Past

It was shown by Balasubramanya that any acylindrically hyperbolic group (a natural generalisation of a hyperbolic group) must act acylindrically and non-elementarily on some quasi-tree. It is therefore sensible to ask to what extent this is true for trees, i.e. given an acylindrically hyperbolic group, does it admit a non-elementary acylindrical action on some simplicial tree? In this talk I will introduce the concepts of acylindrically hyperbolic and acylindrically arboreal groups and discuss some particularly interesting examples of acylindrically hyperbolic groups which do and do not act acylindrically on trees.

We review progress made by our group on soft matter at interfaces and related physics from the nano- to macroscopic lengthscales. Specifically, to capture nanoscale properties very close to interfaces and to establish a link to the macroscale behaviour, we employ elements from the statistical mechanics of classical fluids, namely density-functional theory (DFT). We formulate a new and general dynamic DFT that carefully and systematically accounts for the fundamental elements of any classical fluid and soft matter system, a crucial step towards the accurate and predictive modelling of physically relevant systems. In a certain limit, our DDFT reduces to a non-local Navier-Stokes-like equation that we refer to as hydrodynamic DDFT: an inherently multiscale model, bridging the micro- to the macroscale, and retaining the relevant fundamental microscopic information (fluid temperature, fluid-fluid and wall-fluid interactions) at the macroscopic level.

Work analysing the moving contact line in both equilibrium and dynamics will be presented. This has been a longstanding problem for fluid dynamics with a major challenge being its multiscale nature, whereby nanoscale phenomena manifest themselves at the macroscale. A key property captured by DFT at equilibrium, is the fluid layering on the wall-fluid interface, amplified as the contact angle decreases. DFT also allows us to unravel novel phase transitions of fluids in confinement. In dynamics, hydrodynamic DDFT allows us to benchmark existing phenomenological models and reproduce some of their key ingredients. But its multiscale nature also allows us to unravel the underlying physics of moving contact lines, not possible with any of the previous approaches, and indeed show that the physics is much more intricate than the previous models suggest.

We will close with recent efforts on machine learning and DFT. In particular, the development of a novel data-driven physics-informed framework for the solution of the inverse problem of statistical mechanics: given experimental data on the collective motion of a classical many-body system, obtain the state functions, such as free-energy functionals.

email Hannah Hughes for further information.

We discuss Connes’ quantized calculus on quantum tori and Euclidean spaces, as applications of the recent development of noncommutative analysis.
This talk is based on a joint work in progress with Xiao Xiong and Kai Zeng.
 

Abstract: It’s a classical result by Huber that the number of closed geodesics of length bounded by L on a closed hyperbolic surface S is asymptotic to exp(L)/L as L grows. This result has been generalized in many directions, for example by counting certain subsets of closed geodesics. One such result is the asymptotic growth of those that are homologically trivial, proved independently by both by Phillips-Sarnak and Katsura-Sunada. A homologically trivial curve can be written as a product of commutators, and in this talk we will look at those that can be written as a product of g commutators (in a sense, those that bound a genus g subsurface) and obtain their asymptotic growth. As a special case, our methods give a geometric proof of Huber’s classical theorem. This is joint work with Juan Souto. 

An $n$-vertex graph $G$ is a $C$-expander if $|N(X)|\geq C|X|$ for every $X\subseteq V(G)$ with $|X|< n/2C$ and there is an edge between every two disjoint sets of at least $n/2C$ vertices.

We show that there is some constant $C>0$ for which every $C$-expander is Hamiltonian. In particular, this implies the well known conjecture of Krivelevich and Sudakov from 2003 on Hamilton cycles in $(n,d,\lambda)$-graphs. This completes a long line of research on the Hamiltonicity of sparse graphs, and has many applications.

Joint work with R. Montgomery, D. Munhá Correia, A. Pokrovskiy and B. Sudakov.

High-order tensor methods for solving both convex and nonconvex optimization problems have recently generated significant research interest, due in part to the natural way in which higher derivatives can be incorporated into adaptive regularization frameworks, leading to algorithms with optimal global rates of convergence and local rates that are faster than Newton's method. On each iteration, to find the next solution approximation, these methods require the unconstrained local minimization of a (potentially nonconvex) multivariate polynomial of degree higher than two, constructed using third-order (or higher) derivative information, and regularized by an appropriate power of the change in the iterates. Developing efficient techniques for the solution of such subproblems is currently, an ongoing topic of research,  and this talk addresses this question for the case of the third-order tensor subproblem.

