Past

In this session we discuss techniques to get the most out of your supervision sessions and tips on how to work with different personalities and use your supervisor's skills to your advantage. The session will be run by DPhil students and discussion among students during the session is encouraged.  

Polynomial rings $R[X]$ are a fundamental construction in commutative algebra, under which Hilbert's basis theorem controls a finiteness property: being Noetherian. We will describe the picture for the non-commutative world; this leads us towards other interesting finiteness conditions.

We explore general constraints from unitarity, defect superconformal symmetry and locality of bulk-defect couplings to classify possible superconformal defects in superconformal field theories (SCFT) of spacetime dimensions d>2.  Despite the general absence of locally conserved currents, the defect CFT contains new distinguished operators with protected quantum numbers that account for the broken bulk symmetries.  Consistency with the preserved superconformal symmetry and unitarity requires that such operators arrange into unitarity multiplets of the defect superconformal algebra, which in turn leads to nontrivial constraints on what kinds of defects are admissible in a given SCFT.  We will focus on the case of superconformal lines in this talk and comment on several interesting implications of our analysis, such as symmetry-enforced defect conformal manifolds, defect RG flows and possible nontrivial one-form symmetries in various SCFTs.  

Influenza viruses infect millions of individuals each year and cause a significant amount of morbidity and mortality. Understanding how the virus spreads within the lung, how efficacious host immune control is, and how each influences acute lung injury and disease severity is critical to combat the infection. We used an integrative model-experiment exchange to establish the dynamical connections between viral loads, infected cells, CD8+ T cells, lung injury, and disease severity. Our model predicts that infection resolution is sensitive to CD8+ T cell expansion, that there is a critical T cell magnitude needed for efficient resolution, and that the rate of T cell-mediated clearance is dependent on infected cell density. 
We validated the model through a series of experiments, including CD8 depletion and whole lung histomorphometry. This showed that the infected area of the lung matches the model-predicted infected cell dynamics, and that the resolved area of the lung parallels the relative CD8 dynamics. Additional analysis revealed a nonlinear relation between disease severity, inflammation, and lung injury. These novel links between important host-pathogen kinetics and pathology enhance our ability to forecast disease progression.

Differential equations and neural networks are two of the most widespread modelling paradigms. I will talk about how to combine the best of both worlds through neural differential equations. These treat differential equations as a learnt component of a differentiable computation graph, and as such integrates tightly with current machine learning practice. Applications are widespread. I will begin with an introduction to the theory of neural ordinary differential equations, which may for example be used to model unknown physics. I will then move on to discussing recent work on neural controlled differential equations, which are state-of-the-art models for (arbitrarily irregular) time series. Next will be some discussion of neural stochastic differential equations: we will see that the mathematics of SDEs is precisely aligned with the machine learning of GANs, and thus NSDEs may be used as generative models. If time allows I will then discuss other recent work, such as how the training of neural differential equations may be sped up by ~40% by tweaking standard numerical solvers to respect the particular nature of the differential equations. This is joint work with Ricky T. Q. Chen, Xuechen Li, James Foster, and James Morrill.

We revisit the tears of wine problem for thin films in
water-ethanol mixtures and present a new model for the climbing
dynamics. The new formulation includes a Marangoni stress balanced by
both the normal and tangential components of gravity as well as surface
tension which lead to distinctly different behavior. The combined
physics can be modeled mathematically by a scalar conservation law with
a nonconvex flux and a fourth order regularization due to the bulk
surface tension. Without the fourth order term, shock solutions must
satisfy an entropy condition - in which characteristics impinge on the
shock from both sides. However, in the case of a nonconvex flux, the
fourth order term is a singular perturbation that allows for the
possibility of undercompressive shocks in which characteristics travel
through the shock. We present computational and experimental evidence
that such shocks can happen in the tears of wine problem, with a
protocol for how to observe this in a real life setting.

