Past

This session is particularly aimed at fourth-year and OMMS students who are completing a dissertation this year. The talk will be given by Dr Richard Earl who chairs Projects Committee. For many of you this will be the first time you have written such an extended piece on mathematics. The talk will include advice on planning a timetable, managing the workload, presenting mathematics, structuring the dissertation and creating a narrative, providing references and avoiding plagiarism.

The puzzle-shaped cells that appear in the epidermis of many plants are a striking example of a complex cell shape. Since shape in an organism is often thought to be closely related to its function, it suggests that these unusual shapes must have some functional benefit to the plant. We 
propose that the creation of these complex shapes is an effective strategy to reduce mechanical stress in the cell wall. Although the 
formation of these shapes requires highly anisotropic and non-uniform growth at the sub-cellular level, it appears to be triggered by 
isotropic growth at the organ level. Analysis of cell shape over multiple species is consistent with the idea that the puzzle is in 
response to a developmental constraint, and that the mechanism is like to be conserved among higher plants.

In this talk, we present simple ideas to combine nonparametric approaches based on positive definite kernels with deep learning models. There are many good reasons for bridging these two worlds. On the one hand, we want to provide regularization mechanisms and a geometric interpretation to deep learning models, as well as a functional space that allows to study their theoretical properties (eg invariance and stability). On the other hand, we want to bring more adaptivity and scalability to traditional kernel methods, which are crucially lacking. We will start this presentation by introducing models to represent graph data, then move to biological sequences, and images, showing that our hybrid models can achieves state-of-the-art results for many predictive tasks, especially when large amounts of annotated data are not available. This presentation is based on joint works with Alberto Bietti, Dexiong Chen, and Laurent Jacob.

We analyze the regularity of solutions and discrete-time approximations of extended mean-field control (extended MFC) problems, which seek optimal control of McKean-Vlasov dynamics with coefficients involving mean-field interactions both on the  state and actions, and where objectives are optimized over
open-loop strategies.

We show for a large class of extended MFC problems that the unique optimal open-loop control is 1/2-Hölder continuous in time. Based on the regularity of the solution, we prove that the value functions of such extended MFC problems can be approximated by those with piecewise constant controls and discrete-time state processes arising from Euler-Maruyama time stepping up to an order 1/2 error, which is optimal in our setting. Further, we show that any epsilon-optimal control of these discrete-time problems
converge to the optimal control of the original problems.

To establish the time regularity of optimal controls and the convergence of time discretizations, we extend the canonical path regularity results to general coupled 
McKean-Vlasov forward-backward stochastic differential equations, which are of independent interest.

This is based on join work joint work with C. Reisinger and Y. Zhang.

Daniel M. Anderson

Department of Mathematical Sciences, George Mason University

Applied and Computational Mathematics Division, NIST

Binary and multicomponent alloy solidification occurs in many industrial materials science applications as well as in geophysical systems such as sea ice. These processes involve heat and mass transfer coupled with phase transformation dynamics and can involve the formation of mixed phase regions known as mushy layers.  The understanding of transport mechanisms within mushy layers has important consequences for how these regions interact with the surrounding liquid and solid regions.  Through linear stability analyses and numerical calculations of mathematical models, convective instabilities that occur in solidifying ternary alloys will be explored.  Novel fluid dynamical phenomena that are predicted for these systems will be discussed.

Steklov eigenproblems and their variants (where the spectral parameter appears in the boundary condition) arise in a range of useful applications. For instance, understanding some properties of the mixed Steklov-Neumann eigenfunctions tells us why we shouldn't use coffee cups for expensive brandy. 

In this talk I'll present a high-accuracy discretization strategy for computing Steklov eigenpairs. The strategy can be used to study questions in spectral geometry, spectral optimization and to the solution of elliptic boundary value problems with Robin boundary conditions.

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A link for the talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact @email.

A graded unipotent group U is a unipotent group with a 1PS of automorphisms C^* -- > Aut(U), such that the this 1PS acts on the Lie(U) with all weights positive. Let \hat U be the semi-direct product of U with this 1PS. Let \hat U act linearly on (X,L), a projective variety with a very ample line bundle. With the condition `semistability coincides with stability', and after suitable twist of rational characters, the \hat U-linearisation has a projective geometric quotient, and the invariants are finitely generated. This is a result from \emph{Geometric invariant theory for graded unipotent groups and applications} by G Bérczi, B Doran, T Hawes, F Kirwan, 2018.

