Given a finite dimensional representation \(V/k\) of a group \(G\), we consider the space \(k[V]^G\) of all polynomial functions which are invariant under the action of \(G\). At its heart, invariant theory is the study of \(k[V]^G\) and its interactions with \(k[V]\). We are particularly interested in the situation where \(k[V]\) is free as a \(k[V]^G\)-module, and call such representations cofree. The classification of cofree representations is a motivating problem for a field of research that has been active for over 70 years.

For a Noetherian local ring R of characteristic p, we will study a multiplicity-like object called h-function. It is a function of a real variable s that estimates the asymptotic behavior of the sum of ordinary power and Frobenius power. The h-function of a local ring can be viewed as a mixture of the Hilbert-Samuel multiplicity and the Hilbert-Kunz multiplicity. In this talk, we will prove the existence of h-function and the properties of h-function, including convexity, differentiability and additivity.

Thanks to the Direct Summand Theorem, splinter conditions have emerged as a way of studying singularities in commutative algebra and algebraic geometry. In characteristic zero, work of Kovács (2000) and Bhatt (2012) characterizes rational singularities as derived splinters. In this talk, I will present an analogous characterization of klt singularities by imposing additional conditions on the derived splinter property.

Motivated by recent result of F. Perez and R.R.G. on equality of test ideal of module closure operation and trace ideal, and the well-known result by K.E. Smith that parameter test ideal can never be contained in parameter ideals, we study the obstruction of containment of trace ideals in ideals of finite projective (or injective) dimension. As consequences of our results , we give upper bounds on m-adic order of trace ideals of certain modules. We also prove analogous results for ideal of entries of maps in a free resolution of certain modules.

The problem of computing invariants of natural stacks of curves has a long history, starting from Mumford's seminal paper on the Picard group of the stack of 1-pointed elliptic curves. The Picard group of the stack \(\mathcal{M}_{g,n}\) of \(n\)-pointed smooth curves of genus \(g\geq3\) was later computed over \(\mathbb{C}\) by Harer.