It has long been known that many elliptic partial differential equations can be reformulated as Fredholm integral equations of the second kind on the boundaries of their domains. The kernels of the resulting integral equations are weakly singular, which has historically made their numerical solution somewhat onerous, requiring the construction of detailed and typically sub-optimal quadrature formulas. Recently, a numerical algorithm for constructing generalized Gaussian quadratures was discovered which, given 2n essentially arbitrary functions, constructs a unique n-point quadrature that integrates them to machine precision, solving the longstanding problem posed by singular kernels.

When the domains have corners, the solutions themselves are also singular. In fact, they are known to be representable, to order n, by a linear combination (expansion) of n known singular functions. In order to solve the integral equation accurately, it is necessary to construct a discretization such that the mapping (in the L^2-sense) from the values at the discretization points to the corresponding n expansion coefficients is well-conditioned. In this talk, we present exactly such an algorithm, which is optimal in the sense that, given n essentially arbitrary functions, it produces n discretization points, and for which the resulting interpolation formulas have condition numbers extremely close to one.

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'Quasi-isometric rigidity' in group theory is the slogan for questions of the following nature: let A be some class of groups (e.g. finitely presented groups). Suppose an abstract group H is quasi-isometric to a group in A: does it imply that H is in A? Such statements link the coarse geometry of a group with its algebraic structure. 

Much is known in the case A is some class of lattices in a given Lie group. I will present classical results and outline ideas in their proofs, emphasizing the geometric nature of the proofs. I will focus on one key ingredient, the quasi-flat rigidity, and discuss some geometric objects that come into play, such as neutered spaces, asymptotic cones and buildings. I will end the talk with recent developments and possible generalizations of these results and ideas.

In 1943, Hadwiger conjectured that every graph with no Kt minor is $(t-1)$-colorable for every $t\geq 1$. In the 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t(\log t)^{1/2})$ and hence is $O(t(\log t)^{1/2)}$-colorable. Recently, Norin, Song and I showed that every graph with no $K_t$ minor is $O(t(\log t)^\beta)$-colorable for every $\beta > 1/4$, making the first improvement on the order of magnitude of the $O(t(\log t)^{1/2})$ bound. Here we show that every graph with no $K_t$ minor is $O(t (\log t)^\beta)$-colorable for every $\beta > 0$; more specifically, they are $O(t (\log \log t)^6)$-colorable.

We analyse the eigenvectors of the adjacency matrix of a critical Erdös-Rényi graph G(N,d/N), where d is of order \log N. We show that its spectrum splits into two phases: a delocalized phase in the middle of the spectrum, where the eigenvectors are completely delocalized, and a semilocalized phase near the edges of the spectrum, where the eigenvectors are essentially localized on a small number of vertices. In the semilocalized phase the mass of an eigenvector is concentrated in a small number of disjoint balls centred around resonant vertices, in each of which it is a radial exponentially decaying function. The transition between the phases is sharp and is manifested in a discontinuity in the localization exponents of the eigenvectors. Joint work with Johannes Alt and Raphael Ducatez.

The nilpotent orbits of a Lie algebra play a central role in modern representation theory notably cropping up in the Springer correspondence and the fundamental lemma. Their behaviour when the base field is algebraically closed is well understood, however the p-adic case which arises in the study of admissible representations of p-adic groups is considerably more subtle. Their classification was only settled in the late 90s when Barbasch and Moy ('97) and Debacker (’02) developed an ‘affine Bala-Carter’ theory using the Bruhat-Tits building. In this talk we combine this work with work by Sommers and McNinch to provide a parameterisation of nilpotent orbits over a maximal unramified extension of a p-adic field in terms of so called dual Springer parameters and outline an application of this result to wavefront sets.

