Podcasts about Riemannian

Ep. 158 (Part 1 of 2) | In the 5th dialogue of the A. H. Almaas Wisdom Series, spiritual teacher and author Hameed Ali discusses the dynamic, ever changing, infinitely creative nature of the universe, and explains that our individual souls are in some sense a microcosm of this energy, with endless potentials and possibilities. We can experience creative dynamism, Hameed says, as “a sense of infinite energy, pulsing and throbbing, where we see the whole universe in continual emergence, every moment new.” Although the soul has boundless potential, we tend to take the limited approach that what we already know is the extent of things; the key to loosening the limits we place upon ourselves is to practice inquiry and remain open to all directions of possibilities. Each individual experiences the dynamism in a different way and expresses the potentiality of reality in a different way, says Hameed. When we are in touch with our true nature, we share in the creativity of the divine. In this conversation, Hameed also talks about death: how we can be curious about it, how it is the ultimate in finality, one more possibility of reality, and that he doesn't presume to know it, only that true nature is the source of time and does not die. Life can be experienced like a fountain rather than a flowing river, Hameed relates. And the more our ego structures are released, the more we can open to its beautiful array of endless possibilities. Another profoundly intriguing, subtly humorous, and absolutely enlightening conversation with Hameed Ali. Recorded October 10, 2024.“The soul is a living expression of the fundamental nature of reality. There's no end to the potentiality.”(For Apple Podcast users, click here to view the complete show notes on the episode page.)Topics & Time Stamps – Part 1Introducing the fifth A. H. Almaas Wisdom Series with Hameed Ali, focusing on the soul's infinite potential and the creative dynamism of reality (01:07)The soul has boundless potential, but we tend to take the limited approach that what we know already is the extent of things (04:07)We don't have to look for the boundless possibilities—we just need to be open (08:16)The main tool for fostering this openness is inquiry: what is presenting itself? (10:01)We all share the potential; we are all fundamentally connected (12:16)Reality, true nature, is in constant creative dynamism (13:34)The logos of the integration of spirituality and rational knowing can be applied to every field of knowledge (14:33)Imagine a community of scientists who are all realized spiritually, their inquiry powerfully infused by spiritual understanding (15:50)We are just at the beginning of understanding the physical world (18:22)Just because something is true doesn't mean it's complete (21:55)Einstein's theory of relativity and the Riemannian manifold (26:03)The nondual is never separate from the dual (28:25)Distinguishing between the fundamental nature of pure awareness and the nature of the soul (30:27)The close connection between individual potential and creativity and universal dynamism and creativity (32:24)We can experience creative dynamism: a sense of infinite energy, where we see the universe in continual emergence, every moment new...

Eric Weinstein is a mathematician, economist, science policy expert and a frequent public speaker on a variety of subjects within the sciences. Dr Weinstein was formerly a co-founder of the Sloan Sponsored Science and Engineering Workforce Project at Harvard and the National Bureau of Economic Research, a co-founder and principal of the Natron Group in Manhattan as well as a visiting research fellow at Oxford University in the Mathematical Institute. Since completing a PhD dissertation in the Mathematics Department at Harvard in 1992, he has held research positions in Mathematics, Physics, and Economics departments (at MIT, Hebrew University, and Harvard respectively). He delivered the Special Simonyi Lectures at Oxford University in 2013 putting forth a theory he termed “Geometric Unity” to unify the twin geometries (Riemannian and Ehresmannian) thought to ground the two most fundamental physical theories (General Relativity and the so-called Standard Model of particle theory, respectively). He has been asked to address the National Academy of Sciences on five occasions on the future of scientific and academic research at elite institutions within the United States.