Link: https://teams.microsoft.com/l/meetup-join/19%3ameeting_NzU0ODY5MTUtMzUz…

Veering triangulations are a special class of ideal triangulations with a rather mysterious combinatorial definition. Their importance follows from a deep connection with pseudo-Anosov flows on 3-manifolds. Recently Landry, Minsky and Taylor introduced a polynomial invariant of veering triangulations called the taut polynomial. It is a generalisation of an older invariant, the Teichmüller polynomial, defined by McMullen in 2002.

The aim of my talk is to demonstrate that veering triangulations provide a convenient setup for computations. More precisely, I will use fairly easy arguments to obtain a fairly strong statement which generalises the results of McMullen relating the Teichmüller polynomial to the Alexander polynomial.

I will not assume any prior knowledge on the Alexander polynomial, the Teichmüller polynomial or veering triangulations.

This talk will focus on the following two related problems:
    (1) What is the minimum number of edges in a graph containing all $n$-vertex planar graphs as subgraphs? A simple construction of Babai, Erdos, Chung, Graham, and Spencer (1982) has $O(n^{3/2})$ edges, which is the best known upper bound.
    (2) What is the minimum number of *vertices* in a graph containing all $n$-vertex planar graphs as *induced* subgraphs? Here steady progress has been achieved over the years, culminating in a $O(n^{4/3})$ bound due to Bonamy, Gavoille, and Pilipczuk (2019).
    As it turns out, a bound of $n^{1+o(1)}$ can be achieved for each of these two problems. The two constructions are somewhat different but are based on a common technique. In this talk I will first give a gentle introduction to the area and then sketch these constructions. The talk is based on joint works with Vida Dujmović, Louis Esperet, Cyril Gavoille, Piotr Micek, and Pat Morin.

Averages of multiplicative eigenvalue statistics of Haar distributed unitary matrices are Toeplitz determinants, and asymptotics for these determinants are now well understood for large classes of symbols, including symbols with gaps and (merging) Fisher-Hartwig singularities. Similar averages for Haar distributed orthogonal matrices are Toeplitz+Hankel determinants. Some asymptotic results for these determinants are known, but not in the same generality as for Toeplitz determinants. I will explain how one can systematically deduce asymptotics for averages in the orthogonal group from those in the unitary group, using a transformation formula and asymptotics for certain orthogonal polynomials on the unit circle, and I will show that this procedure leads to asymptotic results for symbols with gaps or (merging) Fisher-Hartwig singularities. The talk will be based on joint work with Gabriel Glesner, Alexander Minakov and Meng Yang.

In this talk I will review the work that has been done by me, N. Gromov, V. Kazakov, G. Korchemsky and G. Sizov on the analysis of fishnet Feynman graphs in a particular scaling limit of $\mathcal N=4$ SYM, a theory dubbed $\chi$FT$_4$. After introducing said theory, in which the Feynman graphs take a very simple fishnet form — in the planar limit — I will review how to exploit integrable techniques to compute these graphs and, consequently, extract the anomalous dimensions of a simple class of operators.

Let $p$ be a prime. Minkowski (1887) gave a bound for the order of a finite $p$-subgroup of the linear group $\mathsf{GL}(n,\mathbf Z)$ as a function of $n$, and this necessarily holds for $p$-subgroups of $\mathsf{GL}(n,\mathbf Q)$ also. A few years ago, Serre asked me whether some analogous result might be obtained for subgroups of $\mathsf{GL}(n,\mathbf C)$ using the methods I employed to obtain optimal bounds for Jordan's theorem.

Bounds can be so obtained and I will explain how but, while Minkowski's bound is achieved, no linear bound (as Serre initially suggested) can be achieved. I will discuss progress on this problem and the issues that arise in seeking an ideal form for the solution.

