KCL Internal Maths Colloquium

17/05 3pm K6.29 (Anatomy Lecture Theatre)

Speaker: Benjamin Doyon

Title: The emergence of hydrodynamics in many-body systems 

Abstract: One of the most important problems of modern science is that of emergence. How do laws of motion emerge at large scales of space and time, from much different laws at small scales? A foremost example is the theory of hydrodynamics. Take molecules in air, which simply follow Newton’s equations. When there are very many of them, these equations becomes untractable; seeking the knowledge of each molecule’s individual trajectory is completely impractical. Happily it is also unnecessary. At our human scale, new, different equations emerge for aggregate quantities: those of hydrodynamics. And these are apparently all we need to know in order to understand the weather! Despite its conceptual significance, the passage from microscopic dynamics to hydrodynamics remains a notorious open problem of mathematical physics. This goes much beyond molecules in air: similar principles hold very generally, such as in quantum gases and spin lattices, where the resulting equations themselves can be very different. In particular, integrable models, where an extensive mathematical structure allows us to make progress, admit an entirely new universality class of hydrodynamic equations. In this talk, I will discuss in a pedagogical and mathematically precise fashion the general problem and principles of hydrodynamics as an emergent theory, and some recent advances in our understanding, including those obtained in integrable models.

14/06 3pm K6.29 (Anatomy Lecture Theatre) 

Speaker: Mehdi Yazdi

Title: Unknot Recognition, Three-dimensional Manifolds, and Algorithms 

Abstract: One of the oldest problems in low-dimensional topology is the unknot recognition problem, posed by Max Dehn in 1910: Is there an algorithm to decide if a given knot can be untangled? You know that this is a challenging problem if you owned a pair of earphones that are tangled! The unknot recognition problem was highlighted by Alan Turing in his last article in 1954, and the first solution was given by Wolfgang Haken in 1961. However, it remains widely open whether there exists a polynomial time algorithm to detect the unknot. The current state-of-the-art is Lackenby’s announcement for a quasi-polynomial time algorithm, which puts it in similar standing to the graph isomorphism problem. I will discuss what is known about the unknot recognition, how it is related to the theory of foliations on three-dimensional manifolds, as well as recent developments on related algorithmic problems.

TBA: Dionysios Anninos

TBA: Francesca Romana Crucinio