Past

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

Standard symmetric α-stable cylindrical processes in Hilbert spaces are the natural generalisation of the analogue processes in Euclidean spaces. However, like standard Brownian motions, standard symmetric α-stable processes in finite dimensions can only be generalised to infinite dimensional Hilbert spaces as cylindrical processes, i.e. processes in a generalised sense (of Gel’fand and Vilenkin (1964) or Segal (1954))  not attaining values in the underlying Hilbert space.

In this talk, we briefly introduce the theory of stochastic integrals with respect to standard symmetric α-stable cylindrical processes. As these processes exist only in the generalised sense, introducing a stochastic integral requires an approach different to the classical one by semi-martingale decomposition. The main result presented in this talk is the existence of a solution to an abstract evolution equation driven by a standard symmetric α-stable cylindrical process. The main tool for establishing this result is a Yosida approximation and an Itô formula for Hilbert space-valued semi-martingales where the martingale part is represented as an integral driven by cylindrical α-stable noise. While these tools are standard in stochastic analysis, due to the cylindrical nature of our noise, their application requires completely novel arguments and techniques.

In this talk I will provide an overview of our current research at the interface of quantum field theory (QFT), Lorentzian geometry and higher categorical structures. I will present operads which encode the rich algebraic structure of QFTs on Lorentzian manifolds and show that in low dimensions their algebras relate to familiar algebraic structures. Our operads share certain similarities with the little disk operads from topology, in particular they involve a homotopical localization at geometric embeddings related to ‘time evolution’. I will show that, in contrast to the topological context, this homotopical localization can be strictified in many important classes of examples, which is loosely speaking due to the 1-dimensional nature of time evolution in Lorentzian geometry. I will conclude by explaining how simple examples of such Lorentzian QFTs can be constructed from a homotopical generalization of the concept of Green’s operators for hyperbolic partial differential equations, which we call Green hyperbolic complexes. Throughout this talk, I will frequently comment on the similarities and differences between our approach, factorization algebras and functorial field theories.

The Steinberg scheme and the equivariant coherent sheaves on it play a very important role in Geometric Representation theory. In this talk we will discuss various t-structures on the equivariant derived category of the Steinberg of importance for Representation theory in positive characteristics. Based on arXiv:2302.05782.

We propose a theory of enumerative invariants for structure groups of type B/C/D, that is, for the orthogonal and symplectic groups. For example, we count orthogonal or symplectic principal bundles on projective varieties, and there is also a quiver analogue called self-dual quiver representations. We discuss two different flavours of these invariants, namely, motivic invariants and homological invariants, the former of which can be used to define Donaldson–Thomas invariants in type B/C/D. We also discuss algebraic structures arising from the relevant moduli spaces, including Hall algebras, Joyce's vertex algebras, and modules for these algebras, which are used to write down wall-crossing formulae for our invariants.

How can we find and apply the best optimization algorithm for a given problem?   This question is as old as mathematical optimization itself, and is notoriously hard: even special cases such as finding the optimal learning rate for gradient descent is nonconvex in general. 

In this talk we will discuss a dynamical systems approach to this question. We start by discussing an emerging paradigm in differentiable reinforcement learning called “online nonstochastic control”. The new approach applies techniques from online convex optimization and convex relaxations to obtain new methods with provable guarantees for classical settings in optimal and robust control. We then show how this methodology can yield global guarantees for learning the best algorithm in certain cases of stochastic and online optimization. 

No background is required for this talk, but relevant material can be found in this new text on online control and paper on meta optimization.

Prof. Elad's Bio

Feeling the winter blues setting in? Come and chill out with us and do some crafts.

Recently, the Newman-Janis shift has been revisited from the angle of scattering amplitudes in terms of the so-called "massive spinor-helicity variables," tracing back to Penrose and Perjés in the 70s. However, well-established results are limited in the same-helicity (self-dual) sector, while a puzzle of spurious poles arises in mixed-helicity sectors. This talk will outline how massive twistor theory can reproduce the same-helicity results while offering a possible solution to the spurious pole puzzle. Firstly, the Newman-Janis shift in the same-helicity sector is derived from a complexified version of the equivalence principle. Secondly, the massive twistor particle is coupled to background fields from bottom-up and top-down perspectives. The former is based on perturbations of symplectic structures in massive twistor space. The latter provides a generalization of Newman-Janis shift in generic backgrounds, which also leads to "curved massive twistor space" and its deformed massive incidence relation. Lastly, the Feynman rules of the first-quantized massive twistor particle and their physical interpretation are briefly discussed. Overall, a significant emphasis is put on the Kähler geometry ("zig-z̄ag structure") of massive twistor space, which eventually connects to a worldsheet structure of the Kerr solution.

