Research Seminar Analysis | Berlin Math Seminars

Analysis

Fri, 30.05.25

Alfonso Garmendia MPI, Bonn

E-symplectic and almost regular Poisson manifolds

Abstract.

Singular symplectic manifolds have been studied to model dynamical systems and classical mechanics for spaces with singularities. One particular example is for manifolds with boundary, the dynamics will be given by a seemingly not smooth symplectic form, but considering only vector fields tangent to the boundary this form is well defined and smooth, therefore this dynamical systems are given by a symplectic form only defined on the dual of an specific class of vector fields (the ones tangent to the boundary). Similarly one can consider symplectic forms on different classes of vector fields. We will consider in this talk classes coming from a singular foliation and compare with the structure that the symplectic form brings on the groupoid that integrates the foliation.

Analysis

Fri, 16.05.25 at 10:00

Harprit Singh Universität Wien

Rough Geometric Integration

Abstract.

Combining ideas from Whitney’s geometric integration theory and rough analysis, we introduce spaces of rough differential k-forms on d-manifolds which are formally given by f=∑IfIdxI where (fI)I belong to a class of genuine distributions of negative regularity. These rough k-forms have several properties desirable of a notion of differential forms: they can be integrated over suitably regular k-manifolds, they form a module under point-wise multiplication with sufficiently regular functions, exterior differentiation as well as the Stokes theorem extend to these spaces, they come with natural embeddings into distribution spaces, they contain classes of form valued distributional random fields. Finally, these spaces unify several previous constructions in the literature. In particular, they generalise spaces of α-flat cochains introduced by Whitney and Harrison, they contain the (rough) k-forms f·dg1∧…∧dgk introduced by Züst using Young integration, and for d=2 and k=1, they are close to the spaces which Chevyrev et al. use to make sense of Yang–Mills connections. Lastly, as a technical tool we introduce a ‘simplicial sewing lemma’, which provides a coordinate invariant formulation of the (known) multi-dimensional sewing lemma. This is a joint work with A. Chandra.

Analysis

Fri, 09.05.25

Nicolai Rothe Technische Universität Berlin

Cosmological solutions to the semiclassical Einstein equation with Minkowski-like vacua

Abstract.

We will discuss some newly found solutions to the full massless semiclassical Einstein equation (SCE) in a cosmological setting (with Λ=0). After a short introduction to the relevant notions, we present the SCE in a particular shape which allows for the construction of certain vacuum states. These states may be viewed as the least possible generalization of the Minkowski vacuum to generic cosmological space-times. In this setting, solving the SCE breaks down into solving a certain ODE which can be approached numerically and, at least generically, we obtain solutions that well fit physical expectations. Moreover, these solutions indicate dark energy as a quantum effect back-reacting on cosmological metrics and, since in our model m=Λ=0, this may not be traced back to the usual, obvious dark-energy/cosmological constant effect of a quantum field. Also we will shortly address some related results obtained in our setting.

Analysis

Wed, 30.04.25 at 12:00

Carlos Perez-Sanchez Universität Heidelberg

Towards random noncommutative geometry

Abstract.

The well-known question whether one can "hear the shape of a drum", posed by Marek Kac, has, also famously, a negative answer constructed by Gordon, Webb and Wolpert. In noncommutative geometry, classical dynamics depends only on the spectrum, in that case, of an operator of Dirac type. If, additionally, algebraic data are provided and some axioms verified - building what is known as spectral triple - this structure does allow to reconstruct a manifold, thus answering a weaker version of Kac's question positively. Spectral triples are relevant in Connes' noncommutative geometric setting, whose path integral quantisation that "averages over noncommutative geometries" shall rely on the concept of ensembles of Dirac operators. This is to be contrasted with a path integral over Riemannian metrics in quantum gravity. In this talk I first explore what an ensemble of noncommutative geometries on a fixed graph is (gauge fields are on, while gravity is still off). Using elements of quiver representation theory we associate a Dirac operator to a quiver representation (in a category that emerges in noncommutative geometry); we derive the constraints that the set of Wilson loops satisfies (generalised Makeenko-Migdal equations); and explore the consequences of the positivity of a certain matrix of Wilson loops ("bootstrap"). In the special case that our graph is a rectangular lattice and our physical action quartic, we obtain Wilsonian lattice Yang-Mills theory, hence the terminology. Unsurprisingly, our ensembles (for an arbitrary graph) boil down to integrating noncommutative polynomials against a product Haar measure on unitary groups. The classical aspects of this theory were constructed in [2401.03705], and the loop equations in [2409.03705].

