Past

Every topological space is metrisable once the symmetry axiom is abandoned and the codomain of the metric is allowed to take values in a suitable structure tailored to fit the topology (and every completely regular space is similarly metrisable while retaining symmetry). This result was popularised in 1988 by Kopperman, who used value semigroups as the codomain for the metric, and restated in 1997 by Flagg, using value quantales. In categorical terms, each of these constructions extends to an equivalence of categories between the category Top and a category of all L-valued metric spaces (where L ranges over either value semigroups or value quantales) and the classical \epsilon-\delta notion of continuous mappings. Thus, there are (at least) two metric formalisms for topology, raising the questions: 1) is any of the two actually useful for doing topology? and 2) are the two formalisms equally powerful for the purposes of topology? After reviewing Flagg's machinery I will attempt to answer the former affirmatively and the latter negatively. In more detail, the two approaches are equipotent when it comes to point-to-point topological consideration, but only Flagg's formalism captures 'higher order' topological aspects correctly, however at a price; there is no notion of product of value quantales. En route to establishing Flagg's formalism as convenient, it will be shown that both fine and coarse variants of homology and homotopy arise as left and right Kan extensions of genuinely metrically constructed functors, and a topologically relevant notion of tensor product of value quantales, a surrogate for the non-existent products, will be described. 


 

The Czech lands were the most industrial part of the Austrian-Hungarian monarchy, broken up at the end of the WW1. As such, Czechoslovakia inherited developed industry supported by developed system of tertiary education, and Czech and German universities and technical universities, where the first chairs for applied mathematics were set up. The close cooperation with the Skoda company led to the establishment of joint research institutes in applied mathematics and spectroscopy in 1929 (1934 resp.).

The development of industry was followed by a gradual introduction of social insurance, which should have helped to settle social contracts, fight with pauperism and prevent strikes. Social insurance institutions set up mathematical departments responsible for mathematical and statistical modelling of the financial system in order to ensure its sustainability. During the 1920s and 1930s Czechoslovakia brought its system of social insurance up to date. This is connected with Emil Schoenbaum, internationally renowned expert on insurance (actuarial) mathematics, Professor of the Charles University and one of the directors of the General Institute of Pensions in Prague.

After the Nazi occupation in 1939, Czech industry was transformed to serve armament of the Wehrmacht and the social system helped the Nazis to introduce the carrot and stick policy to keep weapons production running up to early 1945. There was also strong personal discontinuity, as the Jews and political opponents either fled to exile or were brutally persecuted.

Both in the real and in the p-adic case, I will talk about recent results about C^r-parameterizations and their diophantine applications.  In both cases, the dependence on r of the number of parameterizing C^r maps plays a role. In the non-archimedean case, we get as an application new bounds for rational points of bounded height lying on algebraic varieties defined over finite fields, sharpening the bounds by Sedunova, and making them uniform in the finite field. In the real case, some results from joint work with Pila and Wilkie, and also beyond this work, will be presented, 
in relation to several questions raised by Yomdin. The non-archimedean case is joint work with Forey and Loeser. The real case is joint work with Pila and Wilkie, continued by my PhD student S. Van Hille.  Some work with Binyamini and Novikov in the non-archimedean context will also be mentioned. The relations with questions by Yomdin is joint work with Friedland and Yomdin. 

We present the linearized metrizability problem in the context of parabolic geometries and subriemannian geometry, generalizing the metrizability problem in projective geometry studied by R. Liouville in 1889. We give a general method for linearizability and a classification of all cases with irreducible defining distribution where this method applies. These tools lead to natural subriemannian metrics on generic distributions of interest in geometric control theory.

(Joint work with Coralia Cartis) The problem of finding the most extreme value of a function, also known as global optimization, is a challenging task. The difficulty is associated with the exponential increase in the computational time for a linear increase in the dimension. This is known as the ``curse of dimensionality''. In this talk, we demonstrate that such challenges can be overcome for functions with low effective dimensionality --- functions which are constant along certain linear subspaces. Such functions can often be found in applications, for example, in hyper-parameter optimization for neural networks, heuristic algorithms for combinatorial optimization problems and complex engineering simulations.
We propose the use of random subspace embeddings within a(ny) global minimisation algorithm, extending the approach in Wang et al. (2013). We introduce a new framework, called REGO (Random Embeddings for GO), which transforms the high-dimensional optimization problem into a low-dimensional one. In REGO, a new low-dimensional problem is formulated with bound constraints in the reduced space and solved with any GO solver. Using random matrix theory, we provide probabilistic bounds for the success of REGO, which indicate that this is dependent upon the dimension of the embedded subspace and the intrinsic dimension of the function, but independent of the ambient dimension. Numerical results demonstrate that high success rates can be achieved with only one embedding and that rates are for the most part invariant with respect to the ambient dimension of the problem.
 

