Aalto courses in Algebra, Combinatorics and Number Theory

[Engström Alexander]

Dear all,

We hope you have had a great summer! The fall semester at Aalto is getting closer, and presumably you have all started planning what courses to take this academic year. There are four great courses scheduled this fall in algebra, combinatorics and number theory. Take one, two, three or all four of them!

Best regards,
Alexander Engström and Camilla Hollanti


Sep 9 - Oct 19

MSE1991: Enumerative Combinatorics, Jonathan Browder (ass. Ragnar Freij)
Enumerative combinatorics is the art of counting. Problems of enumeration arise in every branch of mathematics and their solutions are often subtle and deep. This course will give a comprehensive introduction to enumeration, starting with basic techniques and continuing to include permutations, generating functions, partially ordered sets, the principle of inclusion-exclusion, and Möbius inversion.
Some background in Abstract Algebra may be helpful but is not required.
Lectures: M 10:15-12:00, W 10:15-12:00 in Y228a
Exercises: T 12:15-14:00, Th 16:15-18:00 in Y228a

MSE1990: Elliptic Curve Chryptography, David Karpuk
Elliptic curves are central to modern Number Theory. This course will introduce students to applications of elliptic curves by exploring related cryptographic algorithms. We will cover, among other things, Lenstra’s factorization algorithm, the elliptic curve discrete logarithm problem, and pairing-based cryptography. The theoretical aspects of elliptic curves will then be explored in detail during the Spring.
We will introduce the necessary basics of elliptic curves from scratch, and no prior knowledge of either elliptic curves or cryptography is necessary. Students who have taken any one of Abstract Algebra, Number Theory, Galois Theory, or Cryptology should be well-prepared
Lectures: T 14:15-16:00, Th 10:15-12:00 in Y228a
Exercises: F 12:15-14:00 in Y228a


Oct 28 - Dec 7

MSE1992: Algebraic Geometry II, Alexander Engström (ass. Emanuele Ventura)
In classical algebraic geometry one starts off in geometry with affine varieties and constructs their algebraic counterparts: finitely generated, nilpotent-free rings over an algebraically closed field. In contemporary algebraic geometry we start instead with commutative rings with identity and then move back to geometry to find the affine schemes. They are beautiful geometric objects allowing geometrically intuitive arguments to be applied rigorously everywhere abstract structures are present.
Students who have taken the first course in algebraic geometry or commutative algebra are well prepared. Although algebraic geometry is rumored to be layer on layer of abstruse definitions, it’s broad with most parts accessible quite fast for those with some mathematical maturity.
Lectures: M 10:15-12:00, W 10:15-12:00 in Y228a
Exercises: T 12:15-14:00, Th 16:15-18:00 in Y228a

MSE1990: Algebraic Number Theory I, Iván Blanco-Chacón (ass. Tuomas Tajakka)
The aim of this course is to study the proof of Fermat's last Theorem for a certain family of prime numbers: the regular primes. This will be the pretext for introducing the main tools and ideas, such as the ideal class group and the unit group. We will also comment on some recent applications of Algebraic Number Theory to wireless communications, which will be studied in more detail in Algebraic Number Theory II.
Prerequisites: Linear Algebra and Galois Theory
Lectures: T 14:15-16:00, Th 10:15-12:00 in Y228a
Exercises: F 12:15-14:00 in Y228a


Each course gives 5 credits.