Past

The infinitely wide neural network has been proven a useful and manageable mathematical model that enables the understanding of many phenomena appearing in deep learning. One example is the convergence of random deep networks to Gaussian processes that enables a rigorous analysis of the way the choice of activation function and network weights impacts the training dynamics. In this paper, we extend the seminal proof of Matthews (2018) to a larger class of initial weight distributions (which we call "pseudo i.i.d."), including the established cases of i.i.d. and orthogonal weights, as well as the emerging low-rank and structured sparse settings celebrated for their computational speed-up benefits. We show that fully-connected and convolutional networks initialized with pseudo i.i.d. distributions are all effectively equivalent up to their variance. Using our results, one can identify the Edge-of-Chaos for a broader class of neural networks and tune them at criticality in order to enhance their training.

There has been considerable interest in the problem of whether every metric space of bounded geometry coarsely embeds into a uniformly convex Banach space due to the work of Kasparov and Yu that established a connection between such embeddings and the Novikov conjecture. Brown and Guentner were able to prove that a metric space with bounded geometry coarsely embeds into a reflexive Banach space. Kalton significantly extended this result to stable metric spaces and asked whether these classes are coarsely equivalent, i.e. whether every reflexive Banach space coarsely embeds into a stable metric space. Baudier introduced the notion of upper stability, a relaxation of stability, for metric spaces as a new invariant to distinguish reflexive spaces from stable metric spaces. In this talk, we show that in fact, every reflexive space is upper stable and also establish a connection of upper stability to the asymptotic structure of Banach spaces. This is joint work with F. Baudier and Th. Schlumprecht.

One of the more common ways to study a residually finite group (or its profinite completion) is via breaking it down into a graph of groups in some way. The descriptions of this theory generally found in the literature are highly algebraic and difficult to digest. I will present alternative, more geometric, definitions and perspectives on these theories based on properties of virtually free groups and their profinite completions.

The proximal Galerkin finite element method is a high-order, nonlinear numerical method that preserves the geometric and algebraic structure of bound constraints in infinitedimensional function spaces. In this talk, we will introduce the proximal Galerkin method and apply it to solve free-boundary problems, enforce discrete maximum principles, and develop scalable, mesh-independent algorithms for optimal design. The proximal Galerkin framework is a natural consequence of the latent variable proximal point (LVPP) method, which is an stable and robust alternative to the interior point method that will also be introduced in this talk.

In particular, LVPP is a low-iteration complexity, infinite-dimensional optimization algorithm that may be viewed as having an adaptive barrier function that is updated with a new informative prior at each (outer loop) optimization iteration. One of the main benefits of this algorithm is witnessed when analyzing the classical obstacle problem. Therein, we find that the original variational inequality can be replaced by a sequence of semilinear partial differential equations (PDEs) that are readily discretized and solved with, e.g., high-order finite elements. Throughout the talk, we will arrive at several unexpected contributions that may be of independent interest. These include (1) a semilinear PDE we refer to as the entropic Poisson equation; (2) an algebraic/geometric connection between high-order positivity-preserving discretizations and an infinite-dimensional Lie group; and (3) a gradient-based, bound-preserving algorithm for two-field density-based topology optimization.

The complete latent variable proximal Galerkin methodology combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry and can potentially lead to new synergies among these areas as well as within variational and numerical analysis. This talk is based on [1].

Keywords: pointwise bound constraints, bound-preserving discretization, entropy regularization, proximal point

Mathematics Subject Classifications (2010): 49M37, 65K15, 65N30

References  [1] B. Keith, T.M. Surowiec. Proximal Galerkin: A structure-preserving finite element method for pointwise bound constraints arXiv preprint arXiv:2307.12444 2023.

Brown University Email address: @email

Simula Research Laboratory Email address: @email

In this talk, I will present new results addressing two rather well-known problems on the embeddability of planar graphs on point-sets in the plane. The first problem, often attributed to Mohar, asks for the asymptotics of the minimum size of so-called universal point sets, i.e. point sets that simultaneously allow straight-line embeddings of all planar graphs on $n$ vertices. In the first half of the talk I will present a family of point sets of size $O(n)$ that allow straight-line embeddings of a large family of $n$-vertex planar graphs, including all bipartite planar graphs. In the second half of the talk, I will present a family of $(3+o(1))\log_2(n)$ planar graphs on $n$ vertices that cannot be simultaneously embedded straight-line on a common set of $n$ points in the plane. This significantly strengthens the previously best known exponential bound.

I will discuss a KLT relation of closed string amplitudes into open string amplitudes for closed string states carrying winding and momentum in toroidal compactifications. The goal is to introduce an interesting D-brane set up in the target space in order to accommodate both quantum numbers of the closed string. I will then discuss KLT factorization of amplitudes for winding closed strings in the presence of a critical Kalb-Ramond field and the relevance of this work for nonrelativistic string theory when taking the zero Regge limit. 

We consider the homogenisation problem for the ϕ4/2 equation on the torus T² , i.e. the behaviour as ϵ → 0 of the solutions to the equations suggestively written

∂_tu_ϵ − ∇ · A(x/ϵ, t/ϵ² )∇uϵ = −u³_ϵ + ξ

where ξ denotes space-time white noise and A : T ² × R is uniformly elliptic, periodic and H¨older continuous. Based on joint work with M. Hairer

High-dimensional data is often assumed to be distributed near a smooth manifold. But should we really believe that? In this talk I will introduce HADES, an algorithm that quickly detects singularities where the data distribution fails to be a manifold.

By using hypothesis testing, rather than persistent homology, HADES achieves great speed and a strong statistical foundation. We also have a precise mathematical theorem for correctness, proven using optimal transport theory and differential geometry. In computational experiments, HADES recovers singularities in synthetic data, road networks, molecular conformation space, and images.

Paper link: https://arxiv.org/abs/2311.04171
Github link: https://github.com/uzulim/hades
 

The seminar concerns the study of evolution equations on graphs, motivated by applications in data science and opinion dynamics. We will discuss graph analogues of the continuum nonlocal-interaction equation and interpret them as gradient flows with respect to a graph Wasserstein distance, using Benamou--Brenier formulation. The underlying geometry of the problem leads to a Finslerian gradient flow structure, rather than Riemannian, since the resulting distance on graphs is actually a quasi-metric. We will address the existence of suitably defined solutions, as well as their asymptotic behaviour when the number of vertices converges to infinity and the graph structure localises. The two limits lead to different dynamics. From a slightly different perspective, by means of a classical fixed-point argument, we can show the existence and uniqueness of solutions to a larger class of nonlocal continuity equations on graphs. In this context, we consider general interpolation functions of the mass on the edges, which give rise to a variety of different dynamics. Our analysis reveals structural differences with the more standard Euclidean space, as some analogous properties rely on the interpolation chosen. The latter study can be extended to equations on co-evolving graphs. The talk is based on works in collaboration with G. Heinze (Augsburg), L. Mikolas (Oxford), F. S. Patacchini (IFP Energies Nouvelles), A. Schlichting (University of Münster), and D. Slepcev (Carnegie Mellon University). 

In this talk I will give a brief introduction to the field of post-quantum (PQ) cryptography, introducing a few of the most popular computational hardness assumptions. Second, I will give an overview of a recent work of mine on PQ electronic voting. I’ll finish by presenting a short selection of ‘exotic’ cryptographic constructions that I think are particularly hot at the moment (no, not blockchain). The talk will be definitionally light since I expect the area will be quite new to many and I hope this will make for a more engaging introduction.

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.

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).

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.

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.

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.