Past

Global symmetries greatly enrich the phase diagram of quantum many-body systems. As well as symmetry-breaking phases, symmetry-protected topological (SPT) phases have symmetric ground states that cannot be connected to a trivial state without a phase transition. There can also be symmetry-enriched critical points between these phases of matter. I will demonstrate these phenomena in phase diagrams constructed using the N-state chiral clock family of spin chains.  [Based on joint work with Paul Fendley and Abhishodh Prakash.]

Choose your favourite connected graph $\Delta$ and shade a subset $J$ of its vertices. The intersection arrangement associated to the data $(\Delta, J)$ is a collection of real hyperplanes in dimension $|Jc|$, first defined by Iyama and Wemyss. This construction involves taking the classical Coxeter arrangement coming from $\Delta$ and then setting all variables indexed by $J$ to be zero. It turns out that for many choices of $J$ the chambers of the intersection arrangement admit a nice combinatorial description, along with a wall crossing rule to pass between them. I will start by making all this precise before discussing my work to classify tilings of the hyperbolic plane arising as intersection arrangements. This has applications to local notions of stability conditions using the tilting theory of contracted preprojective algebras.

Frontiers in Quantitative Finance is brought to you by the Oxford Mathematical and Computational Finance Group and sponsored by CitiGroup and Mosaic SmartData.

Abstract
This paper parsimoniously generalizes the VIX variance index by constructing model-free factor portfolios that replicate skewness and higher moments. It then develops an infinite series to replicate option payoffs in terms of the stock, bond, and factor returns. The truncated series offers new formulas that generalize the Black-Scholes formula to hedge variance and skewness risk.


About the speaker
Steve Heston is Professor of Finance at the University of Maryland. He is known for his pioneering work on the pricing of options with stochastic volatility.
Steve graduated with a double major in Mathematics and Economics from the University of Maryland, College Park in 1983, an MBA in 1985 followed by a PhD in Finance in 1990. He has held previous faculty positions at Yale, Columbia, Washington University, and the University of Auckland in New Zealand and worked in the private sector with Goldman Sachs in Fixed Income Arbitrage and in Asset Management Quantitative Equities.

Does the limit construction for inverse systems of first-order structures preserve elementary equivalence? I will give sufficient conditions for when this is the case. Using Karp's theorem, we explain the connection between a syntactic and formal-semantic approach to inverse limits of structures. We use this to give a simple proof of van den Dries' AKE theorem (in ZFC), a general AKE theorem for mixed characteristic henselian valued fields with no assumptions on ramification. We also recall a seemingly forgotten result of Feferman, that can be interpreted as a "saturated" AKE theorem in positive characteristic: given two elementarily equivalent $\aleph_1$-saturated fields $k$ and $k'$, the formal power series rings $k[[t]]$ and $k'[[t]]$ are elementarily equivalent as well. We thus hope to popularise some ideas from categorical logic.

The 12th of Hilbert's 23 problems posed in 1900 asks for an explicit description of abelian extensions of a given base field. Over the rationals, this is given by the exponential function, and over imaginary quadratic fields, by meromorphic functions on the complex upper half plane.  Darmon and Vonk's theory of rigid meromorphic cocycles, or "RM theory", includes conjectures giving a $p$-adic solution over real quadratic fields. These turn out to be closely linked to purely topological questions about intersections of geodesics in the upper half plane, and to $p$-adic deformations of Hilbert modular forms. I will explain an extension of results of Darmon, Pozzi and Vonk proving some of these conjectures, and some ongoing work concerning analogous results on Shimura curves.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.

The classical finite element method uses piecewise-polynomial function spaces satisfying continuity and boundary conditions. Hybrid finite element methods, by contrast, drop these continuity and boundary conditions from the function spaces and instead enforce them weakly using Lagrange multipliers. The hybrid approach has several numerical and implementational advantages, which have been studied over the last few decades.

In this talk, we show how the hybrid perspective has yielded new insights—and new methods—in structure-preserving numerical PDEs. These include multisymplectic methods for Hamiltonian PDEs, charge-conserving methods for the Maxwell and Yang-Mills equations, and hybrid methods in finite element exterior calculus.