Multibody interactions can reveal higher-order dynamical effects that are not captured by traditional two-body network models. We derive and analyze models for consensus dynamics on hypergraphs, where nodes interact in groups rather than in pairs. Our work reveals that multibody dynamical effects that go beyond rescaled pairwise interactions can appear only if the interaction function is nonlinear, regardless of the underlying multibody structure. As a practical application, we introduce a specific nonlinear function to model three-body consensus, which incorporates reinforcing group effects such as peer pressure. Unlike consensus processes on networks, we find that the resulting dynamics can cause shifts away from the average system state. The nature of these shifts depends on a complex interplay between the distribution of the initial states, the underlying structure, and the form of the interaction function. By considering modular hypergraphs, we discover state-dependent, asymmetric dynamics between polarized clusters where multibody interactions make one cluster dominate the other.

Building on these results, we generalise the model allowing for interactions within hyper edges of any cardinality and explore in detail the role of involvement and stubbornness on polarisation.

What is fairness, and to what extent is it practically achievable? I'll talk about a simple mathematical model under which one might hope to understand such questions. Red and blue points occur as independent homogeneous Poisson processes of equal intensity in Euclidean space, and we try to match them to each other. We would like to minimize the sum of a some function (say, a power, $\gamma$) of the distances between matched pairs. This does not make sense, because the sum is infinite, so instead we satisfy ourselves with minimizing *locally*. If the points are interpreted as agents who would like to be matched as close as possible, the parameter $\gamma$ encodes a measure of fairness - large $\gamma$ means that we try to avoid occasional very bad outcomes (long edges), even if that means inconvenience to others - small $\gamma$ means everyone is in it for themselves.
    In dimension 1 we have a reasonably complete picture, with a phase transition at $\gamma=1$. For $\gamma<1$ there is a unique minimal matching, while for $\gamma>1$ there are multiple matchings but no stationary solution. In higher dimensions, even existence is not clear in all cases.

In this talk I will explain how theories with local symmetries are treated in perturbative Algebraic Quantum Field Theory (pAQFT). The main mathematical tool used here is the Batalin Vilkovisky (BV) formalism. I will show how the perturbative Master Ward Identity can be applied in this formalism to make sense of the renormalised Quantum Master Equation. I will also comment on perspectives for a non-perturbative formulation.

Given an extension L/K of number fields, we say that the Hasse norm principle (HNP) holds if every non-zero element of K which is a norm everywhere locally is in fact a global norm from L. If L/K is cyclic, the original Hasse norm theorem states that the HNP holds. More generally, there is a cohomological description (due to Tate) of the obstruction to the HNP for Galois extensions. In this talk, I will present work (joint with Rachel Newton) developing explicit methods to study this principle for non-Galois extensions. As a key application, I will describe how these methods can be used to characterize the HNP for extensions whose normal closure has Galois group A_n or S_n. I will additionally discuss some recent generalizations of these methods to study the Hasse principle and weak approximation for multinorm equations as well as consequences in the statistics of these local-global principles.

The risk and return profiles of a broad class of dynamic trading strategies, including pairs trading and other statistical arbitrage strategies, may be characterized in terms of excursions of the market price of a portfolio away from a reference level. We propose a mathematical framework for the risk analysis of such strategies, based on a description in terms of price excursions, first in a pathwise setting, without probabilistic assumptions, then in a Markovian setting.

We introduce the notion of δ-excursion, defined as a path which deviates by δ from a reference level before returning to this level. We show that every continuous path has a unique decomposition into δ-excursions, which is useful for scenario analysis of dynamic trading strategies, leading to simple expressions for the number of trades, realized profit, maximum loss and drawdown. As δ is decreased to zero, properties of this decomposition relate to the local time of the path. When the underlying asset follows a Markov process, we combine these results with Ito's excursion theory to obtain a tractable decomposition of the process as a concatenation of independent δ-excursions, whose distribution is described in terms of Ito's excursion measure. We provide analytical results for linear diffusions and give new examples of stochastic processes for flexible and tractable modeling of excursions. Finally, we describe a non-parametric scenario simulation method for generating paths whose excursion properties match those observed in empirical data.

