These seminars (unless otherwise stated) will take place on Tuesdays at 12pm via Zoom on an almost weekly basis.

## 19 January 2021

**Zoom Link: https://ucl.zoom.us/j/97330142890**

**Speaker: Jakob Stein**

Supervisor: Dr L Foscolo

##### Title: Bubbling and the BPST Instanton

**Abstract:**

An explicit solution to the instanton equations in dimension four was constructed by Belavin, Polyakov, Schwarz, and Tyupkin (BPST). While these authors were motivated by physics, instantons have since been used to construct invariants of 4-manifolds, and we will use their solution to discuss some more general instanton phenomena, such as bubbling. This should hopefully be a gentle introduction to these ideas, but time-permitting we will mention the less well-understood higher-dimensional story, in particular for Calabi-Yau manifolds in dimension six.

## Wednesday 3 March 2021 at 1pm

**Zoom Link: https://ucl.zoom.us/j/97330142890**

**Speaker: Peter Hearnshaw**

Supervisor: Prof L Parnovski

##### Title: Structure of atoms and molecules : regularity of the density matrix

**Abstract:**

Ever since 1926 with the discovery of the Schrodinger equation the problem of understanding chemical structure became the problem of understanding a single mathematical equation. But as Paul Dirac famously said the equation is too complex to be solved. Almost 100 years later many ingenious methods have been discovered to break apart this problem numerically. However it's still notoriously difficult and obtaining numerical solutions remains one of the biggest uses of supercomputers today.

We take a step back and consider what this equation means in the context of analysis. In particular we ask what regularity is possessed by the wavefunction solutions. The smoothness or roughness of the solutions has far reaching consequences on the effectiveness of the numerical methods.

Accessible to all! I will begin with a gentle introduction and move on to our results.

## 9 March 2021

**NO SEMINAR**

## 16 March 2021 at 1pm

**Zoom Link: https://ucl.zoom.us/j/97330142890**

**Speaker: Yohance Osborne**

Supervisor: Dr I Smears

##### Title: Differential Inclusions: A Light Introduction to Theory and Applications

**Abstract:**

Science and industry provides an immense class of problems that can be modelled by Differential Inclusions. The fundamental reason for this is that many models of real-world phenomena involve empirical parameters that come with a degree of uncertainty, such as the rate of virus spread on pairwise interactions featuring in a Susceptible-Infected model, for example. One advantage of being able to formulate a model problem as a differential inclusion, which takes into account the model’s uncertain parameters, is that qualitative features of the original problem can sometimes be derived from the inclusion model regardless of whether or not the uncertain parameters are particularly "badly behaved".

Our presentation will give a light introduction to the theory and applications of Differential Inclusions. In particular, we will highlight a few basic results on the existence of solutions to ordinary differential inclusions involving set-valued maps with convex images. This will then be followed by a discussion of Filippov's treatment of ODEs with discontinuous right-hand side. We will conclude the talk with a brief survey of questions that one can generally ask about the set of admissible solutions associated with a differential inclusion and mention other special topics of interest.

## 23 March 2021

**Zoom Link: https://ucl.zoom.us/j/97330142890**

**Speaker: Ivan Solonenko**

Supervisor: Prof Jürgen Berndt (LSGNT, based at King's College London)

##### Title: The story of isoparametric hypersurfaces in space forms

**Abstract:**

Isoparametric hypersurfaces are certain constant mean curvature hypersurfaces that come in families providing an orbit-like foliation of the ambient space. They were introduced at the beginning of the 20th century where they arose in geometrical optics as wavefronts travelling at a constant speed at each moment. First attempts to understand these objects were made by Somigliana and Segre who showed that in R^3 the list of such surfaces is exhausted by planes, spheres, and cylinders; Segre later generalised this result to R^n. In a series of articles published in 1938-40, Élie Cartan launched a full-scale attack on isoparametric hypersurfaces in space forms. He proved that they have constant principal curvatures and classified them in real hyperbolic spaces. He then attempted to conquer the remaining spherical case only to discover how unimaginably more complex the situation is there. Bizarrely, the number g of distinct principal curvatures of an isoparametric hypersurface in S^n can only be equal to 1, 2, 3, 4, or 6. Cartan managed to handle the first three cases but the question for g = 4 and 6 remained open for 70 years and the classification has only been completed recently.

The story of isoparametric hypersurfaces in space forms and specifically spheres was a century-long drama and one of the big conundrums in differential geometry. In this talk I will present a brief overview of the problem and try to show how it gives rise to an intricate interplay of Riemannian geometry, the theory of symmetric spaces, algebraic topology, and commutative algebra.