Past

Let $P$ be a random polynomial of degree $d$ such that the leading and constant coefficients are 1 and the rest of the coefficients are independent random variables taking the value 0 or 1 with equal probability. Odlyzko and Poonen conjectured that $P$ is irreducible with probability tending to 1 as $d$ grows.  I will talk about an on-going joint work with Emmanuel Breuillard, in which we prove that GRH implies this conjecture. The proof is based on estimates for the mixing time of random walks on $\mathbb{F}_p$, where the steps are given by the maps $x \rightarrow ax$ and $x \rightarrow ax+1$ with equal probability.

Title: Generalized McKean-Vlasov stochastic control problems.

Abstract: I will consider McKean-Vlasov stochastic control problems 
where the cost functions and the state dynamics depend upon the joint 
distribution of the controlled state and the control process. First, I 
will provide a suitable version of the Pontryagin stochastic maximum 
principle, showing that, in the present general framework, pointwise 
minimization of the Hamiltonian with respect to the control is not a 
necessary optimality condition. Then I will take a different 
perspective, and present a variational approach to study a weak 
formulation of such control problems, thereby establishing a new 
connection between those and optimal transport problems on path space.

The talk is based on a joint project with J. Backhoff-Veraguas and R. Carmona.

Elastic gridshells arise from the buckling of an initially planar grid of rods. Architectural elastic gridshells first appeared in the 1970’s. However, to date, only a limited number of examples have been constructed around the world, primarily due to the challenges involved in their structural design. Yet, elastic gridshells are highly appealing: they can cover wide spans with low self-weight, they allow for aesthetically pleasing shapes and their construction is typically simple and rapid. A more mundane example is the classic pasta strainer, which, with its remarkably simple design, is a must-have in every kitchen.

This talk will focus on the geometry-driven nature of elastic gridshells. We use a geometric model based on the theory of discrete Chebyshev nets (originally developed for woven fabric) to rationalize their actuated shapes. Validation is provided by precision experiments and rod-based simulations. We also investigate the linear mechanical response (rigidity) and the non-local behavior of these discrete shells under point-load indentation. Combining experiments, simulations, and scaling analysis leads to a master curve that relates the structural rigidity to the underlying geometric and material properties. Our results indicate that the mechanical response of elastic gridshells, and their underlying characteristic forces, are dictated by Euler's elastica rather than by shell-related quantities. The prominence of geometry that we identify in elastic gridshells should allow for our results to transfer across length scales: from architectural structures to micro/nano–1-df mechanical actuators and self-assembly systems.

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In this talk we will introduce and analyse a class of robust numerical methods for nonlocal possibly nonlinear diffusion and convection-diffusion equations. Diffusion and convection-diffusion models are popular in Physics, Chemistry, Engineering, and Economics, and in many models the diffusion is anomalous or nonlocal. This means that the underlying “particle" distributions are not Gaussian, but rather follow more general Levy distributions, distributions that need not have second moments and can satisfy (generalised) central limit theorems. We will focus on models with nonlinear possibly degenerate diffusions like fractional Porous Medium Equations, Fast Diffusion Equations, and Stefan (phase transition) Problems, with or without convection. The solutions of these problems can be very irregular and even possess shock discontinuities. The combination of nonlinear problems and irregular solutions makes these problems challenging to solve numerically.
The methods we will discuss are monotone finite difference quadrature methods that are robust in the sense that they “always” converge. By that we mean that under very weak assumptions, they converge to the correct generalised possibly discontinuous generalised solution. In some cases we can also obtain error estimates. The plan of the talk is: 1. to give a short introduction to the models, 2. explain the numerical methods, 3. give results and elements of the analysis for pure diffusion equations, and 4. give results and ideas of the analysis for convection-diffusion equations. 
 

In 2012 Eskin, Fisher and Whyte proved there was a locally finite vertex transitive graph which was not quasi-isometric to any connected locally finite Cayley Graph. This motivates the study of vertex transitive graphs from a geometric group theory point of view. We will discus how concepts and problems from group theory generalise to this setting. Constructing one framework in which problems can be framed so that techniques from group theory can be applied. This is work in progress with Agelos Georgakopoulos.

TBA

Part of the series 'What do historians of mathematics do?'

