All 2024 Courses

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.

Number Theory studies the structure of the integers and the natural numbers. In addition to classical topics (prime numbers, congruences, quadratic residues, etc.) there is an emphasis on algorithmic questions and in particular on applications to cryptography.

• Divisibility and prime numbers
• Congruences
• The multiplicative group of $\mathbb{Z}/m$
• Quadratic residues
• Continued fractions
• Algebraic numbers and algebraic integers

• 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

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. Affine and projective spaces, affine and projective maps, Segre and Veronese embeddings, Desargues’s Theorem, Pappus’s Theorem, cross-ratio, projective duality
2. Plane curves: rational curves, linear systems of curves, conics and the Butterfly Theorem, Pascal’s Theorem, Chasles’s Theorem, the group structure on a planar cubic, Bezout’s Theorem
3. Affine algebraic varieties: Hilbert’s Basis Theorem, Zariski topology, irreducible components, Hilbert’s Nullstellensatz, the correspondence between the ideals and the algebraic sets, morphisms and rational maps between affine algebraic varieties
4. Projective varieties: graded rings and homogeneous ideals, the projective correspondence, morphisms, blow-ups, birational equivalence and rational varieties, Grassmannians
5. The basics of dimension theory
6. The basics of smoothness
7. Cubic surfaces and 27 lines. If time permits, other topics will be discussed such as abstract algebraic varieties, Chevaley’s Theorem, Riemann-Roch Theorem and its applications.

The purpose of the course is to provide students with the ability to deal with mathematical problems in a variety of subjects by becoming familiar with common strategies for solving mathematical problems. The course requires active participation of the students during class and includes both group and individual work. The meetings will be conducted as a seminar where initially a classical problem and its solution will be presented. The strategy for solving problems arising from the solution will be discussed and then the participants will be challenged to use this strategy with specific examples. In addition, problems/riddles given as weekly homework will be discussed. We will cover a variety of techniques for solving problems: exploiting parity, pigeonhole principle, checking extreme cases, double counting, the method of geometric transformations in dealing with sophisticated geometric problems, methods of Dynamic programming, the principle of induction and Fermat’s descent method for treating Diophantine equations. The method of generating functions.. Probabilistic considerations and their uses.

This course serves as an introductory course to ergodic theory. The objective of the course is to present Furstenberg’s proof of Szemeredi’s theorem, which states that every positive density subset of integers contains arbitrarily long arithmetic progressions. In order to provide a dynamical proof of this result, Furstenberg developed a “Correspondence Principle” and Structure Theorems for measure-preserving transformations. These results are now considered fundamental concepts in ergodic theory.

Topics:

• Some Topological dynamics and van der Waerden’s Theorem
• Measure preserving transformations and Poincare recurrence
• Ergodicity, weak mixing, factors and conditional expectations
• Furstenberg’s Correspondence Principle
• Ergodic proof for Szemeredi’s Theorem

in this course we will build methods from probability theory and apply them to study geometric questions regarding finitely generated groups. we will ultimately aim to provide a proof of Gromov’s celebrated theorem: a finitely generated group is virtually nilpotent if and only if it has polynomial growth. we will also bring up open questions for further research.

Topics:

1. conditional expectation and martingales
2. random walk on groups
3. Cayley graphs
4. entropy
5. harmonic functions
6. unitary actions
7. nilpotent and solvable groups
8. Milnor-Wolf theorem
9. Gromov’s theorem ** time permitting:
10. bounded harmonic functions
11. Choquet-Deny theorem
12. positive harmonic functions

The course is designed for M.Sc. students. The prerequisite is “Introduction to Topology” 201-1-0091. Also recommended is the course “Complex Analysis”. Excellent B.Sc. students can participate upon approval of lecturer.

The aim of the course is to give the students a working knowledge of the tools of algebraic topology. The point of view is that algebraic topology is essential for the deep understanding of many modern areas of mathematics. Many examples shall be provided.

Topics:

1. Review of material from topology and algebra.
2. Categories and functors.
3. Homotopy.
4. Fundamental group.
5. Covering spaces.
6. Brouwer and Jordan Theorems in the plane.
7. Local systems and representations of the fundamental group.
8. Complexes and homology.
9. Singular homology.
10. The theorems of Brouwer, Jordan and Lefchetz.

1. Recalling prior material. Rings (including noncommutative), ideals, modules and bimodules, exact sequences, infinite direct sums and products, tensor products of modules and rings.
2. Categories and functors. Morphisms of functors, equivalences. Linear categories and linear functors. Exactness of functors.
3. Special modules. Projective, injective and flat modules.
4. Morita Theory. Equivalences of module categories realized as tensor products.
5. Complexes of modules. Operations on complexes, homotopies, the long exact cohomology sequence.
6. Resolutions. Projective, injective and flat resolutions – existence and uniqueness.
7. Left and right derived functors. The general theory. Tor and Ext functors.
8. Applications to commutative algebra. Some local and global theorems, involving $Tor$ and $Ext$ functors. Derived completion and torsion functors.
9. Sheaf cohomology. A survey of the role of homological algebra in geometry.
10. Nonabelian cohomology. A survey of classification theorems: Galois cohomology, vector bundles.

Bibliography

1. R. Hartshorne, “Algebraic Geometry”, Springer-Verlag, New-York, 1977.
2. P.J. Hilton and U. Stammbach, “A Course in Homological Algebra”, Springer, 1971.
3. S. Maclane, “Homology”, Springer, 1994.
4. J. Rotman, “An Introduction to Homological Algebra”, Academic Press, 1979.
5. L.R. Rowen, “Ring Theory” (Student Edition), Academic Press, 1991.
6. C. Weibel, “An introduction to homological algebra”, Cambridge Univ. Press, 1994.
7. M. Kashiwara and P. Schapira, Sheaves on Manifolds, Springer, 1990.
8. The Stacks Project, an online reference, J.A. de Jong (Editor). (9) A. Yekutieli, “Derived Categories”, Cambridge Univ. Press, 2019. Free prepublication version. (10) Course notes, to be uploaded every week to the course web page