In this talk, which should be accessible to all undergraduates, we will explain the title, and explain how number theory understands interesting numbers and relations between them by building appropriate mathematical structures, from algebra, geometry, analysis.

We will discuss potential future directions in the interface between matroids, coding theory and syzygies. This will be a platform to share thoughts on this topic.

Greedy weights of linear codes measure the minimal support cardinality of chains of subcodes for a certain lexicographic order. We show how the problem of finding such weights can be studied by inspecting the poset of flats of the generator matroid, or better: the (dual) poset of cycles for the parity check matroid. The terms of a minimal resolution of the Stanley-Reisner ring of the independence complex of the parity check corresponds to the points of the poset. Optimal paths in the poset diagram, and optimal chains of non-zero component maps in a minimal resolution of the Stanley-Reisner ring. correspond to greedy weights Such maps can be constructed using poset homology, or studied indirectly, through the Taylor complex, which also provides a resolution, although not minimal.

Let C be a smooth projective curve over an algebraically closed field k. Fix r, d ≥ 1. Let Q(r, d) be the quot scheme parametrizing degree d torsion quotients of the trivial vector bundle of rank r. Then we show that the universal kernel bundle over C × Q(r, d) and its pushforward to Q(r, d) are stable with respect to certain natural ample bundles over C × Q(r, d) and Q(r, d) respectively.

In this talk, we discuss the stabilization of a fluid-structure interaction system of a viscous, compressible fluid and rigid ball in a three dimension space with an external force acting on the ball produced by a spring and a damper connecting the centre of the ball to a fixed point $h_1$. We prove global-in-time existence result for this model under the condition that the initial velocities are small and the center of mass of the body is sufficiently near to $h_1$. In this scenario, we discuss the asymptotic behaviour of the whole system.

We will introduce Higgs bundles on curves, discuss the Hitchin fibration and its fibers in terms of spectral curves.

We will introduce Higgs bundles on curves, discuss the Hitchin fibration and its fibers in terms of spectral curves.

(During the week 03 Nov – 08 Nov, 2019) We continue to study smooth morphisms. The purpose of these two lectures is : (1) Definitions and Basic properties of the Module of Käher differentials. (2) Prove a criterion for when a quotient of a formally smooth algebra is again formally smooth. (3) Prove a characterisation of when a morphism is smooth called the J a c o b i a n c r i - t e r i o n. This is a preparation for the proof of Cohen’s structure theorem for complete local rings in the inequicharacteritic case.

We define and give elementary properties of triangulated categories. We also give an application of triangulated categories to linkage theory in commutative algebra.

(During the week 03 Nov – 08 Nov, 2019) We continue to study smooth morphisms. The purpose of these two lectures is : (1) Definitions and Basic properties of the Module of Käher differentials. (2) Prove a criterion for when a quotient of a formally smooth algebra is again formally smooth. (3) Prove a characterisation of when a morphism is smooth called the J a c o b i a n c r i - t e r i o n. This is a preparation for the proof of Cohen’s structure theorem for complete local rings in the inequicharacteritic case.