Proudly sponsored by PyMC Labs, the Bayesian Consultancy. Book a call, or get in touch!My Intuitive Bayes Online Courses1:1 Mentorship with meChanging perspective is often a great way to solve burning research problems. Riemannian spaces are such a perspective change, as Arto Klami, an Associate Professor of computer science at the University of Helsinki and member of the Finnish Center for Artificial Intelligence, will tell us in this episode.He explains the concept of Riemannian spaces, their application in inference algorithms, how they can help sampling Bayesian models, and their similarity with normalizing flows, that we discussed in episode 98.Arto also introduces PreliZ, a tool for prior elicitation, and highlights its benefits in simplifying the process of setting priors, thus improving the accuracy of our models.When Arto is not solving mathematical equations, you'll find him cycling, or around a good board game.Our theme music is « Good Bayesian », by Baba Brinkman (feat MC Lars and Mega Ran). Check out his awesome work at https://bababrinkman.com/ !Thank you to my Patrons for making this episode possible!Yusuke Saito, Avi Bryant, Ero Carrera, Giuliano Cruz, Tim Gasser, James Wade, Tradd Salvo, William Benton, James Ahloy, Robin Taylor,, Chad Scherrer, Zwelithini Tunyiswa, Bertrand Wilden, James Thompson, Stephen Oates, Gian Luca Di Tanna, Jack Wells, Matthew Maldonado, Ian Costley, Ally Salim, Larry Gill, Ian Moran, Paul Oreto, Colin Caprani, Colin Carroll, Nathaniel Burbank, Michael Osthege, Rémi Louf, Clive Edelsten, Henri Wallen, Hugo Botha, Vinh Nguyen, Marcin Elantkowski, Adam C. Smith, Will Kurt, Andrew Moskowitz, Hector Munoz, Marco Gorelli, Simon Kessell, Bradley Rode, Patrick Kelley, Rick Anderson, Casper de Bruin, Philippe Labonde, Michael Hankin, Cameron Smith, Tomáš Frýda, Ryan Wesslen, Andreas Netti, Riley King, Yoshiyuki Hamajima, Sven De Maeyer, Michael DeCrescenzo, Fergal M, Mason Yahr, Naoya Kanai, Steven Rowland, Aubrey Clayton, Jeannine Sue, Omri Har Shemesh, Scott Anthony Robson, Robert Yolken, Or Duek, Pavel Dusek, Paul Cox, Andreas Kröpelin, Raphaël R, Nicolas Rode, Gabriel Stechschulte, Arkady, Kurt TeKolste, Gergely Juhasz, Marcus Nölke, Maggi Mackintosh, Grant Pezzolesi, Avram Aelony, Joshua Meehl, Javier Sabio, Kristian Higgins, Alex Jones, Gregorio Aguilar, Matt Rosinski, Bart Trudeau, Luis Fonseca, Dante Gates, Matt Niccolls, Maksim Kuznecov, Michael Thomas, Luke Gorrie, Cory Kiser and Julio.Visit https://www.patreon.com/learnbayesstats to unlock exclusive Bayesian swag ;)Takeaways:- Riemannian spaces offer a way to improve computational efficiency and accuracy in Bayesian inference by considering the curvature of the posterior distribution.- Riemannian spaces can be used in Laplace approximation and Markov chain Monte Carlo...

