Past

One occasionally encounters computational problems that work just fine on ill-posed inputs, even though they should not. One example is polynomial eigenvalue problems, where standard algorithms such as QZ can find a desired solution to instances with infinite condition number to machine precision, while being completely oblivious to the ill-conditioning of the problem. One explanation is that, intuitively, adversarial perturbations are extremely unlikely, and "for all practical purposes'' the problem might not be ill-conditioned at all. We analyse perturbations of singular polynomial eigenvalue problems and derive methods to bound the likelihood of adversarial perturbations for any given input in different stochastic models.

Joint work with Vanni Noferini
 

Understanding how shifts of multiplicative functions correlate with each other is a central question in multiplicative number theory. A well-known conjecture of Elliott predicts that there should be no correlation between shifted multiplicative functions unless the functions involved are ‘pretentious functions’ in a certain precise sense. The Elliott conjecture implies as a special case the famous Chowla conjecture on shifted products of the Möbius function.

In the last few years, there has been a lot of exciting progress on the Chowla and Elliott conjectures, and we give an overview of this. Nearly all of the previously obtained results have concerned correlations that are weighted logarithmically, and it is an interesting question whether one can remove these logarithmic weights. We show that one can indeed remove logarithmic averaging from the known results on the Chowla and Elliott conjectures, provided that one restricts to almost all scales in a suitable sense.

This is joint work with Terry Tao.

Morse theory explores the topology of a smooth manifold $M$ by looking at the local behaviour of a fixed smooth function $f : M \to \mathbb{R}$. In this talk, I will explain how we can construct ordinary homology by looking at the flow of $\nabla f$ on the manifold. The talk should serve as an introduction to Morse theory for those new to the subject. At the end, I will state a new(ish) proof of the functoriality of Morse homology.

Welfare economics argues that competitive markets lead to efficient allocation of resources. The classical theorems are based on the Walrasian market model which assumes the existence of market clearing prices. The emergence of such prices remains debatable. We replace the Walrasian market model by double auctions and show that the conclusions of welfare economics remain largely the same. Double auctions are not only a more realistic description of real markets but they explain how equilibrium prices and efficient allocations emerge in practice. 

Caustics are places where the light intensity diverges, and where the wave front has a singularity. We use a self-similar description to derive the detailed spatial structure of a cusp singularity, from where caustic lines originate. We also study singularities of higher order, which have their own, uniquely three-dimensional structure. We use this insight to study shock formation in classical compressible Euler dynamics. The spatial structure of these shocks is that of a caustic, and is described by the same similarity equation.

$$
\def\curl#1{\left\{#1\right\}}
\def\vek#1{\mathbf{#1}}
$$
lll-posed problems arise often in the context of scientific applications in which one cannot directly observe the object or quantity of interest. However, indirect observations or measurements can be made, and the observable data $y$ can be represented as the wanted observation $x$ being acted upon by an operator $\mathcal{A}$. Thus we want to solve the operator equation \begin{equation}\label{eqn.Txy} \mathcal{A} x = y, \end{equation} (1) often formulated in some Hilbert space $H$ with $\mathcal{A}:H\rightarrow H$ and $x,y\in H$. The difficulty then is that these problems are generally ill-posed, and thus $x$ does not depend continuously on the on the right-hand side. As $y$ is often derived from measurements, one has instead a perturbed $y^{\delta}$ such that ${y - y^{\delta}}_{H}<\delta$. Thus due to the ill-posedness, solving (1) with $y^{\delta}$ is not guaranteed to produce a meaningful solution. One such class of techniques to treat such problems are the Tikhonov-regularization methods. One seeks in reconstructing the solution to balance fidelity to the data against size of some functional evaluation of the reconstructed image (e.g., the norm of the reconstruction) to mitigate the effects of the ill-posedness. For some $\lambda>0$, we solve \begin{equation}\label{eqn.tikh} x_{\lambda} = \textrm{argmin}_{\widetilde{x}\in H}\left\lbrace{\left\|{b - A\widetilde{x}} \right\|_{H}^{2} + \lambda \left\|{\widetilde{x}}\right\|_{H}^{2}} \right\rbrace. \end{equation} In this talk, we discuss some new strategies for treating discretized versions of this problem. Here, we consider a discreditized, finite dimensional version of (1), \begin{equation}\label{eqn.Axb} Ax =  b \mbox{ with }  A\in \mathbb{R}^{n\times n}\mbox{ and } b\in\mathbb{R}^{n}, \end{equation} which inherits a discrete version of ill conditioning from [1]. We propose methods built on top of the Arnoldi-Tikhonov method of Lewis and Reichel, whereby one builds the Krylov subspace \begin{equation}
\mathcal{K}_{j}(\vek A,\vek w) = {\rm span\,}\curl{\vek w,\vek A\vek w,\vek A^{2}\vek w,\ldots,\vek A^{j-1}\vek w}\mbox{ where } \vek w\in\curl{\vek b,\vek A\vek b}
\end{equation}
and solves the discretized Tikhonov minimization problem projected onto that subspace. We propose to extend this strategy to setting of augmented Krylov subspace methods. Thus, we project onto a sum of subspaces of the form $\mathcal{U} + \mathcal{K}_{j}$ where $\mathcal{U}$ is a fixed subspace and $\mathcal{K}_{j}$ is a Krylov subspace. It turns out there are multiple ways to do this leading to different algorithms. We will explain how these different methods arise mathematically and demonstrate their effectiveness on a few example problems. Along the way, some new mathematical properties of the Arnoldi-Tikhonov method are also proven.

