lectures on Modules and Rings. Springer GTM.
• Rotman, J. (2008). An Introdution to Homological Algebra (2nd Edition). Springer Universitext
• Wisbauer, R. (2011). Foundations of Module and Ring Theory. Available at ht

Credit Hours - 3

This course aims to provide an introduction to Differentiable Manifolds and the tools for performing calculus on these objects; tangent vectors and differential forms. We will see how concepts like the derivative in Rn is extended to a smooth n-dimensional manifold. Topics include: manifold, submanifold, differentiability of maps between manifolds, the tangent space, the tangent bundle and the tangent functor. Vector bundle. The exterior algebra, the notion of a differentiable form on a manifold, singular n-chains and integration of a form over a chain. Partition of unity. Application to Stokes' theorem.

Reading List:

• Jones, A., Gray, A., & Hutton, R. (1987).  Manifolds and Mechanics Aust. Math Soc. Lecture series (2), Cambridge University Press.
• Lee, J. M. (2012). Introdution to Smooth Manifolds. Springer GTM
• Spivak, M. (2015).  Calculus on Manifolds. Addison-Wesley
• Spivak, M. (1999). A Comprehensive Introduction to Differential Geometry (3rd Edition). Vol 1, , Publish or Perish
• Tu, L.W.  (2011). An Introduction to Manifolds. Universitext

Credit Hours - 3

Algebra of sets, measurable sets and functions, measures and their construction (in particular Lebesgue measure), measure spaces. Integration, convergence theorems (Fatou’s Lemma, Monotone Convergence Theorem, Lebesgue Dominated Convergence Theorem). Lebesgue spaces, elementary inequalities, modes of convergence. Product measures and Fubini’s theorem.
Generalisation of the Riemann (R) integral (eg Kurzweil-Henstock (KH) integral). Lebesgue (L) integral. Relationship between the KH-integrable, L-integrable and R-integrable functions.   

Reading List:

• Bass, R. Real Analysis for Graduate Students, (3rd Edition).
• Cannarsa, P. & D'Aprile, T. D. (2007). Lecture notes on measure and functional analysis. https://www.mat.uniroma2.it/~cannarsa/cam_0607.pdf     
• Rudin, W. (1966). Real and ComplexAnalysis available  at             http://ruangbacafmipa.staff.ub.ac.id/files/2012/02/Real-and-Complex-Analysis-by-Walter Rudin.pdf
• Stein, E., & Shakarchi, R. (2005). Real analysis:measure theory, integration and Hilbert spaces Princeton Lectures in Analysis.
• Yee, L. P. & Vyborny, R. (2000). The integral: an easy approach after Kurzweil and Henstock. Aust Math Soc Lecture series 14. Cambridge University Press

Credit Hours - 3

This is a second course in the development of the mathematical foundations for the application of the electromagnetic model to various problems. Magnetostatics: steady currents, heating affect and magnetic field, magnetic vector potential, magnetic properties of matter, dipoles, induced magnetism, permanent magnetism. Time-varying fields: electromagnetic induction. Differential form of Faraday’s law, energy in electromagnetic fields. Maxwell’s equations and their consequences Poynting vector; electromagnetic potentials formation of electrodynamics.

Reading List:

• Chirgwin, B. H., Plumpton, C., & Kilmister, C.W. (1972).  Elementary Electromagnetic Theory. Vols. II.  and III.     Pergamon Press.           
• Fleisch, D. (2008). A Student's guide to Electromagnetic Theory. Cambridge.
• Griffiths, D.J. (2014). Introduction to Electrodynamics. Pearson Educational
• Jackson, J.D. (1962). Classical Electrodynamics. Wiley and Sons
• Reitz, J. R., Milford. F. J., & Christy, R.W. (1979). Foundations of Electromagnetic Theory (3rd  Edition). Narosa Pub. House
  
   

Credit Hours - 3

In this course we focus on 3 types of biological phenomena to be modelled, namely single species population dynamics, interacting species and molecular dynamics. In single species population dynamics we will use difference equations: graphical analysis, fixed points and linear stability analysis. First order systems of ordinary differential equations: logistic equation, steady states, linearisation, and stability. And we will examine applications to 

harvesting and fisheries. For interacting species we examine systems of difference equations (host-parasitoid systems)and systems of ordinary differential equation (predator-prey and competition models) Finally, we will consider biochemical kinetics: Michaelis-Menten kinetics and metabolic pathways: activation and inhibition.

