Past

R. Schoen has conjectured that an asymptotically flat Riemannian n-manifold (M,g) with non-negative scalar curvature is isometric to Euclidean space if it admits a non-compact area-minimizing hypersurface. This has been confirmed by O. Chodosh and M. Eichmair in the case where n=3. In this talk, I will present recent work with M. Eichmair where we confirm this conjecture in the case where 3<n<8 and the area-minimizing hypersurface arises as the limit of large isoperimetric hypersurfaces. By contrast, we show that a large part of spatial Schwarzschild of dimension 3<n<8 is foliated by non-compact area-minimizing hypersurfaces.

In 1989, Mazur defined the deformation functor associated to a residual Galois representation, which played an important role in the proof by Wiles of the modularity theorem. This was used as a basis over which many mathematicians constructed variations both to further specify it or to expand the contexts where it can be applied. These variations proved to be powerful tools to obtain many strong theorems, in particular of modular nature. In this talk I will give an overview of the deformation theory of Galois representations and describe two variants of Mazur's functor that allow one to properly deform reducible residual representations (which is one of the shortcomings of Mazur's original functor). Namely, I will present the theory of determinant-laws initiated by Bellaïche-Chenevier on the one hand, and an idea developed by Calegari-Emerton on the other.
If time permits, I will also describe results that seem to indicate a possible comparison between the two seemingly unrelated constructions.

One can think of the stabilisation of an ∞-category as the ∞-category of objects that admit infinite deloopings. For example, the ∞-category of spectra is the stabilisation of the ∞-category of homotopy types. Costabilisation is the opposite notion of stabilisation, where we are interested in objects that allow infinite desuspensions. It is easy to see that the costabilisation of the ∞-category of homotopy types is trivial. Fix a prime number p. In this talk I will show that the costablisation of the ∞-category of T(h)-local spectral Lie algebras is equivalent to the ∞-category of T(h)-local spectra, where T(h) denotes a p-local telescope spectrum of height h. A key ingredient of the proof is to relate spectral Lie algebras to (spectral) Eₙ algebras via Koszul duality.
 

We consider stochastic differential equations (SDEs) driven by fractional Brownian motion with Hurst parameter less than 1/2. The drift is a measurable function of time and space which belongs to a certain Lebesgue space. Under subcritical regime, we show that a strong solution exists and is unique in path-by-path sense. When the noise is formally replaced by a Brownian motion, our results correspond to the strong uniqueness result of Krylov and Roeckner (2005). Our methods forgo standard approaches in Markovian settings and utilize Lyons' rough path theory in conjunction with recently developed tools. Joint work with Toyomu Matsuda and Oleg Butkovsky.

After a brief introduction to the semiregularity maps of Severi, Kodaira and Spencer, and Bloch, I will focus on the Buchweitz-Flenner semiregularity map and on its importance for the deformation theory of coherent sheaves.
The subject of this talk is the construction of a lifting of each component of the Buchweitz-Flenner semiregularity map to an L-infinity morphism between DG-Lie algebras, which allows to interpret components of the semiregularity map as obstruction maps of morphisms of deformation functors.

As a consequence, we obtain that the semiregularity map annihilates all obstructions to deformations of a coherent sheaf on a complex projective manifold. Based on a joint work with R. Bandiera and M. Manetti.

Deep learning has revolutionised many scientific fields and so it is no surprise that state-of-the-art solutions to several inverse problems also include this technology. However, for many inverse problems (e.g. in medical imaging) stability and reliability are particularly important.

Furthermore, unlike other image analysis tasks, usually only a fairly small amount of training data is available to train image reconstruction algorithms.

Thus, we require tailored solutions which maximise the potential of all ingredients: data, domain knowledge and mathematical analysis. In this talk we discuss a range of such hybrid approaches and will encounter along the way connections to various topics like generative models, convex optimization, differential equations and equivariance.

Speaker: Ximena Laura Fernandez
Title: Let it Be(tti): Topological Fingerprints for Audio Identification

Abstract: Ever wondered how music recognition apps like Shazam work or why they sometimes fail? Can Algebraic Topology improve current audio identification algorithms? In this talk, I will discuss recent collaborative work with Spotify, where we extract low-dimensional homological features from audio signals for efficient song identification despite continuous obfuscations. Our approach significantly improves accuracy and reliability in matching audio content under topological distortions, including pitch and tempo shifts, compared to Shazam.

Talk based on the work: https://arxiv.org/pdf/2309.03516.pdf
 

Speaker: Brett Kolesnik
Title: Coxeter Tournaments

Abstract: We will present ongoing joint work with three Oxford PhD students: Matthew Buckland (Stats), Rivka Mitchell (Math/Stats) and Tomasz Przybyłowski (Math). We met last year as part of the course SC9 Probability on Graphs and Lattices. Connections with geometry (the permutahedron and generalizations), combinatorics (tournaments and signed graphs), statistics (paired comparisons and sampling) and probability (coupling and rapid mixing) will be discussed.

I will discuss various aspects of multi-parameter persistence related to representation theory and decompositions into indecomposable summands, based on joint work with Magnus Botnan, Steffen Oppermann, Johan Steen, Luis Scoccola, and Benedikt Fluhr.