In particular, we propose the CQR algorithmic framework, for minimizing a nonconvex Cubic multivariate polynomial with  Quartic Regularisation, by sequentially minimizing a sequence of local quadratic models that also incorporate both simple cubic and quartic terms. The role of the cubic term is to crudely approximate local tensor information, while the quartic one provides model regularization and controls progress. We provide necessary and sufficient optimality conditions that fully characterise the global minimizers of these cubic-quartic models. We then turn these conditions into secular equations that can be solved using nonlinear eigenvalue techniques. We show, using our optimality characterisations, that a CQR algorithmic variant has the optimal-order evaluation complexity of $O(\epsilon^{-3/2})$ when applied to minimizing our quartically-regularised cubic subproblem, which can be further improved in special cases.  We propose practical CQR variants that judiciously use local tensor information to construct the local cubic-quartic models. We test these variants numerically and observe them to be competitive with ARC and other subproblem solvers on typical instances and even superior on ill-conditioned subproblems with special structure.

Iwasawa algebras are completions of group algebras for p-adic Lie groups, and have applications for studying the representations of these groups. It is an ongoing project to study the prime ideals, and more generally the two-sided ideals, of these algebras.

In the case of Iwasawa algebras corresponding to a simple Lie algebra with a Chevalley basis, we aim to prove that all non-zero two-sided ideals have finite codimension. To prove this, it is sufficient to show faithfulness of modules arising from highest-weight modules for the corresponding Lie algebra.

I have proved two main results in this direction: firstly, I proved the faithfulness of generalised Verma modules over the Iwasawa algebra. Secondly, I proved the faithfulness of all infinite-dimensional highest-weight modules in the case where the Lie algebra has type A. In this talk, I will outline the methods I used to prove these cases.

We discuss timelike surfaces of finite size in general relativity and the initial boundary value problem. We consider obstructions with the standard Dirichlet problem, and conformal version with improved properties. The ensuing dynamical features are discussed with general cosmological constant.

I will give an overview of a recent method introduced by P. Duch to solve some subcritical singular SPDEs, in particular the stochastic quantisation equation for scalar fields. 

In this seminar, I will talk about a notion of entropy of arithmetic functions and some properties of this entropy.  This notion was introduced to study Sarnak's Moebius Disjointness Conjecture.

I will speak about a central question in higher algebra (aka brave new algebra), namely which rings or schemes admit 'higher models', that is lifts to the sphere spectrum. This question is in some sense very classical, but there are many open questions. These questions are closely related to questions about higher versions of prismatic cohomology and delta ring, asked e.g. by Scholze and Lurie. Concretely we will consider the case of group algebras and explain how to understand maps between lifts of group algebras to the sphere spectrum. The results we present are joint with Carmeli and Yuan and on the prismatic side with Antieau and Krause.

Rough path theory provides a framework for the study of nonlinear systems driven by highly oscillatory (deterministic) signals. The corresponding analysis is inherently distinct from that of classical stochastic calculus, and neither theory alone is able to satisfactorily handle hybrid systems driven by both rough and stochastic noise. The introduction of the stochastic sewing lemma (Khoa Lê, 2020) has paved the way for a theory which can efficiently handle such hybrid systems. In this talk, we will discuss how this can be done in a general setting which allows for jump discontinuities in both sources of noise.

For a compact Lie group $G$, its massless Coulomb branch algebra is the $G$-equivariant Borel-Moore homology of its based loop space. This algebra is the same as the algebra of regular functions on the BFM space. In this talk, we will explain how this algebra acts on the equivariant symplectic cohomology of Hamiltonian $G$-manifolds when the symplectic manifolds are open and convex. This is a generalization of the closed case where symplectic cohomology is replaced with quantum cohomology. Following Teleman, we also explain how it relates to the Coulomb branch algebra of cotangent-type representations. This is joint work with Eduardo González and Dan Pomerleano.

Data that have an intrinsic network structure can be found in various contexts, including social networks, biological systems (e.g., protein-protein interactions, neuronal networks), information networks (computer networks, wireless sensor networks),  economic networks, etc. As the amount of graphical data that is generated is increasingly large, compressing such data for storage, transmission, or efficient processing has become a topic of interest. 

