Past

TBA

Let $A$ be drawn uniformly at random from the set of all $n \times n$ symmetric matrices with entries in $\{-1,1\}$. We show that $A$ is singular with probability at most $e^{-cn}$ for some absolute constant $c>0$, thereby resolving a well-known conjecture. This is joint work with Marcelo Campos, Marcus Michelen and Julian Sahasrabudhe.
 

Let $G$ be a finitely presented residually finite group. We are interested in the growth of size of the torsion of $H^{ab}$ as a function of $|G:H|$ where $H$ ranges over normal subgroups of finite index in $G$. It is easy to see that this grows at most exponentially in terms of $|G:H|$. Of particular interest is the case when $G$ is an arithmetic hyperbolic 3-manifold group and $H$ ranges over its congruence subgroups. Proving exponential lower bounds on the torsion appears to be difficult and in this talk I will focus on the situation of finitely presented pro-$p$ groups.

In contrast with abstract groups I will show that in finitely presented pro-$p$ groups torsion in the abelianizations can grow arbitrarily fast. The examples are rather 'large' pro-$p$ groups, in particular they are virtually Golod-Shafarevich. When we restrict to $p$-adic analytic groups the torsion growth is at most polynomial.

Several problems of great importance in the study of glaciers and ice sheets involve processes of attachment and reattachment of the ice from the bedrock. Consider, for example, an ice sheet sliding from the continent into the ocean, where it goes afloat. Another example is that of subglacial cavitation, a fundamental mechanism in glacial sliding where the ice detaches from the bedrock along the downstream area of an obstacle. Such problems are generally modelled as a viscous Stokes flow with a free boundary and contact boundary conditions. In this talk, I will present a framework for solving such problems numerically. I will start by introducing the mathematical formulation of these viscous contact problems and the challenges that arise when trying to approximate them numerically. I will then show how, given a stable scheme for the free boundary equation, one can build a penalty formulation for the viscous contact problem in such a way that the resulting algorithm remains stable and robust.

Title: Measuring association with Wasserstein distances

Abstract: Let π ∈ Π(μ, ν) be a coupling between two probability measures μ and ν on a Polish space. In this talk we propose and study a class of nonparametric measures of association between μ and ν, which we call Wasserstein correlation coefficients. These coefficients are based on the Wasserstein distance between ν and the disintegration of π with respect to the first coordinate. We also establish basic statistical properties of this new class of measures: we develop a statistical theory for strongly consistent estimators and determine their convergence rate in the case of compactly supported measures μ and ν. Throughout our analysis we make use of the so-called adapted/bicausal Wasserstein distance, in particular we rely on results established in [Backhoff, Bartl, Beiglböck, Wiesel. Estimating processes in adapted Wasserstein distance. 2020]. Our approach applies to probability laws on general Polish spaces.

In a world of polynomial Pell’s equations, where the integers are replaced by polynomials with complex coefficients, and its smallest solution is used to generate all other solutions $(u_{n},v_{n})$, $n\in\mathbb{Z}$. One junior number theory group will embark on a journey in search of the properties of the factors of $v_{n}(t)$. There will be Galois extensions, there will be estimations and of course there will be loglogs.

We show that the existence of hyperbolic manifolds fibering over the circle is not a phenomenon confined to dimension 3 by exhibiting some examples in dimension 5. More generally, there are hyperbolic manifolds with perfect circle-valued Morse functions in all dimensions $n\le 5$. As a consequence, there are hyperbolic groups with finite-type subgroups that are not hyperbolic.

The main tool is Bestvina - Brady theory enriched with a combinatorial game recently introduced by Jankiewicz, Norin and Wise. These are joint works with Battista, Italiano, and Migliorini.

TBA

It is well-known that the spectral data of the Gaudin model associated to a finite semisimple Lie algebra is encoded by the differential data of certain flat connections associated to the Langlands dual Lie algebra on the projective line with regular singularities, known as oper/Gaudin correspondence. Recently, some progress has been made in understanding the correspondence associated with affine Lie algebras. I will present a physical perspective from Kondo line defects, physically describing a local impurity chirally coupled to the bulk 2d conformal field theory. The Kondo line defects exhibit interesting integrability properties and wall-crossing behaviors, which are encoded by the generalized monodromy data of affine opers. In the physics literature, this reproduces the known ODE/IM correspondence. I will explain how the recently proposed 4d Chern Simons theory provides a new perspective which suggests the possibility of a physicists’ proof. 

