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.

I will give an overview on results about the higher ramification theory of finite Galois extensions, mainly, but not only, those of prime degree, for arbitrary valuations. I willtalk about ramification groups, ramification ideals, ramification jumps, norms and traces of these ideals, Kähler differentials and their annihilators, and Dedekind differents. Several of these objects are used for the classification of defects of Galois defect extensions of prime degree.

In a 2014 paper, Cutkosky proved a volume equals multiplicity formula for the multiplicity of an m_R-primary ideal. We will discuss a generalization of this result to the epsilon multiplicity.

Let \(R\) be a commutative Noetherian domain with identity and finite Krull dimension \(d\). If \(R\) is non-singular, then for every ideal \(I \subseteq R\) and \(n \in \mathbb{N}\), the symbolic power \(I^{(dn)}\) is contained in the ordinary power \(I^n\). This property is known as the Uniform Symbolic Topology Property, reflecting a uniform comparison between symbolic and ordinary powers of ideals in \(R\).

What is the most singular possible (reduced) hypersurface in positive characteristic? One answer to this question comes from finding a lower bound on an invariant called the F-pure threshold of a polynomial in terms of its degree.

In this talk we discuss the existence and the importance of asymptotic colengths for families of \(m\)-primary ideals in a Noetherian local ring \((R,m)\).  We explore various families such as weakly graded families, weakly \(p\)-families and weakly inverse \(p\)-families and discuss a new analytic method to prove the existence of limits.