Past

Do you feel unable to explain why maths are cool? Are you looking for fun and affordable theorems for your non-mathematician friends? This is your topic.

This talk aims to be a rigorous introduction to Social Choice Theory, a sub-branch of Game Theory with natural applications to economics, sociology and politics that tries to understand how to determine, based on the personal opinions of all individuals, the collective opinion of society. The goal is to prove the three famous and pessimistic impossibility theorems: Arrow's theorem, Gibbard's theorem and Balinski-Young's theorem. Our blunt conclusion will be that, unfortunately, there are no ideally fair social choice systems. Is there any hope yet?

 Lipschitz (or H\"older) spaces $C^\delta, \, k< \delta <k+1$, $k\in\mathbb{N}_0$, are the set of functions that are more regular than the $\mathcal{C}^k$ functions and less regular than the $\mathcal{C}^{k+1}$ functions. The classical definitions of H\"older classes involve  pointwise conditions for the functions and their derivatives.  This implies that to prove   regularity results for an operator among these spaces  we need its pointwise expression.  In many cases this can be a rather involved formula, see for example the expression of $(-\Delta)^\sigma$  in (Stinga, Torrea, Regularity Theory for the fractional harmonic oscilator, J. Funct. Anal., 2011.)

In  the 60's of last century, Stein and Taibleson, characterized bounded H\"older functions via some integral estimates of the Poisson semigroup, $e^{-y\sqrt{-\Delta}},$ and of  the Gauss semigroup, $e^{\tau{\Delta}}$. These kind of semigroup descriptions allow to obtain regularity results for fractional operators in these spaces in a more direct way.

 In this talk we shall see that we can characterize H\"older spaces adapted to other differential operators $\mathcal{L}$ by means of semigroups and that these characterizations will allow us to prove the boundedness of some fractional operators, such as $\mathcal{L}^{\pm \beta}$, Riesz transforms or Bessel potentials, avoiding the long, tedious and cumbersome computations that are needed when the pointwise expressions are handled.

Mirror symmetry, in a crude formulation, is usually presented as a correspondence between curve counting on a Calabi-Yau variety X, and some invariants extracted from a mirror family of Calabi-Yau varieties. After the physicists Bershadsky-Cecotti-Ooguri-Vafa (henceforth BCOV), this is organised according to the genus of the curves in X we wish to enumerate, and gives rise to an infinite recurrence of differential equations. In this talk, I will give a general introduction to these problems, and present a rigorous mathematical formulation of the BCOV conjecture at genus one, in terms of a lifting of the Grothendieck-Riemann-Roch. I will explain the main ideas of the proof of the conjecture for Calabi-Yau hypersurfaces in projective space, based on the Riemann-Roch theorem in Arakelov geometry. Our results generalise from dimension 3 to arbitrary dimensions previous work of Fang-Lu-Yoshikawa.
 

This is joint work with G. Freixas and C. Mourougane.

We extend the lightning Laplace solver (Gopal and Trefethen, SINUM 2019) to unbounded domains and to the biharmonic equation. Illustrating the high accuracy of such methods, we get beautiful contour plots of Moffatt eddies.

I'll give a general introduction to tensor-triangular geometry, the algebraic study of tensor-triangulated categories as they appear in topology, geometry and representation theory. Then I'll discuss an elementary idea, that of a "field" in this theory, and explain what we currently know about them.

A subset $D$ of a finite cyclic group $\mathbb{Z}/m\mathbb{Z}$ is called a "perfect difference set" if every nonzero element of $\mathbb{Z}/m\mathbb{Z}$ can be written uniquely as the difference of two elements of $D$. If such a set exists, then a simple counting argument shows that $m=n^2+n+1$ for some nonnegative integer $n$. Singer constructed examples of perfect difference sets in $\mathbb{Z}/(n^2+n+1)\mathbb{Z}$ whenever $n$ is a prime power, and it is an old conjecture that these are the only such $n$ for which $\mathbb{Z}/(n^2+n+1)\mathbb{Z}$ contains a perfect difference set. In this talk, I will discuss a proof of an asymptotic version of this conjecture.

Many computational techniques in data science involve the factorization of a data matrix into the product of two or more structured matrices. Examples include PCA, which relies on computing an SVD, recommendation systems, which leverage non-negative matrix factorization, infilling missing entries with low rank matrix completion, and finding sparse representations via dictionary learning. In our work we study a new matrix factorization problem, involving the recovery of $\textbf{A}$ and $\textbf{X}$ from $\textbf{Y} := \textbf{A}\textbf{X}$ under the following assumptions; $\textbf{A}$ is an $m \times n$ sparse binary matrix with a fixed number $d$ of nonzeros per column and $\textbf{X}$ is an $n \times N$ sparse real matrix whose columns have $k$ nonzeros and are dissociated. This setup is inspired and motivated by similar models studied in the dictionary learning literature as well as potential connections both with stochastic block models and combinatorial compressed sensing. In this talk we present a new algorithm, EBR, for solving this problem, as well as recovery guarantees in the context of a particular probabilistic data model. Using the properties of expander graphs we are able to show, under certain assumptions, that with just $N = \textit{O}( \log^2(n))$ samples then EBR recovers the factorization up to permutation with high probability. 