Here's a masterclass from Jonathan Gorard.One of the most compelling results to come out of the Wolfram Physics is Jonathan's derivation of the Einstein equations from the hypergraph.Whenever I hear anyone criticize the Wolfram model for bearing no relation to reality, I tell them this: Jonathan Gorard has proved that general relativity can be derived from the hypergraph.In this excerpt from our conversation, Jonathan describes how making just three reasonable assumptions – causal invariance, asymptotic dimension preservation and weak ergodicity – allowed him to derive the vacuum Einstein equations from the Wolfram model.In other words, the structure of space-time in the absence of matter more or less falls out of the hypergraph.And making one further assumption – that particles can be treated as localized topological obstructions – allowed Jonathan to derive the non-vacuum Einstein equations from the Wolfram model.In other words, the structure of space-time in the presence of matter, too, falls out of the hypergraph.It's difficult to overstate the importance of this result.At the very least, we can say that the Wolfram model is consistent with general relativity.To state it more strongly: we no longer need to take general relativity as a given; instead, we can derive it from Wolfram Physics.—Jonathan's seminal paper on how to derive general relativity Some Relativistic and Gravitational Properties of the Wolfram Model; also published in Complex Systems Jonathan Gorard Jonathan Gorard at The Wolfram Physics Project Jonathan Gorard at Cardiff University Jonathan Gorard on Twitter The Centre for Applied Compositionality The Wolfram Physics Project People mentioned by JonathanAlfred GrayResearch mentioned by Jonathan The volume of a small geodesic ball of a Riemannian manifold by Alfred Gray Tubes by Alfred Gray Concepts mentioned by Jonathan Hausdorff dimension Geodesic balls, tubes & cones Ricci scalar curvature Ricci curvature tensor Einstein equations Einstein–Hilbert action Relativistic Lagrangian density Causal graph Tensor rank Trace From A Project to find the Fundamental Theory of Physics by Stephen Wolfram: Dimension Curvature Images Spinning and chargend black hole with accretion disk by Simon Tyran, Vienna (Симон Тыран) licensed under CC BY-SA 4.0 Альфред Грэй в Греции by AlionaKo licensed under CC BY-SA 3.0 —The Last Theory is hosted by Mark Jeffery, founder of the Open Web MindI release The Last Theory as a video too! Watch here.Kootenay Village Ventures Inc.

Link to bioRxiv paper: http://biorxiv.org/cgi/content/short/2023.07.18.549603v1?rss=1 Authors: Namgung, J. Y., Park, Y. J., Park, Y., Kim, C. Y., Park, B.-y. Abstract: Body mass index (BMI) is an indicator of obesity, and recent neuroimaging studies have demonstrated inter-individual variations in BMI to be associated with altered brain structure and function. However, how the structure-function correspondence is altered according to BMI is under-investigated. In this study, we combined structural and functional connectivity using Riemannian optimization with varying diffusion time parameters and assessed their association with BMI. First, we simulated functional connectivity from structural connectivity and generated low-dimensional principal gradients of the simulated functional connectivity across diffusion times, where low and high diffusion times indirectly reflected mono- and polysynaptic communication. We found the most apparent cortical hierarchical organization differentiating between low-level sensory and higher-order transmodal regions in the middle of the diffusion time, indicating that the hierarchical organization of the brain may reflect the intermediate mechanisms of mono- and polysynaptic communications. Associations between the simulated gradients and BMI revealed the strongest relationship when the hierarchical structure was most evident. Moreover, the functional gradient-BMI association map showed significant correlations with the cytoarchitectonic measures of the microstructural gradient and moment features, indicating that BMI-related functional connectome alterations were remarkable in higher-order cognitive control-related brain regions. Finally, transcriptomic association analysis provided potential biological underpinnings, specifying gene enrichment in the striatum, hypothalamus, and cortical cells. Our findings provide evidence that structure-function correspondence is strongly coupled with BMI when hierarchical organization is most apparent, and the associations are related to the multiscale properties of the brain, leading to an advanced understanding of the neural mechanisms related to BMI. Copy rights belong to original authors. Visit the link for more info Podcast created by Paper Player, LLC

Link to bioRxiv paper: http://biorxiv.org/cgi/content/short/2022.11.13.516334v1?rss=1 Authors: Moon, J., Chau, T. Abstract: Background: Brain-computer interfaces (BCIs) can offer solutions to communicative impairments induced by conditions such as locked-in syndrome. While covert speech-based BCIs have garnered interest, a major issue facing their clinical translation is the collection of sufficient volumes of high signal-to-noise ratio (SNR) examples of covert speech signals which can typically induce fatigue in users. Fortuitously, investigations into the linkage between covert speech and speech perception have revealed spatiotemporal similarities suggestive of shared encoding mechanisms. Here, we sought to demonstrate that an electroencephalographic cross-condition machine learning model of speech perception and covert speech can successfully decode neural speech patterns during online BCI scenarios. Methods: In the current study, ten participants underwent a dyadic protocol whereby participants perceived the audio of a randomly chosen word and then subsequently mentally rehearsed it. Eight words were used during the offline sessions and subsequently narrowed down to three classes for the online session (two words, rest). The modelling was achieved by estimating a functional mapping derived from speech perception and covert speech signals of the same speech token (features were extracted via a Riemannian approach). Results: While most covert speech BCIs deal with binary and offline classifications, we report an average ternary and online BCI accuracy of 75.3% (60% chance-level), reaching up to 93% in select participants. Moreover, we found that perception-covert modelling effectively enhanced the SNR of covert speech signals correlatively to their high-frequency correspondences. Conclusions: These findings may pave the way to efficient and more user-friendly data collection for passively training such BCIs. Future iterations of this BCI can lead to a combination of audiobooks and unsupervised learning to train a non-trivial vocabulary that can support proto-naturalistic communication. Copy rights belong to original authors. Visit the link for more info Podcast created by Paper Player, LLC

