**Michaelmas**

**Solving hard optimization problems with “magic dust of light and matter”**, by *Professor Natalia Berloff*

*Tuesday 28th November, 6pm*

Abstract:

Modern supercomputers can only deal with a small subset of optimization problems when the dimension of the function to be minimised is small or when the underlying structure of the problem allows it to find the optimal solution quickly even for a function of large dimensionality. Even a hypothetical quantum computer, if realised, offers at best the quadratic speed-up for the “brute-force” search for the global minimum. What if instead of moving along the mountainous terrain in search of the lowest point, one fills the landscape with a magical dust that only shines at the deepest level, becoming an easily detectable marker of the solution?

Our “magic dust” is created by shining a laser at stacked layers of selected atoms such as gallium, arsenic, indium, and aluminium. The electrons in these layers absorb and emit light of a specific colour. Polaritons are ten thousand times lighter than electrons and may achieve sufficient densities to form a new state of matter known as a Bose-Einstein condensate, where the quantum phases of polaritons synchronise and create a single macroscopic quantum object that can be detected through photoluminescence measurements.

To create a potential landscape that corresponds to the function to be minimised and to force polaritons to condense at its lowest point we focused on a particular type of optimisation problem, but a type that is general enough so that any other hard problem can be related to it, namely minimisation of the XY model which is one of the most fundamental models of statistical mechanics. We have shown that we can create polaritons at vertices of an arbitrary graph: as polaritons condense, the quantum phases of polaritons arrange themselves in a configuration that corresponds to the absolute minimum of the objective function.

** Utility and Preference**, by Dr Michael Tehranchi

*Thursday 14th November, 6pm*

How do you make decisions when faced with faced with randomness? This talk will describe the Von Neumann – Morgenstern preference theory of expected utility, as well as some behavioural criticisms of its axioms. We will also discuss the structural consequences of this expected utility theory to models of financial markets.

**Knots, Graphs, and Polynomials**, by Dr Jacob Rasmussen

*Wednesday 17th October, 6pm*

To a mathematician, a knot is a smooth injective map from the circle into 3- dimensional space. (Think of a knot tied in a piece of string, but with the ends of the string taped together.) Two knots are the same if we can smoothly deform one into the other without the knot crossing through itself. I’ll explain how to tell different knots apart using elementary algebra and describe some relations between knots and the theory of planar graphs.