We develop random graph models where graphs are generated by connecting not only pairs of vertices by edges but also larger subsets of vertices by copies of small atomic subgraphs of arbitrary topology. This allows the for the generation of graphs with extensive numbers of triangles and other network motifs commonly observed in many real world networks. More specifically we focus on maximum entropy ensembles under constraints placed on the counts and distributions of atomic subgraphs and derive general expressions for the entropy of such models. We also present a procedure for combining distributions of multiple atomic subgraphs that enables the construction of models with fewer parameters. Expanding the model to include atoms with edge and vertex labels we obtain a general class of models that can be parametrized in terms of basic building blocks and their distributions that includes many widely used models as special cases. These models include random graphs with arbitrary distributions of subgraphs, random hypergraphs, bipartite models, stochastic block models, models of multilayer networks and their degree corrected and directed versions. We show that the entropy for all these models can be derived from a single expression that is characterized by the symmetry groups of atomic subgraphs.

We will be interested in the structure of random typical minimal factorizations of the n-cycle into transpositions, which are factorizations of $(1,\ldots,n)$ as a product of $n-1$ transpositions. We shall establish a phase transition when a certain amount of transpositions have been read one after the other. One of the main tools is a limit theorem for two-type Bienaymé-Galton-Watson trees conditioned on having given numbers of vertices of both types, which is of independent interest. This is joint work with Valentin Féray.

I discuss a ``running vacuum cosmological model'' of a string-inspired
Universe, in which gravitational anomalies play an important role, in
inducing, through condensates of primordial gravitational waves, an early de
Sitter inflationary phase, during which constant (in cosmic time)
backgrounds of the antisymmetric (Kalb-Ramond (KR)) tensor field of the
massless bosonic string multiplet remain undiluted until the exit from
inflation and well into the subsequent radiation era. During the radiation
phase, such backgrounds, which violate spontaneously Lorentz and CPT
symmetry, induce lepton asymmetry (Leptogenesis) in models involving
right-handed neutrinos. Chiral matter is generated in the model at the exit
phase of inflation, and this leads to the cancellation of gravitational
anomalies in the post inflationary universe. During the radiation era, non
perturbative effects can also be held responsible for the generation of a

potential for the gravitational axion, associated in (3+1)-dimensions with
the field strength of the KR field, which can thus play the role of a Dark
Matter component. In the talk, I discuss the underlying formalism and argue
in favour of the consistency of a theory with gravitational anomalies in the
early Universe. I connect the energy density of such a universe with that of
the so called ``running-vacuum model'' in which the vacuum energy density is
expressed in terms of even powers of the Hubble parameter, which in general
depends on cosmic time. The gravitational-wave condensate induces a term in
the energy density  proportional to the fourth-power of the Hubble parameter
H^4 , which is responsible for the early de Sitter phase, during which the
Hubble parameter is approximately a constant. I also discuss briefly a
connection of this string inspired model with the Swampland and weak gravity
conjectures and explain how consistency with such conjectures is achieved,
despite the fact that the model is compatible with slow-roll inflationary
phenomenology.

In 1983, Faltings proved Mordell's conjecture on the finiteness of $K$-points on curves of genus >1 defined over a number field $K$ by proving the finiteness of isomorphism classes of isogenous abelian varieties over $K$. The "first" major step from Mordell's conjecture to what Faltings did came 15 years earlier when Parshin showed that a certain conjecture of Shafarevich would imply Mordell's conjecture. In this talk, I'll focus on motivating and sketching Parshin's argument in an accessible manner and provide some heuristics on how to get from Faltings' finiteness statement to the Shafarevich conjecture.

Unlike the classical Cauchy problem in general relativity, which has been well-understood since the pioneering work of Y. Choquet-Bruhat (1952), the initial boundary value problem for the Einstein equations still lacks a comprehensive treatment. In particular, there is no geometric description of the boundary data yet known, which makes the problem well-posed for general timelike boundaries. Various gauge-dependent results have been established. Timelike boundaries naturally arise in the study of massive bodies, numerics, AdS spacetimes. I will give an overview of the problem and then present recent joint work with Jacques Smulevici that treates the special case of a totally geodesic boundary.

