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:**

- Real Analysis
- Complex Analysis
- Partial Differential Equations

**Group B:**

- Algebra
- Function Fields
- Projective Geometry
- Finite Geometry
- Commutative Algebra
- Algebraic Curves
- 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:*

- Metric spaces; completeness, compactness and connectedness. Continuous functions. The contraction mapping theorem. Ascoli- Arzela theorem.
- The Lebesgue measure. General measures. Convergence theorems. Decomposition theorems, Hahn decomposition, Radon-Nikodym theorem. Product measure, Fubini and Tonelli's theorems.
- 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:*

- 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.
- The Argument Principle: The index of a closed curve, general form of Cauchy's theorem, Residue theorem, The Argument Principle, Rouche's theorem.
- The Maximum Modulus Principle: The Maximum Modulus Principle, Schwarz Lemma, One-to-one holomorphic mappings of the unit disc onto itself, Mobius transformations.
- Zeros and Poles of Analytic Functions: Runge's theorem, Meromorphic functions, Infinite products, Weierstrass Factorization theorem.
- 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:*

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

- 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.
- 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.
- 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:*

- 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.
- 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:*

- 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.
- 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.
- Classical polar spaces: polarities, classical polar spaces, orthogonal groups, symplectic groups, unitary groups, Witt’s theorem
- 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:*

- projective planes
- affine planes
- mutually orthogonal latin squares
- projective spaces over finite fields
- ovals and ovoids
- arcs and caps
- hyperovals
- blocking sets
- linear sets
- finite classical groups
- finite generalised quadrangles
- non-desarguesian projective planes
- spreads
- translation planes
- finite classical polar spaces
- theory of linear codes
- maximum distance separable codes
- maximum rank metric codes
- diagram geometry
- Tits buildings
- links with quantum coding theory
- equiangular lines
- 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:*

- Ring and ideals
- Modules
- Local properties of rings
- Chain conditions
- Noetherian rings
- Artin rings
- Graded rings
- Tensor and Hom Functors
- Primary decompositions
- Constructions of Free resolutions
- Cohen-Macaulay rings
- Regular sequences and depth
- 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:*

- Affine and Projective Space; Affine and Projective Algebraic Sets ,The Ideal of a Set of Points
- The Hilbert Basis Theorem
- Irreducible Components of an Algebraic Set, Algebraic Subsets of the Plane
- Hilbert’s Nullstellensatz
- Projective and Affine Varieties, Coordinate Rings, Polynomial Maps, Coordinate Changes, Rational Functions and Local Rings
- Local Properties of Plane Curves (Multiple Points and Tangent Lines, Multiplicities and Local Rings, Intersection Numbers) and Projective Plane Curves
- 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:*

- Number Fields and their Extensions
- The Field of Algebraic Numbers
- Norms, Traces and Characteristic Polynomials
- Embeddings of Number Fields into the complex field
- The Primitive Element Theorem
- Rings of Integers in Number Fields
- Discriminants and Bases
- Integral Bases
- Unique Factorization of Ideals
- The Dedekind-Kummer Theorem
- Ideal Class Group and Class Number
- 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