2020–21–B

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

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.

An introduction to applications of algebra and number theory in the field of cryptography. In particular, the use of elliptic curves in cryptography is studied in great detail.

• Introduction to cryptography and in particular to public key systems, RSA, Diffie-Hellman, ElGamal.
• Finite filelds, construction of all finite fields, efficient arithmetic in finite fields.
• Elliptic curves, the group law of an elliptic curve, methods for counting the number of points of an elliptic curves over a finite field: Baby-step giant step, Schoof’s method.
• Construction of elliptic curves based cryptographic systems.
• Methods for prime decomposition, the elliptic curves method, the quadratic sieve method.
• Safety of public key cryptographic methods.

• 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

1. Rings and ideals (revisited and expanded).
2. Modules, exact sequences, tensor products.
3. Noetherian rings and modules over them.
4. Hilbert’s basis theorem.
5. Finitely generated modules over PID.
6. Hilbert’s Nullstellensatz.
7. Affine varieties.
8. Prime ideals and localization. Primary decomposition.
9. Discrete valuation rings.

1. The Fourier transform: convolutions, the inversion formula, Plancherel’s theorem, Hermite functions, tempered distributions. The Poisson summation formula. The Fourier transform in R^n.
2. The Laplace transform. Connections with convolutions and the Fourier transform. Laguerre polynomials. Applications to ODE’s. Uniqueness, Lerch’s theorem.
3. Classification of the second order PDE: elliptic, hyperbolic and parabolic equations, examples of Laplace, Wave and Heat equations.
4. Elliptic equations: Laplace and Poisson equations, Dirichlet and Neumann boundary value problems, Poisson kernel, Green’s functions, properties of harmonic functions, Maximum principle
5. Analytical methods for resolving partial differential equations: Sturm-Liouville problem and the method of separation of variables for bounded domains, applications for Laplace, Wave and Heat equations including non-homogenous problems. Applications of Fourier and Laplace transforms for resolving problems in unbounded domains.

Bibliography

1. Stein E. and Shakarchi R., Fourier analysis, Princeton University Press, 2003.
2. Korner T.W., Fourier analysis, Cambridge University Press, 1988.
3. Katznelson Y., An Introduction to Harmonic Analysis, Dover publications. 4. John, Partial differential equations, Reprint of the fourth edition. Applied Mathematical Sciences, 1. Springer-Verlag, New York, 1991.
4. Evans Lawrence C. Partial Differential Equations, Second Edition.
5. Gilbarg D.; Trudinger N. S. Elliptic partial differential equations of second order, Reprint of the 1998 edition. Classics in Mathematics. Springer-Ver lag, Berlin, 2001.
6. Zauderer E. Partial differential equations of applied mathematics, Second edition. Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1989. xvi+891 pp. ISBN: 0-471-61298-7.

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.

The aim of the course is to expose students to key events in the history of mathematics throughout history from the point of view of modern mathematics and, if possible, to link these events to the content studied within the framework of the degree in mathematics. The study will include getting to know the names and histories of major mathematicians throughout history and discussing their contributions to the development of the various branches of mathematics as we know them today. Alongside this there will be a discussion of the development of ideas and concepts in mathematics over the generations to the present day.

1. Basic Algebraic Structures: rings, modules, algebras, the center, idempotents, group rings

2. Division Rings: the Hamiltonian quaternions, generalized quaternion algebras, division algebras over $\mathbb{F}_q$, $\mathbb{C}$, $\mathbb{R}$, $\mathbb{Q}$ (theorems of Frobenius and Wedderburn), cyclic algebras, the Brauer–Cartan–Hua theorem

3. Simplicity and semi-simplicity: simplicity of algebraic structures, semi-simple modules, semi-simple rings, Maschke’s theorem

4. The Wedderburn–Artin Theory: homomorphisms and direct sums, Schur’s lemma, the Wedderburn–Artin structure theorem, Artinian rings

5. Introduction to Group Representations: representations and characters, applications of the Wedderburn–Artin theory, orthogonality relations, dimensions of irreducible representations, Burnside’s theorem

6. Tensor Products: tensor products of modules and algebras, scalar extensions, the Schur index, simplicity and center of tensor products, the Brauer group, the Skolem–Noether theorem, the double centralizer theorem, maximal fields in algebras, reduced norm and trace, crossed products

The course will present the fundamentals of Shelah’s pcf theory and some of its many applications in cardinal arithmetic, infinite combinatorics, general topology and Boolean algebra.

Pcf theory deals with the spectrum of possible cofinalities of reduced products of small sets of regular cardinals. The pcf theorem asserts the existence of a sequence of generators for the set of possible cofinalities.

The most well known application of the theory are Shelah’s bound on the powers of strong limit singular cardinals, most notably on the power of the first singular cardinal, which actually follows from the more informative bound on the covering number of countable sets. These results will be presented in full. Other applications which will be given are the construction of topological spaces, embeddability of Boolean algebras, extensions of measures and bounds on coloring numbers of graphs.

The general question that leads the course is: what can we deduce about a group when studying random walks on it.

Stationary dynamics is a branch of Ergodic theory that focuses on measurable group actions arising from random walks. The main object studied is the Furstenberg-Poisson boundary.

Applications of the theory can be found in rigidity theory, and recently connections with operator theory have been established

Topics:

• Brief introduction to Ergodic theory: Borel spaces, factors, compact models. Probability measure preserving actions and measure class preserving actions. Stationary measures.
• Random Walks: Markov chains, Martingale convergence theorem, Random walks on groups, Furstenberg-Poisson boundary. Choquet-Deny theorem, amenable groups. Entropy. Realization of the Furstenberg Poisson boundary.
• Applications to Rigidity: Margulis’ Normal Subgroup Theorem, and Bader-Shalom’s theorem (IRS rigidity).

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