Megaphysics II:An Explanation of Nature The Equation of Everything in Terms of Cosmology, Strings and Relativity by Dr. Mitchell Wick This book is a comprehensive mathematical exposition of how one-inch-long equation space-time is directly proportional to space and inversely proportional to mass can be used to explain all phenomena of nature. It explains the equation in terms of metric tensors and in terms of quantum mechanics and demonstrate how they equal each other and the equation of everything, which in tensors is Riemannian 4 space = Mintkowski 3 space - 1/2 space-time curvature metric called gravity/Ricci tensor of inertial mass all multiplied by 1/c^2 or if substituting the Ricci Tensor (c^2) by the energy density of matter completes the equation. A new mathematical operator is introduced called the spiral operator, which describes space-time in black holes and the total curvature of the universe is mathematically determined. Black energy and dark matter are shown how they interrelate and relate to what caused the big bang, and there is a theory as to what occurred prior to the big bang. Also a mathematical mechanism is given showing a rotational component in the quantum bubble prior to the big bang and how it fits with nature. The book applies some comments made by other physicists about compactification (rolled or curled up dimensions) to show the compactified circle of Type IIa string theory can be applied to arc length as space-time, the radius as the sum of all Riemann Forces and the angle as space-time curvature. Relativity and cosmology are explained, as well as quantum mechanics, including the Schrodinger equation, and how many more conn-compactified dimensions can exist besides the postulated twenty-six in string theory. M Theory is also explained, along with membranes and how they interrelate to energy and matter. The Hawking paradox is solved using Schwarzchild space-time and much more. Dr. Wick is a retired physician who, besides being a board-certified physician, had undergraduate studies in chemistry, including research in Polymers at the University of South Florida. Collegiate material for a bachelor’s in chemistry included physics through quantum mechanics, kinetics, and thermodynamics prior to attending medical school in 1976 at Kirksville College of Osteopathic Medicine. He had also published two prior books on physics, which are Megaphysics: A New Look at the Universe”(2003) and The Equation of Everything (2014). He had residency training in Parkview Hospital in radiology and nuclear medicine, and besides being board certified in family practice, he was a diplomat of the American Academy of Pain Management and American Association of HIV Medicine. This author is and has been in Who’s Who in the World, Who’s Who in Science and Engineering, A Dictionary of International Biography (53rd edition, IBC Cambridge Press), and 2000 Outstanding Intellectuals of the 21st Century (40th edition, IBC Cambridge Press, 2007). He was nominated as International Scientist of the Year in 2006 by IBC. https://www.authorhouse.com/en-gb/bookstore/bookdetails/717285-Megaphysics-II-An-Explanation-of-Nature http://www.bluefunkbroadcasting.com/root/twia/mitchwickparch.mp3

Computational complexity is a quantum information concept that recently has found applications in holography. I will consider quantum computational complexity for n qubits using Nielsen's geometrical approach. In the definition of complexity there is a big amount of arbitrariness due to the choice of the penalty factors, which parameterize the cost of the elementary computational gates. In order to reproduce desired features in holography, negative sectional curvatures are required. With the simplest choice of penalties, this is achieved at the price of singular curvatures in the large n limit. I will consider a choice of penalties in which negative curvatures can be obtained in a smooth way. I will also talk about the relation between operator and state complexities, framing the discussion in the language of Riemannian submersions. Finally, I'll discuss conjugate points for a large number of qubits in the unitary space and I'll provide a strong indication that maximal complexity scales exponentially with the number of qubits in a certain regime of the penalties space.

