Past

In certain business environments, it is essential to the success of the business that workers stick closely to their plans and are not distracted, diverted or stopped. A warehouse is a great example of this for businesses where customers order goods online and the merchants commit to delivery dates.  In a warehouse, somewhere, a team of workers are scheduled to pick the items which will make up those orders and get them shipped on time.  If the workers do not deliver to plan, then orders will not be shipped on time, reputations will be damaged, customer will be lost and companies will go out of business.

StayLinked builds software which measures what these warehouse workers do and measures the factors which cause them to be distracted, diverted or stopped.  We measure whenever they start or end a task or process (e.g. start an order, pick an item in an order, complete an order). Some of the influencing factors we measure include the way the worker interacts with the device (using keyboard, scanner, gesture), navigates through the application (screens 1-3-4-2 instead of 1-2-3-4), the performance of the battery (dead battery stops work), the performance of the network (connected to access point or not, high or low latency), the device types being used, device form factor, physical location (warehouse 1, warehouse 2), profile of worker, etc.

We are seeking to build a configurable real-time mathematical model which will allow us to take all these factors into account and confidently demonstrate a measure of their impact (positive or negative) on the business process and therefore on the worker’s productivity. We also want to alert operational staff as soon as we can identify that important events have happened.  These alerts can then be quickly acted upon and problems resolved at the earliest possible opportunity.

In this project, we would like to collaborate with the maths faculty to understand the appropriate mathematical techniques and tools to use to build this functionality.  This product is being used right now by our customers so it would also be a great opportunity for a student to quickly see the results of their work in action in a real-world environment.

Quiver varieties are an attractive research topic of many branches of contemporary mathematics - (geometric) representation theory, (hyper)Kähler differential geometry, (symplectic) algebraic geometry and quantum algebra.

In the talk, I will define different types of quiver varieties, along with some interesting examples. Afterwards, I will focus on Nakajima quiver varieties (hyperkähler moduli spaces obtained from framed-double-quiver representations), stating main results on their topology and geometry. If the time permits, I will say a bit about the symplectic topology of them.

Let k be a field of characteristic zero and K=k((t)). Semi-algebraic sets over K are boolean combinations of algebraic sets and sets defined by valuative inequalities. The associated Grothendieck ring has been studied by Hrushovski and Kazhdan who link it via motivic integration to the Grothendieck ring of varieties over k. I will present a morphism from the former to the Grothendieck ring of motives of rigid analytic varieties over K in the sense of Ayoub. This allows to refine the comparison by Ayoub, Ivorra and Sebag between motivic Milnor fibre and motivic nearby cycle functor.
 

Let k be a field of characteristic zero and K=k((t)). Semi-algebraic sets over K are boolean combinations of algebraic sets and sets defined by valuative inequalities. The associated Grothendieck ring has been studied by Hrushovski and Kazhdan who link it via motivic integration to the Grothendieck ring of varieties over k. I will present a morphism from the former to the Grothendieck ring of motives of rigid analytic varieties over K in the sense of Ayoub. This allows to refine the comparison by Ayoub, Ivorra and Sebag between motivic Milnor fibre and motivic nearby cycle functor.
 

Brain convolutions are specificity of mammals. Varying in intensity according to the animal species, it is measured by an index called the gyrification index, ratio between the effective surface of the cortex compared to its apparent surface. Its value is closed to 1 for rodents (smooth brain), 2.6 for new-borns and 5 for dolphins.  For humans, any significant deviation is a signature of a pathology occurring in fetal life, which can be detected now by magnetic resonance imaging (MRI). We propose a simple model of growth for a bilayer made of the grey and white matter, the grey matter being in cortical position. We analytically solved the Neo-Hookean approximation in the short and large wavelength limits. When the upper layer is softer than the bottom layer (possibly, the case of the human brain), the selection mechanism is dominated by the physical properties of the upper layer. When the anisotropy favours the growth tangentially as for the human brain, it decreases the threshold value for gyri formation. The gyrification index is predicted by a post-buckling analysis and compared with experimental data. We also discuss some pathologies in the model framework.

Several optimization problems combine nonlinear constraints with the integrality of a subset of variables. For an important class of problems  called Mixed Integer Second-Order Cone Optimization (MISOCO), with applications in facility location, robust optimization, and finance, among others, these nonlinear constraints are second-order (or Lorentz) cones.

For such problems, as for many discrete optimization problems, it is crucial to understand the properties of the union of two disjoint sets of feasible solutions. To this end, we apply the disjunctive programming paradigm to MISOCO and present conditions under which the convex hull of two disjoint sets can be obtained by intersecting the feasible set with a specially constructed second-order cone. Computational results show that such cone has a positive impact on the solution of MISOCO problems.

The Landau-Lifshitz-Gilbert equation (LLG) is a continuum model describing the dynamics for the spin in ferromagnetic materials. In the first part of this talk we describe our work concerning the properties and dynamical behaviour of the family of self-similar solutions under the one-dimensional LLG-equation.  Motivated by the properties of this family of self-similar solutions, in the second part of this talk we consider the Cauchy problem for the LLG-equation with Gilbert damping and provide a global well-posedness result provided that the BMO norm of the initial data is small.  Several consequences of this result will be also given.