The last remaining open problem from Erdős and Rényi's original paper on random graphs is the following: for q at least 3, what is the largest d so that the random graph G(n,d/n) is q-colorable with high probability?  A lot of interesting work in probabilistic combinatorics has gone into proving better and better bounds on this q-coloring threshold, but the full answer remains elusive.  However, a non-rigorous method from the statistical physics of glasses - the cavity method - gives a precise prediction for the threshold.  I will give an introduction to the cavity method, with random graph coloring as the running example, and describe recent progress in making parts of the method rigorous, emphasizing the role played by tools from extremal combinatorics.  Based on joint work with Amin Coja-Oghlan, Florent Krzakala, and Lenka Zdeborová.

Decomposition (aka unital 2-Segal) spaces are simplicial ∞-groupoids with a certain exactness property: they take pushouts of active (end-point preserving) along inert (distance preserving) maps in the simplicial category Δ to pullbacks. They encode the information needed for an 'objective' generalisation of the notion of incidence (co)algebra of a poset, and motivating examples include the decomposition spaces for (derived) Hall algebras, the Connes-Kreimer algebra of trees and Schmitt's algebra of graphs. In this talk I will survey recent activity in this area, including some work in progress on a categorification of (Hopf) bialgebroids.
This is joint work with Imma Gálvez and Joachim Kock.
 

Structure from Motion (SfM) is a problem which asks: given photos of an object from different angles, can we reconstruct the object in 3D? This problem is important in computer vision, with applications including urban planning and autonomous navigation. A key part of SfM is bundle adjustment, where initial estimates of 3D points and camera locations are refined to match the images. This results in a high-dimensional nonlinear least-squares problem. In this talk, I will discuss how dimensionality reduction methods such as block coordinates and sketching can be used to improve solver scalability for bundle adjustment problems.

For a nonlinear elliptic system coming from a nonlinear Schroedinger system, the interaction between components is represented by a symmetric matrix. The construction of possibly lower energy nontrivial solutions and the complete description of dependence of the solutions on the matrix are quite challenging tasks. Especially, we are interested in the case that intra-species interaction forces are fixed and inter-species forces are very large, that is, the diagonal part of the symmetric matric is fixed and the non-diagonal entries are very large. In this case, depending on the network between components by repulsive or attractive forces, several different types of patterns may appear. I would like to explain our recent studies on the problem with three components and touch a possible exploration on the general n-components problem.

Topological field theories give rise to a wealth of algebraic structures, extending
the E_n algebra expressing the "topological OPE of local operators". We may interpret these algebraic structures as defining (slightly noncommutative) algebraic varieties and stacks, called moduli stacks of vacua, and relations among them. I will discuss some examples of these structures coming from the geometric Langlands program and their applications. Based on joint work with Andy Neitzke and Sam Gunningham. 

We will discuss the percolation model on the hexagonal grid. In 2001 S. Smirnov proved conformal invariance of its scaling limit through the use of a tricky auxiliary combinatorial construction.

We present a more conceptual approach, implying that the construction in question can be thought of as geometrically natural one.

The main goal of the talk is to make it believable that not all nice and useful objects in the field have been already found.

No background is required.

We present sharp gradient estimates for the solution of the filtering equation and report on its applications in a high order cubature method for the nonlinear filtering problem.

A C^k-extension of a smooth and connected Lorentzian manifold (M,g) is an isometric embedding of M into a proper subset of a connected Lorentzian manifold (N,h) of the same dimension, where the Lorentzian metric h is C^k regular. If no such extension exists, then we say that (M,g) is C^k-inextendible. The study of low-regularity inextendibility criteria for Lorentzian manifolds is motivated by the strong cosmic censorship conjecture in general relativity.

The Schwarzschild spacetime is manifestly inextendible as a Lorentzian manifold with a C^2 regular metric. In this talk I will describe how one
proves the stronger statement that the maximal analytic Schwarzschild spacetime is inextendible as a Lorentzian manifold with a continuous metric.

In the last years complex networks tools contributed to provide insights on the structure of research, through the study of collaboration, citation and co-occurrence networks. The network approach focuses on pairwise relationships, often compressing multidimensional data structures and inevitably losing information. In this paper we propose for the first time a simplicial complex approach to word co-occurrences, providing a natural framework for the study of higher-order relations in the space of scientific knowledge. Using topological methods we explore the conceptual landscape of mathematical research, focusing on homological holes, regions with low connectivity in the simplicial structure. We find that homological holes are ubiquitous, which suggests that they capture some essential feature of research practice in mathematics. Holes die when a subset of their concepts appear in the same article, hence their death may be a sign of the creation of new knowledge, as we show with some examples. We find a positive relation between the dimension of a hole and the time it takes to be closed: larger holes may represent potential for important advances in the field because they separate conceptually distant areas. We also show that authors' conceptual entropy is positively related with their contribution to homological holes, suggesting that polymaths tend to be on the frontier of research.

In this talk I will discuss how computational algebraic geometry and topology can be useful for studying questions arising in systems biology. In particular I will focus on the problem of comparing models and data through the lens of computational algebraic geometry and statistics. I will provide concrete examples of biological signalling systems that are better understood with the developed methods.

