Mathematical Institute, Oxford

From the finite Fourier transform to topological quantum field theory -- Bruce Bartlett

Abstract: In 1979, Auslander and Tolimieri wrote the influential "Is computing with the finite Fourier transform pure or applied mathematics?".  It was a homage to the indivisibility of our two subjects, by demonstrating the interwoven nature of the finite Fourier transform, Gauss sums, and the finite Heisenberg group.  My talk is intended as a new chapter in this story. I will explain how all these topics come together yet again in 3-dimensional topological quantum field theory, namely Chern-Simons theory with gauge group U(1).

Defects in liquid crystals: mathematical approaches -- Giacomo Canevari

Abstract: Liquid crystals are matter in an intermediate state between liquids and crystalline solids.  They are composed by molecules which can flow, but retain some form of ordering.  For instance, in the so-called nematic phase the molecules tend to align along some locally preferred directions.  However, the ordering is not perfect, and defects are commonly observed.

The mathematical theory of defects in liquid crystals combines tools from different fields, ranging from topology - which provides a convenient language to describe the main properties of defects -to calculus of variations and partial differential equations.  I will compare a few mathematical approaches to defects in nematic liquid crystals, and discuss how they relate to each other via asymptotic analysis.

Writing is a part of any career in science or mathematics. I will make some remarks about the role writing has played in my life and the role it might play in yours.

Derived geometry and approximations - Pavel Safronov

Derived geometry has been developed to address issues arising in geometry from a consideration of spaces with intrinsic symmetry or some singular spaces arising as complicated intersections.  It has been successful both in pure mathematics and theoretical physics where derived geometric structures appear in quantum gauge field theories such as the theory of quantum electrodynamics.  Recently Lurie has developed a transparent approach to deformation theory, i.e. the theory of approximations of algebraic structures, using the language of derived algebraic geometry.  I will motivate the theory on a basic example and explain one of the theorems in the subject.

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How magnets and mathematics can help solve the current water crisis - Ian Griffiths

Although water was once considered an almost unlimited resource, population growth, drought and contamination are straining our water supplies.  Up to 70% of deaths in Bangladesh are currently attributed to arsenic contamination, highlighting the essential need to develop new and effective ways of purifying water.

Since arsenic binds to iron oxide, magnets offer one such way of removing arsenic by simply pulling it from the water.  For larger contaminants, filters with a spatially varying porosity can remove particles through selective sieving mechanisms.

Here we develop mathematical models that describe each of these scenarios, show how the resulting models give insight into the design requirements for new purification methods, and present methods for implementing these ideas with industry.