Tuesday, 14 January 2024, 4-5pm Location: A1/3 Physics Building (Google Maps location)

• Speaker: Francis Brown (Oxford)
  
  Title: Graph homology, Voronoi complexes and values of the zeta function.
  
  Abstract:
I will introduce each of the three classical topics in the title from a completely elementary point of view.  Even though they lie in distant fields of mathematics, namely combinatorics, differential geometry and number theory, they are very closely related by some deep theorems and conjectures.

Wednesday, 5 February 2024, 4-5pm Location: Garwood LT (Google Maps location)

• Internal Colloquium
  Speaker: Philip Pearce (UCL)
  
  Title: Multi-scale mathematical modelling of biological systems
  
  Abstract:
Biological tissues typically consist of many cells, which interact physically and chemically. Theoretical models for such systems can use a discrete approach, in which each cell is resolved explicitly, or a continuum approach, in which tissue dynamics are captured via coarse-grained or macroscopic variables that usually satisfy partial differential equations. In this talk I'll give some recent examples of models for biological systems that combine these approaches to understand how cell properties and dynamics affect biological function. I will focus on two applications, one in microbiology (chemotaxis of swarms) and one in physiology (blood flow in sickle cell disease); in both cases the macroscopic dynamics can be modelled using the framework of fluid mechanics..

Tuesday, 25 February 2024, 4-5pm Location: A1/3 Physics Building Google Maps location)

• Speaker: Sung Jin Oh (Berkeley)
  
  Title: Integral formulas for under/overdetermined differential operators and applications
  
  Abstract:
Underdetermined differential operators arise naturally in diverse areas of physics and geometry, including the divergence-free condition for incompressible fluids, the linearized scalar curvature operator in Riemannian geometry, and the constraint equations in general relativity. The duals of underdetermined operators, which are overdetermined, also play a significant role. In this talk, I will present recent joint work with Philip Isett (Caltech), Yuchen Mao (UC Berkeley), and Zhongkai Tao (UC Berkeley) that introduces a novel approach to constructing integral solution/representation formulas (i.e., right-/left-inverses) for a broad class of under/overdetermined operators. They are optimally regularizing and have prescribed support properties (e.g., produce compactly supported solutions for compactly supported forcing terms). A key feature of our approach is a simple algebraic condition on the principal symbol that implies the applicability of our method. This condition simplifies and unifies various treatments of related problems in the literature. If time permits, I will discuss applications to studying the flexibility of initial data sets in general relativity.

Internal Colloquium
March 12, 4-5pm Location: Bentham House LG11 ()

• Speaker: Dario Beraldo (UCL)
  
  Title: Proof of the Deligne–Milnor Conjecture
  
  Abstract:
Understanding the topology of algebraic varieties is a fundamental topic in geometry. For smooth varieties, classical results provide well-established answers (we will review some of these results). A celebrated theorem of Milnor extends these ideas to singular varieties, describing how the topology changes as a 1-parameter family of smooth varieties acquires an isolated singularity. In 1968, Deligne proposed an arithmetic analogue of Milnor’s formula, known as the Deligne–Milnor conjecture. This formula lies at the crossroad of algebraic geometry, number theory and topology. I will present a recent solution of this conjecture, relying on ideas from topological quantum field theory. This is based on joint work with Massimo Pippi.

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You'll find the old colloquium schedules here.