Trading of financial instruments has largely moved away from floor trading and onto electronic exchanges. Orders to buy and sell are queued at these exchanges in a limit-order book. While a full analysis of the dynamics of a limit-order book requires an understanding of strategic play among multiple agents, and is thus extremely complex, so-called zero-intelligence Poisson models have been shown to capture many of the statistical features of limit-order book evolution. These models can be addressed by traditional queueing theory techniques, including Laplace transform analysis. In this work, we demonstrate in a simple setting that another queueing theory technique, approximating the Poisson model by a diffusion model identified as the limit of a sequence of scaled Poisson models, can also be implemented. We identify the diffusion limit, find an embedded semi-Markov model in the limit, and determine the statistics of the embedded semi-Markov model. Along the way, we introduce and study a new type of process, a generalization of skew Brownian motion that we call two-speed Brownian motion.

I will discuss recent progress on the study of homological duality properties of complex algebraic manifolds, with a view towards the projective Singer-Hopf conjecture. (Joint work with Y. Liu and B. Wang.)

A vast class of 4d SCFTs can be engineered by wrapping a stack of M5-branes on a Riemann surface. These SCFTs can exhibit a variety of global symmetries, continuous or discrete, including both ordinary (0-form) symmetries, as well as generalized (higher-form) symmetries. In this talk, I will focus on discrete and higher-form symmetries in setups with M5-branes on a smooth Riemann surface. Adopting a holographic point of view, a crucial role is played by topological mass terms in 5d supergravity (similar to BF terms in four dimensions). I will discuss how the global symmetries of the boundary 4d theory are inferred from the 5d topological terms, and how one can compute 4d ‘t Hooft anomalies involving discrete and/or higher-form symmetries.

Martin Gallauer (North): "Algebraic algebraic geometry"
If a space is described by algebraic equations, its algebraic invariants are endowed with additional structure. I will illustrate this with some simple examples, and speculate on the meaning of the title of my talk.

Zhaohe Dai (South): "Two-dimensional material bubbles"
Two-dimensional (2D) materials are a relatively new class of thin sheets consisting of a single layer of covalently bonded atoms and have shown a host of unique electronic properties. In 2D material electronic devices, however, bubbles often form spontaneously due to the trapping of air or ambient contaminants (such as water molecules and hydrocarbons) at sheet-substrate interfaces. Though they have been considered to be a nuisance, I will discuss that bubbles can be used to characterize 2D materials' bending rigidity after the pressure inside being well controlled. I will then focus on bubbles of relatively large deformations so that the elastic tension could drive the radial slippage of the sheet on its substrate. Finally, I will discuss that the consideration of such slippage is vital to characterize the sheet's stretching stiffness and gives new opportunities to understand the adhesive and frictional interactions between the sheet and various substrates that it contacts.
 

Motivated by string dualities we propose topological gravity as the early phase of our universe.  The topological nature of this phase naturally leads to the explanation of many of the puzzles of early universe cosmology.  A concrete realization of this scenario using Witten's four dimensional topological gravity is considered.  This model leads to the power spectrum of CMB fluctuations which is controlled by the conformal anomaly coefficients $a,c$.  In particular the strength of the fluctuation is controlled by $1/a$ and its tilt by $c g^2$ where $g$ is the coupling constant of topological gravity.  The positivity of $c$, a consequence of unitarity, leads automatically to an IR tilt for the power spectrum.   In contrast with standard inflationary models, this scenario predicts $\mathcal{O}(1)$ non-Gaussianities for four- and higher-point correlators and the absence of tensor modes in the CMB fluctuations.

In this talk I will present an efficient algorithm to produce a provably dense sample of a smooth compact algebraic variety. The procedure is partly based on computing bottlenecks of the variety. Using geometric information such as the bottlenecks and the local reach we also provide bounds on the density of the sample needed in order to guarantee that the homology of the variety can be recovered from the sample.