An informal session for DPhil students, ECRs and undergraduates with an interest in probability. The aim is to gain exposure to areas outside of your own research interests in an informal and accessible way.

We will talk about the Cauchy problem of the three-dimensional isentropic compressible Navier-Stokes equations. When viscosity coefficients are given as a constant multiple of density's power, based on some analysis  of  the nonlinear structure of this system, by introducing some new variables and the initial layer compatibility conditions, we identify the class of initial data admitting a local regular solution with far field vacuum and  finite energy  in some inhomogeneous Sobolev spaces, which solves an open problem of degenerate viscous flow partially mentioned by Bresh-Desjardins-Metivier (2006, Anal. Simi. Fluid Dynam.),  Jiu-Wang-Xin (2014, JMFM) and so on. Moreover, in contrast to the classical well-posedness theory in the case of  the constant viscosity,   we show   that one can not obtain any global classical solution whose $L^\infty$  norm of $u$ decays to zero as time $t$ goes to infinity under the assumptions on the conservation laws of total mass and momentum.

"Fibre theorems" in the style of Quillen's fibre lemma are versatile tools used to study the topology of partially ordered sets. In this talk, I will formulate two of them and explain how these can be used to determine the homotopy type of the complex of (conjugacy classes of) free factors of a free group.
The latter is joint work with Radhika Gupta (see https://arxiv.org/abs/1810.09380).

TBA

In this talk we will introduce quantifier elimination and give various examples of theories with this property. We will see some very useful applications of quantifier elimination to algebra and geometry that will hopefully convince you how practical this property is to other areas of mathematics.

 
We study continuous theories of classes of finite dimensional Hilbert spaces expanded by 
a finite family (of a fixed size) of unitary operators. 
Infinite dimensional models of these theories are called pseudo finite dimensional dynamical Hilbert spaces. 
Our main results connect decidability questions of these theories with the topic of approximations of groups by metric groups. 

In the classical theory of fluid mechanics, a linear relationship between the stress and rate of strain is often assumed. Even when this relationship is non-linear, it is typically formulated in terms of an explicit relation. Implicit constitutive theories provide a theoretical framework that generalises this, allowing a, possibly multi-valued, implicit constitutive relation. Since it is not possible to solve explicitly for the stress in the constitutive relation, a more natural approach would be to include the stress as a fundamental unknown in the formulation of the problem. In this talk I will present a formulation with this feature and a proof of convergence of the finite element approximations to a solution of the original problem.

A fundamental task in numerical computation is the solution of large linear systems. The conjugate gradient method is an iterative method which offers rapid convergence to the solution, particularly when an effective preconditioner is employed. However, for more challenging systems a substantial error can be present even after many iterations have been performed. The estimates obtained in this case are of little value unless further information can be provided about the numerical error. In this paper we propose a novel statistical model for this numerical error set in a Bayesian framework. Our approach is a strict generalisation of the conjugate gradient method, which is recovered as the posterior mean for a particular choice of prior. The estimates obtained are analysed with Krylov subspace methods and a contraction result for the posterior is presented. The method is then analysed in a simulation study as well as being applied to a challenging problem in medical imaging.