Please note that this will be held at Tsuzuki Lecture Theatre, St Annes College, Oxford.

Please note that you will need to register for this event via https://www.eventbrite.co.uk/e/qbiox-colloquium-trinity-term-2018-ticke…

Cancer causing mutations must become permanently fixed within tissues. 

Please note that this will be held at Tsuzuki Lecture Theatre, St Annes College, Oxford.

Please note that you will need to register for this event via https://www.eventbrite.co.uk/e/qbiox-colloquium-trinity-term-2018-ticke…

1600-1645 - Philip Maini
1645-1705 - Edward Morrissey
1705-1725 - Heather Harrington
1725-1800 - Drinks and networking

The talks will be followed by a drinks reception.

Tickets can be obtained from https://www.eventbrite.co.uk/e/qbiox-colloquium-trinity-term-2018-ticke….
(As ever, tickets are not necessary, but they do help in judging catering requirements.)

PHILIP MAINI

Does mathematics have anything to do with biology? In this talk, I will review a number of interdisciplinary collaborations in which I have been involved over the years that have coupled mathematical modelling with experimental studies to try to advance our understanding of processes in biology and medicine. Examples will include somatic evolution in tumours, collective cell movement in epithelial sheets, cell invasion in neural crest, and pattern formation in slime mold. These are examples where verbal reasoning models are misleading and insufficient, while mathematical models can enhance our intuition.

EDWARD MORRISEY

Fixation and spread of somatic mutations in adult human colonic epithelium Cancer causing mutations must become permanently fixed within tissues. I will describe how, by visualizing somatic clones, we investigated the means and timing with which this occurs in the human colonic epithelium. Modelling the effects of gene mutation, stem cell dynamics and subsequent lateral expansion revealed that fixation required two sequential steps. First, one of around seven active stem cells residing within each colonic gland has to be mutated. Second, the mutated stem cell has to replace neighbours to populate the entire gland. This process takes many years because stem cell replacement is infrequent (around once every 9 months). Subsequent clonal expansion due to gland fission is also rare for neutral mutations. Pro-oncogenic mutations can subvert both stem cell replacement to accelerate fixation and clonal expansion by gland fission to achieve high mutant allele frequencies with age. The benchmarking and quantification of these behaviours allows the advantage associated with different gene specific mutations to be compared and ranked irrespective of the cellular mechanisms by which they are conferred. The age related mutational burden of advantaged mutations can be predicted on a gene-by-gene basis to identify windows of opportunity to affect fixation and limit spread.

HEATHER HARRINGTON

Comparing models with data using computational algebra In this talk I will discuss how computational algebraic geometry and topology can be useful for studying questions arising in systems biology. In particular I will focus on the problem of comparing models and data through the lens of computational algebraic geometry and statistics. I will provide concrete examples of biological signalling systems that are better understood with the developed methods.

It is at first sight surprising that a minimizer of an integral of the calculus of variations may make the integrand infinite somewhere.

This talk will discuss some examples of this phenomenon, how it can be related to material defects, and related open questions from nonlinear elasticity and the theory of liquid crystals.

In this talk, I will review a number of interdisciplinary collaborations in which I have been involved over the years that have coupled mathematical
modelling with experimental studies to try to advance our understanding of processes in biology and medicine. Examples will include somatic evolution in
tumours, collective cell movement in epithelial sheets, cell invasion in neural crest, and pattern formation in slime mold. These are examples where
verbal reasoning models are misleading and insufficient, while mathematical models can enhance our intuition.

Please note that this will be held at Tsuzuki Lecture Theatre, St Annes College, Oxford.

Please note that you will need to register for this event via https://www.eventbrite.co.uk/e/qbiox-colloquium-trinity-term-2018-ticke…

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In the late 1880's Goursat investigated what we now call rigid local systems, classically described as linear differential equations without accessory parameters. In this talk I will discuss some arithmetic and geometric aspects of certain particular cases of Goursat's in rank four. For example, I will discuss what are likely to be all cases where the monodromy group is finite. This is joint work with Danylo Radchenko.

Markoff triples are integer solutions  of the equation $x^2+y^2+z^2=3xyz$ which arose in Markoff's spectacular and fundamental work (1879) on diophantine approximation and has been henceforth ubiquitous in a tremendous variety of different fields in mathematics and beyond.  After reviewing some of these, we will discuss  joint work with Bourgain and Sarnak on the connectedness of the set of solutions of the Markoff equation modulo primes under the action of the group generated by Vieta involutions, showing, in particular,  that for almost all primes the induced graph is connected.  Similar results for composite moduli enable us to establish certain new arithmetical properties of Markoff numbers, for instance the fact that almost all of them are composite.
Time permitting, we will also discuss recent joint work with Magee and Ronan on the asymptotic formula for integer points on Markoff-Hurwitz surfaces  $x_1^2+x_2^2 + \dots + x_n^2 = x_1 x_2 \dots x_n$, giving an interpretation for the exponent of growth in terms of certain conformal measure on the projective space.