Snow densification and meltwater refreezing store water in alpine regions and transform snow into ice on the surface of glaciers. Despite their importance in determining snow-water equivalent and glacier-induced sea level rise, we still lack a complete understanding of the physical mechanisms underlying snow compaction and the infiltration of meltwater into snowpacks. Here we (i) analyze snow compaction experiments as a promising direction for determining the rheology of snow though its many stages of densification and (ii) solve for the motion of refreezing fronts and for the temperature increase due to the release of latent heat, which we compare to temperature observations from the Greenland Ice Sheet (Humphrey et al., 2012). In the first part, we derive a mixture theory for compaction and air flow through the porous snow (cf. Hewitt et al. 2016) to compare against laboratory data (Wang and Baker, 2013). We find that a plastic compaction law explains experimental results. Taking standard forms for the permeability and effective pressure as functions of the porosity, we show that this compaction mode persists for a range of densities and overburden loads (Meyer et al., 2020). We motivate the second part of the talk by the observed melting at high elevations on the Greenland Ice Sheet, which causes the refreezing layers that are observed in ice cores. Our analysis shows that as surface temperatures increase, the capacity for meltwater storage in snow decreases and surface runoff increases leading to sea level rise (Meyer and Hewitt, 2017). Together these studies provide a holistic picture for how snow changes through compaction and the role of meltwater percolation in altering the temperature and density structure of surface snow.

Multi-modal data sets are growing rapidly in single cell genomics, as well as other fields in science and engineering. We introduce MultiMAP, an approach for dimensionality reduction and integration of multiple datasets. MultiMAP embeds multiple datasets into a shared space so as to preserve both the manifold structure of each dataset independently, in addition to the manifold structure in shared feature spaces. MultiMAP is based on the rich mathematical foundation of UMAP, generalizing it to the setting of more than one data manifold. MultiMAP can be used for visualization of multiple datasets as well as an integration approach that enables subsequent joint analyses. Compared to other integration for single cell data, MultiMAP is not restricted to a linear transformation, is extremely fast, and is able to leverage features that may not be present in all datasets. We apply MultiMAP to the integration of a variety of single-cell transcriptomics, chromatin accessibility, methylation, and spatial data, and show that it outperforms current approaches in run time, label transfer, and label consistency. On a newly generated single cell ATAC-seq and RNA-seq dataset of the human thymus, we use MultiMAP to integrate cells across pseudotime. This enables the study of chromatin accessibility and TF binding over the course of T cell differentiation.

Wieler has shown that every irreducible Smale space with totally disconnected stable sets is a solenoid (i.e., obtained via a stationary inverse limit construction). Through examples I will discuss how this allows one to compute the K-theory of the stable algebra, S, and the stable Ruelle algebra, S\rtimes Z. These computations involve writing S as a stationary inductive limit and S\rtimes Z as a Cuntz-Pimsner algebra. These constructions reemphasize the view point that Smale space C*-algebras are higher dimensional generalizations of Cuntz-Krieger algebras. The main results are joint work with Magnus Goffeng and Allan Yashinski.

We show how traders use immediate execution limit orders (IELOs) to liquidate a position when the time between a trade attempt and the outcome of the attempt is random, i.e., there is latency in the marketplace and latency is random. We frame our model as a delayed impulse control problem in which the trader controls the times and the price limit of the IELOs she sends to the exchange. The contribution of the paper is twofold: (i) Our paper is the first to study an optimal liquidation problem that accounts for random delays, price impact, and transaction costs. (ii) We introduce a new type of impulse control problem with stochastic delay, not previously studied in the literature. We characterise the value functions as the solution to a coupled system of a Hamilton-Jacobi-Bellman quasi-variational inequality (HJBQVI) and a partial differential equation. We use a Feynman-Kac type representation to reduce the system of coupled value functions to a non-standard HJBQVI, and we prove existence and uniqueness of this HJBQVI in a viscosity sense. Finally, we implement the latency-optimal strategy and compare it with three benchmarks:  (i)  optimal execution with deterministic latency, (ii) optimal execution with zero latency, (iii) time-weighted average price strategy. We show that when trading in the EUR/USD currency pair, the latency-optimal strategy outperforms the benchmarks between ten USD per million EUR traded and ninety USD per million EUR traded.

