Past

After a review of Batalin-Vilkovisky formalism and homotopy algebras, we discuss how these structures emerge in quantum field theory and gravity. We focus then on the application of these sophisticated mathematical tools to scattering amplitudes (both tree- and loop-level) and to the understanding of the dualities between gauge theories and gravity, highlighting generalizations of old results and presenting new ones.

The Ricci flow on a surface is an intrinsic evolution of the metric converging to a constant curvature metric within the conformal class. It can be seen as an infinite-dimensional gradient flow. We introduce a natural 'Langevin' version of that flow, thus constructing an SPDE with invariant measure expressed in terms of Liouville Conformal Field Theory.
Joint work with Hao Shen (Wisconsin).

In this talk I will give an introduction to random multiplicative functions, and cover the recent developments in this area. I will also explain how RMF's are connected to some of the important open problems in Analytic Number Theory.

The main result is the existence of smooth, properly embedded 3-discs in S¹ × D³ that are not smoothly isotopic to {1} × D³. We describe a 2-variable Laurent polynomial invariant of 3-discs in S¹ × D³. This allows us to show that, when taken up to isotopy, such 3-discs form an abelian group of infinite rank. Joint work with David Gabai.

We describe how Smith theory applies in the setting of Hamiltonian Floer homology filtered by the action functional, and provide applications to questions regarding Hamiltonian diffeomorphisms, including the Hofer-Zehnder conjecture on the existence of infinitely many periodic points and a question of McDuff-Salamon on Hamiltonian diffeomorphisms of finite order.

All five-dimensional non-abelian gauge theories have a U(1)U(1)I​U(1) global symmetry associated with instantonic particles. I will describe a mixed ’t Hooft anomaly between this and other global symmetries of  the theory, namely the one-form center symmetry or ordinary flavor symmetry for theories with fundamental matter. I will explore some general dynamical properties of the candidate phases implied by the anomaly, and apply our results to supersymmetric gauge theories in five dimensions, analysing the symmetry enhancement patterns occurring at their conjectured RG fixed points.

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.

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.

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.

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 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.

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.

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.

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.

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.

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.