An advanced course exploring a selection of research topics in Computational Mathematics, as well as tools for automating various aspects of the field.
· Haskell: typed lambda calculus, basic category theory, infinite objects, polymorphism and type classes
· Mathematica: language and libraries, visualization techniques, the Wolfram Alpha online resource
· Tools for Data Type Transformations and Symbolic Computation - slides
· Packages and tools for Combinatorial Generation of sets, permutations, partitions, trees, graphs etc.
· Fundamental Theorem of Arithmetic, Primes, Factoring algorithms
· Finite permutation groups, cycle representations, Lehmer codes, maximal order in the symmetric group
· Recursion equations, Some Famous Integer Sequences
· Stern-Brocot and Calkin-Wilf trees, bijections between N and Q
· Computing integer and set partitions
· Modular arithmetic, Discrete Logarithms, Finite Fields, Block ciphers
· Pseudo-Random Generators, Automorphisms of N and Stream Ciphers
· Homomorphic Encryption with applications to election privacy and confidential cloud computing
· Inductive Definitions, Peano’s Axioms, Presburger arithmetic, ZF-set theory
· Using proof assistants to model simple axiom systems
· Goedel’s Theorems, Incompleteness, Halting Problem, Turing hierarchy, Weak arithmetics
· Ordinals: arithmetic, tree representations, applications to termination analysis
· Goedel Numberings of pairs, finite sequences, trees, graphs, term algebras
· Bijective mappings between natural numbers and rational numbers, Calkin-Wolf and Stern-Brocot trees
· S, K combinators and other minimalist Turing equivalent formalisms
· Kolmogorov-Chaitin complexity, self-delimiting codes
· SAT-solvers, DPLL, Schaeffer’s Dichotomy Theorem, Polynomial algorithms for 2SAT and HornSat
· The Post Lattice, clones, NP-completeness
· Circuit Synthesis Algorithms
· A selections of Graph Theoretical Algorithms
· Application: Subgraph Isomorphism and Information Retrieval
· Bijective Base-N numbering systems
· Conway’s Surreal Numbers
· Symbolic Arithmetic Computations and Arbitrary Length Integers
· Hilbert’s Problems
· The Collatz Conjecture – equivalent formulations, generalizations, fractal representations, Goodstein’s Theorem
· The Riemann Conjecture and equivalent formulations – connecting Number Theory and Complex Analysis
· Predicate Logic and P-NP separation