There are many examples of thin-film flows in fluid dynamics, and in many cases similarity solutions are possible. In the typical, well-known case the thin-film shape is described by a nonlinear partial differential equation in two independent variables (say x and t), which upon recognition of a similarity variable, reduces the problem to a nonlinear ODE. In this talk I describe work we have done on 1) Marangoni-driven spreading on pre-wetted films, where the thickness of the pre-wetted film affects the dynamics, and 2) the drainage of a film on a vertical substrate of finite width. In the latter case we find experimentally a structure to the film shape near the edge, which is a function of time and two space variables. Analysis of the corresponding thin-film equation shows that there is a similarity solution, collapsing three independent variables to one similarity variable, so that the PDE becomes an ODE. The solution is in excellent agreement with the experimental measurements.

We have various ways of describing the extent to which two countably infinite groups are "the same." Are they isomorphic? If not, are they commensurable? Measure equivalent? Quasi-isometric? Orbit equivalent? W*-equivalent? Von Neumann equivalent? In this expository talk, we will define these notions of equivalence, discuss the known relationships between them, and work out some examples. Along the way, we will describe recent joint work with Ishan Ishan and Jesse Peterson.

Any binary classifier (or score-function) can be used to define a dissimilarity between two distributions of points with positive and negative labels. Actually, many well-known distribution-dissimilarities are classifier-based dissimilarities: the total variation, the KL- or JS-divergence, the Hellinger distance, etc. And many recent popular generative modelling algorithms compute or approximate these distribution-dissimilarities by explicitly training a classifier: eg GANs and their variants. After a brief introduction to these classifier-based dissimilarities, I will focus on the influence of the classifier's capacity. I will start with some theoretical considerations illustrated on maximum mean discrepancies --a weak form of total variation that has grown popular in machine learning-- and then focus on deep feed-forward networks and their vulnerability to adversarial examples. We will see that this vulnerability is already rooted in the design and capacity of our current networks, and will discuss ideas to tackle this vulnerability in future.

In this talk, we introduce a new preconditioner for the large, structured systems appearing in implicit Runge–Kutta time integration of parabolic partial differential equations. This preconditioner is based on a block LDU factorization with algebraic multigrid subsolves for scalability.

We compare our preconditioner in condition number and eigenvalue distribution, and through numerical experiments, with others in the literature. In experiments run with implicit Runge–Kutta stages up to s = 7, we find that the new preconditioner outperforms the others, with the improvement becoming more pronounced as the spatial discretization is refined and as temporal order is increased.

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.

I will present a nonlinear version of the open mapping principle which applies to constant-coefficient PDEs which are both homogeneous and weak* stable. An example of such a PDE is the Jacobian equation. I will discuss the consequences of such a result for the Jacobian and its relevance towards an answer to a long-standing problem due to Coifman, Lions, Meyer and Semmes. This is based on joint work with Lukas Koch and Sauli Lindberg.

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I present a general nonlinear open mapping principle suited to applications to scale-invariant PDEs in regularity regimes where the equations are stable under weak* convergence. As an application I show that, for any $p < \infty$, the set of initial data for which there are dissipative weak solutions in $L^p_t L^2_x$ is meagre in the space of solenoidal L^2 fields. This is based on joint work with A. Guerra (Oxford) and S. Lindberg (Aalto).

The Zilber-Pink conjecture predicts how large the intersection of a d-dimensional subvariety of an abelian variety/algebraic torus/Shimura variety/... with the union of special subvarieties of codimension > d can be (where the definition of "special" depends on the setting). In joint work with Fabrizio Barroero, we have reduced this conjecture for complex abelian varieties to the same conjecture for abelian varieties defined over the algebraic numbers. In work in progress, we introduce the notion of a distinguished category, which contains both connected commutative algebraic groups and connected mixed Shimura varieties. In any distinguished category, special subvarieties can be defined and a Zilber-Pink statement can be formulated. We show that any distinguished category satisfies the defect condition, introduced as a useful technical tool by Habegger and Pila. Under an additional assumption, which makes the category "very distinguished", we show furthermore that the Zilber-Pink statement in general follows from the case where the subvariety is defined over the algebraic closure of the field of definition of the distinguished variety. The proof closely follows our proof in the case of abelian varieties and leads also to unconditional results in the moduli space of principally polarized abelian surfaces as well as in fibered powers of the Legendre family of elliptic curves.

