I first review basic results of Koszul cohomology. More precisely, I consider the effect of projections and hyperplane section methods on sygyzies of sections rings. I also discuss Green's duality theorem and Green's vanishing theorem. Finally, I present several questions on Koszul cohomology.

Recently, Okounkov bodies have become a very interesting and useful tool to understand the positivity of divisors (or line bundles). In the first lecture, we review the construction and basic properties of Okounkov bodies following Lazasfeld-Mustata. In the second lecture, we study some of the recent results on Okounkov bodies.

Recently, Okounkov bodies have become a very interesting and useful tool to understand the positivity of divisors (or line bundles). In the first lecture, we review the construction and basic properties of Okounkov bodies following Lazasfeld-Mustata. In the second lecture, we study some of the recent results on Okounkov bodies.

I will study spheres in the context of A^1-homotopy theory.  In particular, I will try to explain why the A^1-homotopy of spheres is algebro-geometrically very rich and has bearing on a number of concrete problems.  Again, to keep the discussion concrete, I will mention applications to vector bundles on smooth varieties.  

I will describe the A^1-homotopy category and basic constructions therein.  Motivated by geometry and classical homotopy theory, it is natural to attempt to study homotopy of maps between two smooth varieties by simply replacing the unit interval by A^1.  I will explain how this ``naive" notion of homotopy is problematic in general, but still a very useful guide in a number of situations of interest.  To keep the discussion concrete, I will focus on the study of vector bundles on smooth affine varieties.  

We give an introduction of our study on nonsingular projective 3-folds of general type whose geometric genus is 2 or 3. The main aspect is the characterization of the birationality of Phi_6 and Phi_5 respectively.

A quasi-phantom category is an admissible category in the bounded derived category of a smooth projective variety having trivial Hochschild homology and finite Grothendieck group. If, in addition,
the Grothendieck group vanishes, then we call such a category a phantom. In these talks I will first give an introduction to derived categories, semiorthogonal decompositions etc., before explaining how
the first example of a quasi-phantom category, namely in the bounded derived category of the classical Godeaux surface, was constructed. To conclude I will describe some of the subsequent developments and discuss possible questions.

After initiated by the work of Böhning, Graf von Bothmer, and Sosna, there have been enumerous results on exceptional collection of maximal length on surfaces of general type. In this talk, we explain our recent result on exceptional collection of maximal length on the surfaces with Kodaira dimension 1. Also, we prove that the orthogonal complement of the collection is nonzero phantom. This is a joint work with Yongnam Lee.

A quasi-phantom category is an admissible category in the bounded derived category of a smooth projective variety having trivial Hochschild homology and finite Grothendieck group. If, in addition,
the Grothendieck group vanishes, then we call such a category a phantom. In these talks I will first give an introduction to derived categories, semiorthogonal decompositions etc., before explaining how
the first example of a quasi-phantom category, namely in the bounded derived category of the classical Godeaux surface, was constructed. To conclude I will describe some of the subsequent developments and discuss possible questions.

A quasi-phantom category is an admissible category in the bounded derived category of a smooth projective variety having trivial Hochschild homology and finite Grothendieck group. If, in addition,
the Grothendieck group vanishes, then we call such a category a phantom. In these talks I will first give an introduction to derived categories, semiorthogonal decompositions etc., before explaining how
the first example of a quasi-phantom category, namely in the bounded derived category of the classical Godeaux surface, was constructed. To conclude I will describe some of the subsequent developments and discuss possible questions.