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

Hierarchical stabilisation, allows us to define topological invariants for data starting from metrics to compare persistence modules. In this talk I will highlight the variety of metrics that can be constructed in an axiomatic way, via so called Noise Systems. The focus will then be on one invariant obtained through hierarchical stabilisation, the Stable Rank, which the TDA group at KTH has been studying in the last years. In particular I will address the problem of using this invariant on noisy and heterogeneous data. Lastly, I will illustrate the use of stable ranks on real data within a project on microglia morphology description, in collaboration with S. Siegert’s group, K. Hess and L. Kanari. 

Let $K$ be an algebraically closed field. Given three elements of some Lie algebra over $K$, we say that these elements form an $sl_2$-triple if they generate a subalgebra which is a homomorphic image of $sl_2(K).$ In characteristic 0, the Jacobson-Morozov theorem provides a bijection between the orbits of nilpotent elements of the Lie algebra and the orbits of $sl_2$-triples. In this talk I will discuss the progress made in extending this result to fields of characteristic $p$. In particular, I will focus on the results in classical Lie algebras, which can be found as subsets of $gl_n(K)$.

In this talk, I will give an overview of our multi-scale models that we have developed to study a number of aspects of the immune response to infection.  Scales that we explore range from molecular to the whole-host scale.  We are also able to study virtual populations and perform simulated clinical trials. We apply these approaches to study Tuberculosis, the disease caused by inhalation of the bacteria, Mycobacterium tuberculosis. It has infected 2 billion people in the world today, and kills 1-2 million people each year, even more than COVID-19. Our goal is to aid in understanding infection dynamics, treatment and vaccines to improve outcomes for this global health burden. I will discuss our frameworks for multi-scale modeling, and the analysis tools and statistical approaches that we have honed to better understand different outcomes at different scales.

We construct smooth flows with surgery that approximate weak mean curvature flows with only spherical and neck-pinch singularities. This is achieved by combining the recent work of Choi-Haslhofer-Hershkovits, and Choi-Haslhofer-Hershkovits-White, establishing canonical neighbourhoods of such singularities, with suitable barriers to flows with surgery. A limiting argument is then used to control these approximating flows. We demonstrate an application of this surgery flow by improving the entropy bound on the low-entropy Schoenflies conjecture.

We develop a new online algorithm for optimizing over the stationary distribution of stochastic differential equation (SDE) models. The algorithm optimizes over the parameters in the multi-dimensional SDE model in order to minimize the distance between the model's stationary distribution and the target statistics. We rigorously prove convergence for linear SDE models and present numerical results for nonlinear examples. The proof requires analysis of the fluctuations of the parameter evolution around the unbiased descent direction under the stationary distribution. Bounds on the fluctuations are challenging to obtain due to the online nature of the algorithm (e.g., the stationary distribution will continuously change as the parameters change). We prove bounds on a new class of Poisson partial differential equations, which are then used to analyze the parameter fluctuations in the algorithm. This presentation is based upon research with Ziheng Wang.
 

The roots of a function represented by its Chebyshev expansion are known to be the eigenvalues of the so-called colleague matrix, which is a Hessenberg matrix that is the sum of a symmetric tridiagonal matrix and a rank 1 perturbation. The rootfinding problem is thus reformulated as an eigenproblem, making the computation of the eigenvalues of such matrices a subject of significant practical interest. To obtain the roots with the maximum possible accuracy, the eigensolver used must posess a somewhat subtle form of stability.

In this talk, I will discuss a recently constructed algorithm for the diagonalization of colleague matrices, satisfying the relevant stability requirements.  The scheme has CPU time requirements proportional to n^2, with n the dimensionality of the problem; the storage requirements are proportional to n. Furthermore, the actual CPU times (and storage requirements) of the procedure are quite acceptable, making it an approach of choice even for small-scale problems. I will illustrate the performance of the algorithm with several numerical examples.

