Abstract:  As is well known, every rational representation of a finite group $G$ can be realized over $\mathbb{Z}$, that is, the corresponding $\mathbb{Q}G$-module $V$ admits a $\mathbb{Z}$-form. Although $\mathbb{Z}$-forms are usually far from being unique, the famous Jordan--Zassenhaus Theorem shows that there are only finitely many $\mathbb{Z}$-forms of any given $\mathbb{Q}G$-module, up to isomorphism. Determining the precise number of these isomorphism classes or even explicit representatives is, however, a hard task in general. In this talk we shall be concerned with the case where $G$ is the symmetric group $\mathfrak{S}_n$ and $V$ is a simple $\mathbb{Q}\mathfrak{S}_n$-module labelled by a hook partition. Building on work of Plesken and Craig we shall present some results as well as open problems concerning the construction of the
integral forms of these modules. This is joint work with Tommy Hofmann from Kaiserslautern.

Deformation quantization modules or DQ-modules where introduced by M. Kontsevich to study the deformation quantization of complex Poisson varieties. It has been advocated that categories of DQ-modules should provide invariants of complex symplectic varieties and in particular a sort of complex analog of the Fukaya category. Hence, it is natural to aim at describing the functors between such categories and relate them with categories appearing naturally in algebraic geometry. Relying, on methods of homotopical algebra, we obtain an analog of Orlov representation theorem for functors between categories of DQ-modules and relate these categories to deformations of the category of quasi-coherent sheaves.
 

This meeting is being held in celebration of Prof Philip Maini's 60th birthday. Prof Maini has been an internationally leading researcher in mathematical biology for decades. He is currently the Director of the Wolfson Centre for Mathematical Biology, a position he has held since 1998. In the past 20 years he has grown the group significantly. He has established countless interdisciplinary collaborations, has over 400 publications in numerous areas of mathematical biology, with major contributions in mathematical modelling of tumours, wound healing and embryonic pattern formation. He has been elected Fellow of the Royal Society (FRS), Fellow of the Academy of Medical Sciences (FMedSci), and Foreign Fellow of the Indian National Science Academy (FNA). He has served or is serving on editorial board of a large number of journals, and was Editor-in-Chief of the Bulletin of Mathematical Biology [2002-15]. And yet his service to the community cannot be captured just by numbers and titles. Anyone who has met him and worked with him cannot but notice and be touched by his unfailing generosity and the many sacrifices he has made and continues to make day in and day out to help students, early career researchers, and fellow faculty alike.

This meeting provides an opportunity to celebrate Prof Maini's many accomplishments; to thank him for all of his sacrifices; and to bring together the large number of researchers – mathematicians, biologists, physiologists, and clinicians – that he has worked with and interacted with over the years. More broadly, the meeting provides a unique opportunity to reflect on mathematical biology, to provide perspectives on the trajectory of a field that was scarcely recognised and had very few dedicated researchers in the days of Prof Maini's own DPhil; yet a field that has grown tremendously since then. Much of this growth is attributable to the work of Prof Maini, so that today the value of mathematics in biology is increasingly recognized by biologists and clinicians, and with theoretical predictions of mathematical models having cemented a role in advancing biological understanding. 

Speakers

David Sumpter, Uppsala University (Public lecture), Derek Moulton, University of Oxford, Hans Othmer, Minnesota University, Jen Flegg, University of Melbourne, Jim Murray, University of Washington, Jonathan Sherratt, Heriot-Watt University, Kevin Painter, Heriot-Watt University, Linus Schumacher, University of Edinburgh, Lucy Hutchinson, Roche, Mark Chaplain, University of St Andrews, Mark Lewis, University of Alberta, Mary Myerscough, University of Sydney, Natasha Martin, University of Bristol, Noemi Picco, Swansea University, Paul Kulesa, Stowers Institute for Medical Research, Ruth Baker, University of Oxford, Santiago Schnell, University of Michigan, Tim Pedley, University of Cambridge

Organising committee

Ruth Baker (University of Oxford)

Derek Moulton (University of Oxford)

Helen Byrne (University of Oxford)

Santiago Schnell (University of Michigan)

Mark Chaplain (University of St Andrews)

We adapt convergent look-ahead and backward finite difference formulas to compute future eigenvectors and eigenvalues of piecewise smooth time-varying matrix flows $A(t)$. This is based on the Zhang Neural Network model for time-varying problems and uses the associated error function

$E(t) =A(t)V(t)−V(t)D(t)$

with the Zhang design stipulation

$\dot{E}(t) =−\eta E(t)$.

