Past

I will review recent work on N=(2,2) boundary conditions of 3d
N=4 theories which mimic isolated massive vacua at infinity. Subsets of
local operators supported on these boundary conditions form lowest
weight Verma modules over the quantised bulk Higgs and Coulomb branch
chiral rings. The equivariant supersymmetric Casimir energy is shown to
encode the boundary ’t Hooft anomaly, and plays the role of lowest
weights in these modules. I will focus on a key observable associated to
these boundary conditions; the hemisphere partition function, and apply
them to the holomorphic factorisation of closed 3-manifold partition
functions and indices. This yields new “IR formulae” for partition
functions on closed 3-manifolds in terms of Verma characters. I will
also discuss ongoing work on connections to enumerative geometry, and
the construction of elliptic stable envelopes of Aganagic and Okounkov,
in particular their physical manifestation via mirror duality
interfaces.

This talk is based on 2010.09741 and ongoing work with Mathew Bullimore
and Samuel Crew.

Speaker: Elena Gal (4pm)

Title: Associativity and Geometry

Abstract: An operation # that satisfies a#(b#c)=(a#b)#c is called "associative". Associativity is "common" - if we are asked to give an example of operation we are more likely to come up with one that has this property. However if we dig a bit deeper we encounter in geometry, topology and modern physics many operations that are not associative "on the nose" but rather up to an equivalence. We will talk about how to describe and work with this higher associativity notion.

Speaker: Alexandre Bovet (4:30pm)

Title: Investigating disinformation in social media with network science

Abstract:
While disinformation and propaganda have existed since ancient times, their importance and influence in the age of
social media is still not clear.  We investigate the spread of disinformation and traditional misinformation in Twitter in the context of the 2016 and 2020 US presidential elections. We analyse the information diffusion networks by reconstructing the retweet networks corresponding to each type of news and the top news spreaders of each network are identified. Our investigation provides new insights into the dynamics of news diffusion in Twitter, namely our results suggests that disinformation is governed by a different diffusion mechanism than traditional centre and left-leaning news. Centre and left leaning traditional news diffusion is driven by a small number of influential users, mainly journalists, and follow a diffusion cascade in a network with heterogeneous degree distribution which is typical of diffusion in social networks, while the diffusion of disinformation seems to not be controlled by a small set of users but rather to take place in tightly connected clusters of users that do not influence the rest of Twitter activity. We also investigate how the situation evolved between 2016 and 2020 and how the top news spreaders from the different news categories have driven the polarization of the Twitter ideological landscape during this time.

Reducing a chain complex (whilst preserving its homotopy-type) using algebraic Morse theory ([1, 2, 3]) gives the same end-result as Gaussian elimination, but AMT does it only on certain rows/columns and with several pivots (in all matrices simultaneously). Crucially, instead of doing costly row/column operations on a sparse matrix, it computes traversals of a bipartite digraph. This significantly reduces the running time and memory load (smaller fill-in and coefficient growth of the matrices). However, computing with AMT requires the construction of a valid set of pivots (called a Morse matching).

In [4], we discover a family of Morse matchings on any chain complex of free modules of finite rank. We show that every acyclic matching is a subset of some member of our family, so all maximal Morse matchings are of this type.

Both the input and output of AMT are chain complexes, so the procedure can be used iteratively. When working over a field or a local PID, this process ends in a chain complex with zero matrices, which produces homology. However, even over more general rings, the process often reveals homology, or at least reduces the complex so much that other algorithms can finish the job. Moreover, it also returns homotopy equivalences to the reduced complexes, which reveal the generators of homology and the induced maps $H_{*}(\varphi)$. 

We design a new algorithm for reducing a chain complex and implement it in MATHEMATICA. We test that it outperforms other CASs. As a special case, given a sparse matrix over any field, the algorithm offers a new way of computing the rank and a sparse basis of the kernel (or null space), cokernel (or quotient space, or complementary subspace), image, preimage, sum and intersection subspace. It outperforms built-in algorithms in other CASs.