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 Mathematics for the mind: network dynamical systems for neurodegenerative disease pathology

Travis Thompson

Can mathematics understand neurodegenerative diseases?  The modern medical perspective on neurological diseases has evolved, slowly, since the 20th century but recent breakthroughs in medical imaging have quickly transformed medicine into a quantitative science.  Today, mathematical modeling and scientific computing allow us to go farther than observation alone.  With the help of  computing, experimental and data-informed mathematical models are leading to new clinical insights into how neurodegenerative diseases, such as Alzheimer's disease, may develop in the human brain.  In this talk, I will overview my work in the construction, analysis and solution of data and clinically-driven mathematical models related to AD pathology.  We will see that mathematical modeling and scientific computing are indeed indispensible for cultivating a data-informed understanding of the brain, AD and for developing potential treatments.

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Simplified battery models via homogenisation  

Robert Timms

Lithium-ion batteries (LIBs) are one of the most popular forms of energy storage for many modern devices, with applications ranging from portable electronics to electric vehicles. Improving both the performance and lifetime of LIBs by design changes that increase capacity, reduce losses and delay degradation effects is a key engineering challenge. Mathematical modelling is an invaluable tool for tackling this challenge: accurate and efficient models play a key role in the design, management, and safe operation of batteries. Models of batteries span many length scales, ranging from atomistic models that may be used to predict the rate of diffusion of lithium within the active material particles that make up the electrodes, right through to models that describe the behaviour of the thousands of cells that make up a battery pack in an electric vehicle. Homogenisation can be used to “bridge the gap” between these disparate length scales, and allows us to develop computationally efficient models suitable for optimising cell design.

Orbifolds are a generalisation of manifolds which allow group actions to enter the picture. The most basic examples of orbifolds are quotients of manifolds by (non-free) finite group actions.
I will give an introduction to orbifolds, recalling a number of philosophically different but mathematically equivalent definitions. For starters, I will try to convince you that "a space locally modelled on a quotient of R^n by a finite group" is misleading. I will draw many pictures of orbifolds, make the connection to complexes of groups, and explain the definition of a map of orbifolds. In the process, I hope to demystify the definition of the orbifold fundamental group, the orbifold Euler characteristic and orbifold cohomology.

I will discuss 3d N=4 supersymmetric gauge theories compactified on an elliptic curve, and how this set-up physically realises recent mathematical results on the equivariant elliptic cohomology of symplectic resolutions. In particular, I will describe the Berry connection for supersymmetric ground states, and in doing so connect the elliptic cohomology of the Higgs branch with spectral data of doubly periodic monopoles. I will show that boundary conditions, via a consideration of boundary ’t Hooft anomalies, naturally represent elliptic cohomology classes. Finally, if I have time, I will discuss mirror symmetry/symplectic duality in our framework, and physically recover concepts in elliptic cohomology such as the mother function, and the elliptic stable envelopes of Aganagic-Okounkov.

This talk will be based on https://arxiv.org/abs/2109.10907 with Mathew Bullimore.

Many PDE models encode fundamental physical, geometric and topological structures. These structures may be lost in discretisations, and preserving them on the discrete level is crucial for the stability and efficiency of numerical methods. The finite element exterior calculus (FEEC) is a framework for constructing and analysing structure-preserving numerical methods for PDEs with ideas from topology, homological algebra and the Hodge theory. 

In this seminar, we present the theory and applications of FEEC. This includes analytic results (Hodge decomposition, regular potentials, compactness etc.), Hodge-Laplacian problems and their structure-preserving finite element discretisation, and applications in electromagnetism, fluid and solid mechanics. Knowledge on geometry and topology is not required as prerequisites.

References:

1. Arnold, D.N.: Finite Element Exterior Calculus. SIAM (2018) 

2. Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numerica 15, 1 (2006) 

3. Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus: from Hodge theory to numerical stability. Bulletin of the American Mathematical Society 47(2), 281–354 (2010) 

4. Arnold, D.N., Hu, K.: Complexes from complexes. Foundations of Computational Mathematics (2021)

Twisted K-theory is an enrichment of topological K-theory that allows local coefficient systems called twists. For spaces and twists equipped with an action by a group, equivariant twisted K-theory provides an even finer invariant. Equivariant twists over Lie groups gained increasing importance in the subject due to a result by Freed, Hopkins and Teleman that relates the corresponding K-groups to the Verlinde ring of the associated loop group. From the point of view of homotopy theory only a small subgroup of all possible twists is considered in classical treatments. In this talk I will discuss a construction that is joint work with David Evans and produces interesting examples of non-classical twists over the Lie groups SU(n) and over tori constructed from exponential functors. They arise naturally as Fell bundles and are equivariant with respect to the conjugation action of the group on itself. For the determinant functor our construction reproduces the basic gerbe over SU(n) used by Freed, Hopkins and Teleman.

