Tue, 11 May 2021

15:30 - 16:30
Virtual

### How many stable equilibria will a large complex system have?

Boris Khoruzhenko
(Queen Mary University London)
Further Information

Meeting links will be sent to members of our mailing list (https://lists.maths.ox.ac.uk/mailman/listinfo/random-matrix-theory-anno…) in our weekly announcement on Monday.

Abstract

In 1972 Robert May argued that (generic) complex systems become unstable to small displacements from equilibria as the system complexity increases. His analytical model and outlook was linear. I will talk about a “minimal” non-linear extension of May’s model – a nonlinear autonomous system of N ≫ 1 degrees of freedom randomly coupled by both relaxational (’gradient’) and non-relaxational (’solenoidal’) random interactions. With the increasing interaction strength such systems undergo an abrupt transition from a trivial phase portrait with a single stable equilibrium into a topologically non-trivial regime where equilibria are on average exponentially abundant, but typically all of them are unstable, unless the dynamics is purely gradient. When the interaction strength increases even further the stable equilibria eventually become on average exponentially abundant unless the interaction is purely solenoidal. One can investigate these transitions with the help of the Kac-Rice formula for counting zeros of random functions and theory of random matrices applied to the real elliptic ensemble with some of the mathematical problems remaining open. This talk is based on collaborative work with Gerard Ben Arous and Yan Fyodorov.

Fri, 12 Mar 2021

15:00 - 16:00
Virtual

### Chain complex reduction via fast digraph traversal

Leon Lampret
(Queen Mary University London)
Abstract

Reducing a chain complex (whilst preserving its homotopy-type) using algebraic Morse theory ([1, 2, 3]) gives the same end-result as Gaussian elimination, but AMT does it only on certain rows/columns and with several pivots (in all matrices simultaneously). Crucially, instead of doing costly row/column operations on a sparse matrix, it computes traversals of a bipartite digraph. This significantly reduces the running time and memory load (smaller fill-in and coefficient growth of the matrices). However, computing with AMT requires the construction of a valid set of pivots (called a Morse matching).

In [4], we discover a family of Morse matchings on any chain complex of free modules of finite rank. We show that every acyclic matching is a subset of some member of our family, so all maximal Morse matchings are of this type.

Both the input and output of AMT are chain complexes, so the procedure can be used iteratively. When working over a field or a local PID, this process ends in a chain complex with zero matrices, which produces homology. However, even over more general rings, the process often reveals homology, or at least reduces the complex so much that other algorithms can finish the job. Moreover, it also returns homotopy equivalences to the reduced complexes, which reveal the generators of homology and the induced maps $H_{*}(\varphi)$.

We design a new algorithm for reducing a chain complex and implement it in MATHEMATICA. We test that it outperforms other CASs. As a special case, given a sparse matrix over any field, the algorithm offers a new way of computing the rank and a sparse basis of the kernel (or null space), cokernel (or quotient space, or complementary subspace), image, preimage, sum and intersection subspace. It outperforms built-in algorithms in other CASs.

References

[1]  M. Jöllenbeck, Algebraic Discrete Morse Theory and Applications to Commutative Algebra, Thesis, (2005).

[2]  D.N. Kozlov, Discrete Morse theory for free chain complexes, C. R. Math. 340 (2005), no. 12, 867–872.

[3]  E. Sköldberg, Morse theory from an algebraic viewpoint, Trans. Amer. Math. Soc. 358 (2006), no. 1, 115–129.

[4]  L. Lampret, Chain complex reduction via fast digraph traversal, arXiv:1903.00783.

Mon, 02 Mar 2020
16:00
L4

### Improved convergence of low entropy Allen-Cahn flows to mean curvature flow and curvature estimates

Shengwen Wang
(Queen Mary University London)
Abstract

The parabolic Allen-Cahn equations is the gradient flow of phase transition energy and can be viewed as a diffused version of mean curvature flows of hypersurfaces. It has been known by the works of Ilmanen and Tonegawa that the energy densities of the Allen-Cahn flows converges to mean curvature flows in the sense of varifold and the limit varifold is integer rectifiable. It is not known in general whether the transition layers have higher regularity of convergence yet. In this talk, I will report on a joint work with Huy Nguyen that under the low entropy condition, the convergence of transition layers can be upgraded to C^{2,\alpha} sense. This is motivated by the work of Wang-Wei and Chodosh-Mantoulidis in elliptic case that under the condition of stability, one can upgrade the regularity of convergence.

Fri, 03 May 2019

15:00 - 16:00
N3.12

### Persistence of Random Structures

Primoz Skraba
(Queen Mary University London)
Abstract

This talk will cover the connections of persistence with the topology of random structures. This includes an overview of various results from stochastic topology as well as the role persistence ideas  play in the analysis. This will include results on the maximally persistent classes and minimum spanning acycles/generalised trees.

