Past

I'll argue that the celestial holography program looks a lot like the twisted holography program when studied on twistor space.  The chiral algebras in celestial holography can be seen by applying techniques such as Koszul duality to holomorphic theories on twistor space. Along the way, I will discuss the role of one-loop gauge anomalies on twistor space and when they can be cancelled by a Green-Schwarz mechanism.   This is joint work in progress with Natalie Paquette.

The speaker will be on zoom, but for a more interactive experience, some of the audience will watch the seminar in L5.

Group theory is littered with undecidable problems. A classic example is the word problem: there are groups for which there exists no algorithm that can decide if a product of generators represents the trivial element or not. Many problems (the word problem included) are at least semidecidable, meaning that there is a correct algorithm guaranteed to terminate if the answer is "yes", but with no guarantee on how long one has to wait. I will discuss strategies to try and tackle various semidecidable problems computationally using modern solvers for Boolean satisfiability, with the key example being the discovery of a counterexample to the Kaplansky unit conjecture.

Henstock and Macbeath asked in 1953 whether the Brunn-Minkowski inequality can be generalized to nonabelian locally compact groups; questions in the same line were also asked by Hrushovski, McCrudden, and Tao. We obtain here such an inequality and prove that it is sharp for helix-free locally compact groups, which includes real linear algebraic groups, Nash groups, semisimple Lie groups with finite center, solvable Lie groups, etc. If time allows I will also discuss some applications of this result. (Joint with Chieu-Minh Tran and Ruixiang Zhang)

Many real world graphs have edges correlated to the distance between them, but, in an inhomogeneous manner. While the Chung-Lu model and geometric random graph models both are elegant in their simplicity, they are insufficient to capture the complexity of these networks. For instance, the Chung-Lu model captures the inhomogeneity of the nodes but does not address the geometric nature of the nodes and simple geometric models treat names homogeneously.

In this talk, we develop a generalized geometric random graph model that preserves many graph-theoretic aspects of these models. Notably, each node is assigned a weight based on its desired expected degree; nodes are then adjacent based on a function of their weight and geometric distance. We will discuss the mathematical properties of this model. We also test the validity of this model on a graphical representation of the Drosophila Medulla connectome, a natural real-world inhomogeneous graph where spatial information is known.

This is joint work with Susama Agarwala, Johns Hopkins, Applied Physics Lab.

arXiv link: https://arxiv.org/abs/2109.00061

Organoids are three–dimensional multicellular tissue constructs. When cultured in vitro, they recapitulate the structure, heterogeneity, and function of their in vivo counterparts. As awareness of the multiple uses of organoids has grown, e.g. in drug discovery and personalised medicine, demand has increased for low–cost and efficient methods of producing them in a reproducible manner and at scale. We are working in collaboration with the biotechnology company Cellesce, who develop bioprocessing systems for the expansion of organoids at scale. Part of their technology includes a bioreactor, which utilises flow of culture media to enhance nutrient delivery to the organoids and facilitate the removal of waste metabolites. A key priority is ensuring uniformity in organoid size and reproducibility; qualities that depends on the bioreactor design and operating conditions. A complete understanding of the system requires knowledge of the spatial and temporal information regarding flow and the resulting oxygen and metabolite concentrations throughout the bioreactor. However, it is impractical to obtain this data empirically, due to the highly–controlled environment of the bioreactor posing difficulties for online real–time monitoring of the system. Thus, we exploit a mathematical modelling approach, to provide spatial as well as temporal information.

In the bioreactor, organoids are seeded as single cells in a layer of hydrogel. We present a general model for the nutrient and waste metabolite concentrations in the hydrogel and organoid regions of the bioreactor. Resolving for the millions of organoids within the hydrogel is computationally expensive and infeasible. Hence, we take a mathematical homogenisation approach to understand how the behaviour of the organoids on the microscale influences the macroscale behaviour in the hydrogel layer. We consider the case of growing organoids, with a temporally and spatially dependent radii, and exploit the separation of scales to systematically derive an effective macroscale model for metabolite transport. We explore some canonical problems to understand our homogenised system.

The recent progress of applying QFT methods to classical GR has provided a new perspective on the Kerr black hole solution. Its leading gravitational interactions are known to involve an infinite tower of spin-induced multipoles with unit coupling constants. In this talk, I will present a novel form of the classical worldline action that implements these multipole interactions within a single worldsheet integral, which is inspired by the Newman-Janis shift relationship of the Kerr and Schwarzschild solutions. I will also discuss connections to our recently discovered ability to model such interactions using a certain family of scattering amplitudes, as well as a simple double-copy property hidden within. 

