Dominik Lukeš from the AI Competency Centre will give an introductory survey of AI in relation to programming.
Dehn surgery is a method of building three-dimensional manifolds that is ubiquitous throughout low-dimensional topology. I will give an introduction to Dehn surgery and discuss recent work with M. Kegel on the uniqueness of Dehn surgery descriptions of 3-manifolds. To do this, I will discuss the reason that Dehn surgery is so prominent - namely that it interacts very well with many structures, such as the geometry and gauge theory of 3-manifolds. (I will do my very best to assume very little background knowledge.)
We study optimal investment and consumption in an incomplete stochastic factor model for a power utility investor on the infinite horizon. When the state space of the stochastic factor is finite, we give a complete characterisation of the well-posedness of the problem and provide an efficient numerical algorithm for computing the value function. When the state space is a (possibly infinite) open interval and the stochastic factor is represented by an Ito diffusion, we develop a general theory of sub- and supersolutions for second-order ordinary differential equations on open domains without boundary values to prove existence of the solution to the Hamilton-Jacobi-Bellman (HJB) equation along with explicit bounds for the solution. By characterising the asymptotic behaviour of the solution, we are also able to provide rigorous verification arguments for various models, including the Heston model. Finally, we link the discrete and continuous setting and show that that the value function in the diffusion setting can be approximated very efficiently through a fast discretisation scheme.