Sztochasztika

1. Combinatorial Number Theory
Applications of the Ramsey theory in Number Theory, the theorems of Schur and Van der Waerden, the Hales-Jewett theorem. Szemerédi's theorems about arithmetic progressions. The Sieve methods and its applications. The Brun sieve, The Larger sieve. Schnirelmann density, the set of primes is a basis. Kneser’s theorem, Mann’s theorem. Primitive sequences. Additive and multiplicative Sidon sequences. Applications of algebraic tools in Combinatorial Number Theory. Applications of the polynomial method and the Combinatorial Nullstellensatz. Erdős-Ginzburg-Ziv theorem and its generalizations.
Recommended books:
A. Sárközy, C. Pomerance: Combinatorial Number Theory, In: Handbook of Combinatorics I., 20th Chapter; MIT Press, 1995.
A. Geroldinger, I. Ruzsa: Combinatorial Number Theory and Additive Group Theory; Advanced Courses in Mathematics-CRM Barcelona, 2009.
R.L. Graham, B.L. Rothschild, J.H. Spencer: Ramsey-Theory; 2nd ed., Wiley & Sons, 1990.
H. Halberstam, H.E. Richert: Sieve methods; Dover, 2011.
H. Halberstam, K.F. Roth: Sequences; Springer, 1983.
T. Tao, V.H. Vu: Additive Combinatorics; Cambridge University Press, 2010.

2. Additive Number Theory
The structures of sumsets and difference sets. Ruzsa distance, additive energy. The theorem of Plünnecke and its applications. Balog-Szemerédi-Gowers theorem. Freiman homomorphism, the theorems of Freiman and their applications. Erdős-Fuchs theorem. The Hardy-Littlewood method and its applications. Waring problem, Goldbach's conjecture, Vinogradov’s theorem, Roth’s theorem.
Recommended books: A. Geroldinger, I. Ruzsa: Combinatorial Number Theory and Additive Group Theory; Advanced Courses in Mathematics-CRM Barcelona, 2009.
H. Halberstam, K.F. Roth: Sequences; Springer, 1983.
M.B. Nathanson: Additive Number Theory I.: The Classical Bases; Springer, 1996.
M.B. Nathanson: Additive Number Theory II.: Inverse Problems and the Geometry of Sumsets; Springer, 1996. T. Tao, V.H. Vu: Additive Combinatorics; Cambridge University Press, 2010.
R.C. Vaughan: The Hardy-Littlewood Method; Cambridge University Press, 1997.

3. Analytic Number Theory
Arithmetic functions. Modern prime number theory, the Prime Number Theorem. Primes is arithmetic progressions. Addditive and multiplicative characters, Dirichlet characters, Dirichlet L-functions. The Riemann zeta function, Riemann hypothesis. Exponential sums, the method of Weyl and Van der Corput. Kloostermann sums. Probabilistic number theory. Turán-Kubilius inequality, Erdős-Kac theorem.
Recommended books:
H. Davenport: Multiplicative Number Theory; Springer, 2000.
A. Ivic: The Riemann Zeta-Function; Dover, 2003.
A.A. Karatsuba: Basic Analytic Number Theory; Springer, 1993.
E. Kowalski, H. Iwaniec: Analytic Number Theory; American Math. Soc., 2004.
H.L. Montgomery, R.C. Vaughan: Multiplicative Number Theory I.: Classical Theory; Cambridge University Press, 2007.
S.J. Patterson: An Introduction to the Theory of the Riemann Zeta-Function; Cambridge University Press, 1988. G. Tenenbaum: Introduction to Analytic and Probabilistic Number Theory; Cambridge University Press, 1995.

4. Algebraic Number Theory
Algebraic numbers, algebraic integers. Dedekind rings, ideals, factorization, connections to the fundamnetal theorem of arithmetic. Fractional ideals, integral closure, discriminant. Quadratic and cyclotomic fields. Class group, the class number is finite. Dirichlet’s unit theorem. Ramification theory, $p$-adic numbers, $p$-adic valuations. Diophantine equations.
Recommended books:
H. Cohen: Number Theory: Volume I.: Tools and Diophantine Equations; Springer, 2007. H. Cohen: Number Theory: Volume II.: Analytic and Modern Tools; Springer, 2007.
A. Fröhlich, M.J. Taylor: Algebraic Number Theory; Cambridge University Press, 1991.
G.J. Janusz: Algebraic Number Fields; American Math. Soc., 2005.
D. Marcus: Number Fields; Springer, 1977.
J. Neukirch: Algebraic Number Theory; Springer, 2010. P. Ribenboim: Classical Theory of Algebraic Numbers; Springer, 2001.

5. Modular forms:
Definition of a modular form, cusp form. Eisenstein series, Poincaré series. Hecke operators, Hecke characters, Hilbert modular forms. Rankin-Selberg method. Artin L-functions. The Langlands functoriality.
Recommended books:
D. Bump: Automorphic Forms and Representations; Cambridge University Press, 1998.
F. Diamond, J. Shurman: A First Course in Modular Forms; Springer, 2005.
H. Iwaniec: Topics in Classical Automorphic Forms; AMS, 1997.
T. Miyake: Modular Forms; Springer, 2006.
G. Shimura: Introduction to the Arithmetic Theory of Automorphic Functions; Princeton University Press, 1994.
G. Shimura: Modular Forms: Basics and Beyond; Springer, 2012.

6. Class field theory:
Cohomology of groups, Tate group. Cohomology of profinite groups, Galois cohomology: Additive theory, Hilbert's theorem 90, Brauer groups. Local class field theory. The local reciprocity law, the local existence theorem. The Brauer group of a local field. Abel extensions of local fields, ramified subgroups and conductors. Global class field theory. Idéles and ideal classes, the cohomology of ideal classes. Artin's reciprocity law.
Recommended books:
E. Artin, J. Tate: Class Field Theory; AMS Chelsea, 2009.
J. Neukirch: Algebraic Number Theory; Springer, 2010.
J. Neukirch: Class Field Theory; Springer, 2013.
J. W.S. Cassels, A. Fröhlich: Algebraic Number Theory; Academic Press, 2010.
G.J. Janusz: Algebraic Number Fields; American Math. Soc., 2005.
A. Weil: Basic Number Theory; Springer, 1995.

7. Number Theory in Cryptography:
Protocols in Public Key Cryptography: Diffie-Hellman principle, the RSA scheme, the ElGamal algorithm. Primality tests. Fermat, Solovay-Strassen, Miller-Rabin primality tests.The AKS algorithm. Factorization methods. Quadratic sieve, the Number field sieve. Discrete logarithm problem, index calculus algorithm. Applications of elliptic curves for encryption, primality testing, factorization and computation of the discrete logarithm.
Recommended books:
H. Cohen: A Course in Computational Algebraic Number Theory; Springer, 1993.
N. Koblitz: A Course in Number Theory and Criptography; Springer, 1994.
A.J. Menzes, P.C. van Oorschot, S.A. Vanstone, Handbook of Applied Cryptography; CRC Press, 1997.
R.A. Mollin: RSA and Public Key Cryptography; Chapman & Hall/CRC, 2003.
A.K. Lenstra, H.W.Jr. Lenstra: The Development of the Number Field Sieve; Springer, 1993.
D.R. Stinson: Cryptography: Theory and Practice; CRC Press, 2006.
L.C. Washington: Elliptic Curves: Number Theory and Cryptography; Chapman &Hall/CRC, 2008.