Let $(A,\m)$ be an excellent normal domain of dimension two. We define an $\m$-primary ideal $I$ to be a $p_g$-ideal if the Rees algebra $A[It]$ is a \CM \ normal domain. When $A$ contains an algebraically closed field $k \cong A/\m$ then Okuma, Watanabe and Yoshida proved that $A$ has $p_g$-ideals and furthermore product of two $p_g$-ideals is a $p_g$ ideal. In this talk we show that if $A$ is an excellent normal domain of dimension two containing a field $k \cong A/\m$ of characteristic zero then also $A$ as $p_g$-ideals. Furthermore product of two $p_g$-ideals is $p_g$.

We discuss ongoing work where we relate Brill-Noether theory on a finite graph to homological invariants of certain modules associated to it. Our approach resembles Stanley's commutative algebraic approach to enumeration of magic squares.

is attached.

In this talk, we will discuss the controllability of wave equation with variable coefficients. The main objectives are to understand the proof of the observability inequality and how the constants in the inequality depend on the coefficients of the wave equation.

We consider the semigroup rings $S = k[t^{a_i}| 1\le i\le n] \subset k[t]$ of embedding dimension $n$ over a field $k$. We write $S = k[x_1, \ldots, x_n]/I_{a_1, \ldots, a_n}$ and explicitly construct the minimal free resolutions of $S$ over $k[x_1, \ldots, x_n]$ when ${a_1, \ldots, a_n}$ are special and derive formulae for the invariants such as Betti Numbers, Cohen-Macaulay type, Frobenius numbers, Hilbert Series and Regularity.

Using Minkowski inequality for mixed multiplicities of ideals we derive an upper bound on the number of generators of m-primary ideals in Cohen-Macaulay local rings. This upper bound implies results of S.S. Abhyankar, E. Becker-D. Eisenbud-D. Rees, I.S. Cohen, D. Rees and J. D. Sally.

In this talk, we will prove the observability inequality for wave equation in the case of localized interior control. Then the control of wave equation with variable coefficients will be discussed.

In this talk, we will prove the observability inequality for wave equation in the case of localized interior control. Then the control of wave equation with variable coefficients will be discussed.

We discuss the problem of local uniformization (resolution of singularities along a valuation). We define the defect of an extension of valued fields and show that it is the only obstruction to local uniformization in positive characteristic. We explain how defect can be seen through lack of finite generation of associated graded rings along the valuation.

We associate a Hilbert function to the exceptional components of a resolution of singularities of a surface singularity, and study its structure. We consider the question of when its associated Poincare' series is rational.