Joint work with Anna Ananova and Rama Cont: https://ssrn.com/abstract=3723980

Operads are tools to encode operations satisfying algebro-homotopic relations. They have proved to be extremely useful tools, for instance for detecting spaces that are iterated loop spaces. However, in many natural examples, composition of operations is only associative up to homotopy and operads are too strict to captured these phenomena. This leads to the notion of infinity operads. While they are a well-established tool, there are few examples of infinity operads in the literature that are not the nerve of an actual operad. I will introduce new topological operad of bracketed trees that can be used to identify and construct natural examples of infinity operads. The key example for this talk will be the normalised cacti model for genus 0 surfaces.

Glueing surfaces along their boundaries defines composition laws that have been used to construct topological field theories and to compute the homology of the moduli space of Riemann surfaces. Normalised cacti are a graphical model for the moduli space of genus 0 oriented surfaces. They are endowed with a composition that corresponds to glueing surfaces along their boundaries, but this composition is not associative. By using the operad of bracketed trees, I will show that this operation is associative up to all higher homotopies and hence that normalised cacti form an infinity operad.

Given a nested pair X and Y of complex projective varieties, there is a single positive integer e which measures the singularity type of X inside Y. This is called the Hilbert-Samuel multiplicity of Y along X, and it appears in the formulations of several standard intersection-theoretic constructions including Segre classes, Euler obstructions, and various other multiplicities. The standard method for computing e requires knowledge of the equations which define X and Y, followed by a (super-exponential) Grobner basis computation. In this talk we will connect the HS multiplicity to complex links, which are fundamental invariants of (complex analytic) Whitney stratified spaces. Thanks to this connection, the enormous computational burden of extracting e from polynomial equations reduces to a simple exercise in clustering point clouds. In fact, one doesn't even need the polynomials which define X and Y: it suffices to work with dense point samples. This is joint work with Martin Helmer.

In this talk I will present an optical theorem for perturbative CFTs, which directly computes the double discontinuity of CFT correlators in terms of the discontinuities of correlators at lower loops or lower points, in analogy to the optical theoreom for scattering amplitudes. I will then discuss the application of this theorem to high-energy scattering of type IIb strings in AdS at one loop and finite 't Hooft coupling. Tidal excitations are taken into account and shown to be efficiently described by an AdS vertex function. The result is related to the 1987 flat space result of Amati, Ciafaloni and Veneziano via the flat space limit in impact parameter space.

I will lead an informal discussion centered on discrete data that need to be specified when reducing 6d relative theories on an internal manifold M and how they determine symmetries of the resulting theory T[M].

In this new session a speaker tells us about how their area of mathematics can be used in different applications.

In this talk, Yuji Nakatsukasa tells us about how random matrix theory can be used in numerical linear algebra. 

Abstract

Randomized SVD is a topic in numerical linear algebra that draws heavily from random matrix theory. It has become an extremely successful approach for efficiently computing a low-rank approximation of matrices. In particular the paper by Halko, Martinsson, and Tropp (SIREV 2011) contains extensive analysis, and has made it a very popular method. The classical Nystrom method is much faster, but only applicable to positive semidefinite matrices. This work studies a generalization of Nystrom's method applicable to general matrices, and shows that (i) it has near-optimal approximation quality comparable to competing methods, (ii) the computational cost is the near-optimal O(mnlog n+r^3) for a rank-r approximation of dense mxn matrices, and (iii) crucially, it can be implemented in a numerically stable fashion despite the presence of an ill-conditioned pseudoinverse. Numerical experiments illustrate that generalized Nystrom can significantly outperform state-of-the-art methods. In this talk I will highlight the crucial role played by a classical result in random matrix theory, namely the Marchenko-Pastur law, and also briefly mention its other applications in least-squares problems and compressed sensing.

In this talk, I will present the theory of combinatorial multivector fields for finite topological spaces, the main subject of my thesis. The idea of combinatorial vector fields came from Forman and emerged naturally from discrete Morse theory. Lately, Mrozek generalized it to the multivector fields theory for Lefschetz complexes. In our work, we simplified and extended it to the finite topological spaces settings. We developed a combinatorial counterpart for dynamical objects, such as isolated invariant sets, isolating neighbourhoods, Conley index, limit sets, and Morse decomposition. We proved the additivity property of the Conley index and the Morse inequalities. Furthermore, we applied persistence homology to study the evolution and the stability of Morse decomposition. In the last part of the talk, I will show numerical results and potential future directions from a data-analysis perspective. 

