Mathematics has provided us with several extremely useful tools to apply in the world beyond mathematics. But it also provides us with mathematicians -- individuals who have trained habits of careful thinking in domains where that is the only way to make progress. This talk will explore some other domains -- such as saying sensible things about the long-term future, or how to identify good actions in the world -- where this style of thinking seems particularly desirable as progress can otherwise be elusive or illusory. It will also consider how a mathematician's curiosity can help to identify important questions.

# Forthcoming Seminars

Please note that the list below only shows forthcoming events, which may not include regular events that have not yet been entered for the forthcoming term. Please see the past events page for a list of all seminar series that the department has on offer.

We derive generalized lower Ricci bounds in terms of signed measures. And we prove associated gradient estimates for the heat flow with Neumann boundary conditions on domains of metric measure spaces obtained through „convexification“ of the domains by means of subtle time changes. This improves upon previous results both in the case of non-convex domains and in the case of convex domains.

We show that homogeneous Einstein metrics on Euclidean spaces are Einstein solvmanifolds, using that they admit periodic, integrally minimal foliations by homogeneous hypersurfaces. For the geometric flow induced by the orbit-Einstein condition, we construct a Lyapunov function based on curvature estimates which come from real GIT.

I will talk about optimal estimates for stochastic integrals

in the case when both rough paths and martingales play a role.

This is an ongoing joint work with Peter Friz (TU Berlin).

The genus of a knot in a 3-manifold is defined to be the minimum genus of a compact, orientable surface bounding that knot, if such a surface exists. We consider the computational complexity of determining knot genus. Such problems have been studied by several mathematicians; among them are the works of Hass--Lagarias--Pippenger, Agol--Hass--Thurston, Agol and Lackenby. For a fixed 3-manifold the knot genus problem asks, given a knot K and an integer g, whether the genus of K is equal to g. In joint work with Lackenby, we prove that for any fixed, compact, orientable 3-manifold, the knot genus problem lies inNP, answering a question of Agol--Hass--Thurston from 2002. Previously this was known for rational homology 3-spheres by the work of Lackenby.

Lipschitz (or H\"older) spaces $C^\delta, \, k< \delta <k+1$, $k\in\mathbb{N}_0$, are the set of functions that are more regular than the $\mathcal{C}^k$ functions and less regular than the $\mathcal{C}^{k+1}$ functions. The classical definitions of H\"older classes involve pointwise conditions for the functions and their derivatives. This implies that to prove regularity results for an operator among these spaces we need its pointwise expression. In many cases this can be a rather involved formula, see for example the expression of $(-\Delta)^\sigma$ in (Stinga, Torrea, Regularity Theory for the fractional harmonic oscilator, J. Funct. Anal., 2011.)

In the 60's of last century, Stein and Taibleson, characterized bounded H\"older functions via some integral estimates of the Poisson semigroup, $e^{-y\sqrt{-\Delta}},$ and of the Gauss semigroup, $e^{\tau{\Delta}}$. These kind of semigroup descriptions allow to obtain regularity results for fractional operators in these spaces in a more direct way.

In this talk we shall see that we can characterize H\"older spaces adapted to other differential operators $\mathcal{L}$ by means of semigroups and that these characterizations will allow us to prove the boundedness of some fractional operators, such as $\mathcal{L}^{\pm \beta}$, Riesz transforms or Bessel potentials, avoiding the long, tedious and cumbersome computations that are needed when the pointwise expressions are handled.