References

[1]  M. Jöllenbeck, Algebraic Discrete Morse Theory and Applications to Commutative Algebra, Thesis, (2005).

[2]  D.N. Kozlov, Discrete Morse theory for free chain complexes, C. R. Math. 340 (2005), no. 12, 867–872.

[3]  E. Sköldberg, Morse theory from an algebraic viewpoint, Trans. Amer. Math. Soc. 358 (2006), no. 1, 115–129.

[4]  L. Lampret, Chain complex reduction via fast digraph traversal, arXiv:1903.00783.

There has been a surge of interest in machine learning in the past few years, and deep learning techniques are more and more integrated into
the way we do quantitative science. A particularly exciting case for deep learning is molecular physics, where some of the "superpowers" of
machine learning can make a real difference in addressing hard and fundamental computational problems - on the other hand the rigorous
physical footing of these problems guides us in how to pose the learning problem and making the design decisions for the learning architecture.
In this lecture I will review some of our recent contributions in marrying deep learning with statistical mechanics, rare-event sampling
and quantum mechanics.

Introduced by Fomin and Zelevinsky in 2002, cluster algebras have become ubiquitous in algebra, combinatorics and geometry. In this talk, I'll introduce the notion of a cluster algebra and present the approach of Kang-Kashiwara-Kim-Oh to categorify a large class of them arising from quantum groups. Time allowing, I will explain some recent developments related to the coherent Satake category.

Learning a representation from data that disentangles different factors of variation is hypothesized to be a critical ingredient for unsupervised learning. Defining disentangling is challenging - a "symmetry-based" definition was provided by Higgins et al. (2018), but no prescription was given for how to learn such a representation. We present a novel nonparametric algorithm, the Geometric Manifold Component Estimator (GEOMANCER), which partially answers the question of how to implement symmetry-based disentangling. We show that fully unsupervised factorization of a data manifold is possible if the true metric of the manifold is known and each factor manifold has nontrivial holonomy – for example, rotation in 3D. Our algorithm works by estimating the subspaces that are invariant under random walk diffusion, giving an approximation to the de Rham decomposition from differential geometry. We demonstrate the efficacy of GEOMANCER on several complex synthetic manifolds. Our work reduces the question of whether unsupervised disentangling is possible to the question of whether unsupervised metric learning is possible, providing a unifying insight into the geometric nature of representation learning.

It is well known that expected signatures can be used as the “moments” of the law of stochastic processes. Inspired by this fact, we introduced higher rank expected signatures to capture the essences of the weak topologies of adapted processes, and characterize the information evolution pattern associated with stochastic processes. This approach provides an alternative perspective on a recent important work by Backhoff–Veraguas, Bartl, Beiglbock and Eder regarding adapted topologies and causal Wasserstein metrics.

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Biological systems provide an inspiration for creating a new paradigm
for materials synthesis. What would it take to enable inanimate material
to acquire the properties of living things? A key difference between
living and synthetic materials is that the former are programmed to
behave as they do, through interactions, energy consumption and so
forth. The nature of the program is the result of billions of years of
evolution. Understanding and emulating this program in materials that
are synthesizable in the lab is a grand challenge. At its core is an
optimization problem: how do we choose the properties of material
components that we can create in the lab to carry out complex reactions?
I will discuss our (not-yet-terribly-successful efforts)  to date to
address this problem, by designing both equiliibrium and kinetic 
properties of materials, using a combination of statistical mechanics,
kinetic modeling and ideas from machine learning.

I will discuss regularity properties for solutions of linear second order non-uniformly elliptic equations in divergence form. Assuming certain integrability conditions on the coefficient field, we obtain local boundedness and validity of Harnack inequality. The assumed integrability assumptions are sharp and improve upon classical results due to Trudinger from the 1970s.

