Past

We study sampling from a target distribution $e^{-f}$ using the unadjusted Langevin Monte Carlo (LMC) algorithm. For any potential function $f$ whose tails behave like $\|x\|^\alpha$ for $\alpha \in [1,2]$, and has $\beta$-H\"older continuous gradient, we derive the sufficient number of steps to reach the $\epsilon$-neighborhood of a $d$-dimensional target distribution as a function of $\alpha$ and $\beta$. Our rate estimate, in terms of $\epsilon$ dependency, is not directly influenced by the tail growth rate $\alpha$ of the potential function as long as its growth is at least linear, and it only relies on the order of smoothness $\beta$.

Our rate recovers the best known rate which was established for strongly convex potentials with Lipschitz gradient in terms of $\epsilon$ dependency, but we show that the same rate is achievable for a wider class of potentials that are degenerately convex at infinity.

 I will review some well-established relationships between four manifolds and vertex algebras that can be deduced from studying the M5-brane worldvolume theory, and outline some of the corresponding results in mathematics which have been understood so far. I will then describe a proposal of Gaiotto-Rapcak to generalize these ideas to the setting of multiple M5 branes wrapping divisors in toric Calabi-Yau threefolds, and explain work in progress on understanding the mathematical implications of this proposal as a complex network of relationships between the enumerative geometry of sheaves on threefolds and the representation theory of affine Lie algebras.

We discuss the formal relationship between the absence of global symmetries and completeness, both in effective field theory and in quantum gravity. In effective field theory, we must broaden our notion of symmetry to include non-invertible topological operators. However, in gravity, the story is simplified as the result of charged gravitational solitons.

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

How can we make maths more accessible, promote its many applications, and encourage more women to enter the field? These are the questions we aim to address with Mathematigals.

Caoimhe Rooney and Jessica Williams met in 2015 at the start of their PhDs in mathematics in Oxford, and in 2020, they co-founded Mathematigals. Mathematigals is an online platform producing content to demonstrate fun mathematical curiosities, showcase ways maths can be used in real life, and promote female mathematicians. Mathematigals primarily produces animated videos that present maths in a way that is engaging to the general public.

In this session, Jess and Caoimhe will talk about their initial motivation to begin Mathematigals, demonstrate the process behind their content creation, and describe their future visions for the platform. The session will end with an opportunity for the audience to provide feedback or ideas to help Mathematigals on their journey to encourage future mathematicians.

Locally analytic representations of $p$-adic Lie groups are of interest in several branches of number theory, for example in the theory of automorphic forms and in the $p$-adic local Langlands program. To better understand these representations, Ardakov-Wadsley introduced a sheaf of infinite order differential operators $\overparen{\mathcal{D}}$ on smooth rigid analytic spaces, which resulted in several Beilinson-Bernstein style localisation theorems. In this talk, we discuss the current research on analogues of a Riemann-Hilbert correspondence for $\overparen{\mathcal{D}}$-modules, and what this has to do with complete convex bornological vector spaces.

High-dimensional approximation of Hamilton-Jacobi-Bellman PDEs – architectures, algorithms and applications

Hamilton-Jacobi Partial Differential Equations (HJ PDEs) are a central object in optimal control and differential games, enabling the computation of robust controls in feedback form. High-dimensional HJ PDEs naturally arise in the feedback synthesis for high-dimensional control systems, and their numerical solution must be sought outside the framework provided by standard grid-based discretizations. In this talk, I will discuss the construction novel computational methods for approximating high-dimensional HJ PDEs, based on tensor decompositions, polynomial approximation, and deep neural networks.

Catheter ablation and antiarrhythmic drug therapy approaches for treatment of atrial fibrillation are sub-optimal. This is in part because it is challenging to predict long-term response to therapy from short-term measurements, which makes it difficult to select optimal patient-specific treatment approaches. Clinical trials identify patient demographics that provide prediction of long-term response to standard treatments across populations. Patient-specific biophysical models can be used to assess novel treatment approaches but are typically applied in small cohorts to investigate the acute response to therapies. Our overall aim is to use machine learning approaches together with patient-specific biophysical simulations to predict long-term atrial fibrillation recurrence after ablation or drug therapy in large populations.

In this talk I will present our methodology for constructing personalised atrial models from patient imaging and electrical data; present results from biophysical simulations of ablation treatment; and finally explain how we are combining these methodologies with machine learning techniques for predicting long-term treatment outcomes.

