Tropical algebraic geometry is in the interface of algebraic and polyhedral geometry with applications to both these topics. We start with a gentle introduction to tropical algebraic geometry. We then focus on the tropical lifting problem and discuss recent progress. Tropical analogues of graph curves play an important role in this study.

Dyer's outer automorphism of PGL(2,Z) induces an involution of the real line, which behaves very much like a kind of modular function. It has some striking properties: it preserves the set of quadratic irrationals sending them to each other in a non-trivial way and commutes with the Galois action on this set. It restricts to an highly non-trivial involution of the set unit of norm +1 of quadratic number fields. It conjugates the Gauss continued fraction map to the so-called Fibonacci map. It preserves harmonic pairs of numbers inducing a duality of Beatty partitions of N. It induces a subtle symmetry of Lebesgue's measure on the unit interval. On the other hand, it has jump discontinuities at rationals though its derivative exists almost everywhere and vanishes almost everywhere. In the talk, I plan to show how this involution arises from a special automorphism of the infinite trivalent tree

Koszul algebras are the algebras over which the resolution of the residue class field is given entirely by linear matrices. This series of talks will be a survey on results obtained about Koszul algebras since they were introduced by Priddy in 1970. In the first talk, We shall see lots of examples of Koszul algebras, and discuss several characterizations of Koszul algebras.

Reed-Muller codes are among the most elementary and most studied codes. Less studied, but equally elementary are their projective counterparts, the protective Reed-Muller codes. Many open questions remain about these codes. Mathematically, a very interesting question is the determination of the generalized Hamming weights. The determination of these weights is equivalent to the determination of the maximum number of common solutions to certain system of polynomial equations. In this talk, I will give an overview of recent work and developments on the theory of generalized Hamming weights of projective Reed-Muller codes. This work was carried out together with Mrinmoy Datta and Sudhir Ghorpade.

We will prove a purely numerical criterion due to M. Artin to test the rationality of a surface singularity. In practice this is the criterion which is used when a rational surface singularity is being considered.

In these lectures we shall introduce motives and present results in Jannsen's paper, which say that the "conjectural" category of motives is semisimple abelian iff the adequate equivalence relation taken is numerical equivalence.

Koszul algebras are the algebras over which the resolution of the residue class field is given entirely by linear matrices. This series of talks will be a survey on results obtained about Koszul algebras since they were introduced by Priddy in 1970. In the first talk, We shall see lots of examples of Koszul algebras, and discuss several characterizations of Koszul algebras.

We will prove a purely numerical criterion due to M. Artin to test the rationality of a surface singularity. In practice this is the criterion which is used when a rational surface singularity is being considered.

In these lectures we shall introduce motives and present results in Jannsen's paper, which say that the "conjectural" category of motives is semisimple abelian iff the adequate equivalence relation taken is numerical equivalence.

Koszul algebras are the algebras over which the resolution of the residue class field is given entirely by linear matrices. This series of talks will be a survey on results obtained about Koszul algebras since they were introduced by Priddy in 1970. In the first talk, We shall see lots of examples of Koszul algebras, and discuss several characterizations of Koszul algebras.