In geophysical and astrophysical flows, we are often interested in understanding the impact of internal waves on the non-wavelike flow. For example, oceanic internal waves generated at the surface and the seafloor transfer energy from the large scale flow to dissipative scales, thereby influencing the global ocean state. A primary challenge in the study of wave-flow interactions is how to separate these processes – since waves and non-wavelike flows can vary on similar spatial and temporal scales in the Eulerian frame. However, in a Lagrangian flow-following frame, temporal filtering offers a convenient way to isolate waves. Here, I will discuss a recently developed method for evolving Lagrangian mean fields alongside the governing equations in a numerical simulation, and extend this theory to allow effective filtering of waves from non-wavelike processes.

Given a pair of metric tensors g_j ≥ g₀ on a Riemannian manifold, M, it is well known that Vol_j(M)≥Vol₀(M). Furthermore, the volumes are equal if and only if the metric tensors are the same, g_j=g₀. Here we prove that if for a sequence g_j, we have g_j≥g₀, Vol_j(M)→Vol₀(M) and diam(M_j) ≤ D then (M,g_j) converges to (M,g₀) in the volume preserving intrinsic flat sense. The previous result will then be applied to prove stability of a class of tori.
 

This talk is based on joint works of myself with: Allen and Sormani (https://arxiv.org/abs/2003.01172), and Cabrera Pacheco and Ketterer (https://arxiv.org/abs/1902.03458).

The Hilbert scheme of d-points on a smooth surface is a well-studied object that still enjoys relatively large interest. We generalize Aldo Conca's Canonical Hilbert-Burch matrices and obtain explicit families of d-points. We show that such descriptions give us Białynicki-Birula cells of the Hilbert scheme for any choice of one-dimensional torus, thus describing the punctual component. This can be potentially applied to the study of singularities of the nested Hilbert scheme of points.

In 2006, Jan Dymara introduced the concept of weighted \(\ell^2\) Betti numbers as a method of computing regular \(\ell^2\) Betti numbers of buildings. This notion of dimension is measured by using Hecke algebras associated to the relevant Coxeter groups. I will briefly introduce buildings and then give a comparison between the regular \(\ell^2\) Betti numbers and the weighted ones.

The organizing principle of Ramsey theory is that in large mathematical structures, there are relatively large substructures which are homogeneous. This is quantified in combinatorics by the notion of Ramsey numbers $r(s,t)$, which denote the minimum $N$ such that in any red-blue coloring of the edges of the complete graph on $N$ vertices, there exists a red complete graph on $s$ vertices or a blue complete graph on $t$ vertices.

While the study of Ramsey numbers goes back almost one hundred years, to early papers of Ramsey and Erdős and Szekeres, the long-standing conjecture of Erdős that $r(s,t)$ has order of magnitude close to $t^{s-1}$ as $t \to \infty$ remains open in general. It took roughly sixty years before the order of magnitude of $r(3,t)$ was determined by Jeong Han Kim, who showed $r(3,t)$ has order of magnitude $t^2/\log t$ as $t \to \infty$. In this talk, we discuss a variety of new techniques, including the modern method of containers, which lead to a proof of the conjecture of Erdős that $r(4,t)$ is of order close to $t^3$.

One of the salient philosophies in our approach is that good Ramsey graphs hide inside pseudorandom graphs, and the long-standing emphasis of tackling Ramsey theory from the point of view of purely random graphs is superseded by pseudorandom graphs. Via these methods, we also come close to determining the well-studied related quantities known as Erdős-Rogers functions and discuss related hypergraph coloring problems and applications.

Joint work in part with Sam Mattheus, Dhruv Mubayi and David Conlon.

The space of measured laminations on a hyperbolic surface is a generalisation of the set of weighted multi curves. The action of the mapping class group on this space is an important tool in the study of the geometry of the surface. 
For orientable surfaces, orbit closures are now well-understood and were classified by Lindenstrauss and Mirzakhani. In particular, it is one of the pillars of Mirzakhani’s curve counting theorems. 
For non-orientable surfaces, the behaviour of this action is very different and the classification fails. In this talk I will review some of these differences and describe mapping class group orbit closures of (projective) measured laminations for non-orientable surfaces. This is joint work with Erlandsson, Gendulphe and Souto.