A classification of indecomposables is infeasible; the category of two-parameter persistence modules has wild representation type. We show [1] that this is still the case if the structure maps in one parameter direction are epimorphisms, a property that is commonly satisfied by degree 0 persistent homology and related to filtered hierarchical clustering. Furthermore, we show [2] that indecomposable persistence modules are dense in the interleaving distance, and that being nearly-indecomposable is a generic property of persistence modules. On the other hand, the two-parameter persistence modules arising from interleaved sets (relative interleaved set cohomology) have a very well-behaved structure [3] that is encoded as a complete invariant in the extended persistence diagram. This perspective reveals some important but largely overlooked insights about persistent homology; in particular, it highlights a strong reason for working at the level of chain complexes, in a derived category [4].

[1] Ulrich Bauer, Magnus B. Botnan, Steffen Oppermann, and Johan Steen, Cotorsion torsion triples and the representation theory of filtered hierarchical clustering, Adv. Math. 369 (2020), 107171, 51. MR4091895

[2] Ulrich Bauer and Luis Scoccola, Generic multi-parameter persistence modules are nearly indecomposable, 2022.

[3] Ulrich Bauer, Magnus Bakke Botnan, and Benedikt Fluhr, Structure and interleavings of relative interlevel set cohomology, 2022.

[4] Ulrich Bauer and Benedikt Fluhr, Relative interlevel set cohomology categorifies extended persistence diagrams, 2022.

Mathematical and computational techniques can improve our understanding of diseases. In this talk, I’ll present ways in which data from cancer patients can be combined with mathematical modelling and used to improve cancer treatments.

Given the variability in individual responses to cancer treatments, agent-based modelling has been a useful technique for accurately capturing cellular behaviours that may lead to stochasticity in patient outcomes. Using a hybrid agent-based model and partial differential equation system, we developed a model for brain cancer (glioblastoma) growth informed by ex-vivo patient samples. Extending the model to capture patient treatment with an oncolytic virus rQNestin, we used our model to propose reasons for treatment failure, which was later confirmed with further patient samples. More recently, we extended this model to investigate the effectiveness of combination treatments (chemotherapy, virotherapy and immunotherapy) informed by individual patient imaging mass cytometry.

This talk hopes to provide examples of ways mathematical and computational modelling can be used to run “virtual” clinical trials with the goal of obtaining more effective treatments for diseases.  

In the context of categorical Langlands, there are many ways in which one could define the notion of n-finitely presented smooth representation. We will explore and compare two different definitions, relating them with the notion of n-coherence for the corresponding Iwasawa ring.

In this talk I will present joint work with Ehud Hrushovski on imaginaries in the ring of adeles and more generally in products and restricted products of structures (including the generalised products of Feferman-Vaught).

We prove a general theorem on weak elimination of imaginaries in products with respect to additional sorts which we deduce from an elimination of imaginaries for atomic and atomless Booleanizations of a theory. This combined with uniform elimination of imaginaries for p-adic numbers in a language with extra sorts as p-adic lattices proved first by Hrushovski-Martin-Rideau and more recently by Hils-Rideau-Kikuchi in a slightly different language, yields weak elimination of imaginaries for the ring of adeles in a language with extra sorts as adelic versions of the p-adic lattices. 

The proofs of the general results on products use Boolean valued model theory, stability theory, analysis of definable groups and liaison groups, and descriptive set theory of smooth Borel equivalence relations including Harrington-Kechris-Louveau and Glimm-Efros dichotomy. 

I will report on current work in progress on the construction of anticyclotomic p-adic L-functions for Rankin--Selberg products. I will explain how by p-adically interpolating the branching law for the spherical pair (U(n)xU(n+1), U(n)) we can construct a p-adic L-function attached to cohomological automorphic representations of U(n) x U(n+1), including anticyclotomic variation. Due to the recent proof of the unitary Gan--Gross--Prasad conjecture, this p-adic L-function interpolates the square root of the central L-value. Time allowing, I will explain how we can extend this result to the Coleman family of an automorphic representation.

We propose a novel framework for exploring weak and $L_2$ generalization errors of algorithms through the lens of differential calculus on the space of probability measures. Specifically, we consider the KL-regularized empirical risk minimization problem and establish generic conditions under which the generalization error convergence rate, when training on a sample of size $n$ , is $\matcal{O}(1/n)$. In the context of supervised learning with a one-hidden layer neural network in the mean-field regime, these conditions are reflected in suitable integrability and regularity assumptions on the loss and activation functions.

We develop an algorithm for parameter-free stochastic convex optimization (SCO) whose rate of convergence is only a double-logarithmic factor larger than the optimal rate for the corresponding known-parameter setting. In contrast, the best previously known rates for parameter-free SCO are based on online parameter-free regret bounds, which contain unavoidable excess logarithmic terms compared to their known-parameter counterparts. Our algorithm is conceptually simple, has high-probability guarantees, and is also partially adaptive to unknown gradient norms, smoothness, and strong convexity. At the heart of our results is a novel parameter-free certificate for the step size of stochastic gradient descent (SGD), and a time-uniform concentration result that assumes no a-priori bounds on SGD iterates.