Reading List:

• Britton, N. F. (2003). Essential Mathematical Biology. Springer.
• De Vries, G., Hillen, T., Lewis, M.,  Muller, J., & Schonfisch, B. (2006) A Course in Mathematical Biology. Quantitative Modeling SIAM.
• Edelstein-Keshet,  L. (1987). Mathematical Models in Biology. Birkhauser.
• Murray, J. D. (2007). Mathematical Biology I, An Introduction. Springer.
• Segel, L.A., & Edelstein-Keshet, (2013).  A primer on mathematical models in biology. SIAM.

Credit Hours - 3

This course aims to use the methods of mathematical analysis and apply them to a special kind of vector space – Function spaces. The course begins with finite dimensional normed vector spaces and treats the following topics: Equivalent norms. Banach spaces. Infinite-dimensional normed vector spaces--Hamel and Schauder bases; separability. Compact linear operators on a Banach space. Complementary subspaces and the open-mapping theorem. Closed Graph theorem. Hilbert spaces. Special subspaces of and and the dual space. The completion of a normed vector space. Reflexive Banach spaces 

Reading List.

• Eidelman, Y., Milman, V., & Tsolomitis, A. (2004). Functional Analysis: An Introduction. AMS
• Kreyszig, E. (2013).  Introductory Functional Analysis. Wiley and Sons
• Lax, P. (2004). Functional Analysis. Wiley-Interscience.
• Rudin, W. (1966). Real and ComplexAnalysis available  at             http://ruangbacafmipa.staff.ub.ac.id/files/2012/02/Real-and-Complex-Analysis-by-Walter Rudin.pdf
• Royden, H. (1988).  Real analysis. Prentice Hall

Credit Hours - 3

This course looks at methods of discretizing and solving differential equations. It begins with the solution of initial value problems for ordinary differential equations. We start with the Euler methods and systematically develop high order solutions for solving problems. The course then develops the concept of finite differences to solve boundary value problems.  In addition, we look at the problem of discretizing partial differential equations in space and time both implicitly and explicitly.

Reading List:

• Burden, R. L., &  Faires, J. D. (2008). Numerical analysis. Cengage Learning (9th Edition).
• Chapra, S. (2008). Applied numerical methods with Matlab for engineers and scientists  (3rd Edition). McGraw Hill.
• Epperson, J. F. (2013). An introduction to numerical methods and analysis (2nd Edition). Wiley.
• Gautschi, W. (2012). Numerical Analysis (2nd Edition). Birkhauser.
• Matthews, J.H., & Fink, K.D. (2014). Numerical methods using Matlab (5th Edition). Pearson. 
• Sauer, T. (2006).  Numerical Analysis. Pearson.

Credit Hours - 3

This course introduces students to the equations govening extremely small particles and their interactions. It introduces new mathematics to model the behavior of such objects. Topics include: The principle of least action, Hamilton's equation, Poisson brackets. Liouville's equation. Canonical transformations. Symmetry and conservation laws. Postulates of quantum mechanics, the wave formalism. Dynamical variables. The Schrodinger equation in one-dimension; free particles in a box, single step and square well potentials. Orbital angular momentum. The 3-dimensional Schrodinger equation; motion in a central force field, the 3-d square well potential, the hydrogenic atom. Heisenberg's equation of motion, harmonic oscillator and angular momentum. 

Reading List:

• Jackson, J. D., (2006).  Mathematics for quantum mechanics- Dover Publications 
• von Neumann, (1955), The Mathematical Foundations of Quantum Mechanics. Princeton Landmarks in Mathematics.
• Shankar, R., (2011). Principles of quantum mechanics. Plenum Press
  Susskind, L., & Friedman, A. (2012). Quantum Mechanics- the theoretical minimum. Basic Books.
• Woit, P. (2016).  Quantum Theory, Groups and Representations:  An Introduction available at http://www.math.columbia.edu/~woit/QM/qmbook.pdf

Credit Hours - 3

The famous problems of squaring the circle, doubling the cube and trisecting an angle captured the imagination of both professional and amateur mathematicians for over two thousand years. Despite the enormous effort and ingenious attempts by these men and women, the problems would not yield to purely geometrical methods. It was only the development of abstract algebra in the nineteenth century which enabled mathematicians to arrive at the surprising conclusion that these problems are impossible.

This course aims to introduce students to the idea of a field in algebra, and investigates properties of algebraic fields, of finite and zero characteristic.