This will be an in-person seminar run in hybrid mode.

 In this talk, we will present the tools of regularity structures to deal with singular stochastic PDEs that involve non-translation invariant differential operators. We describe in particular the renormalized equation for a very large class of spacetime dependent renormalization schemes. Our approach bypasses the previous approaches in the translation-invariant setting. This is joint work with Ismael Bailleul.

We  report  on the theory of evolution equations that combine a strongly nonlinear parabolic character with the presence of fractional operators representing long-range interaction effects, mainly of fractional Laplacian type. Examples include nonlocal porous media equations and fractional p-Laplacian operators appearing in a number of variants. 

Recent work concerns the time-dependent fractional p-Laplacian equation with parameter p>1 and fractional exponent 0<s<1. It is the gradient flow corresponding to the Gagliardo–Slobodeckii fractional energy. Our main interest is the asymptotic behavior of solutions posed in the whole Euclidean space, which is given by a kind of Barenblatt solution whose existence relies on a delicate analysis. The superlinear and sublinear ranges involve different analysis and results. 
 

A nice collection of problems in additive combinatorics focus on analysing solutions to additive equations over sequences that exhibit some flavour of convexity. This, for instance, includes genuine convex sequences as well as images of arbitrary sets under convex functions. In this talk, I will survey some of the literature surrounding these type of questions, along with some motivation from analytic number theory as well as the current best known results towards these problems.

I will talk about joint work with Andrew Lobb related to Toeplitz's square peg problem, which asks whether every (continuous) Jordan curve in the Euclidean plane contains the vertices of a square. Specifically, we show that every smooth Jordan curve contains the vertices of a cyclic quadrilateral of any similarity class. I will describe the context for the result and its proof, which involves symplectic geometry in a surprising way.

A central theme of complex geometry is the relationship between differential-geometric PDEs and algebro-geometric notions of stability. Examples include Hermitian Yang-Mills connections and Kähler-Einstein metrics on the PDE side, and slope stability and K-stability on the algebro-geometric side. I will describe a general framework associating geometric PDEs on complex manifolds to notions of stability, and will sketch a proof showing that existence of solutions is equivalent to stability in a model case. The framework can be seen as an analogue in the setting of varieties of Bridgeland's stability conditions on triangulated categories.

We will review how one can find families of 4d N=1 SCFTs starting from known 6d (1,0) SCFTs. 

Then we will discuss a relation between 6d RG-flows and 4d RG-flows, where the 4d RG-flow relates 4d N=1 models constructed from compactification of 6d (1,0) SCFTs related by the 6d RG-flow. We will show how we can utilize such a relation to find many "Lagrangians" for strongly coupled 4d models. Relating 6d SCFTs to 4d models as mentioned above will result in geometric reasoning behind some 4d phenomena such as dualities and symmetry enhancement.

Such a program generates a large database of known 4d N=1 SCFTs with many interrelations one can use in future efforts to construct 4d N=1 SCFTs from string theory directly.

This session will take place live in L1 and online. A Teams link will be shared 30 minutes before the session begins.

I discuss a novel expansion of superconformal indices of U(N) gauge theories at finite N. When a holographic description is available, the formula expresses the index as a sum over stacks of "giant graviton" branes in the dual string theory. Surprisingly, the expansion turns out to be exact: the sum over strings and branes seems to capture the degeneracy of states expected from saddle geometries such as BPS black holes, while also reproducing the correct degeneracies at lower orders of charges. Based on 2109.02545 with D. Gaiotto.

We will consider modeling shapes and fields via topological and lifted-topological transforms. 

Specifically, we show how the Euler Characteristic Transform and the Lifted Euler Characteristic Transform can be used in practice for statistical analysis of shape and field data. The Lifted Euler Characteristic is an alternative to the. Euler calculus developed by Ghrist and Baryshnikov for real valued functions. We also state a moduli space of shapes for which we can provide a complexity metric for the shapes. We also provide a sheaf theoretic construction of shape space that does not require diffeomorphisms or correspondence. A direct result of this sheaf theoretic construction is that in three dimensions for meshes, 0-dimensional homology is enough to characterize the shape.

Many scientific questions in biology can be formulated as a direct problem:

given a biochemical system, can one deduce some of its properties? 

For example, one might be interested in deducing equilibria of a given intracellular network.  On the other hand, one might instead be interested in designing an intracellular network with specified equilibria. Such scientific tasks take the form of inverse problems:
given a property, can one design a biochemical system that displays this property? 