Dispersive nonlinear partial differential equations can be used to describe a range of physical systems, from water waves to spin states in ferromagnetism. The numerical approximation of solutions with limited differentiability (low-regularity) is crucial for simulating fascinating phenomena arising in these systems including emerging structures in random wave fields and dynamics of domain wall states, but it poses a significant challenge to classical algorithms. Recent years have seen the development of tailored low-regularity integrators to address this challenge. Inherited from their description of physicals systems many such dispersive nonlinear equations possess a rich geometric structure, such as a Hamiltonian formulation and conservation laws. To ensure that numerical schemes lead to meaningful results, it is vital to preserve this structure in numerical approximations. This, however, results in an interesting dichotomy: the rich theory of existent structure-preserving algorithms is typically limited to classical integrators that cannot reliably treat low-regularity phenomena, while most prior designs of low-regularity integrators break geometric structure in the equation. In this talk, we will outline recent advances incorporating structure-preserving properties into low-regularity integrators. Starting from simple discussions on the nonlinear Schrödinger and the Korteweg–de Vries equation we will discuss the construction of such schemes for a general class of dispersive equations before demonstrating an application to the simulation of low-regularity vortex filaments. This is joint work with Yvonne Alama Bronsard, Valeria Banica, Yvain Bruned and Katharina Schratz.

Models for networks that evolve and change over time are ubiquitous in a host of domains including modeling social networks, understanding the evolution of systems in proteomics, the study of the growth and spread of epidemics etc.

This talk will give a brief summary of three recent findings in this area where stochastic approximation techniques play an important role:

1. Understanding the effect and detectability of change point in the evolution of the system dynamics.
2. Reconstructing the initial "seed" that gave rise to the current network, sometimes referred to as Network Archeology.
3. The disparity in the behavior of different centrality measures such as degree and page rank centrality for measuring popularity in settings where there are vertices of different types such as majorities and minorities as well as insight analyzing such problems give for at first sight unrelated issues such as sampling rare groups within the network.

The main goal will to be convey unexpected findings in each of these three areas and in particular the "unreasonable effectiveness" of continuous time branching processes.

I'll give an introduction to a recent theme in the Langlands program over number fields and mixed characteristic local fields (with a much older history over function fields). This is enhancing the traditional 'set-theoretic' Langlands correspondence into something with a more geometric flavour. For example, relating (categories of) representations of p-adic groups to sheaves on moduli spaces of Galois representations. No number theory or 'Langlands' background will be assumed!

The interplay between stochastic processes and optimal control has been extensively explored in the literature. With the recent surge in the use of diffusion models, stochastic processes have increasingly been applied to sample generation. This talk builds on the log transform, known as the Cole-Hopf transform in Brownian motion contexts, and extends it within a more abstract framework that includes a linear operator. Within this framework, we found that the well-known relationship between the Cole-Hopf transform and optimal transport is a particular instance where the linear operator acts as the infinitesimal generator of a stochastic process. We also introduce a novel scenario where the linear operator is the adjoint of the generator, linking to Bayesian inference under specific initial and terminal conditions. Leveraging this theoretical foundation, we develop a new algorithm, named the HJ-sampler, for Bayesian inference for the inverse problem of a stochastic differential equation with given terminal observations. The HJ-sampler involves two stages: solving viscous Hamilton-Jacobi (HJ) partial differential equations (PDEs) and sampling from the associated stochastic optimal control problem. Our proposed algorithm naturally allows for flexibility in selecting the numerical solver for viscous HJ PDEs. We introduce two variants of the solver: the Riccati-HJ-sampler, based on the Riccati method, and the SGM-HJ-sampler, which utilizes diffusion models. Numerical examples demonstrate the effectiveness of our proposed methods. This is an ongoing joint work with Zongren Zou, Jerome Darbon, and George Em Karniadakis.