In this talk, we will survey the movement of mathematical ideas in the 17th century. We will explore, in particular, the mathematical cultures of Paris, Amsterdam, Rome, Cape Town, Goa, Kyoto, Beijing, and London, as well as the journey of mathematical knowledge on a global scale. As it will be an ambitious task to complete a round-the-world history tour in an hour, the focus will be on East Asia. By employing the digital humanities technique, this presentation will use digital media to effectively show historical sources and help the audience imagine the world as a “round” entity when we discuss a global history of mathematics.

I will present a variation of positive model theory which addresses the issues of approximations of conventional geometric structures by sequences of Zariski structures as well as approximation by sequences of finite structures. In particular I am interested in applications to quantum mechanics.

I will report on a progress in defining and calculating oscillating in- tegrals of importance in quantum physics. This is based on calculating Gauss sums of order higher or equal to 2 over rings Z/mZ for very specific m. 

Some natural moduli problems give rise to stacks with infinite stabilizers. I will report on recent work with Dan Edidin where we give a canonical sequence of saturated blow-ups that makes the stabilizers finite. This generalizes earlier work in GIT by Kirwan and Reichstein, and on toric stacks by Edidin-More. Time permitting, I will also mention a recent application to generalized Donaldson-Thomas invariants by Kiem-Li-Savvas.

We construct a space of vector fields that are normal to differentiable curves in the plane. Its basis functions are defined via saddle point variational problems in reproducing kernel Hilbert spaces (RKHSs). First, we study the properties of these basis vector fields and show how to approximate them. Then, we employ this basis to discretise shape Newton methods and investigate the impact of this discretisation on convergence rates.

We prove that the dimension of every poset whose comparability graph has maximum degree $\Delta$ is at most $\Delta\log^{1+o(1)} \Delta$. This result improves on a 30-year old bound of F\"uredi and Kahn, and is within a $\log^{o(1)}\Delta$ factor of optimal. We prove this result via the notion of boxicity. The boxicity of a graph $G$ is the minimum integer $d$ such that $G$ is the intersection graph of $d$-dimensional axis-aligned boxes. We prove that every graph with maximum degree $\Delta$ has boxicity at most $\Delta\log^{1+o(1)} \Delta$, which is also within a $\log^{o(1)}\Delta$ factor of optimal. We also show that the maximum boxicity of graphs with Euler genus $g$ is $\Theta(\sqrt{g \log g})$, which solves an open problem of Esperet and Joret and is tight up to a $O(1)$ factor. This is joint work with Alex Scott (arXiv:1804.03271).

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, which is typically solved using the Gauss-Newton method. In this talk, I will discuss how dimensionality reduction methods such as block coordinates and randomised sketching can be used to improve the scalability of Gauss-Newton for bundle adjustment problems.

Over the past decade, the randomized singular value decomposition (RSVD)
algorithm has proven to be an efficient, reliable alternative to classical
algorithms for computing low-rank approximations in a number of applications.
However, in cases where no information is available on the singular value
decay of the data matrix or the data matrix is known to be close to full-rank,
the RSVD is ineffective. In recent years, there has been great interest in
randomized algorithms for computing full factorizations that excel in this
regime.  In this talk, we will give a brief overview of some key ideas in
randomized numerical linear algebra and introduce a new randomized algorithm for
computing a full, rank-revealing URV factorization.

I will present a procedure for perturbatively constructing the field content of gravitational theories from a convolutive product of two Yang-Mills theories. A dictionary "gravity=YM * YM" is developed, reproducing the symmetries and dynamics of the gravity theory from those of the YM theories. I will explain the unexpected, yet crucial role played by the BRST ghosts of the YM system in the construction of gravitational fields. The dictionary is expected to develop into a solution-generating technique for gravity.
 

Wikipedia has more than 40 million articles in 280 languages. It represents a decent coverage of human knowledge.
Even with its biases it can tell us a lot about what's important for people. London has an article in 238 languages and
Swansea has in 73 languages. Is London more "culturally" important than Swansea? Probably. 
We use this information and look at various factors that could help us model "cultural" importance of a city and hence
try to find the driving force behind sister city relationships.
We also look at creating cultural maps of different cities, finding the artsy/hipster, academic, political neighbourhoods of a city.

Hilbert's 19th problem asks if minimizers of "natural" variational integrals are smooth. For the past century, this problem inspired fundamental regularity results for elliptic and parabolic PDES. It also led to the construction of several beautiful counterexamples to regularity. The dichotomy of regularity vs. singularity is related to that of single PDE (the scalar case) vs. system of PDEs (the vectorial case), and low dimension vs. high dimension. I will discuss some interesting recent counterexamples to regularity in low-dimensional vectorial cases, and outstanding open problems. Parts of this are joint works with A. Figalli and O. Savin.