Reading List:

• Eisenbud, D. (1999). Commutative Algebra with a view towards Algebraic Geometry. Springer GTM.
• Fraleigh, J. B. (2013). A First Course in Abstract Algebra (8thEdition). Addison Wesley.
• Gallian, J. A. (2013).  Contemporary Abstract Algebra (8th Edition). Brooks/Cole
• Jones, A., Morris, S., & Pearson, K.R. (1991). Abstract algebra and famous impossibilities. Universitext, Springer.
• Rotman, J. (2006). A First Course in Abstract Algebra with Applications (3rd Edition). Pearson

Credit Hours - 3

The modern approach to differential geometry uses the language of manifolds. This provides a theory and a variable free notation which frees us from always having to consider the coordinate system. We want to be able to deal with the elements of calculus both invariantly (i.e. independently of the local coordinates) and intrinsically (i.e. independently of the way a geometric object is embedded in Euclidean space). But to appreciate the great contribution to differential geometry made by the theory of manifolds, we will first study classical differential geometry and then a little of the modern approach.

Reading List: 

• Carmo, M. (2016). Differential Geometry of Curves and Surfaces (2nd Edition). Dover. 
• Kuhnel, W. (2006). Differential Geometry -Curves SurfacesManifolds. AMS.
• Millman, R. S., & Parker, G. D. (2007). Elements of Differential Geometry. Pearson
• O'Neill, B. (2012). Elementary Differential Geometry. Academic Press
• Willmore, T. J. (2007). Introduction to Differential Geometry. Oxford University Press

Credit Hours - 3

This is a second course in group theory Topics covered will include: finite groups, Sylow theorems and simple groups. Composition series. We state and prove the Zassenhaus Lemma, the Schreier theorem and the Jordan-Hölder theorem. Direct and semi-direct products. Abelian groups, torsion, torsion-free and mixed abelian groups. Finitely generated group and subgroups. P-groups, nilpotent groups and solvable groups. 

Reading List:

• Dummit, D. S. & Foote, R. M. (2003). Abstract Algebra (3rd Edition). Wiley
• Fraleign, J. B. (2013). A First Course in Abstract Algebra (8th Edition). Addison Wesley.
• Gallian, J. A. (2013).  Contemporary Abstract Algebra (8th Edition). Brooks/Cole
• Judson, T. (2015). Abstract Algebra: Theory and Applications. Open Source available http://abstract.ups.edu/index.html 
• Pinter, C. C. (2010).  A Book of Abstract Algebra (2nd Edition). (Dover Books on Mathematics)

Credit Hours - 3

The objective of this course is to introduce students to complex numbers and functions of a complex variable. We introduce the notions of differentiability (and analyticity) and integrability for a function defined on the complex plane. We also look at ways in which one can integrate complex-valued functions. Elementary topology of the complex plane. Complex functions and mappings. The derivative. Differentiability and analyticity. Harmonic functions. Integrals. Maximum modulus, Cauchy-Gorsat, Cauchy theorems. Applications. Taylor and Laurent series, zeros and poles of a complex function. Residue theorem and consequences. Conformal mapping, analytic continuation.

Reading List:

• Alfors, L.V. (1979). Complex Analysis. McGraw-Hill.
• Saff, E. B. & Snider, A. B. (2013). Fundamentals of Complex Analysis. Pearson Educational
• Stewart, I. N., & Tall, D.O. (2011). Complex Analysis. Cambridge University Press.
• Wunsch, A.D. (2005). Complex Variables with Applications. Pearson Educational
• Zill, D. G., & Shanaban, (2003). A First Course in Complex Analysis. Jones and Bartlett

Credit Hours - 3

Here we think of differentiation as a process of approximating the function f near a, by a linear map. This linear map is called the Fr ́echet derivative of f at a. The main aim of this course is to understand two of the most important theorems for modern analysis: the

Inverse Map Theorem and the Implicit Function Theorem. Other ideas include:

linear and affine maps between normed vector spaces. Limits, continuity, tangency of maps and the derivative as a linear map. Component-wise differentiation, partial derivatives, the Jacobian as the matrix of the linear map. Generalized mean value theorem. 

Reading List:

• Loomis, L., & Sternberg. S. (1990). Advanced calculus. Jones and Bartlett [also at http://www.math.harvard.edu/~shlomo/docs/Advanced_Calculus.pdf]
• McIntyre, M. (2011).   Advanced calculus. Departmental lecture notes
• Rudin, W. (2009). Principles of  Mathematical Analysis available  https://notendur.hi.is/vae11/%C3%9Eekking/principles_of_mathematical_analysis_walter_rudin.pdf
• Spivak, M. (2008). Calculus (4th Edition). Publish or Perish
• Taylor, A. E. & Mann, W. R. (2013).  Advanced Calculus. Wiley