The signature kernel is one of the most powerful measures of similarity for sequences of arbitrary length accompanied with attractive theoretical guarantees from stochastic analysis. Previous algorithms to compute the signature kernel scale quadratically in terms of the length and the number of the sequences. To mitigate this severe computational bottleneck, we develop a random Fourier feature-based acceleration of the signature kernel acting on the inherently non-Euclidean domain of sequences. We show uniform approximation guarantees for the proposed unbiased estimator of the signature kernel, while keeping its computation linear in the sequence length and number. In addition, combined with recent advances on tensor projections, we derive two even more scalable time series features with favourable concentration properties and computational complexity both in time and memory. Our empirical results show that the reduction in computational cost comes at a negligible price in terms of accuracy on moderate-sized datasets, and it enables one to scale to large datasets up to a million time series.

Please click here to read the full paper.

We study the mathematical properties of time-dependent flows of incompressible fluids that respond as an Euler fluid until the modulus of the symmetric part of the velocity gradient exceeds a certain, a-priori given but arbitrarily large, critical value. Once the velocity gradient exceeds this threshold, a dissipation mechanism is activated. Assuming that the fluid, after such an activation, dissipates the energy in a specific manner, we prove that the corresponding initial-boundary-value problem is well-posed in the sense of Hadamard. In particular, we show that for an arbitrary, sufficiently regular, initial velocity there is a global-in-time unique weak solution to the spatially-periodic problem. This is a joint result with Miroslav Bulíček. 

Congruent numbers are natural numbers which are the area of right angled triangles with all rational sides. This talk will investigate conjectures for the density of congruent numbers up to some value $X$. One can phrase the question of whether a natural number is congruent in terms of whether an elliptic curve has non−zero rank. A theorem of Coates and Wiles connects this to whether the $L$-function associated to this elliptic curve vanishes at $1$. We will mention the conjecture of Keating on the asymptotic density based on random matrix considerations, and prove Tunnell’s Theorem, which connects the question of whether a natural number is a congruent number to counting integral points on varieties. Finally, I will hint at some future work I hope to do on non-vanishing of the $L$-functions.

The simplicial volume of a simplicial complex is a topological invariant
related to the growth of the fundamental group, which gives rise to a
semi-norm in homology. In this talk, we introduce the volume entropy
semi-norm, which is also related to the growth of the fundamental group
of simplicial complexes and shares functorial properties with the
simplicial volume. Answering a question of Gromov, we prove that the
volume entropy semi-norm is equivalent to the simplicial volume
semi-norm in every dimension. Joint work with I. Babenko.
 

Optimal transport theory is a natural way to define both a distance and a geometry on the space of probability measures. In settings like graphical causal models (also called Bayes networks or belief networks), the space of probability measures is enriched by an information structure modeled by a directed graph. This talk introduces a variant of optimal transport including such a graphical information structure. The goal is to provide a concept of optimal transport whose topological and geometric properties are well suited for structural causal models. In this regard, we show that the resulting concept of Wasserstein distance can be used to control the difference between average treatment effects under different distributions, and is geometrically suitable to interpolate between different structural causal models.

The Lagrange multipliers method relates critical points on a submanifold with those on an enlarged space. In derived algebraic geometry, we are allowed to consider a more general type of functions called shifted functions and thus a shifted version of the Lagrange multipliers method. If we start with quasi-smooth derived stacks, the Borisov-Joyce-Oh-Thomas virtual Lagrangian cycle of the critical locus coincides with the cosection localized virtual fundamental cycle of the enlarged space. This immediately implies the quantum Lefschetz principle of Chang-Li and an analogous result for branched covers. Based on a joint work with Hyeonjun Park. 

Modern neural network models are trained using fairly standard stochastic gradient optimizers, sometimes employing mild preconditioners. 
A natural question to ask is whether significant improvements in training speed can be obtained through the development of better optimizers. 

In this talk I will argue that this is impossible in the large majority of cases, which explains why this area of research has stagnated. I will go on to identify several situations where improved preconditioners can still deliver significant speedups, including exotic architectures and loss functions, and large batch training. 

