I will be talking about three distinct topics which will serve as preliminaries for the Jones isomorphism theorem which we will discuss in a later talk. Firstly we will talk about some intersection theory and prove the Thom isomorphism theorem. Finally we will define the Hochschild complex for differential graded algebras and it's Hochschild (co) homology.

attached.

The theories of fluid dynamics and solid mechanics are a cornerstone of Engineering, enabling us to make predictions about the way in which materials and structures behave. The continuum-level mathematical formulations which were developed over a century ago allow us to overcome the impracticalities of considering every microscopic particle within the system, and instead consider the material’s macroscopic bulk behaviour. The type of substances usually described in this way are passive, meaning that the drivers of the system are usually externally-applied forces or energy. However, there exist more exotic types of active matter, where the constituent components themselves contain a source of energy. Suspensions of swimming microbes provide an important example. The innumerable cells within the mixture are capable of self-propelling themselves through the suspending medium. Modern micro- and nano-fabrication methods also allow for the creation of artificial microswimmers. The flows generated by self-motile cells leads to fluid-mediated coupling between the swimmers, which can lead to highly-organised collective bulk motions, sometimes referred to as bacterial turbulence or slow turbulence. This self-organisation has also be seen to change the bulk rheology of the suspension, leading to plastic and superfluidic behaviours, some of which may have technological applications. Continuum models developed for passive materials do not perform well for active matter, and so there has been a great deal of interest and interdisciplinary activity in recent years to derive an effective continuum-level description for such systems. In this talk I will outline some of the current challenges, as well as ideas and progress made to-date in this area.

Consider the the multi-Rees algebra \R_R(I_1 \oplus \cdots \oplus I_r) of monomial ideals I_1, . . . , I_r. In this talk, the defining ideal of \R_R(I_1 \oplus \cdots \oplus I_r) will be described explicitly. We will cover Section 1 and Section 2 of the recent paper "Multi-Rees Algebras and Toric Dynamic Systems" ( https://arxiv.org/pdf/1806.08184.pdf) of Cox, Lin and Sosa. We will also see that for any homogeneous ideals J_1, . . . , J_s, the defining ideal of \R_R(J_1 \oplus \cdots \oplus J_s) can be expressed as a contraction of the defining ideal of \R_R(I_1 \oplus \cdots \oplus I_r) for some monomial ideals I_1, . . . , I_r.

In 1973 Richard Stanley solved several conjectures about magic squares proposed by Harsh Anand, V. C. Dumir and Hans Raj Gupta. In his "Green Book" Stanley used the theory of Cohen-Macaulay and Gorenstein rings to solve these conjectures. I will sketch his solution assuming only basic commutative

The Sperner Lemma is a combinatorial lemma that talks about a certain type of coloring (called the Sperner coloring) of a triangulation of a simplex. It has applications in several root-finding and fair-division algorithms. The Brouwer Fixed Point Theorem is a classical theorem that asserts the existence of a fixed point for a continuous function from the unit disc in Euclidean space to itself. There are several proofs for each of these results. In this talk, we will show that both these results are equivalent. We will look at the proof in the case of two dimensions. The general case is similar modulo some more careful book-keeping. [We encourage all MSc, Ph.D and UG students to attend. Note tea and snacks will be served before the talk at 5:00 PM.]

Finite dimensionality of Verma modules and the Weyl character formula.

In this series of two talks, we will begin by discussing some Nullstellensatz-like results when the base field is finite, and outline the proofs. Next, we will discuss a combinatorial approach to determining or estimating the number of common zeros of a system of multivariate polynomials with coefficients in a finite field. Here we will outline a remarkable result of Heijnen and Pellikaan about the maximum number of zeros that a given number of linearly independent multivariate polynomials of a given degree can have over a finite field. A projective analogue of this result about multivariate homogeneous polynomials has been open for quite some time, although there has been considerable progress in the last two decades, and especially in the last few years. We will outline some results and conjectures here, including a recent joint work with Peter Beelen and Mrinmoy Datta.

In 1973 Richard Stanley solved several conjectures about magic squares proposed by Harsh Anand, V. C. Dumir and Hans Raj Gupta. In his "Green Book" Stanley used the theory of Cohen-Macaulay and Gorenstein rings to solve these conjectures. I will sketch his solution assuming only basic commutative algebra.