A Real representation of a $C_2$-graded group $H < G$ ($H$ an index two subgroup) is a complex representation of $H$ with an action of the other coset $G \backslash H$ (“odd" elements) satisfying appropriate algebraic coherence conditions. In this talk I will present three such Real representation theories. In these, each odd element acts as an antilinear operator, a bilinear form or a sesquilinear form (equivalently a linear map to $V$ from the conjugate, the dual, or the conjugate dual of $V$) respectively. I will describe how these theories are related, how representations in each are classified, and how the first generalises the classical representation theory of $H$ over the real numbers - retaining much of its beauty and subtlety.

The classical fractional crystallisation scenario for magma ocean solidification on the Moon suggests that its crust formed by flotation of light anorthite minerals on top of a liquid ocean, which has been used to explain the anorthositic composition of the lunar crust. However, this model points to rapid crustal formation over tens of million years and struggles to predict the age range of primitive ferroan anorthosites from 4.5 and 4.3 Ga. 

Here I will present a new paradigm of slushy magma ocean crystallisation in which crystals are suspended throughout the magma ocean, and the lunar crust forms by magmatic processes over several hundreds of thousand years.

We will then focus on the effects of the particular characteristics of this primary crust on the transport and eruption of magma on the Moon.

 “In this talk, I shall present the past research track passing through quantum mechanical studies of small molecules to biomolecules, to proteome-wide big data analyses and computational genomics. Next, the ongoing research in our group will be presented that builds upon the expertise on different levels of information processing in life (genome, transcriptome, proteins, small molecules), to develop self-consistent “first principles” models in biology with a wide spectrum of usage. The immediate benefits and the targeted processes will be described covering different layers of the central dogma of biology, multigenic diseases and disease driver/passenger mutation predictions."

We present a partial differential equation framework for deep residual neural networks and for the associated learning problem. This is done by carrying out the continuum limits of neural networks with respect to width and depth. We study the wellposedness, the large time solution behavior, and the characterization of the steady states of the forward problem. Several useful time-uniform estimates and stability/instability conditions are presented. We state and prove optimality conditions for the inverse deep learning problem, using standard variational calculus, the Hamilton-Jacobi-Bellmann equation and the Pontryagin maximum principle. This serves to establish a mathematical foundation for investigating the algorithmic and theoretical connections between neural networks, PDE theory, variational analysis, optimal control, and deep learning.

This is based on joint work with Hailiang Liu.

Amy Kent

Multiscale Mathematical Models for Tendon Tissue Engineering

Tendon tissue engineering aims to grow functional tendon in vitro. In bioreactor chambers, cells growing on a solid scaffold are fed with nutrient-rich media and stimulated by mechanical loads. The Nuffield Department of Orthopaedics, Rheumatology and Musculoskeletal Sciences is developing a Humanoid Robotic Bioreactor, where cells grow on a flexible fibrous scaffold actuated by a robotic shoulder. Tendon cells modulate their behaviour in response to shear stresses - experimentally, it is desirable to design robotic loading regimes that mimic physiological loads. The shear stresses are generated by flowing cell media; this flow induces deformation of the scaffold which in turn modulates the flow. Here, we capture this fluid-structure interaction using a homogenised model of fluid flow and scaffold deformation in a simplified bioreactor geometry. The homogenised model admits analytical solutions for a broad class of forces representing robotic loading. Given the solution to the microscale problem, we can determine microscale shear stresses at any point in the domain. In this presentation, we will outline the model derivation and discuss the experimental implications of model predictions.

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Michael Negus

High-Speed Droplet Impact Onto Deformable Substrates: Analysis And Simulations

The impact of a high-speed droplet onto a substrate is a highly non-linear, multiscale phenomenon and poses a formidable challenge to model. In addition, when the substrate is deformable, such as a spring-suspended plate or an elastic sheet, the fluid-structure interaction introduces an additional layer of complexity. We present two modeling approaches for droplet impact onto deformable substrates: matched asymptotics and direct numerical simulations. In the former, we use Wagner's theory of impact to derive analytical expressions which approximate the behaviour during the early stages of the impact. In the latter, we use the open source volume-of-fluid code Basilisk to conduct direct numerical simulations designed to both validate the analytical framework and provide insight into the later times of impact. Through both methods, we are able to observe how the properties of the substrate, such as elasticity, affect the behaviour of the flow. We conclude by showing how these methods are complementary, as a combination of both can lead to a thorough understanding of the droplet impact across timescales.

