I will introduce examples of non-normal very ample toric 3-folds.
Next I will give classes of toric 3-folds whose any ample line bundles are
normally generated. Main subject is how to prove normality of
toric 3-folds admitting surjective morphism onto the projectie line.

Let f be a rational surface automorphism with positive entropy. It is known that the entropy of f is determined by its ``orbit data'', thee positive integers and a permutation sigma in S_3. Under the assumption that there is a curve C such that the closure of f(C - I(f)) = C, one can construct an automorphism with given entropy. We will discuss possible configuration of invariant curves and the construction of an automorphism with given invariant curve. This construction can be done in any dimension (ge 2).

A Fatou set is where the dynamics of a mapping is regular. On interesting kind is a rotation domain, a Fatou component on which the automorphism induces a torus action. In this second part we will discuss a rational surface automorphism with a ``huge'' rotation domain. It is huge in the sense that it contains both a curve of fixed points and isolated fixed points and there is a global linear model for it.

  I will give an estimate of degree of ideals defining toric
varieties
by using the dimension of the varieties, and give a characterization
of toric varieties whose dfining ideals need elements of the highest degree.
And I also talk about higher syzigies of toric varieties.

We give a closed formula for the expression of sums of period polynomials multiplied its associated Hecke eigenform on level N with N square-free. We also show that for N=2, 3, 5 this formula completely determines the Fourier expansions all Hecke eigenforms of all weights on level N. This is joint work with Youngju Choie and Don Zagier.

The theory of algebraic cycles with modulus, such as the additive higher Chow group introduced by Bloch and Esnault and the Chow group with modulus by Binda, Kerz and Saito, is an emerging branch of algebraic cycle theory. The concept "modulus" concerns how cycles behave at the boundary, expressed by a Cartier divisor. In this talk we exhibit how the contravariance (in affine smooth varieties) of these theories can be deduced from a new moving lemma with modulus. We explain what kind of difficulties are caused by the modulus condition when establishing it.

 In this talk, we consider the problem of determining which modular curves can have infinitely many K-rational points when K varies over all the number fields of fixed degree.

We will recall how the regularity of powers of ideals behave

in general. Then we will present results concerning powers of ideals of maximal minors of matrices of linear forms. Joint with WinfiredBruns and Matteo Varbaro.

Let R be a commutative Noetherian ring. It is a classical result that R is regular if and only if it has finite global dimension. In recent years, certain non-commutative rings which are modules-finite over R and has finite global dimension have become objects of intense interests. They can serve as "non-commutative desingularizations" of Spec(R) and have come up in the three-dimensional solution of the Bondal-Orlov conjecture, higher Auslander-Reiten theory and non-commutative minimal model program. Despite all that attention, these objects remain rather mysterious, for examle we do not know fully  when they exist, or what global dimensions can occur. In this talk I will describe some very recent work on these questions. Some of the work are joined with E. Faber, C. Ingalls, O. Iyama, R. Takahashi, I. Shipman and C. Vial. 

We study multigraded ideals with a radical generic initial ideal.
Our main new result is that if a multigraded ideal has a radical multigraded generic initial ideal then the same is true for every multigraded hyperplane section and for every multigradedprojection.
Connection to universal Gr"obner bases for determinantal ideals, Koszul algebras associated to subspaces configurations and to ideals associated to the multiview varieties of Aholt, Sturmfels and Thomas will be discussed. Joint work with Emanuela De Negri and Elisa Gorla.