In this talk, I will give an information theoretic perspective on graph compression. The focus will be on compression limits and their scaling with the size of the graph. For lossless compression, the Shannon entropy gives the fundamental lower limit on the expected length of any compressed representation. 
I will discuss the entropy of some common random graph models, with a particular emphasis on our results on the random geometric graph model. 
Then, I will talk about the problem of compressing a graph with side information, i.e., when an additional correlated graph is available at the decoder. Turning to lossy compression, where one accepts a certain amount of distortion between the original and reconstructed graphs, I will present theoretical limits to lossy compression that we obtained for the Erdős–Rényi and stochastic block models by using rate-distortion theory.

The conference ‘Beyond the Pipeline: Women and Non-binary People in Mathematics Day’ will be held at the University of Oxford on the 17th February 2024. This is a joint event between the Mathematrix and the Mirzakhani societies of the University of Oxford. It is kindly funded by the London Mathematical Society and the Mathematical Institute at the University of Oxford, with additional funding from industry sponsors. 

The metaphor of the 'leaky pipeline' for the decreasing number of women and other gender minorities in Mathematics is problematic and outdated. It conceals the real reasons that women and non-binary people choose to leave Mathematics. This conference, 'Beyond the Pipeline', aims to encourage women and non-binary people to pursue careers in Mathematics, to promote women and non-binary role models, and to create a community of like-minded people. 

Speakers: 

• Brigitte Stenhouse, The Open University
• Mura Yakerson, The University of Oxford
• Vandita Patel, The University of Manchester
• Melanie Rupflin, The University of Oxford
• Christl Donnelly, The University of Oxford

The conference will also include: 

• A panel discussion on careers in and out of academia
• Talks by early-career speakers
• Poster presentations
• 1:1 bookable appointments with our industry sponsors (Cisco, Jane Street, ING, and Optiver)
• Careers stands with our sponsors and the IMA

More information can be found on our website https://www.oxwomeninmaths2024.co.uk/.

This conference is open to everyone regardless of their gender identity. Registration is via the following google form https://forms.gle/cDGaeJCPbBFEPfDB6 and will close when we have reached capacity. We have limited travel funding to support travel to Oxford from within the UK and you can apply for this on the registration form. The deadline for those applying to give a talk and for those applying for travel funding is the 27th January.

If you have any questions email us at @email. 

Conferences and networking are important parts of academic life, particularly early in your academic career.  But how do you make the most out of conferences?  And what are the does and don'ts of networking?  Learn about the answers to these questions and more in this panel discussion by postdocs from across the Mathematical Institute.

The ability of biological matter to move and deform itself is facilitated by microscopic out-of-equilibrium processes that convert chemical energy into mechanical work. In many cases, this mechano-chemical activity takes place on effectively two-dimensional domains formed by, for example, multicellular structures like epithelial tissues or the outer surface of eukaryotic cells, the so-called actomyosin cortex.
We will show in the first part of the talk, that the large-scale dynamics and self-organisation of such structures can be captured by the theory of active fluids. Specifically, using a minimal model of active isotropic fluids, we can rationalize the emergence of asymmetric epithelial tissue flows in the flower beetle during early development, and explain cell rotations in the context of active chiral flows and left-right symmetry breaking that occurs as the model organism C. elegans sets up its body plan.
To develop a more general understanding of such processes, specifically the role of geometry, curvature and interactions with the environment, we introduce in the second part a theory of active fluid surfaces and discuss analytical and numerical tools to solve the corresponding momentum balance equations of curved and deforming surfaces. By considering mechanical interactions with the environment and the fully self-organized shape dynamics of active surfaces, these tools reveal novel mechanisms of symmetry breaking and pattern formation in active matter.

Perverse equivalences, introduced by Chuang and Rouquier, are derived equivalences with a particularly nice combinatorial description. This generalised an earlier construction, with which they proved Broué’s abelian defect group conjecture for blocks of the symmetric groups. Perverse equivalences are of much wider significance in the representation theory of finite dimensional symmetric algebras. Grant has shown that periodic algebras admit perverse autoequivalences. In a similar vein, I will present some perverse equivalences arising from certain periodic modules, with an application to the setting of the symmetric groups.

I am going to discuss main results of my paper "Physics over a finite field and Wick rotation", arxiv 2306.15698. It introduces a structure over a pseudo-finite field which might be of interest in Foundations of Physics. The main theorem establishes an analogue of the polar co-ordinate system in the pseudo-finite field. A stability classification status of the structure is an open question.