My guest today is the great Professor Ludwig Holtmeier, who is a music theorist, musicologist, pianist and the president of the Hochschule für Musik Freiburg! ----- 0:23 What's your musical background? 3:37 Did you have a typical classical training or did you ever improvise as a young musician? 4:18 Do you have perfect or absolute pitch? 4:29 Did you learn music theory and counterpoint as a young musician? 4:59 Moving to Freiburg 7:37 When did you discover partimento? 10:10 How did you break away from Riemannian theory? 12:36 How did you find these older methods? 13:24 Were a lot of treatises lost during WWII? 14:02 What was inside that library of note? 14:45 Continuing the story of how he discovered Fenaroli 15:50 Was it the Fenaroli Regole? 16:10 What date was that manuscript of Fenaroli? 17:12 Why did no one want to teach music theory at the piano at the University of Freiburg? 18:10 How did you fill in the gaps of information in Fenaroli's regole? 20:57 Is Rameau one of the most misunderstood music theorists? 22:46 What are people getting wrong about Rameau? 27:01 How do you integrate partimento theory with the modern curriculum? 28:08 Is it based around Fenaroli or other teachers as well? 28:43 Do most students come to you not knowing anything about partimento? 29:21 What are some of the best exercises you use? 31:50 Are there theories that you exclude at the University? 33:31 How do you balance partimento with other theories in analysis? 34:42 How would you analyze the music of Wagner? 37:08 What do you mean by arabic numerals? 37:52 What is your perspective on Riemann? 43:22 How about Heinrich Schenker? 45:18 What's the best way to learn counterpoint? 47:41 Should Harmony be separated from counterpoint? Should they be 2 separate subjects? 48:50 Do you have a preference for Fixed Do, Moveable Do or Hexachordal Solfeggio? 50:16 On Music Schema 53:46 What is the importance of David Heinichen in music theory in the 18th century? 54:46 Did he codify the methods of the Italians? 55:34 Did Bach use Heinichen's treatise? 57:08 The intriguing unity of the methods of the 18th century 1:00:25 Can partimento and 18th century pedagogy help us understand newer composers like Rachmaninoff, Debussy and Ravel? 1:01:40 What is your advice for parents who will start music lessons for their children? 1:02:49 How can people find you and your work? 1:03:45 Wrapping Up

Uniform Manifold Approximation and Projection (UMAP) is a dimension reduction technique that can be used for visualisation similarly to t-SNE, but also for general non-linear dimension reduction. The algorithm is founded on three assumptions about the data: 1. The data is uniformly distributed on a Riemannian manifold; 2. The Riemannian metric is locally constant (or can be approximated as such); 3. The manifold is locally connected. From these assumptions it is possible to model the manifold with a fuzzy topological structure. The embedding is found by searching for a low dimensional projection of the data that has the closest possible equivalent fuzzy topological structure.

Speakers: Liliana Gorini, John Sigerson (class 4 of 6) The influence of Lyndon LaRouche's ideas in Italy reflects an advancement based on the scientific and artistic revolutions of the 15-century Florentine Renaissance. These advances include our return to natural, scientific musical tuning, as demanded over a century ago by Giuseppe Verdi; Italy’s recent moves to implement LaRouche’s proposal for Glass-Steagall banking legislation; a return to Hamiltonian principles of economic policy; and Italy’s bold leap to join China’s Belt and Road world development movement. At root, however, there is nothing specifically Italian about these advances; Italy is the rich soil bearing the fruits of the Platonic current that rose in Ancient Greece, stretching through Nicolaus of Cusa, Johannes Kepler, the German mathematical physicist Bernhard Riemann, and the musical genius Wilhelm Furtwängler. Furtwängler’s almost single-handed effort to save European musical culture from being utterly destroyed by the British golem Adolf Hitler, later came to be a chief inspiration for LaRouche’s insistence that music unfolds not in sound, but in the Riemannian complex domain.

