Beginning with Wyle's theorem that a bundle induced by the Universal cover is finite (satisfies a polynomial) and its converse by Nori that finite/essentially finite bundles are induced from "coverings/finite principal bundles". We will revisit the etale fundamental group of Grothendieck and the fundamental group scheme by Nori. Later we will glance through other associated group constructions using Tannakian techniques using bundles.

The concept of Hilbert-Samuel polynomial for an m-primary ideal was extended for two m-primary ideals by P. B. Bhattacharya. In other words, the function l(R/I^rJ^s) is given by a polynomial for r, s large. The coefficients appearing in the highest total degree terms in the polynomial are called the mixed multiplicities. These were investigated by B. Teissier (and J. J. Risler) in his Cargese paper. In a series of two talks, we will look at some properties of mixed multiplicities, using superficial elements. These talks aim to cover the preliminaries required for reading the paper 'A generalization of an inequality of Lech relating multiplicity and colength' by C. Huneke, I. Smirnov and J. Validashti.

The concept of Hilbert-Samuel polynomial for an m-primary ideal was extended for two m-primary ideals by P. B. Bhattacharya. In other words, the function l(R/I^rJ^s) is given by a polynomial for r, s large. The coefficients appearing in the highest total degree terms in the polynomial are called the mixed multiplicities. These were investigated by B. Teissier (and J. J. Risler) in his Cargese paper. In a series of two talks, we will look at some properties of mixed multiplicities, using superficial elements. These talks aim to cover the preliminaries required for reading the paper 'A generalization of an inequality of Lech relating multiplicity and colength' by C. Huneke, I. Smirnov and J. Validashti.

The root polytope associated to a subgraph of the complete bipartite graph generalizes the Cartesian product of two simplices. Triangulations of such polytopes is a well studied topic with connections to algebraic geometry and computational algebra. We discuss recent work in characterizing these triangulations. The talk will not assume any background beyond linear algebra.

We will continue our discussion of Hochschild homology from the last talk, defining it for dga's and proceeding to calculate the Hochschild homology of $k[X]/X^2$, the cohomology ring of the sphere. Then we will outline a proof of Jones Isomorphism Theorem using simplicial sets and hence obtain the homology of the loop space of spheres.

This talk will discuss foundational statistical challenges and probabilistic considerations associated with families of stochastic jump models, which often find applications in domains such as genetics, epidemiology, mathematical biology or operations research. I will review Markov jump processes, and by means of common accessible examples, discuss the strong impediments posed by real-world application scenarios to inverse uncertainty quantification tasks. Next, I will discuss current statistical advances linked to structured jump systems along with relevant literature. Through a model exemplar borrowed from queueing theory, I will finally present an approximate and scalable variational Bayesian framework, suitable for uncertainty quantification tasks with a large class of structured processes. The talk will further include examples with applications of the methods introduced.

We will discuss recent work on triangulations of the root polytope associated to a subgraph of the complete bipartite graph.

I will prove the Peter-Weyl theorem for compact topological groups. This talk will be independent of the talks given earlier in the seminar. In particular, no Lie theory is necessary (and nor will it be assumed).

The concept of Hilbert-Samuel polynomial for an m-primary ideal was extended for two m-primary ideals by P. B. Bhattacharya. In other words, the function l(R/I^rJ^s) is given by a polynomial for r, s large. The coefficients appearing in the highest total degree terms in the polynomial are called the mixed multiplicities. These were investigated by B. Teissier (and J. J. Risler) in his Cargese paper. In a series of two talks, we will look at some properties of mixed multiplicities, using superficial elements. These talks aim to cover the preliminaries required for reading the paper 'A generalization of an inequality of Lech relating multiplicity and colength' by C. Huneke, I. Smirnov and J. Validashti.

We shall present a recent result of Davi Cox, K.-N. Lin and G. Sosa which explores the defining equations of multi-Rees algebras of monomial ideals in a polynomial ring over a field.