Analysis

Fri, 25.04.25

online

Coline Emprin École Normale Supérieure de Paris

A prop structure on partitions

Abstract.

PROPs were introduced by Mac Lane in 1965 as a special type of category whose objects are natural numbers, endowed with an additional horizontal composition of morphisms beyond the usual categorical composition. In this talk, I will present a specific PROP structure that emerges from the combinatorics of partitions. The construction of this structure is closely related to the Karoubi envelope of a certain category, which I will introduce along the way. This PROP is of particular interest in the context of functor homology, as its composition corresponds to the Yoneda product of extension groups between exterior power functors. I will conclude by discussing how such constructions can be used to compute extension groups between simple functors defined on free groups. This is joint work with Dana Hunter, Muriel Livernet, Christine Vespa, and Inna Zakharevich.

Analysis

Wed, 16.04.25

Camilo Angulo University of Göttingen

Examples of Poisson manifolds with compactness properties

Abstract.

Poisson geometry lies in the intersection of symplectic geometry, foliation theory and Lie theory. As in each of these areas compactness hypotheses yield a wealth of results, it would be desirable to have a notion of compactness in Poisson geometry that simultaneously subsumes the theory of compact semisimple compact Lie groups and compact symplectic manifolds. This goal has been recently achieved by Crainic, Fernandes and Martinez-Torres, who defined a Poisson manifold of compact type (PMCTs) to be a Poisson manifold whose integrating symplectic groupoid is proper. The wonderful properties of these PMCTs lie in contrast to their relative scarcity. The geometric and topological constraints that go into building a PMCT make their definition rather demanding, and in so, constructing a PMCT beyond the trivial case of a compact symplectic manifold with finite fundamental group has proven a challenging problem. In this talk, after properly explaining the elements that go into play, we explain how by allowing for other geometric structures to "integrate" Poisson manifolds, one can get more examples while preserving most of the compactness properties.

Analysis

Fri, 07.02.25 at 11:00

Stav Zalel University of Cambridge, UK

From spacetime dynamics to Hopf algebras

Abstract.

Causal set theory is an approach to quantum gravity in which spacetime is fundamentally discrete at the Planck scale. In this theory, spacetime takes the form of a locally finite partial order, or "causal set". Understanding how macroscopic phenomena emerge from this microscopic discreteness is an important open problem that is being addressed in variety of ways in the research community. In this talk, I will focus on the Classical Sequential Growth models - models of stochastic posets which give rise to some desirable macroscopic properties - and on their connection with Hopf algebras (based on recent work with Karen Yeats).

Analysis

Fri, 07.02.25 at 10:00

Felix Medwed University of Potsdam

Carnot groups III

Abstract.

This is the last talk of a series of talks on Carnot groups.

Analysis

Fri, 17.01.25 at 11:00

Lena Janshen Georg-August-Universität Göttingen

Asymptotic symmetries in classical gauge theory - bridging between the Hamiltonian and the BV-BFV perspective

Abstract.

Some physical systems are described by a mathematical model that has a redundancy. Such a model is called a gauge theory, where redundancies are described by gauge transformations. When the underlying spacetime for a gauge field theory has a boundary or is non-compact, and the fields satisfy fall-off conditions, some gauge transformations acquire physical significance as asymptotic symmetries. In this talk, I will unravel the mathematical reason for the appearance of asymptotic symmetries. This reason appears naturally in the Hamiltonian formulation of gauge field theory – a setting ideally suited for gauge theories on globally hyperbolic spacetimes. In the first part of my talk I will focus on this approach. The second part of my talk will be dedicated to a modern framework for gauge theories – well suited for quantization – namely the BV-BFV formalism. I will present to you my work on translating the Hamiltonian understanding of asymptotic symmetries into the BV-BFV framework in the case of electromagnetism, and then compare it with already existing notions in the BV-BFV world. I’m passionate about bridging different communities in mathematical physics to bring clarity to the concept of asymptotic symmetries, independent of the framework. I hope I can share some of this clarity with you in this talk.

