Tue, 29 Nov 2022

12:30 - 13:00
C3

Spatial analysis to investigate the emergent dynamics of a cellular automaton model of tumour-immune interactions.

Roisin Stephens
Abstract

Baseline T cell infiltration and the spatial distribution of T cells within a tumour has been found to be a significant indicator of patient outcomes. This observation, coupled with the increasing availability of spatially-resolved imaging data of individual cells within the tumour tissue, motivates the development of mathematical models which capture the spatial dynamics of T cells. Agent-based models allow the simulation of complex biological systems with detailed spatial resolution, and generate rich spatio-temporal datasets. In order to fully leverage the information contained within these simulated datasets, spatial statistics provide methods of analysis and insight into the biological system modelled, by quantifying inherent spatial heterogeneity within the system. We present a cellular automaton model of interactions between tumour cells and cytotoxic T cells, and an analysis of the model dynamics, considering both the temporal and spatial evolution of the system. We use the model to investigate some of the standard assumptions made in these models, to assess the suitability of the models to accurately describe tumour-immune dynamics.

Tue, 15 Nov 2022

12:30 - 13:00
C3

A Hele-Shaw Newton's cradle and Reciprocity in Fluids

Daniel Booth and Matthew Cotton
Abstract

A Hele-Shaw Newton's cradle: Circular bubbles in a Hele-Shaw channel. (Daniel Booth)

We present a model for the motion of approximately circular bubbles in a Hele-Shaw cell. The bubble velocity is determined by a balance between the hydrodynamic pressures from the external flow and the drag due to the thin films above and below the bubble. We find that the qualitative behaviour depends on a dimensionless parameter and is found to agree well with experimental observations.  Furthermore, we show how the effects of interaction with cell boundaries and/or other bubbles also depend on the value of this dimensionless parameter For example, in a train of three identical bubbles travelling along the centre line, the middle bubble either catches up with the one in front or is caught by the one behind, forming what we term a Hele-Shaw Newton's cradle.
 

Reciprocity in Fluids (Matthew Cotton)

Reciprocity is a useful, and often underused, way to calculate integrated quantities when a to solution to a related problem is known. In the remaining time, I will overview these ideas and give some example use cases

Tue, 01 Nov 2022

12:30 - 13:00
C3

Asymptotic Analysis of Deep Residual Networks

Alain Rossier
Abstract

Residual networks (ResNets) have displayed impressive results in pattern recognition and, recently, have garnered considerable theoretical interest due to a perceived link with neural ordinary differential equations (neural ODEs). This link relies on the convergence of network weights to a smooth function as the number of layers increases. We investigate the properties of weights trained by stochastic gradient descent and their scaling with network depth through detailed numerical experiments. We observe the existence of scaling regimes markedly different from those assumed in neural ODE literature. Depending on certain features of the network architecture, such as the smoothness of the activation function, one may obtain an alternative ODE limit, a stochastic differential equation (SDE) or neither of these. Furthermore, we are able to formally prove the linear convergence of gradient descent to a global optimum for the training of deep residual networks with constant layer width and smooth activation function. We further prove that if the trained weights, as a function of the layer index, admit a scaling limit as the depth increases, then the limit has finite 2-variation.

Tue, 18 Oct 2022
12:30
C3

Recovering scattering distributions from covariance-map images of product distributions

Brady Metherall
Abstract

Molecules can be broken apart with a high-powered laser or an electron beam. The position of charged fragments can then be detected on a screen. From the mass to charge ratio, the identity of the fragments can be determined. The covariance of two fragments then gives us the projection of a distribution related to the initial scattering distribution. We formulate the mathematical transformation from the scattering distribution to the covariance distribution obtained from experiments. We expand the scattering distribution in terms of basis functions to obtain a linear system for the coefficients, which we use to solve the inverse problem. Finally, we show the result of our method on three examples of test data, and also with experimental data.

Tue, 25 Oct 2022
14:00
C3

Nonbacktracking spectral clustering of nonuniform hypergraphs

Dr. Phil Chodrow
(Department of Computer Science, Middlebury College)

Note: we would recommend to join the meeting using the Zoom client for best user experience.

