Graduate

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Students should complete the written part consisting of three subject exams during the first five semesters of their Ph.D. program.

The exam subjects are:

Group A:

1. Real Analysis
2. Complex Analysis
3. Partial Differential Equations

Group B:

1. Algebra
2. Function Fields
3. Projective Geometry
4. Finite Geometry
5. Commutative Algebra
6. Algebraic Curves
7. Algebraic Number Theory

The student is required to take a written exam on Real Analysis from Group A and Algebra from Group B. S/he decides on another exam subject from the above list, One of the exam subjects may be from another University Program; in this case the approval of the advisor and the Program coordinator are required.

Exam Subjects

A1) Real Analysis

Topics:

1. Metric spaces; completeness, compactness and connectedness. Continuous functions. The contraction mapping theorem. Ascoli- Arzela theorem.
2. The Lebesgue measure. General measures. Convergence theorems. Decomposition theorems, Hahn decomposition, Radon-Nikodym theorem. Product measure, Fubini and Tonelli's theorems.
3. Normed spaces. Open mapping, Closed graph theorems, Hahn Banach theorem. Uniform Boundedness principle, weak topologies. Linear Operators. Hilbert spaces.

Sources

• Classical Analysis. J. Marsden, M. Hoffman, Freeman.( 1)
• Real Analysis. W. Rudin, (1, 2 and 3)
• Real Analysis. H. L. Royden (1, 2 and 3)
• Introductory Real Analysis. A. N. Kolmogorov, S. V. Fomin (Dover Books) (1, 2 and 3)
• Real Analysis. G. Folland (1, 2 and 3)

A2) Complex Analysis

Topics:

1. Elementary Properties of Analytic Functions: Power series expansions, Complex line integrals, Complex differentiation, Cauchy-Riemann equations, Cauchy's theorem and Integral Formula, Open mapping theorem, Classification of isolated singularities, Laurent expansions, Calculus of residues.
2. The Argument Principle: The index of a closed curve, general form of Cauchy's theorem, Residue theorem, The Argument Principle, Rouche's theorem.
3. The Maximum Modulus Principle: The Maximum Modulus Principle, Schwarz Lemma, One-to-one holomorphic mappings of the unit disc onto itself, Mobius transformations.
4. Zeros and Poles of Analytic Functions: Runge's theorem, Meromorphic functions, Infinite products, Weierstrass Factorization theorem.
5. Analytic Continuation: Analytic continuation along a path, Monodromy theorem

Sources:

• L. V. Ahlfors, Complex Analysis, McGraw-Hill Inc., 1966.
• J. B. Conway, Functions of One Complex Variable, Springer - Verlag, 1978.
• W. Rudin, Real and Complex Analysis, McGraw-Hill Inc., 1966.

A3) Partial Differential Equations

Topics:

1. First order PDE's, characteristics. The Cauchy-Kowalevski theorem. Classification.
2. Hyperbolic equations (the wave equation);solution formulas, characteristics, Cauchy and initial/boundary value problems, energy method.
3. Elliptic equations (Laplace equation); maximum principles, fundamental solutions, Green's function, Poisson's formula, solution of the Dirichlet's problem.
4. Parabolic equations (the heat equation); fundamental solution, maximum principle, energy methods.

Sources:

• Partial Differential Equations: F. John
• Partial Differential Equations: L. Evans
• Partial Differential Equations: J. Wloka

B1) Algebra

Topics:

1. Groups: Groups, subgroups, normal subgroups, cosets, quotient groups, Lagrange’s theorem, cyclic groups, homomorphisms, isomorphism theorems, symmetric, alternating and dihedral groups, direct products, free abelian groups, finitely generated abelian groups, action of a group on a set, Sylow theorems.
2. Rings: Rings, subrings, homomorphisms, ideals, prime and maximal ideals, quotient rings, isomorphism theorems for rings, direct products and Chinese remainder theorem, ring of quotients and localization, factorization in commutative rings, unique factorization domains, Euclidean domains, polynomial rings, factorization in polynomial rings.
3. Fields: Field extensions: algebraic and transcendental extensions, simple extensions and their characterization, Galois extensions and the fundamental theorem of Galois theory, splitting fields, algebraic closure, separability, normality, fundamental theorem of Galois theory, structure of finite fields, cyclic extensions, cyclotomic extensions.