Analysis

Fri, 17.01.25 at 10:00

Felix Medwed University of Potsdam

The geometry of rough paths 2 - A dive into Carnot groups, Part 2

Abstract.

This second talk will serve as supplementary material and will provide more details on the interesting concepts and background material presented in the first talk, like metric spaces, sub-Finsler geometry, Carnot-Carathéodory spaces, nilpotent Lie groups, more properties and details of Carnot groups or others.

Analysis

Fri, 10.01.25

Christiane Klein University of York

Exploring quantum fields on rotating black holes

Abstract.

Rotating black holes, which are the kind of black holes we observe in our universe, are an excellent stage on which to explore the effects of quantum physics. One possible approach to do so is the algebraic framework of quantum field theory. In this talk, I present recent advances on the mathematical study of quantum fields on rotating black holes. After an introduction of the spacetime and quantum theory under consideration, I will discuss the construction of a physically motivated state, the Unruh state, and the proof of its Hadamard property. I will also demonstrate how a constraint on the black-hole angular momentum in this proof can be removed by a new geometric result. I will also discuss the leading divergence of the stress-energy tensor of the quantum field at the inner horizon of the black hole and its state-independence.

Analysis

Fri, 06.12.24 at 13:30

Felix Medwed University of Potsdam

The geometry of rough paths

Abstract.

Following Chapter 8 of the book "An introduction to infinite dimensional geometry" by Alexander Schmeding (2022), I will discuss the (infinite-dimensional) geometric framework for rough paths and their signature.

Analysis

Fri, 06.12.24 at 11:00

Chenchang Zhu Georg-August-Universität Göttingen

Incomplete category of fibrant objects for higher derived shifted symplectic groupoids

Abstract.

In this talk, based on many previous works, we will introduce a helpful new tool for differential geometers using: higher: to deal with quotient singularities, derived: to heal transversality problems, shifted: for more flexible symplectic situations. At the same time as being as complete as possible, we also make the framework as explicit as possible using groupoid models. We build an incomplete category of fibrant objects (iCFO) for higher derived Lie groupoids. In the end, we will explain in some concrete examples how this theory is used. This is based on a joint work in progress with Miquel Cueca Ten, Florian Dorsch and Reyer Sjamaar.

Analysis

Fri, 06.12.24 at 10:00

Emiel Claasen MPI, Potsdam

From modular graph forms to iterated integrals (Part 3)

Abstract.

Modular Graph Forms (MGFs) are a class of modular forms represented by lattice sums associated to directed simple graphs. They originated from the calculation of graviton amplitudes in type II string theory. MGFs have remarkable mathematical properties such as an intricate network of algebraic and differential relations or the appearance of (conjecturally single-valued) multiple zeta values in their Fourier expansion. In particular, they are conjectured to arise as expansion coefficients of certain generating series dubbed equivariant iterated Eisenstein integrals.

Analysis

Fri, 22.11.24 at 11:00

online

Yunnan Li Guangzhou University

Matched pairs, post-Hopf algebras and the quantum Yang-Baxter equation

Abstract.

Recently, Ferri and Sciandra introduced two equivalent notions, matched pair of actions on a Hopf algebra and Yetter-Drinfeld brace. Any of these objects actually provides a solution of the quantum Yang-Baxter equation, generalizing the construction of Yang-Baxter operators by Lu, Yan and Zhu from braiding operator on a group, and also by Angiono, Galindo and Vendramin from a cocommutative Hopf brace. Later, Sciandra ingeniously proposed one more equivalent notion, namely Yetter-Drinfeld post-Hopf algebra, as a non-cocommutative generalization of post-Hopf algebra formerly introduced by Sheng, Tang and me, and most remarkably it provides a sub-adjacent structure as cocommutative post-Hopf algebra does. In this talk, I intend to review these works first, and then discuss some related problems.

Analysis

Fri, 15.11.24 at 13:00

Emiel Claasen MPI, Potsdam

From modular graph forms to iterated integrals (Part 2)

Abstract.