In this work, study the mean first saturation time (MFST), a generalization to the mean first passage time, on networks and show an application to the 2015 Burundi refugee crisis. The MFST between a sink node j, with capacity s, and source node i, with n random walkers, is the average number of time steps that it takes for at least s of the random walkers to reach a sink node j. The same concept, under the name of extreme events, has been studied in previous work for degree biased-random walks [2]. We expand the literature by exploring the behaviour of the MFST for node-biased random walks [1] in Erdős–Rényi random graph and geographical networks. Furthermore, we apply MFST framework to study the distribution of refugees in camps for the 2015 Burundi refugee crisis. For this last application, we use the geographical network of the Burundi conflict zone in 2015 [3]. In this network, nodes are cities or refugee camps, and edges denote the distance between them. We model refugees as random walkers who are biased towards the refugee camps which can hold s_j people. To determine the source nodes (i) and the initial number of random walkers (n), we use data on where the conflicts happened and the number of refugees that arrive at any camp under a two-month period after the start of the conflict [3]. With such information, we divide the early stage of the Burundi 2015 conflict into two waves of refugees. Using the first wave of refugees we calibrate the biased parameter β of the random walk to best match the distribution of refugees on the camps. Then, we test the prediction of the distribution of refugees in camps for the second wave using the same biased parameters. Our results show that the biased random walk can capture, to some extent, the distribution of refugees in different camps. Finally, we test the probability of saturation for various camps. Our model suggests the saturation of one or two camps (Nakivale and Nyarugusu) when in reality only Nyarugusu camp saturated.

[1] Sood, Vishal, and Peter Grassberger. ”Localization transition of biased random walks on random
networks.” Physical review letters 99.9 (2007): 098701.
[2] Kishore, Vimal, M. S. Santhanam, and R. E. Amritkar. ”Extreme event-size fluctuations in biased
random walks on networks.” arXiv preprint arXiv:1112.2112 (2011).
[3] Suleimenova, Diana, David Bell, and Derek Groen. ”A generalized simulation development approach
for predicting refugee destinations.” Scientific reports 7.1 (2017): 13377.

A dedicated search of the CMB sky, driven by implications of conformal
cyclic cosmology (CCC), has revealed a remarkably strong signal, previously
unobserved, of numerous small regions in the CMB sky that would appear to be
individual points on CCC's crossover 3-surface from the previous aeon, most
readily interpreted as the conformally compressed Hawking radiation from
supermassive black holes in the previous aeon, but difficult to explain in
terms of the conventional inflationary picture.

We consider solutions to the two-dimensional incompressible Euler system with only integrable vorticity, thus with possibly locally infinite energy. With such regularity, we use the recently developed theory of Lagrangian flows associated to vector fields with gradient given by a singular integral in order to define Lagrangian solutions, for which the vorticity is transported by the flow. We prove strong stability of these solutions via strong convergence of the flow, under the only assumption of $L^1$ weak convergence of the initial vorticity. The existence of Lagrangian solutions to the Euler system follows for arbitrary $L^1$ vorticity. Relations with previously known notions of solutions are shown.

In directed algebraic topology, a topological space is endowed 
with an extra structure, a selected subset of the paths called the 
directed paths or the d-structure. The subset has to contain the 
constant paths, be closed under concatenation and non-decreasing 
reparametrization. A space with a d-structure is a d-space.
If the space has a partial order, the paths increasing wrt. that order 
form a d-structure, but the circle with counter clockwise paths as the 
d-structure is a prominent example without an underlying partial order.
Dipaths are dihomotopic if there is a one-parameter family of directed 
paths connecting them. Since in general dipaths do not have inverses, 
instead of fundamental groups (or groupoids), there is a fundamental 
category. So already at this stage, the algebra is less desirable than 
for topological spaces.
We will give examples of what is currently known in the area, the kind 
of methods used and the problems and questions which need answering - in 
particular with applications in computer science in mind.
 

The physics literature has for a long time posited a connection between the geometry of continuous random fields and discrete percolation models. Specifically the excursion sets of continuous fields are considered to be analogous to the open connected clusters of discrete models. Recent work has begun to formalise this relationship; many of the classic results of percolation (phase transition, RSW estimates etc) have been proven in the setting of smooth Gaussian fields. In the first part of this talk I will summarise these results. In the second I will focus on the number of excursion set components of Gaussian fields in large domains and discuss new results on the mean and variance of this quantity.