In this episode we’re joined by Nina Miolane, researcher and lecturer at Stanford University. Nina and I recently spoke about her work in the field of geometric statistics in machine learning. Specifically, we discuss the application of Riemannian geometry, which is the study of curved surfaces, to ML. Riemannian geometry can be helpful in building machine learning models in a number of situations including in computational anatomy and medicine where it helps Nina create models of organs like the brain and heart. In our discussion we review the differences between Riemannian and Euclidean geometry in theory and practice, and discuss several examples from Nina’s research. We also discuss her new Geomstats project, which is a python package that simplifies computations and statistics on manifolds with geometric structures. The full show notes for this episode can be found at twimlai.com/talk/196.

Moritz Gruber hat an unserer Fakultät eine Doktorarbeit zu isoperimetrischen Problemstellungen verteidigt und spricht mit Gudrun Thäter über sein Forschungsgebiet. Ein sehr bekanntes Beispiel für ein solches Problem kommt schon in der klassische Mythologie (genauer in Vergils Aeneis) als Problem der Dido vor. Vergil berichtet, dass Dido als Flüchtling an Afrikas Küste landete und sich so viel Land erbat, wie sie mit der Haut eines Rindes umspannen kann. Was zunächst wie ein winziges Fleckchen Erde klingt, wurde jedoch durch einen Trick groß genug, um die Stadt Karthago darauf zu gründen: Dido schnitt die Tierhaut in eine lange Schnur. Das mathematische Problem, dass sich ihr anschließend stellte und das als Didos oder isoperimetrisches Problem bezeichnet wird ist nun: Welche Fläche mit einem Umfang gleich der vorgegebenen Schnurlänge umfasst den größten Flächeninhalt? Natürlich wird dieses Problem zunächst etwas idealisiert in der Euklidischen Ebene gestellt und nicht in der konkreten Landschaft Karthagos. Es ist ein schwieriges Problem, denn man kann nicht alle Möglichkeiten ausprobieren oder einfach die Fälle durchkategorisieren. Andererseits liegt die Vermutung sehr nahe, dass der Kreis die Lösung ist, denn man kann sich schnell überzeugen, dass Symmetrien ausgenutzt werden können, um die eingeschlossene Fläche zu maximieren. Der Kreis hat unendlich viele Symmetrieachsen und schöpft diese Konstruktion deshalb gut aus. Trotzdem war ein stringenter Beweis erst im 18. Jh. mit den bis dahin entwickelten Methoden der Analysis möglich. Unter anderem mussten Verallgemeinerungen des Ableitungsbegriffes verstanden worden sein, die auf dieses Optimierungsproblem passen. Moritz Gruber interessiert sich für Verallgemeinerungen von isoperimetrischen Problemen in metrischen Räume, die in der Regel keinen Ableitungsbegriff haben. Die einzige Struktur in diesen Räumen ist der Abstand. Eine Möglichkeit, hier Aussagen zu finden ist es, das Verhalten für große Längen zu untersuchen und das Wachstum von Flächen in Abhängigkeit vom Wachstum des Umfangs zu charakterisieren. Naheliegend ist eine Approximation durch umschriebene und einbeschriebene Quadrate als obere und untere Schranke für die Fläche, die tatsächlich umschlossen und nicht so einfach berechnet werden kann. Außerdem interessieren ihn Verallgemeinerung auf Lie-Gruppen. Sie sind gleichzeitig differenzierbare Mannigfaltigkeit und Gruppe. Die Gruppenverknüpfung und Inversenbildung ist kompatibel mit der glatten Struktur. Sogenannte nilpotente Lie-Gruppen sind den kommutativen (d.h. abelschen) Gruppen am nächsten und bieten ihm die Möglichkeit, dort Ergebnisse zu erhalten. Die Übertragung der isoperimetrischen Probleme und mathematischen Methoden in höhere Dimensionen ergibt sehr viel mehr Möglichkeiten. In der Regel sind hier sind die unteren Schranken das schwierigere Problem. Eine Möglichkeit ist der Satz von Stokes, weil er Maße auf dem Rand und im Inneren von Objekten vernküpfen kann. Literatur und weiterführende Informationen M.R. Bridson: The geometry of the word problem In Invitations to Geometry and Topology, ed. by M.R. Bridson & S.M. Salomon, Oxord Univ. Press 2002. L. Capogna, D. Danielli, S.C. Pauls & J.T. Tyson: An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem. Birkhäuser Progress in Math. 259, 2007. M. Gruber: Isoperimetry of nilpotent groups (Survey). Frontiers of Mathematics in China 11 2016 1239–1258. Schnupperkurs über metrische Geometrie Podcasts L. Schwachhöfer: Minimalflächen, Gespräch mit G. Thäter im Modellansatz Podcast, Folge 118, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2016. P. Schwer: Metrische Geometrie, Gespräch mit G. Thäter im Modellansatz Podcast, Folge 102, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2016.