Modular Graph Forms (MGFs) are a class of modular forms represented by lattice sums associated to directed simple graphs. They originated from the calculation of graviton amplitudes in type II string theory. MGFs have remarkable mathematical properties such as an intricate network of algebraic and differential relations or the appearance of (conjecturally single-valued) multiple zeta values in their Fourier expansion. In particular, they are conjectured to arise as expansion coefficients of certain generating series dubbed equivariant iterated Eisenstein integrals.

Analysis

Fri, 15.11.24 at 11:15

online

Martin Peev Imperial College

Renormalising Non-Commutative Singular PDEs

Abstract.

When attempting to construct QFTs that include Fermions using the methods of Stochastic Quantisation, one is naturally forced to consider noncommutative stochastic PDEs. I shall show how to formulate SPDEs driven by noncommutative noises in terms of algebra-valued singular PDEs. Furthermore, I will describe how one can renormalise the singular products appearing in such equations for a set of algebras – including free probability – interpolating between Fermions and Bosons by appropriately modifying their topologies. This talk will be based on joint work with Ajay Chandra and Martin Hairer.

Analysis

Fri, 08.11.24 at 14:00

Sumati Surya Raman Research Institute, Bengalore

Order and Number: the causal set approach to quantum gravity

Abstract.

In this talk I will sketch in broad lines a proposal for quantum space time, which is built purely on discrete, order theoretic principles. A key motivation is the Lorentzian character of spacetime, where the causal structure is a partially ordered set. The talk will be a basic introduction, with the aim of introducing the mathematicians in the audience to the fascinating world of discrete Lorentzian geometry.

Analysis

Fri, 08.11.24

online

Lucas Broux MPIMS, Leipzig

Malliavin calculus in regularity structures

Abstract.

This talk will be concerned with some aspects of the renormalization of the $\Phi^4$ stochastic partial differential equation, in the singular but subcritical (also called super-renormalizable) range. I will first try to describe how what in regularity structures is called a "model", here indexed by multi-indices, naturally arises from considering the "geometry" of the solution manifold. The notion of model is central in regularity structures and one of the crucial tasks is to robustly estimate it. I would then like to give some insights into the proof of the estimates, where the use of a spectral gap assumption plays an important role (based on joint work with Felix Otto and Markus Tempelmayr).

Analysis

Wed, 30.10.24

Emiel Claasen MPI, Potsdam

From modular graph forms to iterated integrals (Part 1)

Abstract.

Modular Graph Forms (MGFs) are a class of modular forms represented by lattice sums associated to directed simple graphs. They originated from the calculation of graviton amplitudes in type II string theory. MGFs have remarkable mathematical properties such as an intricate network of algebraic and differential relations or the appearance of (conjecturally single-valued) multiple zeta values in their Fourier expansion. In particular, they are conjectured to arise as expansion coefficients of certain generating series dubbed equivariant iterated Eisenstein integrals. In this first of two talks, I will introduce the MGFs, talk about their appearance in string theory, and set the stage for their systematic conversion into their iterated integral representations.

Analysis

Fri, 25.10.24 at 14:00

Johannes Blümlein DESY

Analytic Integration Methods in Quantum Field Theory

Abstract.

A survey is given on the present status of analytic calculation methods and the mathematical structures of zero- and single scale Feynman amplitudes which emerge in higher order perturbative calculations in the Standard Model of elementary particles, its extensions and associated model field theories, including effective field theories of different kind. The methods include symbolic summation methods, symbolic solutions of differential equations, different classes of higher transcendental functions and special numbers. A main characteristics of the objects studied is their representation as iterative integrals, including higher transcendental letters in the associated alphabets.

Analysis

Fri, 25.10.24 at 11:00

Luisa Herrmann University of Potsdam

Euler's polyhedron formula applied to Kepler-Poinsot-polyhedra

Abstract.