Hermitian matrix models with non-trivial covariance will be introduced. The Kontsevich Model is the prime example, which was used to prove Witten's conjecture about the generating function of intersection numbers of the moduli space $\overline{\mathcal{M}}_{g,n}$. However, we will discuss these models in a different direction, namely as a quantum field theory. As a formal matrix model,  the correlation functions of these models have a unique combinatorial/perturbative interpretation in the sense of Feynman diagrams. In particular, the additional structure (in comparison to ordinary quantum field theories) gives the possibility to compute exact expressions, which are resummations of infinitely many Feynman diagrams. For the easiest topologies, these exact expressions (given by implicitly defined functions) will be presented and discussed. If time remains, higher topologies are discussed by a connection to Topological Recursion.

TBA

In the event of food-borne disease outbreaks, conventional epidemiological approaches to identify the causative food product are time-intensive and often inconclusive. Data-driven tools could help to reduce the number of products under suspicion by efficiently generating food-source hypotheses. We frame the problem of generating hypotheses about the food-source as one of identifying the source network from a set of food supply networks (e.g. vegetables, eggs) that most likely gave rise to the illness outbreak distribution over consumers at the terminal stage of the supply network. We introduce an information-theoretic measure that quantifies the degree to which an outbreak distribution can be explained by a supply network’s structure and allows comparison across networks. The method leverages a previously-developed food-borne contamination diffusion model and probability distribution for the source location in the supply chain, quantifying the amount of information in the probability distribution produced by a particular network-outbreak combination. We illustrate the method using supply network models from Germany and demonstrate its application potential for outbreak investigations through simulated outbreak scenarios and a retrospective analysis of a real-world outbreak.

In this leisure talk I will show how a sum of squares decomposition problem can be transformed to a problem of semi-definite optimization. Then the practicality of such reformulations will be discussed, illustrated by an explicit example of Artin's solutions to Hilberts 17th problem. Finally I will show how a numerical solution could be turned into a mathematically certified one, using the order structure on the cone of sums of squares.
The talk requires no pre-requisite knowledge of neither optimization or programming and only undergraduate mathematics.

In his very first note on noncommutative differential geometry, Connes
showed that the position and momentum operators on the line could be used to
construct constant curvature connections over an irrational noncommutative

2-torus $\mathcal{A}_\theta$. When $\theta$ is a quadratic irrationality,
this yields, in particular, constant curvature connections on non-trivial
noncommutative line bundles---is there an underlying monopole on some
non-trivial noncommutative principal $U(1)$-bundle? We use this case study
to illustrate how approaches to quantum principal bundles introduced by
Brzeziński–Majid and Đurđević, respectively, can be fruitfully synthesized
to reframe classical gauge theory on quantum principal bundles in terms of
synthesis of total spaces (as noncommutative manifolds) from vertical and
horizontal geometric data.

TBA

The Jacquet-Langlands correspondence gives a relationship between automorphic representations on $GL_2$ and its twisted forms, which are the unit groups of quaternion algebras. Writing this out in more classical language gives a combinatorial way of producing the eigenvalues of Hecke operators acting on modular forms. In this talk, we will first go over notions of modular forms and quaternion algebras, and then dive into an explicit example by computing some eigenvalues of the lowest level quaternionic modular form of weight $2$ over $\mathbb{Q}$.

Classical approaches to optimal portfolio selection problems are based 
on probabilistic models for the asset returns or prices. However, by 
now it is well observed that the performance of optimal portfolios are 
highly sensitive to model misspecifications. To account for various 
type of model risk, robust and model-free approaches have gained more 
and more importance in portfolio theory. Based on a rough path 
foundation, we develop a model-free approach to stochastic portfolio 
theory and Cover's universal portfolio. The use of rough path theory 
allows treating significantly more general portfolios in a model-free 
setting, compared to previous model-free approaches. Without the 
assumption of any underlying probabilistic model, we present pathwise 
Master formulae analogously to the classical ones in stochastic 
portfolio theory, describing the growth of wealth processes generated 
by pathwise portfolios relative to the wealth process of the market 
portfolio, and we show that the appropriately scaled asymptotic growth 
rate of Cover's universal portfolio is equal to the one of the best 
retrospectively chosen portfolio. The talk is based on joint work with 
Andrew Allan, Christa Cuchiero and Chong Liu.