Often, one tries to understand the behaviour of non-commutative random variables or of von Neumann algebras through matricial approximations. In some cases, such as when appealing to the determinant conjecture or investigating the soficity of a group, it is important to find approximations by matrices with good algebraic conditions on their entries (e.g., being integers). On the other hand, the most common tool for generating asymptotic independence -- conjugating with random unitaries -- often destroys such delicate structure.

 I will speak on recent joint work with de Santiago, Hayes, Jekel, Kunnawalkam Elayavalli, and Nelson, where we investigate graph products (an interpolation between free and tensor products) and conjugation of matrix models by large structured random permutations. We show that with careful control of how the permutation matrices are chosen, we can achieve asymptotic graph independence with amalgamation over the diagonal matrices. We are able to use this fine structure to prove that strong $1$-boundedness for a large class of graph product von Neumann algebras follows from the vanishing of the corresponding first $L^2$-Betti number. The main idea here is to show that a version of the determinant conjecture holds as long as the individual algebras have generators with approximations by matrices with entries in the ring of integers of some finite extension of Q satisfying some conditions strongly reminiscent of soficity for groups.

A plethora of stochastic models used in particular in mathematical finance, but also population genetics and physics, stems from the class of affine and polynomial processes. The history of these processes is on the one hand closely connected with the important concept of tractability, that is a substantial reduction of computational efforts due to special structural features, and on the other hand with a unifying framework for a large number of probabilistic models. One early instance in the literature where this unifying affine and polynomial point of view can be applied is Lévy's stochastic area formula. Starting from this example,  we present a guided tour through the main properties and recent results, which lead to signature stochastic differential equations (SDEs). They constitute a large class of stochastic processes, here driven by Brownian motions, whose characteristics are entire or real-analytic functions of their own signature, i.e. of iterated integrals of the process with itself, and allow therefore for a generic path dependence. We show that their prolongation with the corresponding signature is an affine and polynomial process taking values in subsets of group-like elements of the extended tensor algebra. Signature SDEs are thus a class of stochastic processes, which is universal within Itô processes with path-dependent characteristics and which allows - due to the affine theory - for a relatively explicit characterization of the Fourier-Laplace transform and hence the full law on path space.

We show the primes have level of distribution 66/107 using triply well-factorable weights. This gives the highest level of distribution for primes in any setting, improving on the prior record level 3/5 of Maynard. We also extend this level to 5/8, assuming Selberg's eigenvalue conjecture. As a result, we obtain new upper bounds for twin primes and for Goldbach representations of even numbers $a$. For the Goldbach problem, this is the first use of a level of distribution beyond the 'square-root barrier', and leads to the greatest improvement on the problem since Bombieri--Davenport from 1966.

We will discuss the physics of M-theory compactifications onto G2-orbifolds of the type that can be desingularised via the method of Joyce and Karigiannis i.e. orbifolds where one has a singular locus of A1 singularities that admits a nowhere-vanishing (Z2-twisted) harmonic 1-form. Interestingly, there are topologically distinct desingularisations of such orbifolds which we show can be physically interpreted as different branches of the 4d vacuum moduli space of the arising gauge theories: Coulomb and Higgs branches. The results suggest generalisations of the results of Joyce and Karigiannis to G2-orbifolds with more diverse ADE singularities and higher order twists. As a bonus, we also get an isomorphism between the moduli space of flat connections on flat compact 3-manifolds and the moduli space of Ricci flat metrics on the G2-orbifolds. We will briefly discuss this. Based on 2309.12869 and 2312.12311.

As machine learning algorithms get integrated into the decision-making process of companies and organizations, insurance products are being developed to protect their providers from liability risk. Algorithmic liability differs from human liability since it is based on data-driven models compared to multiple heterogeneous decision-makers and its performance is known a priori for a given set of data. Traditional actuarial tools for human liability do not consider these properties, primarily focusing on the distribution of historical claims. We propose, for the first time, a quantitative framework to estimate the risk exposure of insurance contracts for machine-driven liability, introducing the concept of algorithmic insurance. Our work provides ML model developers and insurance providers with a comprehensive risk evaluation approach for this new class of products. Thus, we set the foundations of a niche area of research at the intersection of the literature in operations, risk management, and actuarial science. Specifically, we present an optimization formulation to estimate the risk exposure of a binary classification model given a pre-defined range of premiums. Our approach outlines how properties of the model, such as discrimination performance, interpretability, and generalizability, can influence the insurance contract evaluation. To showcase a practical implementation of the proposed framework, we present a case study of medical malpractice in the context of breast cancer detection. Our analysis focuses on measuring the effect of the model parameters on the expected financial loss and identifying the aspects of algorithmic performance that predominantly affect the risk of the contract.

