Uppsala University: Department of Mathematics
Analysis, PDEs and Applications
Stockholm Mathematics Centre (SMC)
Archipelagic perspectives on mathematics, physics
and perceptible spectra of reality
The aim of the workshop is to bring together mathematicians and physicists and to exchange interests and experience across fields.
Monday 30 August - Friday 3 September, 2021
Djurönäset Conference Hotel, Djurö, Stockholm archipelago
Douglas Lundholm (Uppsala University) organizer
Erik Lindgren (Uppsala University) advisory
Funding generously provided by
Göran Gustafsson Foundation
John Andersson (KTH Stockholm)
Benny Avelin (Uppsala)
Luca Fresta (Bonn)
Gerald Goldin (Rutgers)
Jorge Larana-Aragon (Stockholm)
Simon Larson (Caltech / Chalmers)
Per Moosavi (ETH Zurich)
Viktor Qvarfordt (Stockholm)
Julien Ricaud (Paris)
Lukas Rødland (Stockholm)
Lars Svensson (KTH Stockholm)
Oscar Wandery (Stockholm)
The following is a preliminary schedule for the workshop:
Speaker: John Andersson (KTH Royal Institute of Technology)
Title: Some Heuristics For Mathematical Crack Growth
Speaker: Benny Avelin (Uppsala University)
Title: Approximation of BV functions using neural networks
In this talk, I will focus on a recent result together with Vesa Julin, concerning the approximation of functions of Bounded Variation (BV) using special neural networks on the unit circle. I will present the motivation for studying these special networks, their properties, and hopefully some proofs. Specifically the results we will cover: the closure of the class of neural networks in $L^2$, a uniform approximation result, and a localization result.
Speaker: Luca Fresta (Bonn University)
Title: 1+1 D Quantum Field Theories of Anyons
In the talk I will explain how to construct relativistic anyons in 1+1 dimensions by means of the so-called boson-anyon correspondence. In particular, I will describe two interacting QFTs of anyons that generalize the well-known fermionic models named after Luttinger and Schwinger respectively.
As an application, I will discuss the presence of off-diagonal long-range order (ODLRO) by studying the two-point correlation functions and show that the anyonic statistics allows one to get arbitrarily close to ODLRO in the anyonic Luttinger model, but that this possibility is destroyed by the gauge coupling in the anyonic Schwinger model.
This is joint work with P. Moosavi.
Speaker: Gerald A. Goldin (Rutgers University)
Title: First predictions of the "anyon": History and some wider perspectives
The quantum particles or excitations called "anyons", in two space dimensions, may satisfy exchange statistics intermediate to bosons and fermions. Anyons are associated with surface phenomena in the presence of magnetic flux. Theoretical applications range from explaining the quantum Hall effect to the development of quantum computing. Prediction of the "anyon" is often attributed exclusively, and incorrectly, to Frank Wilczek. In fact, three independent predictions, each from a different theoretical perspective, were published by physicists: Jon Magne Leinaas and Jan Myrheim (1977); Gerald Goldin, Ralph Menikoff, and David Sharp (1980, 1981); and subsequently Wilczek (1982). In 2020, experimentalists successfully created such excitations, confirming the early predictions. In this talk I shall review some ideas leading to those predictions that involved fundamental changes in understanding. Then I would like to discuss some wider implications for physics, its teaching, and its presentation to the public. How do we as physicists reach such breakthroughs? Why did the possibility of intermediate statistics elude us for over half a century after bosons and fermions were understood, when the concept now seems so easy? What then led to the independent predictions within a relatively short time of each other? What can we learn from this about the teaching of mathematics and physics? Might the obstacles to securing correct attribution reflect a systemic problem of integrity in science, as many have come to believe - a problem differentially affecting women, racial minorities, and young scientists in developing countries?
Speaker: Jorge Larana-Aragon (Stockholm University)
Title: The good, the bad and the ugly: quantum tunnelling, black holes and Non-Hermitian systems
Black holes are among the most exotic objects in the Universe. In contrast to the classical intuition, they emit thermal radiation at a characteristic temperature upon evaporating. Even though black hole physics has provided key insights into the fundamental nature of spacetime, direct experimental input is missing. However, quantum gravity can be investigated in laboratory setups through analogue models that mimic black hole properties. Among these, Hermitian systems have concentrated most of the attention, despite the advent of non-Hermitian topological phases. The dissipative nature of such systems, where particle numbers and energies are no longer conserved quantities, makes them ideal candidates to simulate black hole radiation. In this talk I will present some recent work where we construct such an analogue model described by a non-hermitian Weyl-like two-band Hamiltonian. In particular, our analogy relates the cone shaped energy degeneracies of the non-Hermitian dissipative system to an observer’s light cone in the vicinity of a Schwarzschild black hole. This relation allows one to identify the black hole radiation with the gain and loss characteristic of the respective analogue system. We explicitly show the presence of Hawking radiation arising from quantum tunnelling processes of quasi-particles in the analogue model. The proposed setup is subject to parity-time symmetry, a feature ubiquitous in optical setups assuring invariance under the combined action of parity and time reversal, which emphasises their high experimental relevance.