By using Malliavin calculus, Bismut type formulas are established for the Lions derivative of , where  0,  is a bounded measurable function,  and  solves a distribution dependent SDE with initial distribution . As applications, explicit estimates are derived for the Lions derivative and the total variational distance between distributions of   solutions with different initial data. Both degenerate and non-degenerate situations are considered. Due to the lack of the semi-group property  and the invalidity of the formula =  , essential difficulties are overcome in the study.

Joint work with Professor Feng-Yu Wang

In recent joint work with Lorenzo Foscolo and Johannes Nordstr\”om we gave an analytic construction of large families of complete circle-invariant $G_2$
holonomy metrics on the total space of circle bundles over a complete noncompact Calabi—Yau 3-fold with asymptotically conical geometry. The
asymptotic models for the geometry of these $G_2$ metrics are circle bundles with fibres of constant length $l$, so-called asymptotically local conical
(ALC) geometry. These ALC $G_2$ metrics can Gromov—Hausdorff collapse with bounded curvature to the given asymptotically conical Calabi—Yau 3-fold as the fibre length $l$ goes to $0$. A natural question is: what happens to these families of $G_2$ metrics as we try to make $l$ large? In general the answer to this question is not known, but in cases with sufficient symmetry we have recently been able to give a complete picture.  

We give an overview of all these results and discuss some analogies with the class of asymptotically locally flat (ALF) hyperkaehler 4-manifolds. In
particular we suggest that a particular $G_2$ metric we construct should be regarded as a $G_2$ analogue of the Euclidean Taub—NUT metric on the complex plane.

Our meeting will be a relaxed opportunity to have informal discussions about issues facing minorities in academia and mathematics over lunch. In particular, if anyone would like to suggest a topic to start a discussion about (either in advance or on the day) then please feel free to do this, and it could be a spring board for organised sessions on the same topics in future terms!

Higgs bundles are pairs of holomorphic vector bundles and holomorphic 1-forms taking values in the endomorphisms of the bundle. Their moduli spaces carry a natural Hyperkahler structure, through which one can study Lagrangian subspaces (A-branes) or holomorphic subspaces (B-branes). Notably, these A and B-branes have gained significant attention in string theory. After introducing Higgs bundles and the associated Hitchin fibration, we shall look at  natural constructions of families of different types of branes, and relate these spaces to the study of 3-manifolds, surface group representations and mirror symmetry.

Plumes are a characteristic feature of convective flow through porous media. Their dynamics are an important part of numerous geological processes, ranging from mixing in magma chambers to the convective dissolution of sequestered carbon dioxide. In this talk, I will discuss models for the spread of convective plumes in a heterogeneous porous environment. I will focus particularly on the effect of thin, roughly horizontal, low-permeability barriers to flow, which provide a generic form of heterogeneity in geological settings, and are a particularly widespread feature of sedimentary formations. With the aid of high-resolution numerical simulations, I will explore how a plume spreads and flows in the presence of one or more of these layers, and will briefly consider the implications of these findings in physical settings.

In this interactive workshop, we'll discuss what mathematicians are looking for in written solutions.  How can you set out your ideas clearly, and what are the standard mathematical conventions?  Please bring a pen or pencil! 

This session is likely to be most relevant for first-year undergraduates, but all are welcome.

Transdifferentiation, the process of converting from one cell type to another without going through a pluripotent state, has great promise for regenerative medicine. The identification of key transcription factors for reprogramming is limited by the cost of exhaustive experimental testing of plausible sets of factors, an approach that is inefficient and unscalable. We developed a predictive system (Mogrify) that combines gene expression data with regulatory network information to predict the reprogramming factors necessary to induce cell conversion. We have applied Mogrify to 173 human cell types and 134 tissues, defining an atlas of cellular reprogramming. Mogrify correctly predicts the transcription factors used in known transdifferentiations. Furthermore, we validated several new transdifferentiations predicted by Mogrify, including both into and out of the same cell type (keratinocytes). We provide a practical and efficient mechanism for systematically implementing novel cell conversions, facilitating the generalization of reprogramming of human cells. Predictions are made available via http://mogrify.net to help rapidly further the field of cell conversion.