Paper Reference: Bertsimas, D. and Orfanoudaki, A., 2021. Pricing algorithmic insurance. arXiv preprint arXiv:2106.00839.

Paper link: https://arxiv.org/pdf/2106.00839.pdf

Graph braid groups are similar to braid groups, except that they are defined as ‘braids’ on a graph, rather than the real plane. We can think of graph braid groups in terms of the discrete configuration space of a graph, which is a CW-complex. One can compute a presentation of a graph braid group using Morse theory. In this talk I will give a few examples on how to compute these presentations in terms of generating circuits of the graph. I will then go through a detailed example of a graph that gives a one-ended braid group.

We introduce a new class of groups with Thompson-like group properties. In the surface case, the asymptotic mapping class group contains mapping class groups of finite type surfaces with boundary. In dimension three, it contains automorphism groups of all finite rank free groups. I will explain how asymptotic mapping class groups act on a CAT(0) cube complex which allows us to show that they are of type F_infinity. 

This is joint work with Javier Aramayona, Kai-Uwe Bux, Jonas Flechsig and Xaolei Wu.

Brown-Erdős-Sós initiated the study of the maximum number of edges in an $n$-vertex $r$-graph such that no $k$ edges span at most $s$ vertices. If $s=rk-2k+2$ then this function is quadratic in $n$ and its asymptotic was previously known for $k=2,3,4$. I will present joint work with Stefan Glock, Jaehoon Kim, Lyuben Lichev and Shumin Sun where we resolve the cases $k=5,6,7$.

Porta and Yue Yu's model of derived analytic geometry takes as its category of basic, or affine, objects the category opposite to simplicial algebras over the entire functional calculus Lawvere theory. This is analogous to Lurie's approach to derived algebraic geometry where the Lawvere theory is the one governing simplicial commutative rings, and Spivak's derived smooth geometry, using the Lawvere theory of C-infinity-rings. Although there have been numerous important applications including GAGA, base-change, and Riemann-Hilbert theorems, these methods are still missing some crucial ingredients. For example, they do not naturally beget a good definition of quasi-coherent sheaves satisfying descent. On the other hand, the Toen-Vezzosi-Deligne approach of geometry relative to a symmetric monoidal category naturally provides a definition of a category of quasi-coherent sheaves, and in two such approaches to analytic geometry using the categories of bornological and condensed abelian groups respectively, these categories do satisfy descent.  In this talk I will explain how to compare the Porta and Yue Yu model of derived analytic geometry with the bornological one. More generally we give conditions on a Lawvere theory such that its simplicial algebras embed fully faithfully into commutative bornological algebras. Time permitting I will show how the Grothendieck topologies on both sides match up, allowing us to extend the embedding to stacks.

This is based on joint work with Oren Ben-Bassat and Kobi Kremnitzer, and follows work of Kremnitzer and Dennis Borisov.

The S-Matrix in flat space is a naturally holographic observable. S-Matrix elements thus contain valuable information about the putative dual CFT. In this talk, I will first introduce some basic aspects of Celestial Holography and then explain how these can be inferred directly from scattering amplitudes. I will then focus on how the singularity structure of amplitudes interplays with traditional CFT structures particularly in the context of the operator product expansion (OPE) of the dual CFT. I will conclude with some discussion about the role played by supersymmetry in simplifying the putative dual CFT.
 

First we will show that the continuum counterpart of the discrete individual-based mechanical model that describes the dynamics of two contiguous cell populations is given by a free-boundary problem for the cell densities.  Then, in addition to interactions, we will consider the microscopic movement of cells and derive a fractional cross-diffusion system as the many-particle limit of a multi-species system of moderately interacting particles.

First we will show that the continuum counterpart of the discrete individual-based mechanical model that describes the dynamics of two contiguous cell populations is given by a free-boundary problem for the cell densities.  Then, in addition to interactions, we will consider the microscopic movement of cells and derive a fractional cross-diffusion system as the many-particle limit of a multi-species system of moderately interacting particles.