If $G$ is an irreducible lattice in a semisimple Lie group, every action of $G$ on a tree has a global fixed point. I will give an elementary discussion of Y. Shalom's proof of this result, focussing on the case of $SL_2(\mathbb{R}) \times SL_2(\mathbb{R})$. Emphasis will be placed on the geometric aspects of the proof and on the importance of reduced cohomology, while other representation theoretic/functional analytic tools will be relegated to a couple of black boxes.

Modular forms holomorphic functions on the upper half of the complex plane, H, invariant under certain matrix transformations of H. The have a very rich structure - they form a graded algebra over C and come equipped with a family of linear operators called Hecke operators. We can also view them as functions on a Riemann surface, which we refer to as a modular curve. It transpires that the integral homology of this curve is equipped with such a rich structure that we can use it to compute modular forms in an algorithmic way. The modular symbols are a finite presentation for this homology, and we will explore this a little and their connection to modular symbols.

Hilbert's Nullstellensatz asserts the existence of a complex point satisfying lying on a given variety, provided there is no (ideal-theoretic) proof to the contrary.
I will describe an analogue for curves (of unbounded degree), with respect to conditions specifying that they lie on a given smooth variety, and have homology class
near a specified ray.   In particular, an analogue of the Lefschetz principle (relating large positive characteristic to characteristic zero) becomes available for such questions.
The proof is very close to a theorem of  Boucksom-Demailly-Pau-Peternell on moveable curves, but requires a certain sharpening.   This is part of a joint project with Itai Ben Yaacov, investigating the logic of the product formula; the algebro-geometric statement is needed for proving the existential closure of $\Cc(t)^{alg}$ in this language.  
 

Hilbert's Nullstellensatz asserts the existence of a complex point satisfying lying on a given variety, provided there is no (ideal-theoretic) proof to the contrary.
I will describe an analogue for curves (of unbounded degree), with respect to conditions specifying that they lie on a given smooth variety, and have homology class
near a specified ray.   In particular, an analogue of the Lefschetz principle (relating large positive characteristic to characteristic zero) becomes available for such questions.
The proof is very close to a theorem of  Boucksom-Demailly-Pau-Peternell on moveable curves, but requires a certain sharpening.   This is part of a joint project with Itai Ben Yaacov, investigating the logic of the product formula; the algebro-geometric statement is needed for proving the existential closure of $\Cc(t)^{alg}$ in this language. 

In this talk we introduce a novel dynamic programming (DP) approximation that exploits the inherent network structure present in revenue management problems. In particular, our approximation provides a new lower bound on the value function for the DP, which enables conservative revenue forecasts to be made. Existing state of the art approximations of the revenue management DP neglect the network structure, apportioning the prices of each product, whereas our proposed method does not: we partition the network of products into clusters by apportioning the capacities of resources. Our proposed approach allows, in principle, for better approximations of the DP to be made than the decomposition methods currently implemented in industry and we see it as an important stepping stone towards better approximate DP methods in practice.

A subset S of initially infected vertices of a graph G is called forcing if we can infect the entire graph by iteratively applying the following process. At each step, any infected vertex which has a unique uninfected neighbour, infects this neighbour. The forcing number of G is the minimum cardinality of a forcing set in G. It was introduced independently as a bound for the minimum rank of a graph, and as a tool in quantum information theory.

The focus of this talk is on the forcing number of the random graph. Furthermore, we will state our bounds on the forcing number of pseudorandom graphs and related problems. The results are joint work with Thomas Kalinowski and Benny Sudakov.

Dimer models with boundary were introduced in joint work with King and Marsh as a natural
generalisation of dimers. We use these to derive certain infinite dimensional algebras and
consider idempotent subalgebras w.r.t. the boundary.
The dimer models can be embedded in a surface with boundary. In the disk case, the
maximal CM modules over the boundary algebra are a Frobenius category which
categorifies the cluster structure of the Grassmannian.

 

One of the key challenges in revenue management is unconstraining demand data. Existing state of the art single-class unconstraining methods make restrictive assumptions about the form of the underlying demand and can perform poorly when applied to data which breaks these assumptions. In this talk, we propose a novel unconstraining method that uses Gaussian process (GP) regression. We develop a novel GP model by constructing and implementing a new non-stationary covariance function for the GP which enables it to learn and extrapolate the underlying demand trend. We show that this method can cope with important features of realistic demand data, including nonlinear demand trends, variations in total demand, lengthy periods of constraining, non-exponential inter-arrival times, and discontinuities/changepoints in demand data. In all such circumstances, our results indicate that GPs outperform existing single-class unconstraining methods.