Sources:

• Algebra: T. W. Hungerford
• Algebra: S. Lang
• Topics in Algebra: I. N. Herstein
• Abstract Algebra: D. S. Dummit, R. M. Foote

B2) Function Fields

Topics:

1. Foundations: Algebraic function fields of one variable, places, valuation ring of a place, discrete valuations, the rational function field and its places, weak approximation theorem, divisors, genus of a function field, canonical divisors, Riemann-Roch theorem, strong approximation theorem, Weierstrass gap theorem, Clifford’s theorem.
2. Extensions of function Fields: Algebraic extensions of function fields, ramification index, relative degree, subrings of function fields, local integral bases, Kummer’s theorem, Hurwitz genus formula, the different and Dedekind’s different theorem, constant field extensions, Galois extensions: Kummer and Artin-Schreier extensions, function fields over finite fields, Hasse-Weil Theorem. 

Sources:

• Algebraic Function Fields and Codes: H. Stichtenoth
• Rational Points on Curves over Finite Fields: H. Niederreiter, C. Xing
• Algebraic Curves over a Finite Field: J. Hirschfeld, G. Korshmaros, F. Torres.  

B3) Projective Geometry

Topics:

1. Projective spaces over fields: homogeneous coordinates, frames, Desargues and Pappus, affine spaces, incidence structures, the hyperplane at infinity, collineations, correlations, polarities, principle of duality, projective groups, perspectivities, projections and quotients, collineations on projective lines, cross ratio.
2. Projective algebraic varieties: algebraic varieties, dimension and degree, quadrics, reguli and spreads, cubic surfaces, Plücker and Klein, hermitian varieties, Veronese varieties, Segre varieties, Grassmann varieties.
3. Classical polar spaces: polarities, classical polar spaces, orthogonal groups, symplectic groups, unitary groups, Witt’s theorem
4. Axiomatic geometry: incidence geometry, projective spaces, projective planes, coordinatisation, translation planes, polar spaces, generalised polygons, Tits buildings.

Sources:

• Coxeter, H.S.M. Projective Geomety (1987)
• Casse, R. Projective Geometry, An Introduction. (2006)
• Pierre Samuel, P. Projective Geometry (1988)
• Hughes and Piper. Projective Planes (1973)

B4) Finite Geometry

Topics:

1. projective planes
2. affine planes
3. mutually orthogonal latin squares
4. projective spaces over finite fields
5. ovals and ovoids
6. arcs and caps
7. hyperovals
8. blocking sets
9. linear sets
10. finite classical groups
11. finite generalised quadrangles
12. non-desarguesian projective planes
13. spreads
14. translation planes
15. finite classical polar spaces
16. theory of linear codes
17. maximum distance separable codes
18. maximum rank metric codes
19. diagram geometry
20. Tits buildings
21. links with quantum coding theory
22. equiangular lines
23. mutually unbiassed bases.

Sources:

• Ball, S. Finite Geometry and Combinatorial Applications (2015)
• Dembowski, P. Finite Geometries (1997)
• Hirschfeld, J.W.P. and Thas, J.A. General Galois Geometries (2016)

B5) Commutative Algebra

Topics:

1. Ring and ideals
2. Modules
3. Local properties of rings
4. Chain conditions
5. Noetherian rings
6. Artin rings
7. Graded rings
8. Tensor and Hom Functors
9. Primary decompositions
10. Constructions of Free resolutions
11. Cohen-Macaulay rings
12. Regular sequences and depth
13. Properties of monomial and binomial ideals in a polynomial ring

Sources:

• Introduction to Commutative Algebra (M. F. Atiyah, I.G. Macdonald), Cohen-Macaulay rings (W. Bruns, J. Herzog)

B6) Algebraic Curves

Topics:

1. Affine and Projective Space; Affine and Projective Algebraic Sets ,The Ideal of a Set of Points
2. The Hilbert Basis Theorem
3. Irreducible Components of an Algebraic Set, Algebraic Subsets of the Plane
4. Hilbert’s Nullstellensatz
5. Projective and Affine Varieties, Coordinate Rings, Polynomial Maps, Coordinate Changes, Rational Functions and Local Rings
6. Local Properties of Plane Curves (Multiple Points and Tangent Lines, Multiplicities and Local Rings, Intersection Numbers) and Projective Plane Curves
7. Bézout’s Theorem

Sources:

• Algebraic curves: an introduction to algebraic geometry Volume 30 of Mathematics lecture note series, Issue 30 of Math Lecture Notes Series, by William Fulton and Richard Weiss
• Plane Algebraic Curves, Volume 15 of American Indian Studies, Issue 15 of Student mathematical library, American Mathematical Society, by Gerd Fischer
• Introduction to Algebraic Curves, Volume 76 of Translations of Mathematical Monographs, Amer. Mathematical Society, 1989, by Phillip A. Griffiths
• A Guide to Plane Algebraic Curves, Volume 46 of Dolciani Mathematical Expositionsz, Dolciani Mathematical Expositions, 2011, by Keith Kendig

B7) Algebraic Number Theory

Topics:

1. Number Fields and their Extensions
2. The Field of Algebraic Numbers
3. Norms, Traces and Characteristic Polynomials
4. Embeddings of Number Fields into the complex field
5. The Primitive Element Theorem
6. Rings of Integers in Number Fields
7. Discriminants and Bases
8. Integral Bases
9. Unique Factorization of Ideals
10. The Dedekind-Kummer Theorem
11. Ideal Class Group and Class Number
12. Groups of Units and Dirichlet's Theorem

Sources:

• Algebraic Number Theory; 1st edition; by Frazer Jarvis; Springer; 2014 ; ISBN: 978-3-319-07544-0.
• Number Fields, 2nd edition; by D. A. Marcus; Springer; 2018; ISBN: 978-3-319-90232-6

Common Outcomes of Our Graduate Programs at SU:

1. Develop the ability to use critical, analytical, and reflective thinking and reasoning

2. Reflect on social and ethical responsibilities in his/her professional life.

3. Gain experience and confidence in the dissemination of project/research outputs

4. Work responsibly and creatively as an individual or as a member or leader of a team and in multidisciplinary environments.

5. Communicate effectively by oral, written, graphical and technological means and have competency in English.

6. Independently reach and acquire information, and develop appreciation of the need for continuously learning and updating.

Mathematics Graduate Program Specific Outcomes:

7. Develop abstract mathematical thinking and mathematical intuition.

8. Demonstrate a broad understanding of several areas of advanced mathematics and of their interrelations.

9. Have knowledge of the fundamental and advanced concepts, principles and techniques from a range of topics.

10. The ability to tackle complex problems, reveal structures and clarify problems, discover suitable analytical and/or numerical methods and interpret solutions.

11. Analyze problems of the area of specialization, plan strategies for their solution, and apply notions and methods of abstract and/or applied mathematics to solve them.

1. Develop and deepen the current and advanced knowledge in the field with original thought and/or research and come up with innovative definitions based on Master's degree qualifications

2. Conceive the interdisciplinary interaction which the field is related with ; come up with original solutions by using knowledge requiring proficiency on analysis, synthesis and assessment of new and complex ideas.

3. Evaluate and use new information within the field in a systematic approach.

4. Develop an innovative knowledge, method, design and/or practice or adapt an already known knowledge, method, design and/or practice to another field; research, conceive, design, adapt and implement an original subject.

5. Critical analysis, synthesis and evaluation of new and complex ideas.

6. Gain advanced level skills in the use of research methods in the field of study.

7. Contribute the progression in the field by producing an innovative idea, skill, design and/or practice or by adapting an already known idea, skill, design, and/or practice to a different field independently.

8. Broaden the borders of the knowledge in the field by producing or interpreting an original work or publishing at least one scientific paper in the field in national and/or international refereed journals.

9. Demonstrate leadership in contexts requiring innovative and interdisciplinary problem solving.

10. Develop new ideas and methods in the field by using high level mental processes such as creative and critical thinking, problem solving and decision making.

11. Investigate and improve social connections and their conducting norms and manage the actions to change them when necessary.

12. Defend original views when exchanging ideas in the field with professionals and communicate effectively by showing competence in the field.

13. Ability to communicate and discuss orally, in written and visually with peers by using a foreign language at least at a level of European Language Portfolio C1 General Level.

14. Contribute to the transition of the community to an information society and its sustainability process by introducing scientific, technological, social or cultural improvements.

15. Demonstrate functional interaction by using strategic decision making processes in solving problems encountered in the field.

16. Contribute to the solution finding process regarding social, scientific, cultural and ethical problems in the field and support the development of these values.