Given a biochemical system, can one embed additional molecular species and reactions into the original system to control some of its properties?
These questions are at the heart of the emerging field of synthetic biology, where it has recently become possible to systematically realize dynamical systems using molecules.  Furthermore, addressing these questions for man-made synthetic systems may also shed light on how evolution has overcome similar challenges for natural systems.  In this talk, I will focus on the inverse problems, and outline some of the results and challenges which are important when biochemical systems are designed and controlled.

We consider the problem faced by a central bank which bails out distressed financial institutions that pose systemic risk to the banking sector. In a structural default model with mutual obligations, the central agent seeks to inject a minimum amount of cash to a subset of the entities in order to limit defaults to a given proportion of entities. We prove that the value of the agent's control problem converges as the number of defaultable agents goes to infinity, and it satisfies  a drift controlled version of the supercooled Stefan problem. We compute optimal strategies in feedback form by solving numerically a forward-backward coupled system of PDEs. Our simulations show that the agent's optimal strategy is to subsidise banks whose asset values lie in a non-trivial time-dependent region. Finally, we study a linear-quadratic version of the model where instead of the losses, the agent optimises a terminal loss function of the asset values. In this case, we are able to give semi-analytic strategies, which we again illustrate numerically. Joint work with Christa Cuchiero and Stefan Rigger.

Randomized NLA methods have recently gained popularity because of their easy implementation, computational efficiency, and numerical robustness. We propose a randomized version of a well-established FEAST eigenvalue algorithm that enables computing the eigenvalues of the Hermitian matrix pencil $(\textbf{A},\textbf{B})$ located in the given real interval $\mathcal{I} \subset [\lambda_{min}, \lambda_{max}]$. In this talk, we will present deterministic as well as probabilistic error analysis of the accuracy of approximate eigenpair and subspaces obtained using the randomized FEAST algorithm. First, we derive bounds for the canonical angles between the exact and the approximate eigenspaces corresponding to the eigenvalues contained in the interval $\mathcal{I}$. Then, we present bounds for the accuracy of the eigenvalues and the corresponding eigenvectors. This part of the analysis is independent of the particular distribution of an initial subspace, therefore we denote it as deterministic. In the case of the starting guess being a Gaussian random matrix, we provide more informative, probabilistic error bounds. Finally, we will illustrate numerically the effectiveness of all the proposed error bounds.

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The organized movement of intracellular material is part of the functioning of cells and the development of organisms. These flows can arise from the action of molecular machines on the flexible, and often transitory, scaffoldings of the cell. Understanding phenomena in this realm has necessitated the development of new simulation tools, and of new coarse-grained mathematical models to analyze and simulate. In that context, I'll discuss how a symmetry-breaking "swirling" instability of a motor-laden cytoskeleton may be an important part of the development of an oocyte, modeling active material in the spindle, and what models of active, immersed polymers tell us about chromatin dynamics in the nucleus.

In this talk I will review some recent results obtained in collaboration with E. Runa and A. Kerschbaum on the one-dimensionality of the minimizers
of a family of continuous local/nonlocal interaction functionals in general dimension. Such functionals have a local term, typically the perimeter or its Modica-Mortola approximation, which penalizes interfaces, and a nonlocal term favouring oscillations which are high in frequency and in amplitude. The competition between the two terms is expected by experiments and simulations to give rise to periodic patterns at equilibrium. Functionals of this type are used  to model pattern formation, either in material science or in biology. The difficulty in proving the emergence of such structures is due to the fact that the functionals are symmetric with respect to permutation of coordinates, while in more than one space dimensions minimizers are one-dimesnional, thus losing the symmetry property of the functionals. We will present new techniques and results showing that for two classes of functionals (used to model generalized anti-ferromagnetic systems, respectively  colloidal suspensions), both in sharp interface and in diffuse interface models, minimizers are one-dimensional and periodic, in general dimension and also while imposing a nontrivial volume constraint.

Forcing axioms spell out the dictum that if a statement can be forced, then it is already true. The P_max axiom (*) goes beyond that by claiming that if a statement is consistent, then it is already true. Here, the statement in question needs to come from a resticted class of statements, and "consistent" needs to mean "consistent in a strong sense". It turns out that (*) is actually equivalent to a forcing axiom, and the proof is by showing that the (strong) consistency of certain theories gives rise to a corresponding notion of forcing producing a model of that theory. Our result builds upon earlier work of R. Jensen and (ultimately) Keisler's "consistency properties".

(This is Part I of a two-part talk.)

Finiteness properties of groups provide various generalisations of the properties "finitely generated" and "finitely presented." We will define different types of finiteness properties and discuss Bestvina-Brady groups as they provide examples of groups with interesting combinations of finiteness properties.