Spring 2016 term
Mar 6-Jul 1, 2016 Exam Period Ends: August 12, 2016
The course is intended for 3rd year undergraduate as well as M.Sc students both in computer science and mathematics. We will touch main topics in the area of discrete geometry. Some of the topics are motivated by the analysis of algorithms in computational geometry, wireless and sensor networks. Some other beautiful and elegant tools are proved to be powerful in seemingly non-related areas such as additive number theory or hard Erdos problems. The course does not require any special background except for basic linear algebra, and a little of probability and combinatorics. During the course many open research problems will be presented.
- Fundamental theorems and basic definitions: Convex sets, convex combinations, separation , Helly’s theorem, fractional Helly, Radon’s theorem, Caratheodory’s theorem, centerpoint theorem. Tverberg’s theorem. Planar graphs. Koebe’s Theorem. A geometric proof of the Lipton-Tarjan separator theorem for planar graphs.
- Geometric graphs: the crossing lemma. Application of crossing lemma to Erdos problems: Incidences between points and lines and points and unit circles. Repeated distance problem, distinct distances problem. Selection lemmas for points inside discs, points inside simplexes. 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. Conflict-free colorings .
- Arrangements : Davenport Schinzel sequences and sub structures in arrangements. Geometric permutations.
- Geometric Ramsey and Turan type theorems: Application of Dilworth theorem, Erdos-Szekeres theorem for convex sets, quasi-planar graphs.
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.
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
- The multiplicative group of
- Quadratic residues
- Continued fractions
- Algebraic numbers and algebraic integers
- Groups, the factor group and the homomorphism theorems, Sylow’s theorems and permutation actions of groups.
- Rings, Integral Domains and Fields. Ideals: maximal and prime. Unique Factorization Domains, Principle Ideal Domains, Euclidean Domains.
- Modules, structure theorems for finitely generated modules over a PID, application to finitely generated abelian groups and to the Jordan Canonical Form.
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.
- 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
- 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 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
- The reflection theorem
- Mostowski collapse theorem
- Absoluteness of formulas
- The constructible universe
- Consistency of the negation of the continuum hypothesis
- Consistency of the negation of the Axiom of choice
This is a first course in modern commutative algebra that provides the background for further study of commutative and homological algebra, algebraic geometry, etc.
- Rings, ideals, and homomorphisms
- Modules, Cayley-Hamilton theorem, and Nakayama’s lemma
- Noetherian rings and modules, Hilbert basis theorem
- Integral extensions, Noether normalization lemma, and Nullstellensatz
- Affine varieties
- Localization of rings and modules
- Primary decomposition theorem
- Discrete valuation rings
- Selected topics
- Miles Reid, Undergraduate Commutative Algebra
- Miles Reid, Undergraduate Algebraic Geometry
- Altman, Kleiman, A Term of Commutative Algebra
Symbolic dynamics is a branch of mathematics that deals with sequences of characters letters or “symbols” form the point of view of dynamical systems. The basic guiding philosophy is that sometimes it is possible to code and understand complicated systems by a sequence of discrete samples. The decimal expansion of real numbers is a simple example of this kind of procedure. Techniques and ideas from symbolic dynamics have found significant applications in data storage and transmission as well as other parts of mathematics. In this course we will introduce basic notions and results in symbolic dynamics, via interesting examples. We will illustrate relations to other fields and relate to the more general frameworks of topological dynamics and ergodic theory. Basic topics to be covered: A brief introduction to topological dynamics Shift spaces and Languages Shifts of finite type and sofic shifts Cellular automata and sliding block codes, endomorphisms and automorphisms of shift spaces. Topological entropy Possible additional topics (subject to time, participants background and participants preferences): Krieger’s topological embedding theorem On the isomorphism problem for shifts of finite type (strong shift equivalence and shift equivalence) Multidimensional shift spaces and shift spaces over countable groups. We will introduce and study the notions of Measure-preserving transformations, Ergodicity, Recurrence The formal prerequisite is basic knowledge of measure theory. The background in other fields (such as functional and harmonic analysis and probability) is not a prerequisite, but will be introduced as needed.
- Commutative algebra via derived categories (regular and CM rings, Grothendieck’s Local Duality, MGM Equivalence, rigid dualizing complexes).
- Geometric derived categories (of sheaves on spaces). Direct and inverse image functors, Grothendieck Duality, Poincaré-Verdier Duality, perverse sheaves).
- Derived categories associated to noncommutative rings (dualizing complexes, tilting complexes and derived Morita theory).
- Derived categories in modern algebraic geometry and modern string theory (a survey).
In a random process, by definition, it is not possible to deterministically predict the next step. However, we will see in this course how to predict rigorously the long term behavior of processes. We will study in this course processes, known as Markov processes, in which the next step depends only on the current position. We will see that these processes are deeply related to electrical networks, and to notions from information theory such as entropy. We will develop techniques of discrete analysis, which are counterparts of classical analysis in the discrete setting. These notions are at the cutting edge of current research methods in these fields
- Courses marked with (*) are required for admission to the M.Sc. program in Mathematics.
- The M.Sc. degree requires the successful completion of at least 2 courses marked (#). See the graduate program for details
- The graduate courses are open to strong undergraduate students who have a grade average of 85 or above and who have obtained permission from the instructors and the head of the teaching committee.
- Please see the detailed undergraduate and graduate programs for the for details on the requirments and possibilities for complete the degree.