2021–22–B

Graphs and sub-graphs, trees, connectivity, Euler tours, Hamilton cycles, matching, vertex and edge colorings, planar graphs, introduction to Ramsey theory, directed graphs, probabilistic methods and linear algebra tools in Graph Theory.

Topological spaces and continuous functions (product topology, quotient topology, metric topology). Connectedness and Compactness. Countabilty Axioms and Separation Axioms (the Urysohn lemma, the Urysohn metrization theorem, Partition of unity). The Tychonoff theorem and the Stone-Cech compactification. Metrization theorems and paracompactness.

• Complex numbers. Analytic functions, Cauchy–Riemann equations.
• Conformal mappings, Mobius transformations.
• Integration. Cauchy Theorem. Cauchy integral formula. Zeroes, poles, Taylor series, Laurent series. Residue calculus.
• The theorems of Weierstrass and of Mittag-Leffler. Entire functions. Normal families.
• Riemann Mapping Theorem. Harmonic functions, Dirichlet problem.

• Fields: basic properties and examples, the characteristic, prime fields
• Polynomials: irreducibility, the Eisenstein criterion, Gauss’s lemma
• Extensions of fields: the tower property, algebraic and transcendental extensions, adjoining an element to a field
• Ruler and compass constructions
• Algebraic closures: existence and uniqueness
• Splitting fields
• Galois extensions: automorphisms, normality, separability, fixed fields, Galois groups, the fundamental theorem of Galois theory.
• Cyclic extensions
• Solving polynomial equations by radicals: the Galois group of a polynomial, the discriminant, the Cardano-Tartaglia method, solvable groups, Galois theorem
• Roots of unity: cyclotomic fields, the cyclotomic polynomials and their irreducibility
• Finite fields: existence and uniqueness, Galois groups over finite fields, primitive elements

Coding Theory investigates error-detection and error-correction. Such errors can occur in various communication channels: satellite communication, cellular telephones, CDs and DVDs, barcode reading at the supermarket, etc. A mathematical analysis of the notions of error detection and correction leads to deep combinatorial problems, which can be sometimes solved using techniques ranging from linear algebra and ring theory to algebraic geometry and number theory. These techniques are in fact used in the above-mentioned communication technologies.

Topics

1. The main problem of Coding Theory
2. Bounds on codes
3. Finite fields
4. Linear codes
5. Perfect codes
6. Cyclic codes
7. Sphere packing
8. Asymptotic bounds

Bibliography:

R. Hill, A First Course in Coding Theory, Clarendon Press, Oxford 1986

Embedded differentiable manifolds with boundary in Euclidean space. The tangent space, normal, vector fields. Orientable manifolds, the outer normal orientation. Smooth partitions of unity. Differential forms on embedded manifolds, the exterior derivative. Integration of differential forms and the generalized Stokes theorem. Classical formulations (gradient, curl and divergence and the theorems of Green, Stokes and Gauss). Closed and exact forms. Conservative vector fields and existence of potentials. Application to exact ordinary differential equations. Introduction to differential geometry: curvature of curves and surfaces in 3 dimensional space, the Gauss map, the Gauss-Bonnet theorem (time permitting).

Ordinary differential equations of first order, existence and uniqueness theorems, linear equations of order $n$ and the Wronskian, vector fields and autonomous equations, systems of linear differential equations, nonlinear systems of differential equations and stability near equilibrium

1. Preliminaries: floating point arithmetic, round-off errors and stability. Matrix norms and the condition number of a matrix.
2. Introduction to numerical solutions for ODE’s:initial value problems, Euler’s method, introduction to multistep methods. Boundary value problems.
3. Numerical solution of linear equations: Gauss elimination with pivoting, LU decomposition. Iterative techniques: Jacobi, Gauss-Seidel, conjugate gradient. Least squares approximation.
4. Numerical methods for finding eigenvalues: Gershgorin circles. The power method. Stability considerations in Gram-Schmidt: Hausholder reflections and Givens rotations. Hessenberg and tridiagonal forms. QR decomposition and the QR algorithm.

1. Introduction: Actions of groups on sets. Induced linear actions. Multilinear algebra.
2. Representations of groups, direct sum. Irreducible representations, semi-simple representations. Schur’s lemma. Irreducible representations of finite abelian groups. Complete reducibility, Machke’s theorem.
3. Equivalent representations. Morphisms between representations. The category of representations of a finite group. A description using the group ring. Multilinear algebra of representations: dual representation, tensor product (inner and outer).
4. Decomposition of the regular representation into irreducible representations. The number of irreducibles is equal to the number of conjugacy classes. Matrix coefficients, characters, orthogonality.
5. Harmonic analysis: Fourier transform on finite groups and the non-commutative Fourier transform.
6. Frobenius divisibility and Burnside $p^aq^b$ theorem.
7. Constructions of representations: induced representations. Frobenius reciprocity. The character of induced representation. Mackey’s formula. Mackey’s method for representations of semi-direct products.
8. Induction functor: as adjoint to restrictions, relation to tensor product. Restriction problems, multiplicity problems, Gelfand pairs and relative representation theory.
9. Examples of representations of specific groups: $SL_2$ over finite fields, Icosahedron group, Symmetric groups.
10. Artin and Brauer Theorems on monomial representations

The aim of the course is to present applications of model theory (a branch of Mathematical Logic) in one or more area of mathematics. The particular direction will be determined by coordination with the students, but might include the following:

• Algebraic theory of differential equations (Galois theory, dimension, classification in dimension 1)
• Model theory of valued fields (imaginaries, integration theory, analytic spaces)
• o-minimality, with applications to arithmetic
• Difference fields and difference equations, applications to dynamics, asymptotic theory of the Frobenius
• Continuous model theory, applications to operator algebras, probability etc. Background from logic and other relevant areas will be covered as necessary.

1. Measure preserving systems for Borel group actions
2. Ergodicity, ergodic decomposition, mixing and weak mixing
3. Minimality and unique ergodicity
4. Mean and pointwise ergodic theorems for a single transformation
5. (*) Joinings
6. (*) Decomposition of a measure relative to a partition and conditional measures
7. (*) Entropy
8. (*) Ergodic theorems for general groups and amenability.

(*) denotes optional topics to be covered according to class level and teacher discretion.

1. Cohomology: definitions, Universal coefficient theorem, Orientation, Poincare duality, cap and cup products, cohomology ring, Kunneth formula
2. Review of (Smooth manifolds, differential forms, orientability, Stokes theorem), degree of the map, Sard theorem, De-Rham cohomology.
3. Isomorphism between De-Rham cohomology and singular cohomology.
4. (If time permits) additional topics according to the instructor preferences

See Course Web Page for details, including updated syllabus and administrative information.