The Junior Algebra and Representation Theory Seminar will kick-off the start of Trinity term with a social event in the common room. Come to catch up with your fellow students and maybe play a board game or two. Afterwards we'll have lunch together.

Minimal surfaces, which are critical points of the area functional, have long been a source of fruitful problems in geometry. In this talk, I will introduce a new approach, primarily coming from a recent paper of M. Struwe, to constructing free boundary minimal discs using a gradient flow of a suitable energy functional. I will discuss the uniqueness of solutions to the gradient flow, including recent work on the uniqueness of weak solutions, and also what is known about the qualitative behaviour of the flow, especially regarding the interpretation of singularities which arise. Time permitting, I will also mention ongoing joint work with M. Rupflin and M. Struwe on extending this theory to general surfaces with boundary.

TBA

I will talk about the behaviour of random maps on surfaces (for example, random triangulations) of given genus, when their size tends to infinity. Such questions can be asked from the viewpoint of the local behaviour (Benjamini-Schramm convergence) or global behaviour (diameter, Gromov Hausdorff convergence), and in both cases, much combinatorics is involved. I will survey the landmark results for the case of fixed genus, and state very recent results in which we manage to address the "high genus" regime, when the genus grows proportionally to the size – for this regime we establish isoperimetric inequalities and prove the long-suspected fact that the diameter is logarithmic with high probability.

Based on joint work with Thomas Budzinski and Baptiste Louf.