As an application of the local boundedness result, we deduce a quenched invariance principle for random walks among random degenerate conductances. If time permits I will discuss further regularity results for nonlinear non-uniformly elliptic variational problems.

I will discuss the multi-sorted structure of analytic covers H -> Y(N), where H is the upper half-plane and Y(N) are the N-level modular curves, all N, in a certain language, weaker than the language applied by Adam Harris and Chris Daw.  We define a certain locally modular reduct of the structure which is called "pure" structure - an extension of the structure of special subvarieties.  
The problem of non-elementary categorical axiomatisation for this structure is closely related to the theory of "canonical models for Shimura curves", in particular, the description of Gal_Q action on the CM-points of the Y(N). This problem for the case of curves is basically solved (J.Milne) and allows the beautiful interpretation in our setting:  the abstract automorphisms of the pure structure on CM-points are exactly the automorphisms induced by Gal_Q.  Using this fact and earlier theorem of Daw and Harris we prove categoricity of a natural axiomatisation of the pseudo-analytic structure.
If time permits I will also discuss a problem which naturally extends the above:  a categoricity statement for the structure of unramified analytic covers H -> X, where X runs over all smooth curves over a given number field.  

Let us say that a theory $T$ in the language of set theory is $\beta$-consistent at $\alpha$ if there is a transitive model of $T$ of height $\alpha$, and let us say that it is $\beta$-categorical at $\alpha$ iff there is at most one transitive model of $T$ of height $\alpha$. Let us also assume, for ease of formulation, that there are arbitrarily large $\alpha$ such that $\mathrm{ZFC}$ is $\beta$-consistent at $\alpha$.

The sentence $\mathrm{VEL}$ has the feature that $\mathrm{ZFC}+\mathrm{VEL}$ is $\beta$-categorical at $\alpha$, for every $\alpha$. If we assume in addition that $\mathrm{ZFC}+\mathrm{VEL}$ is $\beta$-consistent at $\alpha$, then the uniquely determined model is $L_\alpha$, and the minimal such model, $L_{\alpha_0}$, is model of determined by the $\beta$-categorical theory $\mathrm{ZFC}+\mathrm{VEL}+M$, where $M$ is the statement "There does not exist a transitive model of $\mathrm{ZFC}$."

It is natural to ask whether $\mathrm{VEL}$ is the only sentence that can be $\beta$-categorical at $\alpha$; that is, whether, there can be a sentence $\phi$ such that $\mathrm{ZFC}+\phi$ is $\beta$-categorical at $\alpha$, $\beta$-consistent at $\alpha$, and where the unique model is not $L_\alpha$.  In the early 1970s Harvey Friedman proved a partial result in this direction. For a given ordinal $\alpha$, let $n(\alpha)$ be the next admissible ordinal above $\alpha$, and, for the purposes of this discussion, let us say that an ordinal $\alpha$ is minimal iff a bounded subset of $\alpha$ appears in $L_{n(\alpha)}\setminus L_\alpha$. [Note that $\alpha_0$ is minimal (indeed a new subset of $\omega$ appears as soon as possible, namely, in a $\Sigma_1$-definable manner over $L_{\alpha_0+1}$) and an ordinal $\alpha$ is non-minimal iff $L_{n(\alpha)}$ satisfies that $\alpha$ is a cardinal.] Friedman showed that for all $\alpha$ which are non-minimal, $\mathrm{VEL}$ is the only sentence that is $\beta$-categorical at $\alpha$. The question of whether this is also true for $\alpha$ which are minimal has remained open.

In this talk I will describe some joint work with Hugh Woodin that bears on this question. In general, when approaching a "lightface" question (such as the one under consideration) it is easier to first address the "boldface" analogue of the question by shifting from the context of $L$ to the context of $L[x]$, where $x$ is a real. In this new setting everything is relativized to the real $x$: For an ordinal $\alpha$, we let $n_x(\alpha)$ be the first $x$-admissible ordinal above $\alpha$, and we say that $\alpha$ is $x$-minimal iff a bounded subset of $\alpha$ appears in $L_{n_x(\alpha)}[x]\setminus L_{\alpha}[x]$.