What opportunities are available outside of academia? What skills beyond strong academic background are companies looking for to be successful in transitioning to industry? Come along and hear from video technology company V-Nova and Dr Anne Wolfes from the Careers Service to get some invaluable advice on careers outside academia.

In this talk, we discuss two-dimensional Riemann problems in the framework of potential flow
equation and isentropic Euler system. We first review recent results on the existence, regularity and properties of
global self-similar solutions involving transonic shocks for several 2D Riemann problems in the
framework of potential flow equation. Examples include regular shock reflection, Prandtl reflection, and four-shocks
Riemann problem. The approach is to reduce the problem to a free boundary problem for a nonlinear elliptic equation
in self-similar coordinates. A well-known open problem is to extend these results to a compressible Euler system,
i.e. to understand the effects of vorticity. We show that for the isentropic Euler system, solutions have
low regularity, specifically velocity and density do not belong to the Sobolev space $H^1$ in self-similar coordinates.  
We further discuss the well-posedness of the transport equation for vorticity in the resulting low regularity setting.

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Accurate interpretation of medical images is crucial for disease diagnosis and treatment, and AI has the potential to minimize errors, reduce delays, and improve accessibility. The focal point of this presentation lies in a grand ambition: the development of 'Generalist Medical AI' systems that can closely resemble doctors in their ability to reason through a wide range of medical tasks, incorporate multiple data modalities, and communicate in natural language. Starting with pioneering algorithms that have already demonstrated their potential in diagnosing diseases from chest X-rays or electrocardiograms, matching the proficiency of expert radiologists and cardiologists, I will delve into the core challenges and advancements in the field. The discussion will navigate towards the topic of label-efficient AI models: with a scarcity of meticulously annotated data in healthcare, the development of AI systems capable of learning effectively from limited labels has become a key concern. In this vein, I'll delve into how the innovative use of self-supervision and pre-training methods has led to algorithmic advancements that can perform high-level diagnostic tasks using significantly less annotated data. Additionally, I will talk about initiatives in data curation, human-AI collaboration, and the creation of open benchmarks to evaluate the generalizability of medical AI algorithms. In sum, this talk aims to deliver a comprehensive picture of the state of 'Generalist Medical AI,' the advancements made, the challenges faced, and the prospects lying ahead.

In this talk I will review how the BV formalism is used in quantizing theories with local gauge symmetries within the framework of perturbative algebraic quantum field theory. The latter is a mathematically rigorous approach to QFT that combines the locality idea going back to Haag and Kastler with Epstein-Glaser renormalization. In my talk I will also show how these methods can also lead to the construction of a factorization algebra.

The Hecke algebra is an algebraic gadget for studying the smooth complex representations of locally profinite groups. We demonstrate the spherical Hecke algebra of GL(n,F) is commutative and present a combinatorial proof of the Satake isomorphism. We apply this to the classification of spherical representations of GL(2,F).

This talk will cover the greatest hits of pointwise ergodic theory, beginning with Birkhoff's theorem, then Bourgain's work, and finishing with more modern directions.

Popular automated market makers (AMMs) use constant function markets (CFMs) to clear the demand and supply in the pool of liquidity. A key drawback in the implementation of CFMs is that liquidity providers (LPs) are currently providing liquidity at a loss, on average. In this paper, we propose two new designs for decentralised trading venues, the arithmetic liquidity pool (ALP) and the geometric liquidity pool (GLP). In both pools, LPs choose impact functions that determine how liquidity taking orders impact the marginal exchange rate of the pool, and set the price of liquidity in the form of quotes around the marginal rate. The impact functions and the quotes determine the dynamics of the marginal rate and the price of liquidity. We show that CFMs are a subset of ALP; specifically, given a trading function of a CFM, there are impact functions and  quotes in the ALP that replicate the marginal rate dynamics and the execution costs in the CFM. For the ALP and GLP, we propose an optimal liquidity provision strategy where the price of liquidity maximises the LP's expected profit and the strategy depends on the LP's (i) tolerance to inventory risk and (ii) views on the demand for liquidity. Our strategies admit closed-form solutions and are computationally efficient.  We show that the price of liquidity in CFMs is suboptimal in the ALP. Also, we give conditions on the impact functions and the liquidity provision strategy to prevent arbitrages from rountrip trades. Finally, we use transaction data from Binance and Uniswap v3 to show that liquidity provision is not a loss-leading activity in the ALP.

Two areas of current research in Mathematical Gauge Theory are the study of higher dimensional instantons on manifolds with special holonomy (for example, Calabi-Yau three folds, G₂ and Spin(7) manifolds), and low dimensional gauge theories (for example the Kapustin-Witten, Haydys-Witten and ADHM Seiberg-Witten equations). A common feature of these two sets of theories is that the moduli spaces of solutions are in general not compact. In both cases, compactness issues arise because of solutions to a certain non-linear equation called the Fueter equation. In this talk, I'll explain how this non compactness gives a relationship between these high and low dimensional gauge theories.

