Mon, 24 Feb 2020

16:00 - 17:00

How close together are the rational points on a curve?

Netan Dogra
Abstract

Understanding the size of the rational points on a curve of higher genus is one of the major open problems in the theory of Diophantine equations. In this talk I will discuss the related problem of understanding how close together rational points can get. I will also discuss the relation to the subject of (generalised) Wieferich primes.

Mon, 17 Feb 2020

16:00 - 17:00

Random matrices over p-adic numbers

Valerie Kovaleva
Abstract

The goal of this talk is to introduce a way to use the philosophy of Random Matrix Theory to understand, pose, and maybe even solve problems about p-adic matrices.

Mon, 10 Feb 2020

16:00 - 17:00
C1

Periods and the motivic Galois group

Deepak Kamlesh
(Oxford)
Abstract

A long time ago, Grothendieck made some conjectures. This has resulted in some things.

Tue, 18 Feb 2020

12:00 - 13:00
C1

Can we have null models of real networks? Maximum Entropy Random Loopy Graphs

Fabián Aguirre-López
(King's College London)
Abstract

Real networks are highly clustered (large number of short cycles) in contrast with their random counterparts. The Erdős–Rényi model and the Configuration model will generate networks with a tree like structure, a feature rarely observed in real networks. This means that traditional random networks are a poor choice as null models for real networks. Can we do better than that? Maximum entropy random graph ensembles are the natural choice to generate such networks. By introducing a bias with respect to the number of short cycles in a degree constrained graph, we aim to get a random graph model with a tuneable number of short cycles [1,2]. Nevertheless, the story is not so simple. In the same way random unclustered graphs present undesired topology, highly clustered ones will do as well if one is not careful with the scaling of the control parameters relative to the system size. Additionally the techniques to generate and sample numerically from general biased degree constrained graph ensembles will also be discussed. The topological transition has an important impact on the computational cost to sample graphs from these ensembles. To take it one step further, a general approach using the eigenvalues of the adjacency matrix rather than just the number of short cycles will also be presented, [2].

[1] López, Fabián Aguirre, et al. "Exactly solvable random graph ensemble with extensively many short cycles." Journal of Physics A: Mathematical and Theoretical 51.8 (2018): 085101.
[2] López, Fabián Aguirre, and Anthony CC Coolen. "Imaginary replica analysis of loopy regular random graphs." Journal of Physics A: Mathematical and Theoretical 53.6 (2020): 065002.

Mathematics has long played a crucial role in understanding ecological dynamics in a range of ecosystems, from classical models of predators and prey to more recent models of aquatic organisms' interaction with global climate. When we think of ecosystems we usually imagine coral reefs or tropical forests; however, over the past decade a substantial effort has emerged in studying a tiny, yet deadly ecosystem: human tumours. Rather than being a single malignant mass, tumours are living and evolving ecosystems.

Mon, 24 Feb 2020

16:00 - 17:00
L4

$\Gamma$- convergence and homogenisation for a class of degenerate functionals

Federica Dragoni
(Cardiff University)
Abstract

I will present a $\Gamma$-convergence for degenerate integral functionals related to homogenisation problems  in the Heisenberg group. In our  case, both the rescaling and the notion of invariance or periodicity are chosen in a way motivated by the geometry of the Heisenberg group. Without using special geometric features, these functionals would be neither coercive nor periodic, so classic results do not apply.  All the results apply to the more general case of Carnot groups. Joint with Nicolas Dirr, Paola Mannucci and Claudio Marchi.

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