If one interprets the Kepler-Poinsot-polyhedra as the union of several objects and applies Euler's polyhedron formula on their number of vertices, edges and faces, than the result is two. This is also the case for normal connected polyhedra without holes. In my presentation I will explain how to interpret the Kepler-Poinsot-polyhedra as the union of several objects and how to count them. For that purpose, it is helpful to understand what a polygon, whose number of vertices is rational but not an integer could be. This makes it possible to apply Euler's polyhedron formula on the star polyhedra. Even though single polygons, whose number of vertices is not an integer, are not definable, one can count them as faces on two of the four star polyhedra. Therewith one gets the value two by applying Euler's polyhedron formula.

Analysis

Fri, 18.10.24

Elke Rosenberg University of Potsdam

Weyl asymptotics for discrete pseudo-differential operators

Abstract.

For a class of elliptic self-adjoint pseudo-differential operators in a discrete semi-classical setting we give asymptotic estimates for the number of eigenvalues in a fixed interval. Here we assume the related symbols to be periodic with respect to momentum. The associated operator acts on functions on a lattice scaled by a semi-classical parameter. Under the assumption that the boundaries of the spectral Interval are non-critical values of the principal symbol, we show that the exact leading order term for the number of eigenvalues is given by the phase space volume of the pre-image of the interval under the principal symbol.

Analysis

Fri, 21.06.24

Alexander Schmeding University of Trondheim

When the Lie group exponential is bad

Abstract.

From finite dimensional Lie theory, it is well known that the Lie group exponential yields a local diffeomorphism from the Lie algebra onto a unit neighborhood of the Lie group. These exponential coordinates are useful tools to understand the interplay between Lie algebra and Lie group. It is well known that this correspondence breaks down in infinite-dimensional Lie theory. Beyond Banach spaces, the Lie group exponential is in general not a local diffeomorphism. For these non-locally exponential Lie groups the Lie group exponential becomes much more restricted. In this talk we will revisit some of the classical examples of the breakdown of the exponential (e.g. the diffeomorphism group). It turns out that in many known examples this defect can be traced to properties of flows of vector fields. Time permitting, we will show some new results for non-local exponentiality of semidirect products of Lie groups. This is joint work with R. Dahmen and K.-H. Neeb.

Analysis

Fri, 14.06.24

Leonid Ryvkin University of Lyon

Universal central extensions of the Lie algebra of volume-preserving vector fields

Abstract.

The goal of the talk is proving a conjecture of Claude Roger about the universal central extension of the Lie algebra of volume-preserving vector fields. In the beginning we will briefly review the notion of central extensions of Lie algebras and their link to Chevalley-Eilenberg-cohomology. We will then proceed to Rogers conjecture, which lies in the (continuous) infinite-dimensional setting. To solve it we will need a combination of analytical and geometrical methods, and maybe even a bit of representation theory. Based on an ongoing collaboration with Bas Janssens and Cornelia Vizman.

Analysis

Tue, 28.05.24

Rosa Marchesini University of Göttingen

On the homotopy invariance of Lie algebroid cohomology

Abstract.

Several extensively studied cohomology theories play an important role in differential geometry and mathematical physics. Examples are De Rham cohomology, Chevalley-Eilenberg cohomology, BRST cohomology, and then Poisson, Foliated, and Principal de Rham cohomologies. All of these can be elegantly formalized at once using special smooth vector bundles called Lie algebroids. Lie algebroids also arise naturally as an infinitesimal description of Lie groupoids and are therefore a central subject of study in higher Lie theory. After an accessible introduction to Lie algebroids and their cohomology, we propose a notion of homotopy for Lie algebroids and justify it with examples and applications. This is a joint work with my supervisor Madeleine Jotz. We mention some related research projects currently in progress with Ryszard Nest.

Analysis

Fri, 24.05.24

Stefano Ronchi University of Göttingen

Cotangent spaces for higher Lie groupoids and applications

Abstract.

In the same way Lie groupoids can encode local symmetries of a manifold, Lie 2-groupoids are a way to encode higher symmetries. After a short crash course on this topic, we present the construction of a cotangent space for Lie 2-groupoids analogous to the well known one for Lie groupoids. This turns out to have a canonical shifted symplectic structure (that is, symplectic up to homotopy) in the same way the cotangent groupoid is canonically a symplectic groupoid, and the tangent bundle of a manifold is canonically symplectic. This makes our cotangent space a good global model for a class of symplectic Q-manifolds that appear in some TQFTs. We will then discuss various applications, including a definition of hamiltonian actions of Lie 2-groupoids. This talk is based on joint work in progress with Miquel Cueca and Chenchang Zhu.