I plan to survey known facts and open questions about ultrafilters on omega generating (or not generating) ultrafilters in forcing extensions.

Primitive elements are elements that are part of a basis for a free group. We present the classical Whitehead algorithm for the recognition of such elements, and discuss the ideas behind the proof. We also present a second algorithm, more recent and completely different in the approach.

I will describe a general method for comparing the counting functions of determinantal point processes in terms of trace class norm distances between their kernels (and review what all of those words mean). Then I will outline joint work with Elizabeth Meckes using this method to prove a version of a self-similarity property of eigenvalues of Haar-distributed unitary matrices conjectured by Coram and Diaconis.  Finally, I will discuss ongoing work by my PhD student Kyle Taljan, bounding the rate of convergence for counting functions of GUE eigenvalues to the Sine or Airy process counting functions.

Smectic A liquid crystals are of great interest in physics for their striking defect structures, including curvature walls and focal conics. However, the mathematical modeling of smectic liquid crystals has not been extensively studied. This work takes a step forward in understanding these fascinating topological defects from both mathematical and numerical viewpoints. In this talk, we propose a new (two- and three-dimensional) mathematical continuum model for the transition between the smectic A and nematic phases, based on a real-valued smectic order parameter for the density perturbation and a tensor-valued nematic order parameter for the orientation. Our work expands on an idea mentioned by Ball & Bedford (2015). By doing so, physical head-to-tail symmetry in half charge defects is respected, which is not possible with vector-valued nematic order parameter.

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.

I will explain how the representation theory of rational Cherednik algebras interacts with the commutative algebra of certain subspace arrangements arising from the reflection arrangement of a complex reflection group. Potentially, the representation theory allows one to study both qualitative questions (e.g., is the arrangement Cohen-Macaulay or not?) and quantitative questions (e.g., what is the Hilbert series of the ideal of the arrangement, or even, what are its graded Betti numbers?), by applying the tools (such as orthogonal polynomials, Kazhdan-Lusztig characters, and Dirac cohomology) that representation theory provides. This talk is partly based on joint work with Susanna Fishel and Elizabeth Manosalva.

Modular and hierarchical structures are pervasive in real-world complex systems. A great deal of effort has gone into trying to detect and study these structures. Important theoretical advances in the detection of modular, or "community", structures have included identifying fundamental limits of detectability by formally defining community structure using probabilistic generative models. Detecting hierarchical community structure introduces additional challenges alongside those inherited from community detection. Here we present a theoretical study on hierarchical community structure in networks, which has thus far not received the same rigorous attention. We address the following questions: 1) How should we define a valid hierarchy of communities? 2) How should we determine if a hierarchical structure exists in a network? and 3) how can we detect hierarchical structure efficiently? We approach these questions by introducing a definition of hierarchy based on the concept of stochastic externally equitable partitions and their relation to probabilistic models, such as the popular stochastic block model. We enumerate the challenges involved in detecting hierarchies and, by studying the spectral properties of hierarchical structure, present an efficient and principled method for detecting them.

https://arxiv.org/abs/2009.07196 (15 sept.)

