We revisit the Bose-Mesner algebra of the perfect matching association scheme (aka the Hecke algebra of the Gelfand pair (S_2n, H_n), where H_n is the hyperoctahedral group). Our main results are: (1) An algorithm to compute the eigenvalues from symmetric group characters by solving linear equations. (2) Universal formulas, as content evaluations of symmetric functions, for the eigenvalues of fixed orbitals (generalizing a result of Diaconis and Holmes). (3) An inductive construction of the eigenvectors (generalizing a result of Godsil and Meagher).

Gotzmann's regularity theorem establishes a bound on Castelnuovo-Mumford regularity using a binomial representation (the Macaulay representation) of the Hilbert polynomial of a standard graded algebra. Gotzmann's persistence theorem shows that once the Hilbert function of a homogeneous ideal achieves minimal growth then it grows minimally for ever. We start with a proof of Eisenbud-Goto's theorem to establish regularity in terms of graded Betti numbers. Then we discuss Gotzmann's theorems in the language of commutative algebra.

We will give proofs of Cellular approximation and then discuss fibrations and Blaker-Massey Homotopy Excision thorem.

Topological K-theory was one of the first instances of a generalized cohomology theory being used to successfully resolve classical problems involving very concrete objects like vector fields and division algebras. In this talk, we will briefly review some properties of vector bundles, introduce the complex K groups, and discuss some of their properties - including the all-important Bott periodicity theorem.

We will describe the general fiber of the Hitchin fibration for the classical groups.

Tertiary oil recovery is the process which consists in injecting a solvent through a well in an underground oil reservoir, that will mix with the oil and reduce its viscosity, thus enabling it to flow towards a second reservoir. Mathematically, this process is represented by a coupled system of an elliptic equation (for the pressure) and a parabolic equation (for the concentration). The parabolic equation is strongly convection-dominated, and discretising the convection term properly is therefore essential to obtain accurate numerical representations of the solution. One of the possible discretisation techniques for this term involves using characteristic methods, applied on the test functions. This is called the Eulerian-Lagrangian Localised Adjoint Method (ELLAM). In practice, due to the ground heterogeneities, the available grids can be non-conforming and have cells of various geometries, including generic polytopal cells. Along with the non-linear and heterogeneous/anisotropic diffusion tensors present in the model, this creates issues in the discretisation of the diffusion terms. In this talk, we will present a generic framework, agnostic to the specific discretisation of the diffusion terms, to design and analyse ELLAM schemes. Our convergence result applies to a range of possible schemes for the diffusion terms, such as finite elements, finite volumes, discontinuous Galerkin, etc. Numerical results will be presented on various grid geometries.

We will talk about the classifying theorem of principal G-bundles for a topological group G. For every group G, there is a classifying space BG so that the homotopy classes of maps from a space X to BG are in bijective correspondence with the set of isomorphism classes of principal G-bundles over X. We will be outlining the construction, due to Milnor, of a classifying space for any group G.

We revisit the Bose-Mesner algebra of the perfect matching association scheme (aka the Hecke algebra of the Gelfand pair (S_2n, H_n), where H_n is the hyperoctahedral group). Our main results are: (1) An algorithm to compute the eigenvalues from symmetric group characters by solving linear equations. (2) Universal formulas, as content evaluations of symmetric functions, for the eigenvalues of fixed orbitals (generalizing a result of Diaconis and Holmes). (3) An inductive construction of the eigenvectors (generalizing a result of Godsil and Meagher).

We will talk about relative homotopy groups, long exact sequence in homotopy and cellular approximation theorem.

Consider a Noetherian ring R and an ideal I of R. Ratliff asked a question that what happens to Ass(R/I^n) as n gets large ? He was able to answer that question for the integral closure of I. Meanwhile Brodmann answered the original question, and proved that the set Ass(R/I^n) stabilizes for large n. We will discuss the proof of stability of Ass(R/I^n). We will also give an example to show that the sequence is not monotone. The aim of this series of talks to present the first chapter of S. McAdam, Asymptotic prime divisors, Lecture Notes in Mathematics 1023, Springer-Verlag, Berlin, 1983.