[REMOTE TALK]

In this talk, we will propose a new construction of the virtual fundamental classes of quasi-smooth derived schemes using the vanishing cycle complexes. This is based on the dimensional reduction theorem of cohomological Donaldson—Thomas invariants which can be regarded as a variant of the Thom isomorphism. We will also discuss a conjectural approach to construct DT4 virtual classes using the vanishing cycle complexes.

Zoom link: https://us02web.zoom.us/j/86267335498?pwd=R2hrZ1N3VGJYbWdLd0htZzA4Mm5pd…

Computing spectral properties of operators is fundamental in the sciences, with applications in quantum mechanics, signal processing, fluid mechanics, dynamical systems, etc. However, the infinite-dimensional problem is infamously difficult (common difficulties include spectral pollution and dealing with continuous spectra). This talk introduces classes of practical resolvent-based algorithms that rigorously compute a zoo of spectral properties of operators on Hilbert spaces. We also discuss how these methods form part of a broader programme on the foundations of computation. The focus will be computing spectra with error control and spectral measures, for general discrete and differential operators. Analogous to eigenvalues and eigenvectors, these objects “diagonalise” operators in infinite dimensions through the spectral theorem. The first is computed by an algorithm that approximates resolvent norms. The second is computed by building convolutions of appropriate rational functions with the measure via the resolvent operator (solving shifted linear systems). The final part of the talk provides purely data-driven algorithms that compute the spectral properties of Koopman operators, with convergence guarantees, from snapshot data. Koopman operators “linearise” nonlinear dynamical systems, the price being a reduction to an infinite-dimensional spectral problem (c.f. “Koopmania”, describing their surge in popularity). The talk will end with applications of these new methods in several thousand state-space dimensions.

Microscopic green algae show great diversity in structural complexity, and successfully evolved efficient swimming strategies at low Reynolds numbers. Gonium is one of the simplest multicellular algae, with only 16 cells arranged in a flat plate. If the swimming of unicellular organisms, like Chlamydomonas, is nowadays widely studied, it is less clear how a colony made of independent Chlamydomonas-like cells performs coordinated motion. This simple algae is therefore a key organism to model the evolution from single-celled to multicellular locomotion.

In the absence of central communication, how can each cell adapt its individual photoresponse to efficiently reorient the whole algae? How crucial is the distinctive Gonium squared structure?

In this talk, I will present experiments investigating the shape and the phototactic swimming of Gonium, using trajectory tracking and micro-pipette techniques. I will explain our model linking the individual flagella response to the colony trajectory. This eventually emphasises the importance of biological noise for efficient swimming.

The curve graph of a surface of finite type is a fundamental object in the study of its mapping class group both from the metric and the combinatorial point of view. I will discuss joint work with Valentina Disarlo and Thomas Koberda where we conduct a thorough study of curve graphs from the model theoretic point of view, with particular emphasis in the problem of interpretability between different curve graphs and other geometric complexes.   

Given a group G acting on a CAT(0) polygonal complex, X, it is natural to ask whether the structure of X allows us to deduce properties of G. We discuss some recent work on local properties that X may possess which allow us to answer these questions - in many cases we can in fact deduce that the group is a linear group over Z.

I'll explain 'symplectic duality', a surprising relationship between certain pairs of algebraic symplectic manifolds, under which Hamiltonian automorphisms of one are identified with Poisson deformations of the other, and which is ultimately characterized by a Koszul-type equivalence between categories of modules over their filtered quantizations. I'll outline why such relationships are expected from physics in terms of three dimensional mirror symmetry, and rediscover the Coulomb branch construction of Braverman-Finkelberg-Nakajima from this perspective. We'll see that this explicitly constructs the symplectic dual of any variety which is presented as the symplectic reduction of a vector space by a reductive group.
 

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)

Here we present the findings of our summer research project: an 8-week investigation of C*-Algebras. Our aim was to explore when a group is uniquely determined by its reduced group C*-algebra; i.e. when the group is C*-rigid. In particular, we applied techniques from topology, algebra, and analysis to prove C*-rigidity for a certain class of Bieberbach groups.

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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A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact @email.

 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.

___________________________________________________________________________________________________

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 $\mathbb{Q}$.