1. Arithmetic of Finite Fields:
- Permutation polynomials, polynomial factorization.
2. Function Fields and Curves over Finite Fields:
- Rational points, maximal curves, towers of function fields, automorphisms, modular curves, Drinfeld modular curves.
3. Coding Theory:
- Cyclic and quasi-cyclic codes, algebraic geometry codes, asymptotically good codes.
- Sequences and stream ciphers, cryptographically significant functions (bent, plateaued, almost perfect nonlinear), secret sharing schemes.
4. Enumerative Combinatorics and Applications:
- Integer partitions, permutations and permutation statistics. Basic hypergeometric series and their identities. Bijective and sieve methods.