Tuesday, 4 October 2022, 4-5pm Location: South Wing 9 Garwood LT

• Speaker: Anna Maltsev (QMUL)
  
  Title: Spectral properties of quantum graphs
  
  Abstract:
A quantum graph is a set of vertices and edges, like a usual (combinatorial) graph but the edges are taken to be segments of a real line equipped with a Euclidean metric, which allows us to do analysis on graphs. We study the Schrodinger equation on such graphs, with a Kirchoff condition at vertices, which means that outward derivatives at each vertex add up to 0. Quantum graphs not only arise in a multitude of real-world applications (e.g. electrical networks, roads, pipes, neurons, etc.) but also have very interesting mathematical properties. In this talk, which is based on joint work with Evans Harrell, I will discuss eigenfunction localization on quantum graphs and illustrate new connections between spectra of quantum and combinatorial graphs via a “quantum” Ihara's theorem.

Tuesday, 1 November 2022, 4-5pm Location: South Wing 9 Garwood LT

• Speaker: Peter Topping (Warwick)
  
  Title:How is Ricci flow used to solve problems in Differential Geometry and Topology?
  
  Abstract:
Ricci flow is essentially a parabolic partial differential equation that can evolve the shape of a manifold. It has a stunning record of solving open problems in completely different areas of mathematics, as is well known. What is perhaps less well appreciated is how different applications use Ricci flow in completely different ways, particularly over the past 5 years or so. I will try to give an overview of many of these, making time to cover one or two applications in more detail. I will not assume that the audience knows anything about Ricci flow. I will also make an effort to discuss the Differential Geometry in a way that can be understood by non-experts.

Tuesday, December 6 4-5pm Location: South Wing 9 Garwood LT

• Speaker: Felix Schulze (Warwick)
  
  Title: TMean curvature flow with generic initial data and applications
  
  Abstract: Mean curvature flow is the gradient flow of the area functional and constitutes a natural geometric heat equation on the space of hypersurfaces in an ambient Riemannian manifold. It is believed, similar to Ricci Flow in the intrinsic setting, to have the potential to serve as a tool to approach several fundamental conjectures in geometry. The obstacle for these applications is that the flow develops singularities, which one in general might not be able to classify completely. Nevertheless, a well-known conjecture of Huisken states that a generic mean curvature flow should have only spherical and cylindrical singularities. As a fundamental step in this direction Colding-Minicozzi have shown that spheres and cylinders are the only linearly stable singularity models. In this talk we will give an introduction to the problem and discuss recent further progress in joint work with Otis Chodosh, Kyeongsu Choi and Christos Mantoulidis, including applications to the low entropy Schoenflies conjecture.

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You'll find the old colloquium schedules here.