Consider a graded ideal in the polynomial ring in several variables. We shall discuss criterion for the graded ideal and its power to have linear resolution. Then we focus our attention to study linear resolution of monomial ideals. Monomial ideals are the bridge between commutative algebra and the combinatorics. Monomial ideals are also significant because they appear as initial ideals of arbitrary ideals. Since many properties of an initial ideal are inherited by its original ideal, one often adopt this strategy to decipher properties of general ideals. The first talk is meant for covering the preliminary results on resolution and regularity of monomial ideal. The aim of this series of talk is to present the result in ArXiv:1709.05055 .

Perhaps surprisingly, the study of degenerate curves plays a crucial role in our understanding of a general smooth curve. One of the first successes of this idea was the theory of limit linear series developed by Griffiths and Harris which they used to prove the Brill-Noether theorem. The analogous theory for degenerate curves of non-compact type falls in the realm of tropical geometry where it takes the shape of metric graphs (or tropical curves) and divisors on them. This leads to a rich interplay between graph theory and algebraic geometry of curves. After explaining the central ideas we will discuss some applications to Brill-Noether theory and curves of large theta Characteristic.

We continue the study of resolutions of monomial ideals. We start with a short proof of the exactness of the Koszul complex. We then generalize this to free resolutions of any monomial ideal. We'll conclude with the proof of the Hilbert syzygy theorem and some more examples of monomial ideals.

A Higgs bundle on a compact Kahler manifold M consists of a holomorphic vector bundle E together with a holomorphic 1-form with values in End(E), say \phi, such that \phi^\phi = 0 as a 2-form with values in End(E). It turns out that there is a one to one correspondence between irreducible representations of fundamental group of M and stable Higgs bundles on M with vanishing Chern classes. This can be seen as the analogue of the Narasimhan-Seshadri theorem connecting irreducible unitary representations of the fundamental group with stable, flat vector bundles.

Huneke and Itoh independently proved a celebrated result on integral closure of powers of an ideal generated by a regular sequence. As a consequence of this theorem, one can find the Hilbert-Samuel polynomial of the integral closure filtration of I if the normal reduction number is at most 2. We prove Hong and Ulrich's version of the intersection theorem.

We will study free resolutions of monomial ideals via the concept of labelled simplicial complexes due to Bayer, Peeva and Sturmfels. We will derive a criterion for a labelled complex to define a free resolution. As applications, we will obtain the exactness of the Koszul complex and the Hilbert syzygy theorem. If time permits, we will obtain a formula for Betti numbers of a monomial ideal in terms of a corresponding labelled simplicial complex.

Huneke and Itoh independently proved a celebrated result on integral closure of powers of an ideal generated by a regular sequence. As a consequence of this theorem, one can find the Hilbert-Samuel polynomial of the integral closure filtration of I if the normal reduction number is at most 2. We prove Hong and Ulrich's version of the intersection theorem.

Consider a graded ideal in the polynomial ring in several variables. We shall discuss criterion for the graded ideal and its power to have linear resolution. Then we focus our attention to study linear resolution of monomial ideals. Monomial ideals are the bridge between commutative algebra and the combinatorics. Monomial ideals are also significant because they appear as initial ideals of arbitrary ideals. Since many properties of an initial ideal are inherited by its original ideal, one often adopt this strategy to decipher properties of general ideals. The first talk is meant for covering the preliminary results on resolution and regularity of monomial ideal. The aim of this series of talk is to present the result in ArXiv:1709.05055 .

We will cover the material in Chapter 2 in Cassels and Frohlich.

Huneke and Itoh independently proved a celebrated result on integral closure of powers of an ideal generated by a regular sequence. As a consequence of this theorem, one can find the Hilbert-Samuel polynomial of the integral closure filtration of I if the normal reduction number is at most 2. We prove Hong and Ulrich's version of the intersection theorem.