Title: Gotzmann's regularity and persistence theorem - II
Abstract: 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.
Time:
10:30am-11:25am
Location:
Ramanujan Hall
Description:
Speaker: Provanjan Mallick
Title : Asymptotic prime divisors - III
Abstract : 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.
Time:
3:30pm
Location:
Room 216
Description:
Speaker: Rekha Santhanam
We will talk about relative homotopy groups, long exact sequence in
homotopy and cellular approximation theorem.
Time:
11:00am-12:30pm
Location:
Ramanujan Hall
Description:
Title: Eigenvalues and eigenvectors of the perfect matching association
scheme.
Abstract:
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).
Time:
11:00am
Location:
Room 215
Description:
Speaker: Reebhu Bhattacharya
Topic: Universal Bundles and Classifying Spaces
Abstract: 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.
Title: ELLAM schemes for a model of miscible flow in porous medium: design
and analysis.
Abstract: 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.
Time:
3:30pm
Location:
Room 215, Department of Mathematics
Description:
Title: Higgs bundles
Abstract: We will describe the general fiber of the Hitchin fibration
for the classical groups.