 

Consider simple rank-one Lie groups $SO(n, 1)$, $SU(n, 1)$ and $Sp(n ,1)$ ($n>1$). They are the isometry groups of real, complex and quaternionic hyperbolic spaces respectively.

By a result of Kostant, the trivial representation of $Sp(n ,1)$ is isolated in the space of irreducible unitary representations on Hilbert spaces. That is, $Sp(n ,1)$ has Kazhdan’s property (T) which is equivalent to the vanishing of 1st cohomology of the group in all unitary representations. This is in contrast to the case of $SO(n ,1)$ and $SU(n ,1)$ where they have the Haagerup approximation property, a strong negation of property (T).

This dichotomy between $SO(n ,1)$, $SU(n ,1)$ and $Sp(n ,1)$ disappears when we consider so-called uniformly bounded representations on Hilbert spaces. By a result of Cowling in 1980’s, the trivial representation of $Sp(n ,1)$ is no longer isolated in the space of uniformly bounded representations. Moreover, there is a uniformly bounded representation of $Sp(n ,1)$ with non-zero first cohomology group.

The goal of this talk is to describe these facts.

It is expected that complete noncompact Calabi-Yau manifolds are in some sense governed by their asymptotics at infinity. In the maximal volume growth case, the asymptotics at infinity are given by Calabi-Yau cones. We are interested in deformations of such metrics that fix the asymptotic cones at infinity. In the asymptotically conical case, Conlon-Hein proved uniqueness under such deformations. Their method is based on the corresponding linearized problem, namely the study of subquadratic harmonic functions. We generalize their work to the maximal volume growth case, allowing the tangent cones at infinity to have non-isolated singularities. Part of the talk is based on work in progress joint with Gabor Szekelyhidi.

The success of operator splitting techniques for convex optimization has led to an explosion of methods for solving large-scale and non convex optimization problems via convex relaxation. 

This success is at the cost of overlooking direct approaches to operator splitting that embrace some of the more inconvenient aspects of many model problems, namely nonconvexity, non smoothness and infeasibility.  I will introduce some of the tools we have developed for handling these issues, and present sketches of the basic results we can obtain.

The formalism is in general metric spaces, but most applications have their basis in Euclidean spaces.  Along the way I will try to point out connections to other areas of intense interest, such as optimal mass transport.

 In this talk I will discuss an algorithm to piecewise dualise linear quivers into their mirror duals. This applies to the 3d N=4 version of mirror symmetry as well as its recently introduced 4d counterpart, which I will review. The algorithm uses two basic duality moves, which mimic the local S-duality of the 5-branes in the brane set-up of the 3d theories, and the properties of the S-wall. The S-wall is known to correspond to the N=4 T[SU(N)] theory in 3d and I will argue that its 4d avatar corresponds to an N=1 theory called E[USp(2N)], which flows to T[SU(N)] in a suitable 3d limit. All the basic duality moves and S-wall properties needed in the algorithm are derived in terms of some more fundamental Seiberg-like duality, which is the Intriligator--Pouliot duality in 4d and the Aharony duality in 3d.

Superconformal field theories (SCFTs) of Argyres-Dougles type are inherently strongly coupled and provide a window onto remarkable non-perturbative phenomena (such as mutually non-local massless dyons and relevant Coulomb branch operators of fractional dimension). I am going to discuss the first explicit proposal for the holographic duals of a class of SCFTs of Argyres-Douglas type. The theories under examination are realised by a stack of M5-branes wrapped on a sphere with one irregular puncture and one regular puncture. In the dual 11d supergravity solutions, the irregular puncture is realised as an internal M5-brane source.

What do we use metrics on persistent modules for? Is it only to asure  stability of some constructions? 

In my talk I will describe why I care about such metrics, show how to construct a rich space of them and illustrate how  to use

them for analysis. 