Motivated by the advent of machine learning, the last few years saw the return of hardware-supported low-precision computing. Computations with fewer digits are faster and more memory and energy efficient, but can be extremely susceptible to rounding errors. An application that can largely benefit from the advantages of low-precision computing is the numerical solution of partial differential equations (PDEs), but a careful implementation and rounding error analysis are required to ensure that sensible results can still be obtained. In this talk we study the accumulation of rounding errors in the solution of the heat equation, a proxy for parabolic PDEs, via Runge-Kutta finite difference methods using round-to-nearest (RtN) and stochastic rounding (SR). We demonstrate how to implement the numerical scheme to reduce rounding errors and we present \emph{a priori} estimates for local and global rounding errors. Let $u$ be the roundoff unit. While the worst-case local errors are $O(u)$ with respect to the discretization parameters, the RtN and SR error behaviour is substantially different. We show that the RtN solution is discretization, initial condition and precision dependent, and always stagnates for small enough $\Delta t$. Until stagnation, the global error grows like $O(u\Delta t^{-1})$. In contrast, the leading order errors introduced by SR are zero-mean, independent in space and mean-independent in time, making SR resilient to stagnation and rounding error accumulation. In fact, we prove that for SR the global rounding errors are only $O(u\Delta t^{-1/4})$ in 1D and are essentially bounded (up to logarithmic factors) in higher dimensions.

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 propose a subspace Gauss-Newton method for nonlinear least squares problems that builds a sketch of the Jacobian on each iteration. We provide global rates of convergence for regularization and trust-region variants, both in expectation and as a tail bound, for diverse choices of the sketching matrix that are suitable for dense and sparse problems. We also have encouraging computational results on machine learning problems.

We construct a class of Cauchy initial data without (marginally) trapped surfaces whose future evolution is a trapped region bounded by an apparent horizon, i.e., a smooth hypersurface foliated by MOTS. The estimates obtained in the evolution lead to the following conditional statement: if Kerr Stability holds, then this kind of initial data yields a class of scale critical vacuum examples of Weak Cosmic Censorship and the Final State Conjecture. Moreover, owing to estimates for the ADM mass of the data and the area of the MOTS, the construction gives a fully dynamical vacuum setting in which to study the Spacetime Penrose Inequality. We show that the inequality is satisfied for an open region in the Cauchy development of this kind of initial data, which itself is controllable by the initial data. This is joint work with Nikos Athanasiou https://arxiv.org/abs/2009.03704.

Let $r>3$ be an integer and consider the following game on the complete graph $K_n$ for $n$ a multiple of $r$: Two players, Maker and Breaker, alternately claim previously unclaimed edges of $K_n$ such that in each turn Maker claims one and Breaker claims $b$ edges. Maker wins if her graph contains a $K_r$-factor, that is a collection of $n/r$ vertex-disjoint copies of $K_r$, and Breaker wins otherwise. In other words, we consider the $b$-biased $K_r$-factor Maker-Breaker game. We show that the threshold bias for this game is of order $n^2/(r+2)$. This makes a step towards determining the threshold bias for making bounded-degree spanning graphs and extends a result of Allen, Böttcher, Kohayakawa, Naves and Person who resolved the case $r=3$ or $4$ up to a logarithmic factor.
    Joint work with Rajko Nenadov.

I will introduce recent work on the two- and three-dimensional uniform spanning trees (USTs) that establish the laws of these random objects converge under rescaling in a space whose elements are measured, rooted real trees, continuously embedded into Euclidean space. (In the three-dimensional case, the scaling result is currently only known along a particular scaling sequence.) I will also discuss various properties of the intrinsic metrics and measures of the limiting spaces, including their Hausdorff dimension, as well as the scaling limits of the random walks on the two- and three-dimensional USTs. In the talk, I will attempt to emphasise where the differences lie between the two cases, and in particular the additional challenges that arise when it comes to the three-dimensional model.
    The two-dimensional results are joint with Martin Barlow (UBC) and Takashi Kumagai (Kyoto). The three-dimensional results are joint with Omer Angel (UBC) and Sarai Hernandez-Torres (UBC).

A remarkable theorem due to Khovanskii asserts that for any finite subset $A$ of an abelian group, the cardinality of the $h$-fold sumset $hA$ grows like a polynomial for all sufficiently large $h$. However, neither the polynomial nor what sufficiently large means are understood in general. We obtain an effective version of Khovanskii's theorem for any $A \subset \mathbb{Z}$ whose convex hull is a simplex; previously such results were only available for $d = 1$. Our approach also gives information about the structure of $hA$, answering a recent question posed by Granville and Shakan. The work is joint with Leo Goldmakher at Williams College.