Theorem. Assume $M_1^\#$ exists and is fully iterable. There is a sentence $\phi$ in the language of set theory with two additional constants, \r{c} and \r{d}, such that for a Turing cone of $x$, interpreting \r{c} by $x$, for all $a$

1. if $L_\alpha[x]\vDash\mathrm{ZFC}$ then there is an interpretation of \r{d}  by something in $L_\alpha[x]$ such that there is a $\beta$-model of $\mathrm{ZFC}+\phi$ of height $\alpha$ and not equal to $L_\alpha[x]$, and
2. if, in addition, $\alpha$ is $x$-minimal, then there is a unique $\beta$-model of $\mathrm{ZFC}+\phi$ of height $\alpha$ and not equal to $L_\alpha[x]$.

The sentence $\phi$ asserts the existence of an object which is external to $L_\alpha[x]$ and which, in the case where $\alpha$ is minimal, is canonical. The object is a branch $b$ through a certain tree in $L_\alpha[x]$, and the construction uses techniques from the HOD analysis of models of determinacy.

In this talk I will sketch the proof, describe some additional features of the singleton, and say a few words about why the lightface version looks difficult.

The course covers the standard material on nonlinear wave equations, including local existence, breakdown criterion, global existence for small data for semi-linear equations, and Strichartz estimate if time allows.

In the world of finitely generated groups, presentations are a blessing and a curse. They are versatile and compact, but in general tell you very little about the group. Tietze transformations offer much (but deliver little) in terms of understanding the possible presentations of a group. I will introduce a different way of transforming presentations of a group called a Nielsen transformation, and show how topological methods can be used to study Nielsen transformations.

Non-Hermitian random matrices with complex eigenvalues represent a truly two-dimensional (2D) Coulomb gas at inverse temperature beta=2. Compared to their Hermitian counter-parts they enjoy an enlarged bulk and edge universality. As an application to ecology we model large scale data of the approximately 2D distribution of buzzard nests in the Teutoburger forest observed over a period of 20 y. These birds of prey show a highly territorial behaviour. Their occupied nests are monitored annually and we compare these data with a one-component 2D Coulomb gas of repelling charges as a function of beta. The nearest neighbour spacing distribution of the nests is well described by fitting to beta as an effective repulsion parameter, that lies between the universal predictions of Poisson (beta=0) and random matrix statistics (beta=2). Using a time moving average and comparing with next-to-nearest neighbours we examine the effect of a population increase on beta and correlation length.
 

The classical Turán problem asks: given a graph $H$, how many edges can an $4n$-vertex graph have while containing no isomorphic copy of $H$? By viewing $(k+1)$-uniform hypergraphs as $k$-dimensional simplicial complexes, we can ask a topological version (first posed by Nati Linial): given a $k$-dimensional simplicial complex $S$, how many facets can an $n$-vertex $k$-dimensional simplicial complex have while containing no homeomorphic copy of $S$? Until recently, little was known for $k > 2$. In this talk, we give an answer for general $k$, by way of dependent random choice and the combinatorial notion of a trace-bounded hypergraph. Joint work with Jason Long and Bhargav Narayanan.

Often in scattering applications it is advantageous to reformulate the problem as an integral equation, discretize, and then solve the resulting linear system using a fast direct solver. The computational cost of this approach is typically dominated by the work needed to compress the coefficient matrix into a rank-structured format. In this talk, we present a novel technique which exploits the bandlimited-nature of solutions to the Helmholtz equation in order to accelerate this procedure in environments where multiple frequencies are of interest.

This talk is based on joint work with Gunnar Martinsson (UT Austin).