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Edwina Yeo

Modelling of Magnetically Targeted Stem Cell Delivery

Targeting delivery of stem cells to the site of an injury is a key challenge in regenerative medicine. One possible approach is to inject cells implanted withmagnetic nanoparticles into the blood stream. Cells can then be targeted to the delivery site by an external magnetic field. At the injury site, it is of criticalimportance that the cells do not form an aggregate which could significantly occlude the vessel.We develop a model for the transport of magnetically tagged cells in blood under the action of an external magnetic field. We consider a system of blood and stem cells in a single vessel.  We exploit the small aspect ratio of the vessel to examine the system asymptotically. We consider the system for a range of magnetic field strengths and varying strengths of the diffusion coefficient of the stem cells. We explore the different regimes of the model and determine the optimal conditions for the effective delivery of stem cells while minimising vessel occlusion.

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Helen Zha

Mathematical model of a valve-controlled, gravity driven bioreactor for platelet production

Hospitals sometimes experience shortages of donor blood platelet supplies, motivating research into~\textit{in vitro}~production of platelets. We model a novel platelet bioreactor described in Shepherd et al [1]. The bioreactor consists of an upper channel, a lower channel, and a cell-seeded porous collagen scaffold situated between the two. Flow is driven by gravity, and controlled by valves on the four inlets and outlets. The bioreactor is long relative to its width, a feature which we exploit to derive a lubrication reduction of unsteady Stokes flow coupled to Darcy. As the shear stress experienced by cells influences platelet production, we use our model to quantify the effect of varying pressure head and valve dynamics on shear stress.

[1] Shepherd, J.H., Howard, D., Waller, A.K., Foster, H.R., Mueller, A., Moreau, T., Evans, A.L., Arumugam, M., Chalon, G.B., Vriend, E. and Davidenko, N., 2018. Structurally graduated collagen scaffolds applied to the ex vivo generation of platelets from human pluripotent stem cell-derived megakaryocytes: enhancing production and purity. Biomaterials.

Agent-based models are an intuitive, interpretable way to model markets and give us a powerful mechanism to analyze counterfactual scenarios that might rarely occur in historical market data. However, building realistic agent-based models is challenging and requires that we (a) ensure that agent behaviors are realistic, and (b) calibrate the agent composition using real data. In this talk, we will present our work to build realistic agent-based models using a multi-agent reinforcement learning approach. Firstly, we show that we can learn a range of realistic behaviors for heterogeneous agents using a shared policy conditioned on agent parameters and analyze the game-theoretic implications of this approach. Secondly, we propose a new calibration algorithm (CALSHEQ) which can estimate the agent composition for which calibration targets are approximately matched, while simultaneously learning the shared policy for the agents. Our contributions make the building of realistic agent-based models more efficient and scalable.

Scientific machine learning is a burgeoning discipline for mixing machine learning into scientific simulation. Use cases of this field include automated discovery of physical equations and accelerating physical simulators. However, making the analyses of this field automated will require building a set of tools that handle stiff and ill-conditioned models without requiring user tuning. The purpose of this talk is to demonstrate how the methods and tools of scientific machine learning can be consolidated to give a single high performance and robust software stack. We will start by describing universal differential equations, a flexible mathematical object which is able to represent methodologies for equation discovery, 100-dimensional differential equation solvers, and discretizations of physics-informed neural networks. Then we will showcase how adjoint sensitivity analysis on the universal differential equation solving process gives rise to efficient and stiffly robust training methodologies for a large variety of scientific machine learning problems. With this understanding of differentiable programming we will describe how the Julia SciML Software Organization is utilizing this foundation to provide high performance tools for deploying battery powered airplanes, improving the energy efficiency of buildings, allow for navigation via the Earth's magnetic field, and more.

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please send email to @email.