Speaker: Simon Larson (Caltech / Chalmers)
Title: On the spectrum of the Kronig-Penney model in a constant electric field
I will discuss the nature of the spectrum of the one-dimensional Schrödinger operators
– F x +
with F > 0 and two different choices of the coupling constants gn. In the first model gn ≡ λ and we prove that if F ∈ π2 ℚ the spectrum is absolutely continuous away from a discrete set of points. In the second model gn are independent random variables with mean zero and variance λ2. Under weak assumptions on the distribution of the gn we prove that in this setting the spectrum is almost surely pure point if F / λ2 < 1/2 and purely singular continuous if F / λ2 > 1/2. Based on joint work with Rupert Frank.
Speaker: Erik Lindgren (Uppsala University)
Title: ∞-Ground states in the plane
In this talk, I will discuss ∞-Ground states. In particular, I will discuss
∞-Ground states in two dimensional convex domains and report on some recent
results obtained in collaboration with Peter Lindqvist. In a polygon, the points where an ∞-Ground state does not satisfy the ∞-Laplace equation are characterized: they are restricted to lie on specific curves, which are acting as attracting (fictitious) streamlines.
Speaker: Douglas Lundholm (Uppsala University)
I will try to give a mini-intro to some of the concepts discussed at the workshop, so that we might start on (almost) the same page.
Speaker: Per Moosavi (ETH Zurich)
Title: Floquet drives, inhomogeneous CFT, and diffeomorphism representations
Conformal field theory (CFT) is routinely used as low-temperature effective description of quantum many-body systems in equilibrium. Recently, CFT has been used to study such 1+1-dimensional systems also out of equilibrium, and more recently even when there are smooth inhomogeneities, such as gapless spin chains with uniformly and smoothly spatially-varying couplings. The resulting effective description, e.g., of these spin chains, is in the form of inhomogeneous CFT where the velocity is given by a smooth position-dependent function. In this talk, I will present an approach to study such CFTs by exact analytical means based on projective unitary representations of diffeomorphisms. In particular, I will show that this can be used to construct a geometric approach to driven inhomogeneous CFT that establishes a correspondence with classical dynamical systems on the circle. The latter generalizes previous results for a small subfamily of similar systems that used only the finite-dimensional sl(2) algebra to general smooth inhomogeneities that require the full infinite-dimensional Virasoro algebra.
Speaker: Viktor Qvarfordt (Stockholm Mathematics Centre / Sana Labs)
Title: Language models and their perceptible spectra of reality
Speaker: Julien Ricaud (Ecole Polytechnique, Paris)
Title: Spectral Stability in the nonlinear Dirac equation with Soler-type nonlinearity
Speaker: Lukas Rødland (Stockholm University)
Title: What can we learn by classifying non-hermitian topological phases?
I will review some of the basics about classifying symmetry protected topological phases, and talk in details about how this can be generalised to non-hermitian systems. I will focus mostly on how a hermitian system can be mapped to a non-hermitian system in the same symmetry class. In the end I will discuss some potential problems with this classification scheme, which some people have tried to solve with a more homotopical approach.
Speaker: Lars Svensson (KTH Royal Institute of Technology)
Title: A connection between formal systems and algebraic geometry
A relation between formal systems in logic and ideals in polynomial rings in infinite variables is established where consistency corresponds to properness, models to zeroes and completeness to irreducibility. A generalization of the Hilbert Nullstellensatz to infinite number of variables is used to prove the Gödel completeness theorem and some results on infinite graphs.
Speaker: Oscar Wandery (Stockholm University)
Title: The disappearance and reappearance of death-conscious medicine in the west: a game-theory interpretation
The aim of my talk will be to get feedback on a thesis I have for an article I'm writing. At one level (that of high abstraction) it is about ways that working in an institution change how we think, percieve, and act in the world. At another level (that of specific cases) it is about the history of palliative medicine over the past 500 years. I explore how palliative medicine looks today, how it looked in early modern times, and what happened in the 19th century when hosptials became the main site of employment for doctors. I argue that when their livelyhood become dependent on "work organizations" (i.e. they got paid for their labor-power rather than the service they provided on the market) it changed how they looked at and practiced medicine. I suspect that we may be able to understand some of that change with the help of game-theory and ecological-institutional-theory. Although, I have not yet worked out all quirks of this argument, and here is where this talk comes in. I hope to get valueable feedback on this thesis: Does it seem plausible? What can I do to substantiate it? How may I pursue this mathematically? The article itself is in a very early and preliminary state, so any input will be strongly appriciated, as it is likely to influence whether or not this article gets completed, and if so how it looks in the end.
Current info on entry into Sweden: here, here and here
Rail network Stockholm
Bus from Slussen to Djurönäset (after Djuröbron), from Djurönäset to Slussen
(also good to just search Slussen - Djurönäset on SL.SE)