The injection of CO2 into porous subsurface reservoirs is a technological means for removing anthropogenic emissions, which relies on a series of complex porous flow properties. During injection of CO2 small-scale heterogeneities, often in the form of sedimentary layering, can play a significant role in focusing the flow of less viscous CO2 into high permeability pathways, with large-scale implications for the overall motion of the CO2 plume. In these settings, capillary forces between the CO2 and water preferentially rearrange CO2 into the most permeable layers (with larger pore space), and may accelerate plume migration by as much as 200%. Numerous factors affect overall plume acceleration, including the structure of the layering, the permeability contrast between layers, and the playoff between the capillary, gravitational and viscous forces that act upon the flow. However, despite the sensitivity of the flow to these heterogeneities, it is difficult to acquire detailed field measurements of the heterogeneities owing to the vast range of scales involved, presenting an outstanding challenge. As a first step towards tackling this uncertainty, we use a simple modelling approach, based on an upscaled thin-film equation, to create ensemble forecasts for many different types and arrangements of sedimentary layers. In this way, a suite of predictions can be made to elucidate the most likely scenarios for injection and the uncertainty associated with such predictions. 

This presentation will focus on the role of mathematical modelling and predictive toxicology in the safety assessment of chemicals and consumer products. The starting point will be regulatory assessment of chemicals based on their potential for harming human health or the environment. This will set the scene for describing current practices in the development and application of mathematical and computational models. A wide variety of methodological approaches are employed, ranging from relatively simple statistical models to more advanced machine learning approaches. The modelling context also ranges from discovering the underlying mechanisms of chemical toxicity to the safe and sustainable design of chemical products. The main modelling approaches will be reviewed, along with the challenges and opportunities associated with their use.  The presentation will conclude by identifying current research needs, including progress towards a Unified Theory of Chemical Toxicology.

Following on from Christoph's talk last week, I will present a version of the supercooled Stefan problem with noise. I will start by discussing the physical intuition and then give a probabilistic representation of solutions. From there, I will identify a simple relationship between the initial heat profile and a single parameter for how the liquid solidifies, which, if violated, forces the temperature to develop a discontinuity in finite time with positive probability. On the other hand, when the relationship is satisfied, the temperature remains globally continuous with probability one. The work is part of a new preprint that should soon be available on arXiv.

The past few years have been an exciting time for my work related to rational approximation.  This talk will present four developments:

1. AAA approximation (2016, with Nakatsukasa & Sète)
2. Root-exponential convergence and tapered exponential clustering (2020, with Nakatsukasa & Weideman)
3. Lightning (2017-2020, with Gopal & Brubeck)
4. Log-lightning (2020-21, with Nakatsukasa & Baddoo)

Two other topics will not be discussed:

X. AAA-Lawson approximation (2018, with Nakatsukasa)
Y. AAA-LS approximation (2021, with Costa)

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.

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

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".

In this talk, I will discuss the important Grothendieck conjecture which originated Grothendieck-Teichmuller Theory, a bridge between Topology and Number Theory. On the geometric side, there is the study of automorphisms of mapping class groups that satisfy compatibility conditions with respect to subsurface inclusions. On the other side, there is the study of the absolute Galois group of the rationals, one of the most important objects in Number Theory today.
In my talk, I will introduce these objects and discuss the recent progress that has been made in understanding such automorphisms of mapping class groups. No background in Number Theory or Galois Theory is required.

In this talk we will study scattering amplitudes N=4 super-Yang-Mills theory. In this theory, scattering amplitudes are known to be functions of cluster variables of Gr(4,n) and certain algebraic functions of cluster variables. We will give an overview of how this cluster algebraic structure manifests, and will exploit it in an algorithm for computing symbol alphabets by solving matrix equations of the form C.Z = 0 associated with plabic graphs. These matrix equations associate functions on Gr(m,n) to parameterizations of certain cells of Gr_+ (k,n) indexed by plabic graphs. We are able to reproduce all known algebraic functions of cluster variables appearing in known symbol alphabets. We further show that it is possible to obtain all rational symbol letters (in fact all cluster variables) by solving C.Z = 0 if one allows C to be an arbitrary cluster parameterization of the top cell of Gr_+ (n-4,n).