Stochastic analysis on a Riemannian manifold is a well developed area of research in probability theory.

We will discuss some recent developments on stochastic analysis on a manifold whose Riemannian metric evolves with time, a typical case of which is the Ricci flow. Familiar results such as stochastic parallel transport, integration by parts formula, martingale representation theorem, and functional inequalities have interesting extensions from

time independent metrics to time dependent ones. In particular, we will discuss an extension of Beckner’s inequality on the path space over a Riemannian manifold with time-dependent metrics. The classical version of this inequality includes the Poincare inequality and the logarithmic Sobolev inequality as special cases.

I will prove that the knot Floer homology group
HFK-hat(K, g-1) for a genus g fibered knot K is isomorphic to a
variant of the fixed points Floer homology of an area-preserving
representative of its monodromy. This is a joint work with Gilberto
Spano.
 

A C^infinity scheme is a version of a scheme that uses a maximal spectrum. The category of C^infinity schemes contains the category of Manifolds as a full subcategory, as well as being closed under fibre products. In other words, this category is equipped to handle intersection singularities of smooth spaces.

While originally defined in the set up of Synthetic Differential Geometry, C^infinity schemes have more recently been used to describe derived manifolds, for example, the d-manifolds of Joyce. There are applications of this in Symplectic Geometry, such as the describing the moduli space of J-holomorphic forms.

In this talk, I will describe the category of C^infinity schemes, and how this idea can be extended to manifolds with corners. If time, I will mention the applications of this in derived geometry.

The signature of a parametric curve is a sequence of tensors whose entries are iterated integrals, and they are central to the theory of rough paths in stochastic analysis.  For some special families of curves, such as polynomial paths and piecewise-linear paths, their parametrized signature tensors trace out algebraic varieties in the space of all tensors. We introduce these varieties and examine their fundamental properties, while highlighting their intimate connection to the problem of recovering a path from its signature. This is joint work with Peter Friz and Bernd Sturmfels. 

I will overview my research for a general math audience.

 First I will present the biological questions and motivate why systems biology needs computational algebraic biology and topological data analysis. Then I will present the mathematical methods I've developed to study these biological systems. Throughout I will provide examples.

Jan Sbierski

Title: On the unique evolution of solutions to wave equations

Abstract: An important aspect of any physical theory is the ability to predict the future of a system in terms of an initial configuration. This talk focuses on wave equations, which underlie many physical theories. We first present an example of a quasilinear wave equation for which unique predictability in fact fails and then turn to conditions which guarantee predictability. The talk is based on joint work with Felicity Eperon and Harvey Reall.

Andrew Krause

Title: Surprising Dynamics due to Spatial Heterogeneity in Reaction-Diffusion Systems

Abstract: Since Turing's original work, Reaction-Diffusion systems have been used to understand patterning processes during the development of a variety of organisms, as well as emergent patterns in other situations (e.g. chemical oscillators). Motivated by understanding hair follicle formation in the developing mouse, we explore the use of spatial heterogeneity as a form of developmental tuning of a Turing pattern to match experimental observations of size and wavelength modulation in embryonic hair placodes. While spatial heterogeneity was nascent in Turing's original work, much work remains to understand its effects in Reaction-Diffusion processes. We demonstrate novel effects due to heterogeneity in two-component Reaction-Diffusion systems and explore how this affects typical spatial and temporal patterning. We find a novel instability which gives rise to periodic creation, translation, and destruction of spikes in several classical reaction-diffusion systems and demonstrate that this periodic spatiotemporal behaviour appears robustly away from Hopf regimes or other oscillatory instabilities. We provide some evidence for the universal nature of this phenomenon and use it as an exemplar of the mostly unexplored territory of explicit heterogeneity in pattern formation.
 

Classical work of Jeffery from 1922 established how at low Reynolds number, ellipsoids in steady shear flow undergo periodic motion with non-uniform rotation rate, termed 'Jeffery orbits'.  I will present two problems where Jeffery orbits play a critical role in understanding the transport and aggregation of rod-shaped organisms.  I will discuss the trapping of motile chemotactic bacteria in high shear, and the sedimentation rate of negatively buoyant plankton.