Can spatial fungal networks be informative for both ecology and network science?

Filamentous organisms grow as adaptive biological spatial networks. These networks are in a continuous balance of two main forces: exploration of the habitat to acquire scarce resources, and the transport of those resources within the developing network. In addition, the construction of the network has to be kept a low cost while taking into account the risk of damage by predation. Such network optimization is not unique to biological systems, but is relevant to transport networks across many domains. Thus, this collaborative project between FU-Berlin and University of Oxford represents the beginning of a research program that aims at: First, setting up protocols for the use of network analysis to characterize spatial networks formed by both macroscopic and microscopic filamentous organisms (e.g. Fungi), and determining the fitness and ecological consequences of different structure of the networks. Second, extracting biologically-inspired algorithms that lead to optimized network formation in fungi and discuss their utility in other network domains. This information is critical to demonstrate that we have a viable and scalable pipeline for the measurement of such properties as well provide preliminary evidence of the usefulness of studying network properties of fungi.

We consider an analogue of Kontsevich's matrix Airy function where the cubic potential $\mathrm{Tr}(\Phi^3)$ is replaced by a quartic term $\mathrm{Tr}(\Phi^4)$. By methods from quantum field theory we show that also the quartic case is exactly solvable. All cumulants can be expressed as composition of elementary functions with the inverse of another elementary function. For infinite matrices the inversion gives rise to hyperlogarithms and zeta values as familiar from quantum field theory. For finite matrices the elementary functions are rational and should be viewed as branched covers of Riemann surfaces, in striking analogy with the topological recursion of the Kontsevich model. This rationality is strong support for the conjecture that the quartic analogue of the Kontsevich model is integrable.
 

Gauss noted quadratic reciprocity to be among his favourite results, and any undergrad will quickly pick up on just how strange it is despite a plethora of elementary proofs. By 1930, E. Artin had finalized Artin reciprocity which wondrously subsumed all previous generalizations, but was still confined to abelian contexts. An amicable non-abelian reciprocity remains a driving force in number-theoretic research.

In this talk, I'll recount Artin reciprocity and show it implies quadratic and cubic reciprocity. I'll then talk about some candidate non-abelian reciprocities, and in particular, which morals of Artin reciprocity they preserve.

Nonlocal minimal surfaces are hypersurfaces of Euclidean space that minimize the fractional perimeter, a geometric functional introduced in 2010 by Caffarelli, Roquejoffre, and Savin in connection with phase transition problems displaying long-range interactions.

In this talk, I will introduce these objects, describe the most important progresses made so far in their analysis, and discuss the most challenging open questions.

I will then focus on the particular case of nonlocal minimal graphs and present some recent results obtained on their regularity and classification in collaboration with X. Cabre, A. Farina, and L. Lombardini.

The genus of a knot in a 3-manifold is defined to be the minimum genus of a compact, orientable surface bounding that knot, if such a surface exists. We consider the computational complexity of determining knot genus. Such problems have been studied by several mathematicians; among them are the works of Hass--Lagarias--Pippenger, Agol--Hass--Thurston, Agol and Lackenby. For a fixed 3-manifold the knot genus problem asks, given a knot K and an integer g, whether the genus of K is equal to g. In joint work with Lackenby, we prove that for any fixed, compact, orientable 3-manifold, the knot genus problem lies inNP, answering a question of Agol--Hass--Thurston from 2002. Previously this was known for rational homology 3-spheres by the work of Lackenby.

 I will talk about optimal estimates for stochastic integrals
in the case when both rough paths and martingales play a role.

This is an ongoing joint work with Peter Friz (TU Berlin).

We derive generalized lower Ricci bounds in terms of signed measures. And we prove associated gradient estimates for the heat flow with Neumann boundary conditions on domains of metric measure spaces obtained through „convexification“ of the domains by means of subtle time changes. This improves upon previous results both in the case of non-convex domains and in the case of convex domains.
 

We  show that homogeneous Einstein metrics on Euclidean spaces are Einstein solvmanifolds, using that they admit periodic, integrally minimal foliations by homogeneous hypersurfaces. For the geometric flow induced by the orbit-Einstein condition, we construct a Lyapunov function based on curvature estimates which come from real GIT.

