2021–22–A

• Groups as symmetries. Examples: cyclic, dihedral, symmetric and matrix groups.
• Homomorphism. Subgroups and normal subgroups. Quotient groups. Lagrange’s theorem. The isomorphism theorems. Direct products of groups.
• Actions of groups on sets. Cayley’s theorem.
• Group automorphisms.
• Sylow’s theorems. Application: classification of groups of small order.
• Composition series and Jordan–Hoelder theorem. Solvable groups.
• Classification of finite abelian groups, finitely-generated abelian groups.
• Symmetric group and alternating group. The alternating group is simple.
• Rings, maximal and prime ideals, integral domain, quotient ring. Homomorphism theorems.
• Multilinear algebra: Quotient spaces. Tensor products of vector spaces. Action of $S_n$ on tensor powers. Exterior and symmetric algebras. Multilinear forms and determinant.
• Optional topics: group of symmetries of platonic solids, free groups, semidirect products, representation theory of finite groups.

Algebras and sigma-algebras of subsets, the extension theorem and construction of Lebesgue’s measure on the line, general measure spaces, measurable functions and their distribution functions, integration theory, convergence theorems (Egorov’s, relations between convergence in measure and a.e. convergence), the spaces $L_1$ and $L_2$ and their completeness, signed measures, the Radon-Nikodym theorem, measures in product spaces and Fubini’s theorem.

• An axiom system for predicate calculus and the completeness theorem.
• Introduction to model theory: The compactness Theorem, Skolem–Löwenheim Theorems, elementary substructures.
• Decidability and undecidability of theories, Gödel first Incompleteness Theorem.

An introduction to the basic notions of probability theory:

sample spaces limits of events conditional probability independent events sigma algebras, continuous spaces, Lebesgue measure random variables and distributions independence expectation variance and covariance convergence of random variables: almost-sure, in Lp, in probability law of large numbers convergence in law central limit theorem

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

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.

Metric and normed spaces. Equivalence of norms in finite dimensional spaces, the Heine-Borel theorem. Convergence of sequences and series of functions: pointwise, uniform, in other norms. Term-by-term differentiation and integration of series of functions, application to power series. Completeness: completeness of the space of continuous functions on a closed interval and a compact metric space. The Weierstrass $M$-test. The Baire category theorem and applications, bounded linear functionals and the Banach-Steinhaus theorem. Compactness in function spaces and the Arzela-Ascoli theorem. Introduction to Fourier series: Cesaro means, convolutions and Fejer’s theorem. The Weierstrass approximation theorem. $L^2$ convergence. Pointwise convergence, the Dirichlet kernel and Dini’s criterion.

Open, closed and compact sets in Euclidean space. Matrix norms and equivalence of norms. Limits and continuity in several variables. Curves and path connectedness. Partial and directional derivatives, the gradient and differentiability. The implicit, open and inverse function theorems. Largange multipliers. Optimization: the Hessian matrix and critical points. Multivariable Riemann integration: Fubini’s theorem and the change of variables formula.

Basic concepts: Group actions, Cayley graphs and Schreier graphs, The word metric. Quasi isometries. The Milnor-Svarc lemma. Free groups and trees: Group presentations. The groupAut(T),elliptic and hyperbolic elements. The boundary of the tree. Covering theory of graphs and the Nielsen Shcreier theorem. The ping-pong lemma. Free and amalgamated products, HNN extensions. The group PSL_2 (Z)=Z/2Z*Z/3Z and its action on the Farey tree. Some Hyperbolic geometry: Poincare models, their boundaries. Isometry groupsPSL_2 (R),PSL_2 (C), elliptic, hyperbolic, parabolic and loxodromic isometries and the trace. Free subgroups using the ping-pong lemma. Poincare lemma and surface groups as crystallographic hyperbolic groups. The fundamental domain of SL_2 (Z)and the space of lattices. The Farey tessellation and continued fractions. Hyperbolic groups. Gromov hyperbolic spaces and their boundaries. Hyperbolic groups, elliptic and hyperbolic elements. Quasi convex subgroups. Existence of free subgroups using ping-pong. Small cancellation groups. Solvability of the word problem and finite presentability. Existence of many quotients.

• Topological manifolds. The fundamental group and covering spaces. Applications.
• Singular homology and applications.
• Smooth manifolds. Differential forms and Stokes’ theorem, definition of de-Rham cohomology.
• Additional topics as time permits.

1. Doubly periodic functions. Lattices, complex tori. The field of elliptic functions.
2. Riemann surfaces: definitions, maps, the genus, Riemann–Hurwitz formula.
3. Differential forms on a Riemann surface. The Abel–Jacobi map.
4. Local study of holomorphic functions. Points of Riemann surfaces as valuations on the field of meromorphic functions. Classifications of the absolute values on the field of rational numbers. The field of p-adic numbers. The product formula.
5. Algebraic curves over a field. Algebraic curves over C and relation to Riemann surfaces.
6. Rational points. Arithmetic of curves according to the genus. The case of conics (genus 0). Hasse principle. Finite generation of the rational points on elliptic curves (genus 1): the case of Fermat quartic.
7. Modular surfaces and modular forms. Analytic construction of rational points on elliptic curves.

1. Expander graphs and their applications — Equivalent definitions, Mixing Lemma, Alon–Boppana Theorem, Applications, Constructions
2. Kazhdan Property (T) - group representations (short intro.), Cayley and Schrier graphs, Property (T) — definition and properties, examples, construction of expander graphs via property (T)
3. Additional topics (as time allows)

Banach spaces and Hilbert spaces. Basic properties of Hilbert spaces. Topological vector spaces. Banach-Steinhaus theorem; open mapping theorem and closed graph theorem. Hahn-Banach theorem. Duality. Measures on locally compact spaces; the dual of $C(X)$. Weak and weak-$*$ topologies; Banach-Alaoglu theorem. Convexity and the Krein-Milman theorem. The Stone-Weierstrass theorem. Compact operators on Hilbert space. Introduction to Banach algebras and Gelfand theory. Additional topics as time permits.

Course Topics: (as much as time permits)

1. Review of prior material. On rings, ideals and modules (including noncommutative rings).

2. Categories and functors. Emphasis on linear categories. (This topic will be introduced gradually, as we go along.)

3. Universal constructions. Free modules, products, direct sums, polynomial rings.

4. Tensor products. Definition, construction and properties.

5. Exactness. Exact sequences and functors.

6. Special modules. Projective, injective and flat modules.

7. Complexes of modules. Operations on complexes, homotopies, the long exact cohomology sequence.

8. Resolutions. Projective, flat and injective resolutions.

9. Left and right derived functors. Applications to commutative algebra.

10. Further applications of derived functors. Classification problems, extensions.

11. Morita Theory.

(Some of the material might move to the subsequent course “Commutative Algebra”)

For an updated syllabus and course requirements see the course web page