Geometric evolutions can arise as simple models or fundamental building blocks in various applications with moving boundaries and time-dependent domains, such as grain boundaries in materials or deforming cell boundaries. Mesh-based methods require adaptation and smoothing, particularly in the case of strong deformations. We consider finite element schemes based on classical approaches for geometric evolution equations but augmented with the gradient of the Dirichlet energy or a variant of it, which is known to produce a tangential mesh movement beneficial for the mesh quality. We focus on the one-dimensional case, where convergence of semi-discrete schemes can be proved, and discuss two cases. For networks forming triple junctions, it is desirable to keep the impact of any additional, mesh smoothing terms on the geometric evolution as small as possible, which can be achieved with a perturbation approach. Regarding the elastic flow of curves, the Dirichlet energy can serve as a replacement of the usual penalty in terms of the length functional in that, modulo rescaling, it yields the same minimisers in the long run.

Capillary forces acting at the surface of a liquid drop can be strong enough to deform small objects and recent studies have provided several examples of elastic instabilities induced by surface tension. Inspired by the windlass mechanism in spider webs, we present a system where a liquid drop sits on a straight fiber and attracts the fiber which thereby coils inside the drop. We then introduce a 2D extension of the mechanism and build a membrane that can extend/contract by a factor of 20.

The local Kronecker-Weber theorem states that the maximal abelian extension of p-adic numbers Q_p is obtained from this field by adjoining all roots of unity. In 2018, Koenigsmann conjectured that the maximal abelian extension of Q_p is decidable. In my talk, we will discuss Koenigsmann's proposed axiomatisation. In contrast, the maximal unramified extension of Q_p is known to be decidable, admitting a complete axiomatisation by an informed but simple set of axioms (this is due to Kochen). We explain how the question of completeness can be reduced to an Ax-Kochen-Ershov result in residue characteristic 0 by the method of coarsening.

The vacant set of the random walk on the torus undergoes a percolation phase transition at Poissonian timescales in dimensions 3 and higher. The talk will review this phenomenon and discuss recent progress regarding the nature of the transition, both for this model and its infinite-volume limit, the vacant set of random interlacements, introduced by Sznitman in Ann. Math., 171 (2010), 2039–2087. The discussion will lead up to recent progress regarding the long purported equality of several critical parameters naturally associated to the transition. 

Named after mathematical physicists Kubo, Martin, and Schwinger, KMS states are a special class of states on any C$^*$-algebra admitting a continuous action of the real numbers. Unlike in the case of von Neumann algebras, where each modular flow has a unique KMS state, the collection of KMS states for a given flow on a C$^*$-algebra can be quite intricate. In this talk, I will explain what abstract properties these simplices have and show how one can realise such a simplex on various classes of simple C$^*$-algebras.

I will talk about two separate projects where random walks are building a random tree, yielding preferential attachment behaviour from completely local mechanisms.
First, the Tree Builder Random Walk is a randomly growing tree, built by a walker as she is walking around the tree. At each time $n$, she adds a leaf to her current vertex with probability $n^{-\gamma}, \gamma\in(2/3, 1]$, then moves to a uniform random neighbor on the possibly modified tree. We show that the tree process at its growth times, after a random finite number of steps, can be coupled to be identical to the Barabási-Albert preferential attachment tree model. This coupling implies that many properties known for the BA-model, such as diameter and degree distribution, can be directly transferred to our model. Joint work with János Engländer, Giulio Iacobelli, and Rodrigo Ribeiro. Second, we introduce a network-of-networks model for physical networks: we randomly grow subgraphs of an ambient graph (say, a box of $\mathbb{Z}^d$) until they hit each other, building a tree from how these spatially extended nodes touch each other. We compute non-rigorously the degree distribution exponent of this tree in large generality, and provide a rigorous analysis when the nodes are loop-erased random walks in dimension $d=2$ or $d\geq 5$, using a connection with the Uniform Spanning Tree. Joint work with Ádám Timár, Sigurdur Örn Stefánsson, Ivan Bonamassa, and Márton Pósfai. (See https://arxiv.org/abs/2306.01583)

The celebrated Kirchberg-Phillips classification theorem classifies so-called Kirchberg algebras by K-theory. Many examples of Kirchberg algebras can be constructed via the crossed product construction starting from a group action on a compact space. One might ask: When exactly does the crossed product construction produce a Kirchberg algebra? In joint work with Gardella, Geffen, and Naryshkin, we obtained a dynamical answer to this question for a large class of nonamenable groups which we call "groups with paradoxical towers". Our class includes many non-positively curved groups such as acylindrically hyperbolic groups and lattices in Lie groups. I will try to advertise our notion of paradoxical towers, outline how we use it, and pose some open questions.