Analysis

Fri, 17.05.24

Fabrizio Zanello University of Potsdam

A leisurely stroll through the multisymplectic approach to Lagrangian field theories

Abstract.

The mathematical toolbox of multysimplectic geometry was introduced in order to generalize the symplectic formulation of classical mechanics to Lagrangian field theories. After a substantial body of foundational works for first order field theories, though, the program came to a standstill due to technical difficulties. In this talk we give an overview of the motivations, techniques and limitations of the traditional results and present some of the core ideas of a new approach, recently proposed by Blohmann, to overcome those technical difficulties. The talk is based on an ongoing joint project with Antonio Miti.

Analysis

Fri, 10.05.24

Luisa Herrmann University of Potsdam

On a problem of optimal transport under marginal martingale constraints

Abstract.

Based on an article by Mathias Beigelböck and Nicolas Juillet, I will first describe the martingale version of the optimal transport problem after which I will define the convex order and give some examples. I will then discuss the central question as to whether the set of all martingale transport plans is nonempty and present a constructive proof of the existence under certain assumptions.

Analysis

Fri, 03.05.24

Alejandro Peñuela Diaz University of Potsdam

Construction of marginally outer trapped surfaces

Abstract.

In physics, marginally outer trapped surfaces (MOTS) can be understood as quasi-local versions of black hole boundaries. Therefore, these surfaces are crucial to understand black holes and are widely used in black hole simulations. From a mathematical point of view, MOTS are prescribed mean curvature surfaces, that is a generalization of a minimal surface. I will talk about marginally outer trapped surfaces, introduce some of their properties, and explain the challenges when trying to construct them analytically.

Analysis

Fri, 26.04.24

Alberto Bonicelli University of Pavia

Convergence results in the stochastic sine-Gordon model: an algebraic viewpoint

Abstract.

The importance of the sine-Gordon model in 1+1 spacetime dimensions resides in the integrability of the field theory that it describes. A recent result showed how, within the setting of algebraic quantum field theory, this property translates into a convergence result for both the formal series associated to the S-matrix and to the interacting field of the quantum field theory. After introducing an algebraic approach to the perturbative study of singular stochastic PDEs, I will show how an adaptation of the aforementioned convergence results yields convergence of the momenta of the solution to a stochastic version of the sine-Gordon equation. Interestingly enough, our two-step procedure passes through the quantum theory and recollects the stochastic information via the classical limit.

Analysis

Fri, 19.04.24

Leonard Schmitz MPI Leipzig

Free generators and Hoffman's isomorphism for the two-parameter shuffle algebra

Abstract.

Signature transforms have recently been extended to data indexed by two and more parameters. With free Lyndon generators, ideas from B∞-algebras and a novel two-parameter Hoffman exponential, we provide three classes of isomorphisms between the underlying two-parameter shuffle and quasi-shuffle algebras. In particular, we provide a Hopf algebraic connection to the (classical, one-parameter) shuffle algebra over the extended alphabet of connected matrix compositions. This is joint work with Nikolas Tapia.

Analysis

Fri, 12.04.24

Alexander Schmeding NTNU, Trondheim

On manifolds of Lie group valued continuous BV-functions

Abstract.

Functions of bounded variation (BV) with values in a Banach space are a classical topic of analysis with specific applications for example in rough path theory. In the theory of rough paths one considers routinely even BV-functions with values in non-linear spaces such as manifolds and (finite and infinite-dimensional) Lie groups. In this talk we will explain how the well known construction of manifolds of mappings carries over to the world of BV-functions. As a consequence we are able to generalise the construction of current groups to the BV-setting. This also strengthens known regularity properties a la Milnor for Banach Lie groups. Joint work with H. Glöckner and A. Suri (Paderborn).

Analysis

Fri, 09.02.24

Paolo Meda University of Pavia

The Semiclassical Einstein Equations and the Stability of Linearized Solutions

Abstract.

