A construction of Behrend from the 1940s shows that there are subsets of [N] of size N^{1-o(1)} that contain no 3-term APs (also called capsets). For a long time, it was open whether there is such a construction over F_3^n (i.e. a capset in F_3^n of size 3^{n-o(n)}). Recently, building on work of Croot, Lev and Pach, it was shown by Ellenberg and Gijswijit ( https://arxiv.org/abs/1605.09223 ) that such a construction does not exist: i.e. any capset in F_3^n can have size at most c^n for some c < 3. The construction has had several applications already in Combinatorics and Theoretical Computer Science. We will see a proof of the theorem of Ellenberg and Gijswijt

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.

My emphasis in this talk will be on functional analytical tools to the solvability and uniqueness of solutions to the nonhomogeneous boundary value problems, dealing with degenerate PDEs of elliptic type. My aim is to consider possibly general class of weights. In particular, I consider the $B_{p}$-class of weights, introduced by Kufner and Opic, which is much more general class than the commonly studied Muckenhoupt $A_{p}$-class.

This will be a continuation of the overview from the last week. Some details will be briefly recalled from the last time, for continuity and the benefit of new audience if any.

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.

This will be an overview of the results on values of quadratic forms at integer points.

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.