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https://pure.au.dk/portal/en/persons/jan-frahm(cef7c90d-5b0f-46be-afef-93fd42b6b52b)/projects.html
RSS FeedSun, 24 Sep 2023 22:44:32 GMT2023-09-24T22:44:32ZSpectral Correspondences for Trees and Buildings
https://pure.au.dk/portal/en/projects/spectral-correspondences-for-trees-and-buildings(aaadb9a5-7f0d-4915-b1ef-f92e15183694).html
<div style='font-size: 9px;'><div class="rendering rendering_upmproject rendering_short rendering_upmproject_short"><h2 class="title"><a rel="UPMProject" href="https://pure.au.dk/portal/en/projects/spectral-correspondences-for-trees-and-buildings(aaadb9a5-7f0d-4915-b1ef-f92e15183694).html" class="link"><span>Spectral Correspondences for Trees and Buildings</span></a></h2><p><a rel="Person" href="https://pure.au.dk/portal/en/persons/jan-frahm(cef7c90d-5b0f-46be-afef-93fd42b6b52b).html" class="link person"><span>Frahm, J.</span></a> & <a rel="Person" href="https://pure.au.dk/portal/en/persons/christian-arends(5d973059-6424-4c92-92c3-b18d37d81d86).html" class="link person"><span>Arends, C.</span></a></p><p class="period"><span class="date">01/04/2023</span> → <span class="date">31/03/2026</span></p><p class="type"><span class="type_family">Project<span class="type_family_sep">: </span></span><span class="type_classification">Research</span></p></div><div class="rendering rendering_upmproject rendering_detailsportal rendering_upmproject_detailsportal"><div class="projectdescription"><h3 class="subheader">Description</h3><div class="textblock description">In theoretical physics, one often compares classical mechanics with quantum mechanics. While the laws of classical mechanics can be used to describe the movement of macroscopic particles in terms of position and momentum, quantum mechanics provides a model for the behavior of microscopic particles where only the probability of observable quantities is described.<br />While these models are conceptually very different, they can be put into a common mathematical framework which allows to compare quantities from both sides. This framework is based on the geometry of the surrounding space on which the particles live, and the mathematical model for this space is a Riemannian manifold M, generalizing the notion of curves and surfaces to higher dimensions. To every Riemannian manifold M, one can associate a set of classical resonances and a set of quantum resonances, both sets describing the long time behavior of the corresponding physical system. A spectral correspondence is an explicit relationship between classical and quantum resonances.<br />For general Riemannian manifolds M, spectral correspondences are difficult to establish, so one often focuses on manifolds M which are particularly symmetric, also called (locally) symmetric spaces. For locally symmetric spaces of rank one, surprisingly explicit spectral correspondences were recently obtained and are considered a major breakthrough in the field. It seems natural to try to extend these results to locally symmetric spaces of higher rank. However, higher rank spaces are significantly more complicated, and there are both conceptual, geometric and analytical difficulties to overcome.<br />The key idea of this project is to study spectral correspondences in a different geometric framework that is analytically simpler than the one of Riemannian manifolds, but still has analogous geometric features. More precisely, we propose to consider classical and quantum resonances for affine buildings. In contrast to manifolds which require tools from geometry and analysis, affine buildings are defined and studied in terms of algebra and combinatorics and are therefore easier to handle. The<br />simplest affine buildings are trees, i.e. collections of points connected with edges. In this setting, we expect to overcome the difficulties described above, first for trees and then for general affine buildings, and establish spectral correspondences in this framework. We expect that many ideas and techniques can be transferred from buildings to manifolds and vice versa. Moreover, affine buildings are themselves interesting mathematical objects, allowing applications to geometric group theory and number theory, and we further hope to establish new connections to these topics with our results.</div></div><table class="properties"><tbody><tr class="status"><th scope="row">Status</th><td>Active</td></tr><tr class="effective-startend-date"><th scope="row">Effective start/end date</th><td><span class="date">01/04/2023</span> → <span class="date">31/03/2026</span></td></tr></tbody></table></div></div>Sun, 24 Sep 2023 22:40:41 GMThttps://pure.au.dk/portal/en/projects/spectral-correspondences-for-trees-and-buildings(aaadb9a5-7f0d-4915-b1ef-f92e15183694).html2023-09-24T22:40:41ZSymmetry Breaking in Mathematics
https://pure.au.dk/portal/en/projects/symmetry-breaking-in-mathematics(c5ee2c4f-70f6-4236-92b6-9bc4b25745b8).html
<div style='font-size: 9px;'><div class="rendering rendering_upmproject rendering_short rendering_upmproject_short"><h2 class="title"><a rel="UPMProject" href="https://pure.au.dk/portal/en/projects/symmetry-breaking-in-mathematics(c5ee2c4f-70f6-4236-92b6-9bc4b25745b8).html" class="link"><span>Symmetry Breaking in Mathematics</span></a></h2><p><a rel="Person" href="https://pure.au.dk/portal/en/persons/jan-frahm(cef7c90d-5b0f-46be-afef-93fd42b6b52b).html" class="link person"><span>Frahm, J.</span></a>, Weiske, C., <a rel="Person" href="https://pure.au.dk/portal/en/persons/jonathan-ditlevsen(991e3226-ecf8-45d5-b851-bbcc07bcbed3).html" class="link person"><span>Ditlevsen, J.</span></a>, Spilioti, P., <a rel="Person" href="https://pure.au.dk/portal/en/persons/frederik-juul-bangjensen(a11a9d4b-ef98-40bd-8433-fcef6a82660d).html" class="link person"><span>Bang-Jensen, F. J.</span></a> & <a rel="Person" href="https://pure.au.dk/portal/en/persons/quentin-labriet(53b25d69-94c9-4da5-9f04-9e9b87a0c4d9).html" class="link person"><span>Labriet, Q.</span></a></p><p class="period"><span class="date">01/08/2019</span> → <span class="date">31/07/2024</span></p><p class="type"><span class="type_family">Project<span class="type_family_sep">: </span></span><span class="type_classification">Research</span></p></div><div class="rendering rendering_upmproject rendering_detailsportal rendering_upmproject_detailsportal"><div class="projectdescription"><h3 class="subheader">Description</h3><div class="textblock description">"Symmetry breaking" describes the process of passing from a state of high symmetry to one with less symmetry, and it plays an important role in modern theoretical physics. This research project aims at a mathematical description and study of symmetry breaking in terms of representations of reductive Lie groups, with possible applications to questions in partial differential equations and analytic number theory. The grant will fund the recepient as well as one postdoc and two PhD students.</div></div><table class="properties"><tbody><tr class="status"><th scope="row">Status</th><td>Active</td></tr><tr class="effective-startend-date"><th scope="row">Effective start/end date</th><td><span class="date">01/08/2019</span> → <span class="date">31/07/2024</span></td></tr></tbody></table></div></div>Mon, 27 Apr 2020 11:46:47 GMThttps://pure.au.dk/portal/en/projects/symmetry-breaking-in-mathematics(c5ee2c4f-70f6-4236-92b6-9bc4b25745b8).html2020-04-27T11:46:47Z