Finding implementable descriptions of the possible configurations of a knotted DNA molecule has remarkable importance from a biological point of view, and it is a hard and well studied problem in mathematics.
Here we present two newly developed mathematical tools that describe the configuration space of knots and model the action of Type I and II Topoisomerases on a covalently closed circular DNA molecule: the Reidemeister graphs.
We determine some local and global properties of these graphs and prove that in one case the graph-isomorphism type is a complete knot invariant up to mirroring.
Finally, we indicate how the Reidemeister graphs can be used to infer information about the proteins' action.

In this talk I will present recent results obtained within the
framework of perturbative algebraic quantum field theory. This novel
approach to mathematical foundations of quantum field theory allows to
combine the axiomatic framework of algebraic QFT by Haag and Kastler with
perturbative methods. Recently also non-perturbative results have been
obtained within this approach. I will report on these results and present
new perspectives that they open for better understanding of foundations of
QFT.

We consider a couple of problems belonging to Random Geometry, and describe some new analytical challenges they pose for planar PDE's via Beltrami equations. The talk is based on joint work with various people including K. Astala, P. Jones, A. Kupiainen, Steffen Rohde and T. Tao.

In this talk I will construct a reduced tensor product of braided fusion categories containing a symmetric fusion category $\mathcal{A}$. This tensor product takes into account the relative braiding with respect to objects of $\mathcal{A}$ in these braided fusion categories. The resulting category is again a braided fusion category containing $\mathcal{A}$. This tensor product is inspired by the tensor product of $G$-equivariant once-extended three-dimensional quantum field theories, for a finite group $G$.
To define this reduced tensor product, we equip the Drinfeld centre $\mathcal{Z}(\mathcal{A})$ of the symmetric fusion category $\mathcal{A}$ with an unusual tensor product, making $\mathcal{Z}(\mathcal{A})$ into a 2-fold monoidal category. Using this 2-fold structure, we introduce a new type of category enriched over the Drinfeld centre to capture the braiding behaviour with respect to $\mathcal{A}$ in the braided fusion categories, and use this encoding to define the reduced tensor product.
 

In the talk, we discuss how to combine the recurrent neural network with the signature feature set to tackle the supervised learning problem where the input is a data stream. We will apply this method to different datasets, including the synthetic datasets( learning the solution to SDEs ) and empirical datasets(action recognition) and demonstrate the effectiveness of this method.

In this talk, we will revisit the proof of the large deviations principle of Wiener chaoses partially given by Borell, and then by Ledoux in its full form. We show that some heavy-tail phenomena observed in large deviations can be explained by the same mechanism as for the Wiener chaoses, meaning that the deviations are created, in a sense, by translations. More precisely, we prove a general large deviations principle for a certain class of functionals $f_n : \mathbb{R}^n \to \mathcal{X}$, where $\mathcal{X}$ is some metric space, under the probability measure $\nu_{\alpha}^n$, where $\nu_{\alpha} =Z_{\alpha}^{-1}e^{-|x|^{\alpha}}dx$, $\alpha \in (0,2]$, for which the large deviations are due to translations. We retrieve, as an application, the large deviations principles known for the so-called Wigner matrices without Gaussian tails of the empirical spectral measure, the largest eigenvalue, and traces of polynomials. We also apply our large deviations result to the last-passage time which yields a large deviations principle when the weight matrix has law $\mu_{\alpha}^{n^2}$, where $\mu_{\alpha}$ is the probability measure on $\mathbb{R}^+$ with density $2Z_{\alpha}^{-1}e^{-x^{\alpha}}$ when $\alpha \in (0,1)$.

While many cubic fourfolds are known to be rational, it is expected that the very general cubic fourfold is irrational (although none have been
proven to be so). There is a conjecture for precisely which cubics are rational, which can be expressed in Hodge-theoretic terms (by work of Hassett)
or in terms of derived categories (by work of Kuznetsov). The conjecture can be phrased as saying that one can associate a `noncommutative K3 surface' to any cubic fourfold, and the rational ones are precisely those for which this noncommutative K3 is `geometric', i.e., equivalent to an honest K3 surface. It turns out that the noncommutative K3 associated to a cubic fourfold has a conjectural symplectic mirror (due to  Batyrev-Borisov). In contrast to the algebraic side of the story, the mirror is always `geometric': i.e., it is always just an honest K3 surface equipped with an appropriate Kähler form. After explaining this background, I will state a theorem: homological mirror symmetry holds in this context (joint work with Ivan Smith).

I will talk about three-dimensional N=2 supersymmetric gauge theories on a class of Seifert manifold. More precisely, I will compute the supersymmetric partition functions and correlation functions of BPS loop operators on M_{g,p}, which is defined by a circle bundle of degree p over a genus g Riemann surface. I will also talk about four-dimensional uplift of this construction, which computes the generalized index of N=1 gauge theories defined on elliptic fiberation over genus g Riemann surface. We will find that the partition function or the index can be written as a sum over "Bethe vacua” of two-dimensional A-twisted theory obtained by a circle compactification. With this framework, I will show how the partition functions on manifolds with different topologies are related to each other. We will also find that these observables are very useful to study the action of Seiberg-like dualities on co-dimension two BPS operators.