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Coadmissible modules over Frechet-Stein algebras arise naturally in p-adic representation theory, e.g. in the study of locally analytic representations of p-adic Lie groups or the function spaces of rigid analytic Stein spaces. We show that in many cases, the category of coadmissible modules admits an exact and fully faithful embedding into the category of complete bornological modules, also preserving tensor products. This allows us to introduce derived methods to the study of coadmissible modules without forsaking the analytic flavour of the theory. As an application, we introduce six functors for Ardakov-Wadsley's D-cap-modules and discuss some instances where coadmissibility (in a derived sense) is preserved.

Motivated by the work of K.R. Rajagopal, the objective of the talk is to discuss the construction and analysis of numerical approximations to a class of models that fall outside the realm of classical Cauchy elasticity. The models under consideration are implicit and nonlinear, and are referred to as strain-limiting, because the linearised strain remains bounded even when the stress is very large, a property that cannot be guaranteed within the framework of classical elastic or nonlinear elastic models. Strain-limiting models can be used to describe, for example, the behavior of brittle materials in the vicinity of fracture tips, or elastic materials in the neighborhood of concentrated loads where there is concentration of stress even though the magnitude of the strain tensor is limited.

We construct a finite element approximation of a strain-limiting elastic model and discuss the theoretical difficulties that arise in proving the convergence of the numerical method. The analytical results are illustrated by numerical experiments.

The talk is based on joint work with Andrea Bonito (Texas A&M University) and Vivette Girault (Sorbonne Université, Paris).

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Self-similar fragmentation processes are random models for particles that are subject to successive fragmentations. When the index of self-similarity is negative the fragmentations intensify as the masses of particles decrease. This leads to a shattering phenomenon, where the initial particle is entirely reduced to dust - a set of zero-mass particles - in finite time which is what we call the extinction time. Equivalently, these extinction times may be seen as heights of continuous compact rooted trees or scaling limits of heights of sequences of discrete trees. Our objective is to set up precise bounds for the large time asymptotics of the tail distributions of these extinction times.

We consider a nonlinear flow on simplicial complexes related to the simplicial Laplacian, and show that it is a generalization of various consensus and synchronization models commonly studied on networks. In particular, our model allows us to formulate flows on simplices of any dimension, so that it includes edge flows, triangle flows, etc. We show that the system can be represented as the gradient flow of an energy functional, and use this to deduce the stability of various steady states of the model. Finally, we demonstrate that our model contains higher-dimensional analogues of structures seen in related network models.

arXiv link: https://arxiv.org/abs/2010.07421

Coronary heart disease is characterised by the formation of plaque on artery walls, restricting blood flow. If a plaque deposit ruptures, blood clot formation (thrombosis) rapidly occurs with the potential to fatally occlude the vessel within minutes. Von Willebrand Factor (vWF) is a shear-sensitive protein which has a critical role in blood clot formation in arteries. At the high shear rates typical in arterial constrictions (stenoses), vWF undergoes a conformation change, unfolding and exposing binding sites and facilitating rapid platelet deposition. 

To understand the effect of  stenosis geometry and blood flow conditions on the unfolding of vWF and subsequent platelet binding, we developed a continuum model for the initiation of thrombus formation by vWF in an idealised arterial stenosis. In this talk I will discuss modelling proteins in flow using viscoelastic fluid models, the insight asymptotic reductions can offer into this complex system and some of the challenges of studying fast arterial blood flows. 

The propagation of high-frequency electromagnetic waves can be analyzed using the geometrical optics approximation. In the case of large but finite frequencies, the geometrical optics approximation is no longer accurate, and polarization-dependent corrections at first order in wavelength modify the propagation of light in an inhomogenous medium via a spin-orbit coupling mechanism. This effect, known as the spin Hall effect of light, has been experimentally observed. In this talk I will discuss recent work which generalizes the spin Hall effect to the propagation of light and gravitational waves in inhomogenous spacetimes. This is based on joint work with Marius Oancea and Jeremie Joudioux.