Let A be a finite subset of a field F. Define A+A and AA to be the set of pairwise sums and products of elements of A, respectively. We will see a theorem of Bourgain, Katz and Tao that shows that if neither A+A nor AA is much bigger than A, then A must be (in some well-defined sense) close to a subfield of F.

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.

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 https://arxiv.org/abs/1709.05055>.

We'll study free resolutions of monomial ideals via the notion of a labelled simplicial complex. We derive a criterion due to Bayer, Peeva and Sturmels for a labelled simplicial complex to define a free resolution. As consequences, we show that the Koszul complex is exact and prove the Hilbert syzygy theorem.

Let $M$ be a compact Kahler manifold and $\Omega (M)$ the canonical $2$-form on $M$. When $M$ is projective $n$-space $\P^n(\C)$ , $H^2(M,\C)$ is of dimension 1. It follows that for any Kahler metric on the projective space, the cohomology class $[\Omega (M)$ of the canonical $2$-form is a multiple of the (unique up to sign) of a generator of $H^2(M,\Z)$. It is immediate from this that if $M$ is a complex sub-manifold of $\P^n(\C)$ for some $n$, then for the Kahler metric on $M$ induced from one on $\P^n(\C)$, it is clear that $[\Omega(M)] \in $\C \cdot H^2(M, Z)$. Kodaira's theorem is a converse to this fact: If a complex manifold $M$ admits a Kahler metric such that the class of $\Omega(M)$ is a multiple of an integral class, then $M$ can be embedded in some projective space. This result was conjectured by W V D Hodge.

We show that in a finite group G of bouded torsion, any set A \subseteq G such that |A + A| = O(|A|) generates a subgroup H of size O(|A|). We will introduce some standard techniques in additive combinatorics to prove this theorem.

Singularities occur naturally everywhere around us, may it be an eye of a cyclone where there is no wind at all, or the north pole where the different time zones converge. The purpose of this talk is to study these patterns mathematically. Hairy ball theorem precisely states that: An even dimensional sphere does not possess any continuous nowhere vanishing tangent vector field". The basic notions of tangent vector field, fundamental groups, some concepts of point set topology will be discussed (at least intuitively) and then a geometric proof of the theorem will be studied. It will be followed by a few applications in the end.