I will discuss the origin of the choice of global structure
--- or equivalently, the choice for which higher p-form symmetries are
present in the theory --- for various (Lagrangian and non-Lagrangian)
field theories in terms of their realization in IIB and M-theory. I
will explain how this choice on the field theory side can be traced
back to the fact that fluxes in string/M-theory do not commute in the
presence of torsion. I will illustrate how these ideas provide a
stringy explanation for the fact that six-dimensional (2,0) and (1,0)
theories generically have a partition vector (as opposed to a partition
function) and explain how this reproduces the classification of N=4
theories provided by Aharony, Seiberg and Tachikawa. Time permitting, I
will also explain how to use these ideas to obtain the algebra of
higher p-form symmetries for 5d SCFTs arising from M-theory at
arbitrary isolated toric singularities, and to classify global forms
for various 4d theories in the presence of duality defects.

Mathematics has provided us with several extremely useful tools to apply in the world beyond mathematics.  But it also provides us with mathematicians -- individuals who have trained habits of careful thinking in domains where that is the only way to make progress. This talk will explore some other domains -- such as saying sensible things about the long-term future, or how to identify good actions in the world -- where this style of thinking seems particularly desirable as progress can otherwise be elusive or illusory.  It will also consider how a mathematician's curiosity can help to identify important questions.

The collapse of granular columns in a viscous fluid is a common model case for submarine geophysical flows. In immersed granular collapses, dense packings result in slow dynamics and short runout distances, while loose packings are associated with fast dynamics and long runout distances. However, the underlying mechanisms of the triggering and runout, particularly regarding the complex fluid-particle interactions at the pore-scale, are yet to be fully understood. In this study, a three-dimensional approach coupling the Lattice Boltzmann Method and the Discrete Element Method is adopted to investigate the influence of packing density on the collapsing dynamics. The direct numerical simulation of fluid-particle interactions provides evidence of the pore pressure feedback mechanism. In dense cases, a strong arborescent contact force network can form to prevent particles from sliding, resulting in a creeping failure behavior. In contrast, the granular phase is liquefied substantially in loose cases, leading to a rapid and catastrophic failure. Furthermore, hydroplaning can take place in loose cases due to the fast-moving surge front, which reduces the frictional resistance dramatically and thereby results in a longer runout distance. More quantitatively, we are able to linearly correlate the normalized runout distance and the densimetric Froude number across a wide range of length scales, including small-scale numerical/experimental data and large-scale field data.

I will present numerical methods for low-rank matrix and tensor problems that explicitly make use of the geometry of rank constrained matrix and tensor spaces. We focus on two types of problems: The first are optimization problems, like matrix and tensor completion, solving linear systems and eigenvalue problems. Such problems can be solved by numerical optimization for manifolds, called Riemannian optimization methods. We will explain the basic elements of differential geometry in order to apply such methods efficiently to rank constrained matrix and tensor spaces. The second type of problem is ordinary differential equations, defined on matrix and tensor spaces. We show how their solution can be approximated by the dynamical low-rank principle, and discuss several numerical integrators that rely in an essential way on geometric properties that are characteristic to sets of low rank matrices and tensors. Based on joint work with André Uschmajew (MPI MiS Leipzig).

Background: The traditional business models for B2B freight and distribution are struggling with underutilised transport capacities resulting in higher costs, excessive environmental damage and unnecessary congestion. The scale of the problem is captured by the European Environmental Agency: only 63% of journeys carry useful load and the average vehicle utilisation is under 60% (by weight or volume). Decarbonisation of vehicles would address only part of the problem. That is why leading sector researchers estimate that freight collaboration (co-shipment) will deliver a step change improvement in vehicle fill and thus remove unproductive journeys delivering over 20% of cost savings and >25% reduction in environmental footprint. However, these benefits can only be achieved at a scale that involves 100’s of players collaborating at a national or pan-regional level. Such scale and level of complexity creates a massive optimisation challenge that current market solutions are unable to handle (modern route planning solutions optimise deliveries only within the “4 walls” of a single business).

Maths challenge: The mentioned above optimisation challenge could be expressed as an extended version of the TSP, but with multiple optimisation objectives (other than distance). Moreover, besides the scale and multi-agent setup (many shippers, carriers and recipients engaged simultaneously) the model would have to operate a number of variables and constraints, which in addition to the obvious ones also include: time (despatch/delivery dates/slots and journey durations), volume (items to be delivered), transport equipment with respective rate-cards from different carriers, et al. With the possible variability of despatch locations (when clients have multi-warehouse setup) this potentially creates a very-large non-convex optimisation problem that would require development of new, much faster algorithms and approaches. Such algorithm should be capable of finding “local” optimums and subsequently improve them within a very short window i.e. in minutes, which would be required to drive and manage effective inter-company collaboration across many parties involved. We tried a few different approaches eg used Gurobi solver, which even with clustering was still too slow and lacked scalability, only to realise that we need to build such an algorithm in-house.

Ask: We started to investigate other approaches like Simulated Annealing or Gravitational Emulation Local Search but this work is preliminary and new and better ideas are of interest. So in support of our Technical Feasibility study we are looking for support in identification of the best approach and design of the actual algorithm that we’ll use in the development of our Proof of Concept.