The semiclassical formulation of gravity is discussed in the framework of algebraic quantum field theory in curved spacetimes. The main topic of the talk is the Semiclassical Einstein Equations, which describe the backreaction of a quantum scalar field on spacetime geometry. In the first part of the talk, it is shown that an initial-value problem for local solutions of the Semiclassical Einstein Equations can be formulated in cosmological spacetimes, after fixing four initial data on the scale factor of the Universe. In the second part of the talk, it is studied the problem of stability of linearized solutions, using a toy model which mimics the semiclassical equations in cosmological spacetimes. In this case, it is proved that, if the quantum field driving the backreaction is massive, then there are choices of renormalization constants for which linear perturbations with compact spatial support decay for large times, thus indicating stability of the underlying theory [arXiv:2007.14665, 2201.10288]. The content of the talk is based on two papers I wrote during my PhD. I will try to provide a lay introduction to the topic (Semiclassical Einstein Equations + algebraic methods) before discussing the results, to make the topic as accessible to a broad audience as possible.

Analysis

Fri, 02.02.24

David Kern University of Göttingen

Constraint Geometry and Reduction of Dirac Geometry

Abstract.

In this talk we first motivate constraint geometry by recalling Poisson geometry. Afterwards we introduce constraint manifolds as well as constraint vector bundles and prove a constraint Cartan calculus. Finally, we apply it to Dirac manifolds showing their reduction.

Analysis

Fri, 26.01.24

Dominique Manchon Clermont-Ferrand

The free tracial pre-Lie-Rinehart algebra

Abstract.

Tracial Lie-Rinehart algebras are a purely algebraic version of finite-dimensional Lie algebroids, for which the trace is given fibrewise. A tracial Lie-Rinehart algebra is pre-Lie if moreover both torsion and curvature vanish. The simplest example is given by the tangent bundle the smooth manifold R^d, the trace being given by the divergence of vector fields. We shall describe in this talk the free tracial pre-Lie-Rinehart algebra in terms of aromatic rooted trees. Joint work with Gunnar Fløystad and Hans Z. Munthe-Kaas.

Analysis

Fri, 01.12.23

Jean-David Jacques University of Potsdam

Post-Lie algebra of derivations and regularity structures

Abstract.

Post-Lie algebra structures are a generalization of Pre-Lie algebras. They have their roots in geometry and correspond to the algebraic properties satisfied by the covariant derivative in the case of a flat and constant torsion connection. In my talk, I will provide a brief overview of the new theory of regularity structures developed by F. Otto and colleagues, and discuss how post-Lie algebra structures arise in this framework.

Analysis

Fri, 24.11.23

Fabrizio Zanello University of Potsdam

Higher currents for the sine-Gordon model in perturbative Algebraic Quantum Field Theory

Abstract.

First, we review the 2-dimensional sine-Gordon model in Classical Field Theory and derive recursive formulas for the components of an infinite number of conserved currents. Then, we describe the formulation of the sine-Gordon model in perturbative Algebraic Quantum Field Theory and show, in the framework of Epstein-Glaser renormalization, that the interacting components of the currents are super-renormalizable by power counting. Finally, we describe how, under suitable conditions, the formal power series describing the interacting components of the currents are in fact summable.

Analysis

Fri, 17.11.23

Umida Baltaeva Uzbekistan Academy of Sciences

Boundary value problems for loaded partial differential equations with the classical and nonclassical operators

Abstract.

I will introduce loaded equations and their applications to applied problems. Next, I will discuss the investigation of the boundary value problem for the linear loaded partial differential equations and their relation with nonlocal boundary value problems for classical partial differential equations. Further, I investigate the Cauchy-type problem for a loaded parabolic-hyperbolic type equation with Riemann-Liouville fractional differential operator.

Analysis

Fri, 27.10.23

Pablo Linares Imperial College, London

Rough paths based on multi-indices and their algebraic renormalization

Abstract.

Following the approach of Otto et. al., we derive a notion of rough path based on multi-indices suitable for real-valued rough differential equations, and build its associated algebraic renormalization group. The construction is based on well-known results on pre-Lie algebras (Guin-Oudom construction), applied to pre-Lie products that arise from shifting the affine spaces of solutions and equations. We show that these products correspond in the tree-based language to grafting and insertion, respectively.

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