Let $I$ be a proper non-zero ideal in a Noetherian commutative ring $R$. The behavior of many invariants associated to $I^n$ and/or $R/I^n$ is quite mysterious for  initial values of $n$. However, when $n$ is large enough, these invariants become stable. The first place $n_0$ from which such an invariant becomes stable is called the stability index of that invariant. However it is very hard problem to give a bound on the stability index of a particular invariant. Until now, all most all results  in these problems  are obtained only for monomial ideals of a polynomial ring.

In this talk the following invariants are considered: the set of associated primes, depth, the maximal defining degree and the Castelnuovo-Mumford regularity. 

In this talk we give an upper bound for Catelnuovo-Mumford regularity of powers of monomial ideals. Let $I$ be a monomial ideal in a polynomial ring $R$ of $r$ variables over a field $K$. We give an upper bound for $reg I^n$ in the linear form $d(n-1)+e$ where $d,e$ are integers, for all $n$, which can determine explicitly. Moreover, this bound is sharp. 

Each rational function in complex variables defines a locally trivial fibration in a small neighbourhood of its singular point, called Milnor fibration. A basic problem in singularity is to study such Milnor fibration, more precise question is to study the topology of the Milnor fibration, eigenvalue of its monodromy, .... . The aim of this talk is to introduce an algebraic approach to this problem. This is a joint work with K. Takeuchi. 

Given a Henselian and Japanese discrete valuation ring A and a flat and projective A-scheme X, we follow the approach of Biswas and dos Santos [J. Inst. Math. Jussieu 10 (2011), no. 2, 225-234] to introduce a full subcategory of coherent modules on X which is then shown to be Tannakian. We then prove that, under normality of the generic fibre, the associated affine and flat group is pro-finite in a strong sense (so that its ring of functions is a Mittag-Leffler A-module) and that it classifies finite torsors $Q \to X$. This establishes an analogy to Nori’s theory of the essentially finite fundamental group. In addition, we compare our theory with the ones recently developed by Mehta-Subramanian and Antei-Emsalem-Gasbarri. Using the comparison with the former, we show that any quasi-finite torsor $Q\to X$ has a reduction of the structure group to a finite one. 

We define m(Gamma_0(N)) as the minimum of the largest denominator in the cusp set of P across all special fundamental polygons P for Gamma_0(N). For any prime p, we prove that m(Gamma_0(p)) can be expressed using a finite system of quadratic Diophantine equations and inequalities, computable in O(p^2) time complexity. This computation yields freely independent generators of Gamma_0(p) with 0 or p in their (2,1) components, confirming a conjecture of Kulkarni. Similarly, we prove that Gamma_0(N) has freely independent generators with Frobenius norms bounded by O(N) for N=p or N=pq, where p and q are two "close" odd primes.

For N=p^n, while the construction is more intricate, we establish m(Gamma_0(p^n)) to be at most Cp^{n-1}, where C depends solely on p. In the special case when p=2, we show that m(Gamma_0(2^n))=2^{n-1}. This work is a collaboration with Sang-hyun Kim, Mong Lung Lang, and Ser Peow Tan. 

Dissolved oxygen (DO) and biochemical oxygen demand (BOD) are among the most important water quality indicating factors. The advection-reaction (AR) equations or advection-diffusion-reaction (ADR) equations are used as a water quality model which describes the evolution of BOD and DO in a river or stream. In these models, the coefficients representing  the deoxygenation and reaeration rates are not known, and there are some empirical formulas for the reaeration rate, but none for  the deoxygenation one. Furthermore,  it is difficult to confirm the validity of these formulas for an arbitrary river stretch because this coefficient depends on geological factors of the river. In this talk, we consider the inverse problem of determining these coefficients from additional measurements. Based on the Carleman estimates technique, we prove Lipschitz-type stability estimates with respect to data. For numerical computation, the coefficient identification problem is reformulated as an optimization problem using the least-squares method coupled with the adjoint equation method for computing the gradient of the objective functional. Error estimates are derived and numerical examples are provided for demonstrating the performance of the proposed algorithm.

In the first part of the talk, I will briefly introduce random dynamical systems. The second part is to review a literature on the topic of genericity of Lyapunov spectrum. The last part of the talk is devoted to present a new result on genericity of Lyapunov spectrum for bounded random dynamical systems on an infinite dimensional Hilbert space.