2019–20–A

1. Ordered fields: Hilbert’s 17 problem, orderings and preorderings, sums of squares, real closed fields, Artin-Schreier theory
2. Infinite Galois theory: profinite groups, the Galois correspondence in the infinite case
3. Introduction to Galois cohomology

• 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.

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.

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

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.

• 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.

• Fundamental theorems and basic definitions: Convex sets, separation , Helly’s theorem, fractional Helly, Radon’s theorem, Caratheodory’s theorem, centerpoint theorem. Tverberg’s theorem. Planar graphs. Koebe’s Theorem.
• Geometric graphs: the crossing lemma. Application of crossing lemma to Erdos problems: Geometric Incidences, Repeated distance problem, distinct distances problem. Selection lemmas. Counting $k$-sets. An application of incidences to additive number theory.
• Coloring and hiting problems for geometric hypergraphs : $VC$-dimension, Transversals and Epsilon-nets. Weak eps-nets for convex sets. $(p,q)$-Theorem, Conflict-free colorings.
• Arrangements : Davenport Schinzel sequences and sub structures in arrangements.
• Geometric Ramsey and Turan type theorems: Application of Dilworth theorem, Erdos-Szekeres theorem for convex sets, quasi-planar graphs.

Generally speaking, algebraic geometry deals with geometric objects defined by polynomial equations. We will cover topics 1-3 below, and maybe one other topic, depending on the level of the students.

Topics

1. Topics from commutative algebra.
2. Algebraic varieties over an algebraically closed field: definitions and main properties.
3. Curves and their function fields.
4. Intersection Theory in the projective plane.
5. Vector bundles and coherent sheaves.
6. Picard group and projective embeddings.
7. Schemes: definitions, examples and basic properties.

This course concerns the physical notion of phase transition, specifically in the model known as “percolation”.

We will review the main mathematical results regarding percolation and its related counterparts the Ising model, Potts model and Fortuin-Kasteleyn cluster model, starting from works of Ising and Pierles in the beginning of the 20th century and culminating in modern work of Smirnov (for which he was awarded a Fields Medal).

The topics in the course are, time permitting:

1. Percolation on graphs. Definitions and basic properties.
2. Harris’ inequality
3. van den Berg-Kesten inequality, Reimer’s inequality
4. Russo’s formula
5. Burton-Keane Theorem
6. Exponential decay of correlations in sub-critical regime
7. Planar percolation: Russo-Seymour-Welsh theory
8. Planar percolation: the Harris-Kesten theorem.
9. Conformal invariance: Cardy-Smirnov formula on the triangular lattice
10. Percolation on groups
11. Critical percolation on non-amenable groups: BLPS

Course Topics

1. Modules: free modules, exact sequences, tensor products, Hom modules, flatness.
2. Prime ideals and localization: local rings, Nakayama’s Lemma, the spectrum of a ring, dimension and connectedness.
3. Noetherian rings: the Hilbert basis theorem, the Artin-Rees lemma, completion, grading.
4. Dimension theory: the Hilbert nullstellensatz, Noether normalization, transcendence degree.

The course will examine the role of the representation theory of the group G=SL(2) in the mathematics of 20th century. We plan to achieve this by surveying its connections to geometry, algebra, analysis and number theory.

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.