Let $E$ be an elliptic curve over a number field $K$ and $n \geq 2$ an integer. We recall that elements of the $n$-Selmer group of $E/K$ can be explicitly written in terms of certain equations for $n$-coverings of $E/K$. Writing the elements in this way is called conducting an explicit $n$-descent. One of the applications of explicit $n$-descent is in finding generators of large height for $E(K)$ and from this point of view one would like to be able to take $n$ as large as possible. General algorithms for explicit $n$-descent exist but become computationally challenging already for $n \geq 5$. In this talk we discuss combining $n$- and $(n+1)$-descents to $n(n+1)$-descent and the role that invariant theory plays in this procedure.

The handlebody group is defined to be the mapping class group of a handelbody (rel. boundary). It is a subgroup of the mapping class group of the surface of the handlebody, and maps onto the outer automorphism group of its fundamental group (the free group of rank equal to its genus). 

Recently Hainaut and Petersen described a subspace of moduli space forming an orbifold classifying space for the handlebody group, and combined this with work of Chan-Galatius-Payne to construct cohomology classes in the group. I will talk about how one can build on their ideas to define a cocompact EG for the handlebody group inside Teichmüller space. This is a manifold with boundary and comes with a filtration by labelled disk systems which we call the `RGB (red-green-blue) disk complex.' I will describe this filtration, use it to describe the boundary of the manifold, and speculate about potential applications to duality results. Based on work-in-progress with Dan Petersen.

This presentation focuses on the study of the regulartiy of random wavelet series. We first study their belonging to certain functional spaces and we compare these results with long-established results related to random Fourier series. Next, we show how the study of random wavelet series leads to precise pointwise regularity properties of processes like fractional Brownian motion. Additionally, we explore how these series helps create Gaussian processes  with random Hölder exponents.

On homogeneous spaces, solutions to the prescribed Ricci curvature equation coincide with the critical points of the scalar curvature functional subject to a constraint. We provide a complete description of Palais--Smale sequences for this functional. As an application, we obtain new existence results for the prescribed Ricci curvature equation, which enables us to observe previously unseen phenomena. Joint work with Wolfgang Ziller (University of Pennsylvania).

The tremendous success of the Machine Learning paradigm heavily relies on the development of powerful optimization methods, and the canonical algorithm for training learning models is SGD (Stochastic Gradient Descent). Nevertheless, the latter is quite different from Gradient Descent (GD) which is its noiseless counterpart. Concretely, SGD requires a careful choice of the learning rate, which relies on the properties of the noise as well as the quality of initialization.

 It further requires the use of a test set to estimate the generalization error throughout its run. In this talk, we will present a new SGD variant that obtains the same optimal rates as SGD, while using noiseless machinery as in GD. Concretely, it enables to use the same fixed learning rate as GD and does not require to employ a test/validation set. Curiously, our results rely on a novel gradient estimate that combines two recent mechanisms which are related to the notion of momentum.

Finally, as much as time permits, I will discuss several applications where our method can be extended.

to follow

Oxford University Innovation, the University’s commercialisation team, will explain the support they can give to Maths researchers who want to generate commercial impact from their work and expertise. In addition to an overview of consulting, this talk will explain how mathematical techniques and software can be protected and commercialised.

In this possibly speculative talk I will try to outline a way to define analytic D-modules, using (higher) category theory and the ``six operations" on quasicoherent sheaves as the main tools. The aim is to follow the successful approach of Andy Jiang in the algebraic setting, who obtained such a theory without using stacks or formal schemes (as in Gaitsgory-Rozenblyum's approach). By using local cohomology, Jiang was able to avoid enlarging the category of algebras beyond the usual ones. We believe that an analytic variant of local cohomology can be used to recover the Ardakov-Wadsley theory of D-cap modules ``on the nose". (Work in progress).

Kaufman constructed a family of Fourier-decaying measures on the set of badly approximable numbers. Pollington and Velani used these to show that Littlewood’s conjecture holds for a full-dimensional set of pairs of badly approximable numbers. We construct analogous measures that have implications for inhomogeneous diophantine approximation. In joint work with Agamemnon Zafeiropoulos and Evgeniy Zorin, our idea is to shift the continued fraction and Ostrowski expansions simultaneously.