Additionally, we present theoretical and numerical results for a dynamic step size schedule for SGD based on a variant of this idea. On a broad range of vision and language transfer learning tasks our methods performance is close to that of SGD with tuned learning rate. Also, a per-layer variant of our algorithm approaches the performance of tuned ADAM.

This talk is based on papers with Yair Carmon and Maor Ivgi.

We consider degenerate fully nonlinear equations, whose degeneracy rate depends on the gradient of solutions. We work under a Dini-continuity condition on the degeneracy term and prove that solutions are continuously differentiable. Then we frame this class of equations in the context of a free transmission problem. Here, we discuss the existence of solutions and establish a result on interior regularity. We conclude the talk by discussing a boundary regularity estimate; of particular interest is the case of point-wise regularity at the intersection of the fixed and the free boundaries. This is based on joint work with David Stolnicki.

Predictive scoring modelling is a common approach to measure financial health and credit worthiness in banking. Whilst the latter is a key factor in making decisions for lending, evaluating financial health helps to identify vulnerable customers trending towards financial hardship, who need support. The current macroeconomic uncertainties amplify the importance of extensive flexibility in modelling data solutions so that they can remain effective and adaptive to a volatile economic environment. This workshop is focused on discussing relevant techniques and mathematical methodologies which can help modernise traditional scoring models and accelerate innovation. In summary, the problem definition in the banking context is how automation and optimality can be achieved in a multiobjective problem where a subset of existing data features should be selected by relevance and uniqueness, assigned scoring weights by importance and how a pool of customers can be categorised accordingly using their individual scores and auto-adjusting thresholds of risk classification scales. The key challenge is imposed by the mutual dependency of the three sub-problems and their objectives. Introducing or removing constraints in any of them can change the feasibility and optimality of the others and the overall solution. It is common for traditional scoring models to be mainly focused on the predictive accuracy and their setup is often defined and revised manually, following ad-hoc exploratory data analysis and business-led decision making. An automated optimisation of the data features’ selection, scoring weights and classification thresholds definition can achieve respectively: ▪ Precise financial health evaluation and book classification under changing economic climate; ▪ Development of innovative data-driven solutions to enhance prevention from financial hardship and bankruptcies.

In the 17^th century, certainty was still largely organized around heterogeneous categories such as “absolute certainty” and “moral certainty”. “Absolute certainty” was the highest kind of certainty rather than degree and it was limited to metaphysical and mathematical demonstrations. On the other hand, “moral certainty” was a high degree of assent which, even though it was subjective and always fallible, was regarded as sufficient for practical decisions based on empirical evidence. Although this duality between “moral” and “absolute” certainty remained in use well into the 18^th century, its meaning shifted with the emergence of the calculus of probabilities. Probability calculus provided tools for attempts to mathematise “moral certainty” which would have been a contradiction in terms in their classical 17th-century sense.

Jacob Bernoulli's Ars Conjectandi (1713) followed by Thomas Bayes and Richard Price's An Essay towards solving a Problem in the Doctrine of Chances (1763) reshuffled what was before mutually exclusive characteristics of those categories of certainty. Moral certainty became mathematizable and measurable, while absolute certainty would sit in continuity in degree with moral certainty rather than be different in kind. The concept of certainty as a whole is thus redefined as a quantitative continuum.

This transformation lays the conceptual foundations for a new approach to knowledge. Knowledge and even scientific knowledge are no longer defined by a binary model of an absolute exclusion of uncertainty, but rather by the accuracy of measurement of the irreducible uncertainty in all empirical-based knowledge. Such measurement becomes possible thanks to the new tools provided by the emergence of probability calculus.

You may be surprised to see the generalised Riemann hypothesis appear in algorithmic topology. For example, knottedness was originally shown to be in NP under the assumption of GRH.
Where does this condition come from? We will discuss this in the context of 3-sphere recognition, and examine why the approach fails for higher dimensions.

Complex dynamics underlying industrial manufacturing depend in part on multiphase multiphysics, in which fluids and materials interact across orders of magnitude variations in time and space. In this talk, we will discuss the development and application of a host of numerical methods for these problems, including Level Set Methods, Voronoi Implicit Interface Methods, implicit adaptive representations, and multiphase discontinuous Galerkin Methods.  Applications for industrial problems will include modeling how foams evolve, how electro-fluid jetting devices work, and the physics and dynamics of rotary bell spray painting across the automotive industry.

Complex dynamics underlying industrial manufacturing depend in part on
multiphase multiphysics, in which fluids and materials interact across
orders of magnitude variations in time and space. In this talk, we will
discuss the development and application of a host of numerical methods for
these problems, including Level Set Methods, Voronoi Implicit Interface
Methods, implicit adaptive representations, and multiphase discontinuous
Galerkin Methods.  Applications for industrial problems will include modeling
how foams evolve, how electro-fluid jetting devices work, and
the physics and dynamics of rotary bell spray painting across the automotive
industry.

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.

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.

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

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. 

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
 

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

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. 

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.

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.

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).

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.

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.