Sun, 12 May 2019

13:00 - 14:00
L1

### Matt Parker at the Oxford Maths Festival

Matt Parker
(Queen Mary University London)
Further Information

Matt Parker is a stand-up comedian and mathematician. He appears regularly on TV and online and is a presenter on the Discovery Channel. As part of the comedy group Festival of the Spoken Nerd, Matt has toured worldwide and is the first person to use an overhead projector on-stage at the Hammersmith Apollo since Pink Floyd.

Previously a maths teacher, Matt visits schools to talk to students about maths as part of Think Maths and he is involved in the Maths Inspiration shows. He is the Public Engagement in Mathematics Fellow at Queen Mary University of London.

Matt is coming to the Oxford Maths Festival on 12 May and will be signing copies of his new book 'Humble Pi' after his talk. To book a space at this talk, please visit https://mathsfest.web.ox.ac.uk/event/matt-parker. Suitable for ages 16+.

Mon, 29 Jan 2018

14:15 - 15:15
L5

### Compactness results for minimal hypersurfaces with bounded index

Reto Buzano
(Queen Mary University London)
Abstract

First, we will discuss sequences of closed minimal hypersurfaces (in closed Riemannian manifolds of dimension up to 7) that have uniformly bounded index and area. In particular, we explain a bubbling result which yields a bound on the total curvature along the sequence and, as a consequence, topological control in terms of index and area. We then specialise to minimal surfaces in ambient manifolds of dimension 3, where we use the bubbling analysis to obtain smooth multiplicity-one convergence under bounds on the index and genus. This is joint work with Lucas Ambrozio, Alessandro Carlotto, and Ben Sharp

Mon, 06 Feb 2017

16:00 - 17:00
L4

### An Energy Identity for Sequences of Immersions

Huy Nguyen
(Queen Mary University London)
Abstract

In this talk, we will discuss sequences of immersions from 2-discs into Euclidean with finite total curvature where the Willmore energy converges to zero (a minimal surface). We will show that away from finitely many concentration points of the total curvature, the surface converges strongly in $W^{2,2}$.  Furthermore, we have an energy identity for the total curvature, at the concentration points after a blow-up procedure we show that there is a bubble tree consisting of complete non-compact (branched) minimal surfaces of finite total curvature which are quantised in multiples of 4\pi. We will also apply this method to the mean curvature flow, showing that sequences of surfaces that are locally converging to a self-shrinker in L^2 also develop a bubble tree of complete non-compact (branched) minimal surfaces in Euclidean space with finite total curvature quantised in multiples of 4\pi.

Mon, 24 Oct 2016

16:00 - 17:00
L4

### Chern-Gauss-Bonnet formulas for singular non-compact manifold

Reto Buzano
(Queen Mary University London)
Abstract

A generalisation of the classical Gauss-Bonnet theorem to higher-dimensional compact Riemannian manifolds was discovered by Chern and has been known for over fifty years. However, very little is known about the corresponding formula for complete or singular Riemannian manifolds. In this talk, we explain a new Chern-Gauss-Bonnet theorem for a class of manifolds with finitely many conformally flat ends and singular points. More precisely, under the assumptions of finite total Q curvature and positive scalar curvature at the ends and at the singularities, we obtain a Chern-Gauss-Bonnet type formula with error terms that can be expressed as isoperimetric deficits. This is joint work with Huy Nguyen.

Thu, 22 Oct 2015
17:30
L6

### Definability in algebraic extensions of p-adic fields

Angus Macintyre
(Queen Mary University London)
Abstract

In the course of work with Jamshid Derakhshan on definability in adele rings, we came upon various problems about definability and model completeness for possibly infinite dimensional algebraic extensions of p-adic fields (sometimes involving uniformity across p). In some cases these problems had been closely approached in the literature but never  explicitly considered.I will explain what we have proved, and try to bring out many big gaps in our understanding of these matters. This  seems appropriate just over 50 years after the breakthroughs of Ax-Kochen and Ershov.

Mon, 13 Oct 2014

15:45 - 16:45
Oxford-Man Institute

### A-free Groups and Tree-free Groups

IAN CHISWELL
(Queen Mary University London)
Abstract

The idea of A-free group, where A is a discrete ordered abelian group, has been introduced by Myasnikov, Remeslennikov and Serbin. It generalises the construction of free groups. A proof will be outlined that a group is A-free for some A if and only if it acts freely and without inversions on a \lambda-tree, where \lambda is an arbitrary ordered abelian group.

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