State-of-the art experimental data promises exquisite insight into the spatial heterogeneity in tissue samples. However, the high level of detail in such data is contrasted with a lack of methods that allow an analysis that fully exploits the available spatial information. Persistent Homology (PH) has been very successfully applied to many biological datasets, but it is typically limited to the analysis of single species data. In the first part of my talk, I will highlight two novel techniques in relational PH that we develop to encode spatial heterogeneity of multi species data. Our approaches are based on Dowker complexes and Witness complexes. We apply the methods to synthetic images generated by an agent-based model of tumour-immune cell interactions. We demonstrate that relational PH features can extract biological insight, including the dominant immune cell phenotype (an important predictor of patient prognosis) and the parameter regimes of a data-generating model. I will present an extension to our pipeline which combines graph neural networks (GNN) with local relational PH and significantly enhances the performance of the GNN on the synthetic data. In the second part of the talk, I will showcase a noise-robust extension of Reani and Bobrowski’s cycle registration algorithm  (2023) to reconstruct 3D brain atlases of Drosophila flies from a sequence of μ-CT images.

In this talk I will explain how a dictionary between the Bondi-Sachs and the Newman-Penrose formalism can be used to organize the subleading data appearing in the metric for asymptotically-flat spacetimes. In particular, this can be used to show that the higher Bondi aspects can be traded for higher spin charges, and that the latter form a w_infinity algebra.

Morphogen protein gradients play an essential role in the spatial regulation of patterning during embryonic development.  The most commonly accepted mechanism of protein gradient formation involves the diffusion and degradation of morphogens from a localized source. Recently, an alternative mechanism has been proposed, which is based on cell-to-cell transport via thin, actin-rich cellular extensions known as cytonemes. It has been hypothesized that cytonemes find their targets via a random search process based on alternating periods of retraction and growth, perhaps mediated by some chemoattractant. This is an actin-based analog of the search-and-capture model of microtubules of the mitotic spindle searching for cytochrome binding sites (kinetochores) prior to separation of cytochrome pairs. In this talk, we introduce a search-and-capture model of cytoneme-based morphogenesis, in which nucleating cytonemes from a source cell dynamically grow and shrink until making contact with a target cell and delivering a burst of morphogen. We model the latter as a one-dimensional search process with stochastic resetting, finite returns times and refractory periods. We use a renewal method to calculate the splitting probabilities and conditional mean first passage times (MFPTs) for the cytoneme to be captured by a given target cell. We show how multiple rounds of search-and-capture, morphogen delivery, cytoneme retraction and nucleation events lead to the formation of a morphogen gradient. We proceed by formulating the morphogen bursting model as a queuing process, analogous to the study of translational bursting in gene networks. We end by briefly discussing current work on a model of cytoneme-mediated within-host viral spread.

The Hecke category first rose to prominence through the proof of the Kazhdan-Lusztig conjecture. Since then, the Hecke category has proven to be a fundamental object in representation theory with many interesting applications to modular representation theory, quantum groups, knot theory, categorification and diagrammatic algebra. This talk aims to give a gentle introduction to the Hecke category. We will first discuss the geometric incarnation of the Hecke category and how it was used to prove the Kazhdan-Lusztig conjecture. Then, we move on to a more modern approach due to Soergel and Elias-Williamson which is purely algebraic, and we will discuss some recent advances in representation theory based on this approach.

Gotzmann's regularity and persistence theorems provide tools which allow us to find explicit equations for the Hilbert scheme Hilb_P(P^n). A natural next step is to generalise these results to the multigraded Hilbert scheme Hilb_P(X) of a smooth projective toric variety X. In 2003 Maclagan and Smith generalise Gotzmann's regularity theorem to this case. We present new persistence type results for the product of two projective spaces, and time permitting discuss how these may be applied to a more general smooth projective toric variety.

In recent joint work with Pablo Cubides Kovacsics and Jinhe Ye on beautiful pairs in the unstable context, the amalgamation property (AP) for the class of global definable types plays a key role. In the talk, we will first indicate some important cases in which AP holds, and we will then present the construction of examples of theories, obtained in joint work with Rosario Mennuni, where AP fails.

One can get fairly good estimates for primes in short
intervals under the assumption of the Riemann Hypothesis. Weaker
estimates can be shown unconditionally by using a 'zero density
estimate' in place of the Riemann Hypothesis. These zero density
estimates are typically proven by bounding how often a Dirichlet
polynomial can take large values, but have been limited by our
understanding of the number of zeros with real part 3/4. We introduce a
new method to prove large value estimates for Dirichlet polynomials,
which improves on previous estimates near the 3/4 line.

