2020–21–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.

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.

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

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.

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.

In the 1980s A. Grothendieck suggest a project for developing a tame topology that will not suffer from the many counter-examples and pathologies known in classical topology. Nowadays many view the notion of o-minimality as successful fulfillment of this program: in o-minimal fields all (unary) functions are piecewise differentiable (and therefore infinitely differentiable at almost every point); unary functions are piecewise monotone, connectedness is the same as path connectedness and the axiom of choice holds for definable sets. In the o-minimal setting most of the classical differential calculus can be developed, and so are large portion of the theory of Lie groups, algebraic topology and much more. O-minimality plays a key role in real geometry and in recent years had a crucial role in important breakthroughs in Diophantine geometry and in Hodge theory.

In the course we will define o-minimality and develop its basic theory. We will show that real closed fields are o-minimal and discuss – time permitting – some applications.

The goal of the workshop is to examine and provide complimentary material for the course “Geometric Calculus 1” 201.1.1031 (as well as the course “Introduction to Analysis” 201.1.1051). The workshop is given in parallel with Geometric Calculus 1, and the workshop content follows the courses. Part of the main goals of the workshop, especially due to the Corona pandemic, is to improve the student’s teamwork skills. During the workshop, the students will work in small groups and will practice their “mathematical conversation” skills: how to think together, how to find the essence of an idea and how to present a mathematical idea to others.

• 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. Background material in functional analysis and Hilbert space theory; basics of Banach algebras, and Gelfand theory.
2. Basic results: commutative $C^*$-algebras, positivity, ideals and quotients, states and representations, the GNS constuction, the algebra of compact operators.
3. A discussion of some basic examples in the theory (e.g. AF algebras, Toeplitz and Cuntz algebras, irrational rotation algebras, as time permits). Introduction to K-theory.

Further topics as time permits.

1. Riemann surfaces
2. Holomorphic functions of several variables
3. Isolated singularities of holomorphic functions
4. Fundamentals of differential topology
5. Topology of singularities.

See course web page for details

Please email me if you want to attend (even just for the first one or two lectures, to “taste” the course). I will send you the zoom link.

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.