At first glance the Interdistrict shipping problem resembles a transportation problem.  N sources with M destinations with k Stock keeping units (SKU’s); however, we want to solve for the optimal shipping frequency between each node while determining the flow of each SKU across the network.  As the replenishment quantity goes up, the shipping frequency goes down and the inventory holding cost goes up (AWI = Replenishment Qty/2 + SS).  Safety stock also increases as frequency decreases.  The relationship between replenishment quantity and shipping frequency is non-linear (frequency = annual demand/replenishment qty).  The trucks which are used to transfer the product have finite capacity and the cost to drive the truck between 2 locations is constant regardless of how many containers are actually on the truck up to the max capacity.  Each product can have a different footprint of truck capacity.  Cross docking is allowed.  (i.e. a truck may travel from Loc A to loc B carrying products X and Y.  At loc B, the truck unloads product X, picks up product Z, and continues to location C.  The key here is that product Y does not incur any handling costs at Loc B while products X and Z do.)

The objective function seeks to minimize the total costs ( distribution + handling + inventory holding costs)  for all locations, for all SKU’s, while determining how much of each product should flow across each arc such that all demand is satisfied.

We will see some results and conjectures on the zeta and multizeta values in the function field context, and see how they relate to homological-homotopical objects, such as t-motives, iterated extensions, and to Hopf algebras, big Galois representations.

In this talk, I will present an incomplete equilibrium model to determine the price of an annuity.  A finite number of agents receive stochastic income streams and choose between consumption and investment in the traded annuity.  The novelty of this model is its ability to handle running consumption and general income streams.  In particular, the model incorporates mean reverting income, which is empirically relevant but historically too intractable in equilibrium.  The model is set in a Brownian framework, and equilibrium is characterized and proven to exist using a system of fully coupled quadratic BSDEs.  This work is joint with Gordan Zitkovic.

Conical symplectic resolutions are one of the main objects in the contemporary mix of algebraic geometry and representation theory, 

known as geometric representation theory. They cover many interesting families of objects such as quiver varieties and hypertoric

varieties, and some simpler such as Springer resolutions. The last findings [Braverman, Finkelberg, Nakajima] say that they arise

as Higgs/Coulomb moduli spaces, coming from physics. Most of the gadgets attached to conical symplectic resolutions are rather

algebraic, such as their quatizations and $\mathcal{O}$-categories. We are rather interested in the symplectic topology of them, in particular 

finding smooth exact Lagrangians that appear in the central fiber of the (defining) resolution, as they are objects of the Fukaya category.

Many types of patterns emerging spontaneously can be observed in systems involving thin elastic plates and subjected to external or internal stresses (compression, differential growth, shearing, tearing, etc.). These mechanical systems can sometime be seen as model systems for more complex natural systems and allow to study in detail elementary emerging patterns. One of the simplest among such systems is a bilayer composed of a thin plate resting on a thick deformable substrate. Upon slight compression, periodic undulations (wrinkles) with a well-defined wavelength emerge at the level of the thin layer. We will show that, as the compression increases, this periodic state is unstable and that a second order transition to a localized state (fold) occurs when the substrate is a dense fluid.

Since the legendary 1972 encounter of H. Montgomery and F. Dyson at tea time in Princeton, a statistical correspondence of the non-trivial zeros of the Riemann Zeta function with eigenvalues of high-dimensional random matrices has emerged. Surrounded by many deep conjectures, there is a striking analogyto the energy levels of a quantum billiard system with chaotic dynamics. Thanks 
to extensive calculation of Riemann zeros by A. Odlyzko, overwhelming numerical evidence has been found for the quantum analogy. The statistical accuracy provided by an enormous dataset of more than one billion zeros reveals distinctive finite size effects. Using the physical analogy, a precise prediction of these effects was recently accomplished through the numerical evaluation of operator determinants and their perturbation series (joint work with P. Forrester and A. Mays, Melbourne).
 

We will consider an extension of the Eisenberg-Noe model of financial contagion to allow for time dynamics in both discrete and continuous time. Mathematical results on existence and uniqueness of firm wealths under discrete and continuous-time will be provided. The financial implications of time dynamics will be considered, with focus on how the dynamic clearing solutions differ from those of the static Eisenberg-Noe model.
 

In this talk I will present the recent developments in the topic of existence of solutions to the two-fluid systems. I will discuss the application of approach developed by P.-L. Lions and E. Feireisl and explain the limitations of this technique in the context of multi-component flow models. A particular example of such a model is two-fluids Stokes system with single velocity field and two densities, and with an algebraic pressure law closure. The existence result that uses the compactness criterion introduced for the Navier-Stokes system by D. Bresch and P.-E. Jabin will be presented. I will also mention an innovative construction of solutions relying on the G. Crippa and C. DeLellis stability estimates for the transport equation.