Here $E(t)$ decreased exponentially over time for $\eta >0$. It leads to a discrete-time differential equation of the form $P(t_k)\dot{z}(t_k) = q(t_k)$ for the eigendata vector $z(t_k)$ of $A(t_k)$. Convergent high order look-ahead difference formulas then allow us to express $z(t_k+1)$ in terms of earlier discrete $A$ and $z$ data. Numerical tests, comparisons and open questions follow.

I plan to present a brief introduction to optimal control theory (no background knowledge assumed), and discuss a fascinating and oft-forgotten family of problems where the optimal control behaves very strangely; it changes state infinitely often in finite time. This causes havoc in practice, and even more so in the literature.
 

Dimension reduction is an overarching theme in data science: we enjoy finding informative patterns, features or substructures in large, complex data sets. Within the field of network science, an important problem of this nature is to identify core-periphery structure. Given a network, our task is to assign each node to either the core or periphery. Core nodes should be strongly connected across the whole network whereas peripheral nodes should be strongly connected only to core nodes. More generally, we may wish to assign a non-negative value to each node, with a larger value indicating greater "coreness." This type of problem is related to, but distinct from, commumnity detection (finding clusters) and centrality assignment (finding key players), and it arises naturally in the study of networks in social science and finance. We derive and analyse a new iterative algorithm for detecting network core-periphery structure.

Using techniques in nonlinear Perron-Frobenius theory we prove global convergence to the unique solution of a relaxed version of a natural discrete optimization problem. On sparse networks, the cost of each iteration scales linearly with the number of nodes, making the algorithm feasible for large-scale problems. We give an alternative interpretation of the algorithm from the perspective of maximum likelihood reordering of a new logistic core--periphery random graph model. This viewpoint also gives a new basis for quantitatively judging a core--periphery detection algorithm. We illustrate the algorithm on a range of synthetic and real networks, and show that it offers advantages over the current state-of-the-art.

This is joint work with Francesco Tudisco (Strathclyde)

The problem of decomposing a given dataset as a superposition of basic motifs arises in a wide range of application areas, including neural spike sorting and the analysis of astrophysical and microscopy data. Motivated by these problems, we study a "short-and-sparse" deconvolution problem, in which the goal is to recover a short motif a from its convolution with a random spike train $x$. We formulate this problem as optimization over the sphere. We analyze the geometry of this (nonconvex) optimization problem, and argue that when the target spike train is sufficiently sparse, on a region of the sphere, every local minimum is equivalent to the ground truth, up to symmetry (here a signed shift). This characterization obtains, e.g., for generic kernels of length $k$, when the sparsity rate of the spike train is proportional to $k^{-2/3}$ (i.e., roughly $k^{1/3}$ spikes in each length-$k$ window). This geometric characterization implies that efficient methods obtain the ground truth under the same conditions. 

Our analysis highlights the key roles of symmetry and negative curvature in the behavior of efficient methods -- in particular, the role of a "dispersive" structure in promoting efficient convergence to global optimizers without the need to explicitly leverage second-order information. We sketch connections to broader families of benign nonconvex problems in machine learning and signal processing, in which efficient methods obtain global optima independent of initialization. These problems include variants of sparse dictionary learning, tensor decomposition, and phase recovery.

Joint work with Yuqian Zhang, Yenson Lau, Han-Wen Kuo, Dar Gilboa, Sky Cheung, Abhay Pasupathy