11.01.2025, 11:00
– Campus Golm, Building 9, Room 2.22 and via Zoom
Arbeitsgruppenseminar Analysis
Exploring quantum fields on rotating black holes
Christiane Klein (York University, UK)
October 18th 2024 - 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.
October 25th 2024 (11 AM) - 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.
October 25th 2024 (2 PM) - 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.
October 30th 2024 - 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.
November 8th 2024 - Lucas Broux (MPIMS, Leipzig), "Malliavin calculus in regularity structures" (online).
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).
November 8th 2024 (2 PM) - 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.
November 15th 2024 (11:15 AM) - Martin Peev (Imperial College), "Renormalising Non-Commutative Singular PDEs" (online).
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.
November 15th 2024 (1 PM) - 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. 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.
November 22nd 2024 (11 AM) - Yunnan Li (Guangzhou University), "Matched pairs, post-Hopf algebras and the quantum Yang-Baxter equation" (online).
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.
December 6th 2024 (10 AM) - 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.
December 6th 2024 (11 AM) - 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:
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.
December 6th 2024 (1:30 PM) - 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, I will discuss the (infinite-dimensional) geometric framework for rough paths and their signature.
January 10th 2025 - 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.
January 17th 2025 - Lena Janshen (Georg-August-Universität Göttingen), "TBA".
Abstract: TBA.
January 31st 2025 - Tobias Diez (Shanghai Jiao Tong University), "TBA".
Abstract: TBA.