Graph products are a class of groups that 'interpolate' between direct and free products, and generalise the notion of right-angled Artin groups. Given a property that free products (and maybe direct products) are known to satisfy, a natural question arises: do graph products satisfy this property? For instance, it is known that graph products act on tree-like spaces (quasi-trees) in a nice way (acylindrically), just like free products. In the talk we will discuss a construction of such an action and, if time permits, its relation to solving systems of equations over graph products.

In this talk, we shall introduce various identities among partitions of integers, and how these can be expressed via formal power series. In particular, we shall look at the Rogers Ramanujan identities of power series, and discuss possible combinatorial proofs using partitions and Durfree squares.

This is a survey on recent advances in classical decidability issues for local and global fields and for some canonical infinite extensions of those.

In this informal talk, I describe the kinds of functions relevant to scattering amplitudes in perturbative, four-dimensional quantum field theories. In particular, I will argue that generic amplitudes are non-polylogarithmic (beyond one loop), but that there is an upper bound to their geometric complexity. Moreover, I show a veritable `bestiary' of examples which saturate this bound in complexity---including three, all-loop families of integrals defined in massless $\phi^4$ theory which can, at best, be represented as dilogarithms integrated over (2L-2)-dimensional Calabi-Yau manifolds. 

This work deals with exploiting the low-dimensional hierarchical structure of speech signals towards the  goal  of  improving  acoustic  modelling using deep neural networks (DNN).  To this aim the work employ tools from sparsity aware signal processing under novel frameworks to enrich  the  acoustic  information  present  in  DNN posterior features. 

We will discuss some recent results with Martin Bridson about 
Sidki's construction X(G). In particular, if G is a finitely presented
group then X(G) is a finitely presented group. We will discuss as well the
result that if G has polynomial isoperimetric function and the maximal
metabelian quotient of G is virtually nilpotent then X(G) has polynomial
isoperimetric function. Part of the arguments we will use have homological
nature.

Convex and in particular semidefinite relaxations (SDP) for non-convex continuous quadratic optimisation can provide tighter bounds than traditional linear relaxations. However, using SDP relaxations directly in Branch&Cut is impeded by lack of warm starting and inefficiency when combined with other cut classes, i.e. the reformulation-linearization technique. We present a general framework based on machine learning for a strong linear outer-approximation that can retain most tightness of such SDP relaxations, in the form of few strong low dimensional linear cuts selected offline. The cut selection complexity is taken offline by using a neural network estimator (trained before installing solver software) as a selection device for the strongest cuts. Lastly, we present results of our method on QP/QCQP problem instances.

Recovery of reaction pathways under the DCSC framework.

Graphs are a central object of study in various scientific fields, such as discrete mathematics, theoretical computer science and network science. These graphs are typically studied using combinatorial, algebraic or probabilistic methods, each of which highlights the properties of graphs in a unique way. I will discuss a novel approach to study graphs: the simplex geometry (a simplex is a generalized triangle). This perspective, proposed by Miroslav Fiedler, introduces techniques from (simplex) geometry into the field of graph theory and conversely, via an exact correspondence. We introduce the graph-simplex correspondence, identify a number of basic connections between graph characteristics and simplex properties, and suggest some applications as example.

Reference: https://arxiv.org/abs/1807.06475
 

The study of isolated systems has been vastly successful in the context of vanishing cosmological constant, Λ=0. However, there is no physically useful notion of asymptotics for the universe we inhabit with Λ>0.  The full non-linear framework is still under development, but some interesting results at the linearized level have been obtained. I will focus on the conceptual subtleties that arise at the linearized level and discuss the quadrupole formula for gravitational radiation as well as some recent developments.  

A particularly active area of research in modern algebraic number theory is the study of a class of numbers, called periods. In their simplest form, periods are integrals of rational functions over domains defined by rational in equations. They form a ring, which encompasses all algebraic numbers, logarithms thereof and \pi. They arise in the study of modular forms, cohomology and quantum field theory, and conjecturally have a sort of Galois theory.

We will take a whirlwind tour of these numbers, before discussing non-periods. In particular, we will sketch the construction of an explicit non-period, often forgotten about.