Abstract

Spectral methods offer a tractable, global framework for clustering in graphs via eigenvector computations on graph matrices. Hypergraph data, in which entities interact on edges of arbitrary size, poses challenges for matrix representations and therefore for spectral clustering. We study spectral clustering for arbitrary hypergraphs based on the hypergraph nonbacktracking operator. After reviewing the definition of this operator and its basic properties, we prove a theorem of Ihara-Bass type which allows eigenpair computations to take place on a smaller matrix, often enabling faster computation. We then propose an alternating algorithm for inference in a hypergraph stochastic blockmodel via linearized belief-propagation which involves a spectral clustering step, again using nonbacktracking operators. We provide proofs related to this algorithm that both formalize and extend several previous results. We pose several conjectures about the limits of spectral methods and detectability in hypergraph stochastic blockmodels in general, supporting these with in-expectation analysis of the eigeinpairs of our studied operators. We perform experiments with real and synthetic data that demonstrate the benefits of hypergraph methods over graph-based ones when interactions of different sizes carry different information about cluster structure.

Joint work with Nicole Eikmeier (Grinnell) and Jamie Haddock (Harvey Mudd).

Tue, 01 Nov 2022
14:00
C3

Large network community detection by fast label propagation

Dr. Vincent Traag
(Leiden University)
Abstract

Many networks exhibit some community structure. There exists a wide variety of approaches to detect communities in networks, each offering different interpretations and associated algorithms. For large networks, there is the additional requirement of speed. In this context, the so-called label propagation algorithm (LPA) was proposed, which runs in near linear time. In partitions uncovered by LPA, each node is ensured to have most links to its assigned community. We here propose a fast variant of LPA (FLPA) that is based on processing a queue of nodes whose neighbourhood recently changed. We test FLPA exhaustively on benchmark networks and empirical networks, finding that it runs up to 700 times faster than LPA. In partitions found by FLPA, we prove that each node is again guaranteed to have most links to its assigned community. Our results show that FLPA is generally preferable to LPA.

Fri, 07 Oct 2022

12:00 - 13:00
C3

Maximality properties of generalised Springer representations of $\text{SO}(N)$

Ruben La
(University of Oxford)
Abstract

Waldspurger proved maximality and minimality results for certain generalised Springer representations of $\text{Sp}(2n,\mathbb{C})$. We will discuss analogous results for $G = \text{SO}(N,\mathbb{C})$ and sketch their proofs.

Let $C$ be a unipotent class of $G$ and $E$ an irreducible $G$-equivariant local system on $C$. Let $\rho$ be the generalised Springer representation corresponding to $(C,E)$. We call $C$ the support of $\rho$. It is well-known that $\rho$ appears in the top cohomology of a certain variety. Let $\bar\rho$ be the representation obtained by summing the cohomology groups of this variety.

We show that if $C$ is parametrised by an orthogonal partition consisting of only odd parts, then $\bar\rho$ has a unique irreducible subrepresentation $\rho^{\text{max}}$ whose support is maximal among the supports of the irreducible subrepresentations of $\rho^{\text{max}}$. We also show that $\text{sgn}\otimes\rho^{\text{max}}$ is the unique subrepresentation of $\text{sgn}\otimes\bar\rho$ with minimal support. We will also present an algorithm to compute $\rho^{\text{max}}$.

Tue, 28 Jun 2022

14:00 - 15:00
C3

The temporal rich club phenomenon

Nicola Pedreschi
(Mathematical Institute (University of Oxford))
Abstract

Identifying the hidden organizational principles and relevant structures of complex networks is fundamental to understand their properties. To this end, uncovering the structures involving the prominent nodes in a network is an effective approach. In temporal networks, the simultaneity of connections is crucial for temporally stable structures to arise. In this work, we propose a measure to quantitatively investigate the tendency of well-connected nodes to form simultaneous and stable structures in a temporal network. We refer to this tendency as the temporal rich club phenomenon, characterized by a coefficient defined as the maximal value of the density of links between nodes with a minimal required degree, which remain stable for a certain duration. We illustrate the use of this concept by analysing diverse data sets and their temporal properties, from the role of cohesive structures in relation to processes unfolding on top of the network to the study of specific moments of interest in the evolution of the network.

Article link: https://www.nature.com/articles/s41567-022-01634-8

Mon, 06 Jun 2022

16:00 - 17:00
C3

TBA

Nina Zubrilina
(Princeton University)
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