We analyze the steady motion of a viscous incompressible fluid in a two- and three-dimensional channel containing an obstacle through the Navier-Stokes equations under different types of boundary conditions. In the 2D case we take constant non-homogeneous Dirichlet boundary data in a (virtual) square containing the obstacle, and emphasize the connection between the appearance of lift and the unique solvability of Navier-Stokes equations. In the 3D case we consider mixed boundary conditions: the inflow is given by a fairly general datum and the flow is assumed to satisfy a constant traction boundary condition on the outlet. In the absence of external forcing, explicit bounds on the inflow velocity guaranteeing existence and uniqueness of such steady motion are provided after estimating some Sobolev embedding constants and constructing a suitable solenoidal extension of the inlet velocity. In the 3D case, this solenoidal extension is built through the Bogovskii operator and explicit bounds on its Dirichlet norm (in terms of the geometric parameters of the obstacle) are found by solving a variational problem involving the infinity-Laplacian.

The talk accounts for results obtained in collaboration with Filippo Gazzola and Ilaria Fragalà (both at Politecnico di Milano).

I will introduce left-orderable groups and discuss constructions and examples of such groups. I will then motivate studying left-orders under the framework of formal languages and discuss some recent results.

The Burali-Forti paradox suggests that the transfinite cardinals “go on forever,” surpassing any conceivable bound one might try to place on them. The traditional Zermelo-Frankel axioms for set theory fall into a hierarchy of axiomatic systems formulated by reasserting this intuition in increasingly elaborate ways: the large cardinal hierarchy. Or so the story goes. A serious problem for this already naive account of large cardinal set theory is the Kunen inconsistency theorem, which seems to impose an upper bound on the extent of the large cardinal hierarchy itself. If one drops the Axiom of Choice, Kunen’s proof breaks down and a new hierarchy of choiceless large cardinal axioms emerges. These axioms, if consistent, represent a challenge for those “maximalist” foundational stances that take for granted both large cardinal axioms and the Axiom of Choice. This talk concerns some recent advances in our understanding of the weakest of the choiceless large cardinal axioms and the prospect, as yet unrealized, of establishing their consistency and reconciling them with the Axiom of Choice.

A spanning tree of $G$ is a subgraph of $G$ with the same vertex set as $G$ that is a tree. In 1981, McKay proved an asymptotic result regarding the number of spanning trees in random $k$-regular graphs, showing that the number of spanning trees $\kappa_1(G_n)$ in a random $k$-regular graph on $n$ vertices satisfies $\lim_{n \to \infty} (\kappa_1(G_n))^{1/n} = c_{1,k}$ in probability, where $c_{1,k} = (k-1)^{k-1} (k^2-2k)^{-(k-2)/2}$.

In this talk we will discuss a high-dimensional of the matching model for simplicial complexes, known as random Steiner complexes. In particular, we will prove a high-dimensional counterpart of McKay's result and discuss the local limit of such random complexes. 
Based on a joint work with Lior Tenenbaum. 

For 20 years we have known that average values of characteristic polynomials of random unitary matrices provide a good model for moments of the Riemann zeta function.  Now we consider moments of the logarithmic derivative of characteristic polynomials, calculations which are motivated by questions on the distribution of zeros of the derivative of the Riemann zeta function.  Joint work with Emilia Alvarez. 

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please send email to @email.

Let W be the Witt algebra of vector fields on the punctured complex plane, and let Vir be the Virasoro algebra, the unique nontrivial central extension of W.  We discuss work in progress with Alexey Petukhov to analyse Poisson ideals of the symmetric algebra of Vir.

We focus on understanding maximal Poisson ideals, which can be given as the Poisson cores of maximal ideals of Sym(Vir) and of Sym(W).  We give a complete classification of maximal ideals of Sym(W) which have nontrivial Poisson cores.  We then lift this classification to Sym(Vir), and use it to show that if $\lambda \neq 0$, then $(z-\lambda)$ is a maximal Poisson ideal of Sym(Vir).