This is joint work (still in progress) with Larry Guth.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.

Joint work Marta Betcke and Bolin Pan

In photoacoustic tomography (PAT) with flat sensor, we routinely encounter two types of limited data. The first is due to using a finite sensor and is especially perceptible if the region of interest is large relatively to the sensor or located farther away from the sensor. In this talk we focus on the second type caused by a varying sensitivity of the sensor to the incoming wavefront direction which can be modelled as binary i.e. by a cone of sensitivity. Such visibility conditions result, in Fourier domain, in a restriction of the data to a bowtie, akin to the one corresponding to the range of the forward operator but further narrowed according to the angle of sensitivity. 

We show how we can separate the visible and invisible wavefront directions in PAT image and data using a directional frame like Curvelets, and how such decomposition allows for decoupling of the reconstruction involving application of expensive forward/adjoint solvers from the training problem. We present fast and stable approximate Fourier domain forward and adjoint operators for reconstruction of the visible coefficients for such limited angle problem and a tailored UNet matching both the multi-scale Curvelet decomposition and the partition into the visible/invisible directions for learning the invisible coefficients from a training set of similar data.

Liquid crystal elastomers are rubbery solids containing molecular LC rods that align along a common director. On heating, the alignment is disrupted, leading to a substantial (~50%) contraction along the director.  In recent years, there has been a great deal of interest in fabrication LCE sheets with a bespoke alignment pattern. On heating, these patterns generate  corresponding patterns of contraction that can morph a sheet into a bespoke curved surface such as a cone or face. Moreover, LCEs can also be activated by light, either photothermally or photochemically, leading to similarly large contractions. Stimulation by light also introduces an important new possibility: using spatio-temporal patterns of illumination to morph a single LCE sample into a range of different surfaces. Such stimulation can enable non-reciprocal actuation for viscous swimming or pumping, and control over the whole path taken by the sheet through shape-space rather than just the final destination. In this talk, I will start by with an experimental example of a spatio-temporal pattern of illumination being used to actuate an LCE peristaltic pump. I will then introduce a second set of experiments, in which a monodomain sheet morphs first into a cone, an anti-cone and then an array of cones upon exposure to different patterns of illumination. Finally, I will then discuss the general problem of how to choose a pattern of illumination to morph a director-patterned sheet into an arbitrary surface, first analytically for axisymmetric cases, then numerically for low symmetry cases. This last study exceeds our current experimental capacity, but highlights how, with full spatio-temporal control over the stimulation magnitude, one can choreograph an LCE sheet to undergo almost any pattern of morphing.

Consider definable sets over the family of finite fields $\mathbb{F}_q$. Ax proved a quantifier-elimination result for this theory, in a reasonable geometric language. Chatzidakis, Van den Dries and Macintyre showed that to a first-order approximation, the cardinality of a definable set $X$ is definable in a very mild expansion of Ax's theory.  Can such a statement be true of the next higher order approximation, i.e. can we write $|X(\mathbb{F}_q)| = aq^{d} + bq^{d-1/2} + o(q^{d-1/2})$, with $d,a,b$ varying definably with $X$ in a tame theory?    Here $b$ must be viewed as real-valued so continuous logic is needed. I will report on joint work in progress with Will Johnson.

This talk will introduce Khovanov and Knot Floer Homology as tools for studying knots. I will then cover some applications to problems in knot theory including distinguishing embedded surfaces and how they can be used in the context of ribbon concordances. No prior knowledge of either will be necessary and lots of pictures are included.

A group is called W*-superrigid if its group von Neumann algebra completely remembers the original group. In this talk, I will present a recent joint work with Stefaan Vaes in which we generalize W*-superrigidity for groups in two directions. Firstly, we find a class of groups for which W*-superrigidity holds in the presence of a twist by an arbitrary 2-cocycle: the twisted group von Neumann algebra completely remembers both the original group and the 2-cocycle. Secondly, for the same class of groups, the superrigidity also holds up to virtual isomorphism.

If a group quasiisometrically embeds into a finite product of infinite valence trees then a number of things are implied; for example, the group will have finite Assouad-Nagata dimension and finite asymptotic dimension. An even stronger statement is that the group quasiisometrically embeds into a finite product of uniformly bounded valence trees. The research on which groups quasiisometrically embed into finite products of uniformly bounded valence trees is limited, however a notable result of Buyalo, Dranishnikov and Schroeder from 2007 proves that all hyperbolic groups do admit these quasiisometric embeddings. In a recently released preprint, I extend their result to cover groups which are relatively hyperbolic with respect to virtually abelian peripheral subgroups. 

This talk will focus on the ideas at the core of Buyalo, Dranishnikov and Schroeder’s result and the extension that I proved, and in particular I will attempt to provide a general framework for upgrading quasiisometric embeddings into infinite valence trees so that they are now quasiisometric embeddings into uniformly bounded valence trees. The central concept is called a diary which I will define. 

Let $G$ be a $d$-regular graph of growing degree on $n$ vertices. Form a random subgraph $G_p$ of $G$ by retaining edge of $G$ independently with probability $p=p(d)$. Which conditions on $G$ suffice to observe a phase transition at $p=1/d$, similar to that in the binomial random graph $G(n,p)$, or, say, in a random subgraph of the binary hypercube $Q^d$?

We argue that in the supercritical regime $p=(1+\epsilon)/d$, $\epsilon>0$ a small constant, postulating that every vertex subset $S$ of $G$ of at most $n/2$ vertices has its edge boundary at least $C|S|$, for some large enough constant $C=C(\epsilon)>0$, suffices to guarantee likely appearance of the giant component in $G_p$. Moreover, its asymptotic order is equal to that in the random graph $G(n,(1+\epsilon)/n)$, and all other components are typically much smaller.

We also give examples demonstrating tightness of our main result in several key senses.

A joint work with Sahar Diskin, Joshua Erde and Mihyun Kang.

In this talk I will describe the systematic construction of strongly interacting RG fixed points with a finite disorder strength.  Such random-field disorder is quite common in condensed matter experiment, necessitating an understanding of the effects of this disorder on the properties of such fixed points. In the past, such disordered fixed points were accessed using e.g. epsilon expansions in perturbative quantum field theory, using the replica method to treat disorder.  I will show that holography gives an alternative picture for RG flows towards disordered fixed points.  In holography, spatially inhomogeneous disorder corresponds to inhomogeneous boundary conditions for an asymptotically-AdS spacetime, and the RG flow of the disorder strength is captured by the solution to the Einstein-matter equations. Using this construction, we have found analytically-controlled RG fixed points with a finite disorder strength.  Our construction accounts for, and explains, subtle non-perturbative geometric effects that had previously been missed.  Our predictions are consistent with conformal perturbation theory when studying disordered holographic CFTs, but the method generalizes and gives new models of disordered metallic quantum criticality.

We consider a quantum field model with exponential interactions on the two-dimensional torus,  which is called the ${¥rm{exp}(¥Phi)_{2}$-quantum field model or Hoegh-Krohn’s model. In this talk, we discuss the stochastic quantization of this model. Combining key properties of Gaussian multiplicative chaos with a method for singular SPDEs, we construct a unique time-global solution to the corresponding parabolic stochastic quantization equation in the full $L_{1}$-regime $¥vert ¥alpha ¥vert<{¥sqrt{8¥pi}}$ of the charge parameter $¥alpha$. We also identify the solution with an infinite dimensional diffusion process constructed by the Dirichlet form approach. 

The main part of this talk is based on joint work with Masato Hoshino (Osaka University) and  Seiichiro Kusuoka (Kyoto University), and the full paper can be found on https://link.springer.com/article/10.1007/s00440-022-01126-z

We present a general concept that is suitable for studying the stability of equilibria for open systems in continuum thermodynamics. We apply such concept to a generalized Newtonian incompressible heat conducting fluid with prescribed nonuniform temperature on the boundary and with the no-slip boundary conditions for the velocity in three dimensional domain. For large class of constitutive relation for the Cauchy stress, we identify a class of proper solutions converging to the equilibria exponentially in a suitable metric and independently of the distance to equilibria at the initial time. Consequently, the equilibrium is nonlinearly stable and attracts all weak solutions from that class. The proper solutions exist and satisfy entropy (in)equality.