Randomized experiments, or "A/B" tests, remain the gold standard for evaluating the causal effect of a policy intervention or product change. However, experimental settings such as social networks, where users are interacting and influencing one another, violate conventional assumptions of no interference needed for credible causal inference. Existing solutions include accounting for the fraction or count of treated neighbors in a user's network, among other strategies. Yet, there are often a high number of researcher degrees of freedom in specifying network interference conditions and most current methods do not account for the local network structure beyond simply counting the number of neighbors. Capturing local network structures is important because it can account for theories, such as structural diversity and echo chambers. Our study provides an approach that accounts for both the local structure in a user's social network via motifs as well as the assignment conditions of neighbors. We propose a two-part approach. We first introduce and employ "causal network motifs," i.e. network motifs that characterize the assignment conditions in local ego networks; and then we propose a tree-based algorithm for identifying different network interference conditions and estimating their average potential outcomes. We test our method on a real-world experiment on a large-scale network and a synthetic network setting, which highlight how accounting for local structures can better account for different interference patterns in networks.

Preconditioning techniques are widely used for speeding up the iterative solution of systems of linear equations, often by transforming the system into one with lower condition number. Even though the condition number also serves as the determining constant in simple bounds for the numerical error of the solution, simple experiments and bounds show that such preconditioning on the matrix level is not guaranteed to reduce this error. Transformations on the operator level, on the other hand, improve both accuracy and speed of iterative methods as predicted by the change of the condition number. We propose to investigate such methods under a common framework, which we call full operator preconditioning, and show practical examples.

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We study the minimum total weight of a disk triangulation using any number of vertices out of $\{1,..,n\}$ where the boundary is fixed and the $n \choose 3$ triangles have independent rate-1 exponential weights. We show that, with high probability, the minimum weight is equal to $(c+o(1))n-1/2\log n$ for an explicit constant $c$. Further, we prove that, with high probability, the minimum weights of a homological filling and a homotopical filling of the cycle $(123)$ are both attained by the minimum weight disk triangulation. We will discuss a related open problem concerning simple-connectivity of random simplicial complexes, where a similar phenomenon is conjectured to hold. Based on joint works with Itai Benjamini, Eyal Lubetzky, and Zur Luria.

What do random spanning trees, graph embeddings, random walks, simplices and graph curvature have in common? As you may have guessed from the title, they are indeed all intimately connected to the effective resistance on graphs! While originally invented as a tool to study electrical circuits, the effective resistance has proven time and again to be a graph characteristic with a variety of interesting and often surprising properties. Starting from a number of equivalent but complementary definitions of the effective resistance, we will take a stroll through some classical theorems (Rayleigh monotonicity, Foster's theorem), a few modern results (Klein's metricity, Fiedler's graph-simplex correspondence) and finally discuss number of recent developments (variance on graphs, discrete curvature and graph embeddings).

Motivated by an attempt to construct a theory of quantum gravity as a perturbation around some flat background, Penrose has shown that, despite being asymptotically flat, there is an inconsistency between the causal structure at infinity of Schwarzschild and Minkowski spacetimes. This suggests that such a perturbative approach cannot possibly work. However, the proof of this inconsistency is specific to 4 spacetime dimensions. In this talk I will discuss how this result extends to higher (and lower) dimensions. More generally, I will consider examples of how the causal structure of asymptotically flat spacetimes are affected by dimension and by the presence of mass (both positive and negative). I will then show how these ideas can be used to prove a higher dimensional extension of the positive mass theorem of Penrose, Sorkin and Woolgar.

Stochastic quantisation is, broadly speaking, the use of a stochastic differential equation to construct a given probability distribution. Usually this refers to Markovian Langevin evolution with given invariant measure. However we will show that it is possible to construct other kind of equations (elliptic stochastic partial differential equations) whose solutions have prescribed marginals. This connection was discovered in the '80 by Parisi and Sourlas in the context of dimensional reduction of statistical field theories in random external fields. This purely probabilistic results has a proof which depends on a supersymmetric formulation of the problem, i.e. a formulation involving a non-commutative random field defined on a non-commutative space. This talk is based on joint work with S. Albeverio and F. C. de Vecchi.

Sieve methods are analytic tools that we can use to tackle problems in additive number theory. This talk will serve as a gentle introduction to the area. At the end we will discuss recent progress on a variation on the prime $k$-tuples conjecture which involves sums of two squares. No knowledge of sieves is required!