Petra Schwer ist seit Oktober 2014 Juniorprofessorin an unserer Fakultät. Sie arbeitet im Institut für Algebra und Geometrie in der Arbeitsgruppe Metrische Geometrie. Ab Oktober 2016 startet in diesem Institut ein neues Graduiertenkolleg mit dem Titel Asymptotic Invariants and Limits of Groups and Spaces und Petra Schwer freut sich darauf, dort viele mit ihrer Begeisterung anstecken zu können. Ihr Weg in die Algebra war nicht ganz direkt: Sie hat zunächst Wirtschaftsmathematik in Ulm studiert. Ein Wechsel an die Uni Bonn ebnete den Weg ins etwas abstraktere Fahrwasser. Zwei Ausflüge in die Industrie (zwischen Diplom und Promotionszeit und in der Postdoc-Phase) haben ihre Entscheidung für die akademische Mathematik bekräftigt. Im Gegensatz zur Differentialgeometrie, die von Ihrem Ursprung her auf analytischen Methoden und Methoden der Differentialrechnung (wie zum Beispiel des Ableitens) beruht, untersucht die Metrische Geometrie Mengen mit Abstandsfunktion. Darunter fallen auch die klassischen Riemannschen Geometrien, aber auch viel allgemeinere geometrische Strukturen, wie zum Beispiel Gruppen oder Graphen. Eine Metrik ist nichts anderers als eine Funktion, die einen Abstand zwischen zwei Punkten definiert. Die Euklidische Geometrie (in zwei bzw. drei Dimensionen) ist sicher allen aus der Schule bekannt. Sie ist ein Beispiel eines Geometriemodells in der metrischen Geometrie. Euklid versuchte erstmals Geometrie von Ihren Grundbausteinen her zu beschreiben. Er hat sich gefragt: Was ist ein Punkt? Was ist eine Gerade? Wie lässt sich der Abstand eines Punktes zu einer Geraden definieren? Schließlich stellte er eine Liste von grundlegenden Objekten sowie deren Eigenschaften und Beziehungen auf (Axiome genannt) die eine Geometrie erfüllen soll. Diese Axiome sind dabei die Eigenschaften, die sich nicht aus anderen ableiten lassen, also nicht beweisbar sind. Eines dieser Axiome besagte, dass durch einen festen Punkt genau eine Gerade parallel zu einer vorgegebenen anderen Geraden verläuft. Es entbrannte ein Jahrhunderte dauernder Streit darüber, ob sich dieses Parallelenaxiom aus den anderen aufgestellten Axiomen ableiten lässt, oder ob man diese Eigenschaft als Axiom fordern muss. Sehr viel später wurde klar, dass der Streit durchaus einen wichtigen und tief liegenden Aspekt unserer Anschauungsgeometrie berührte. Denn es wurden gleich mehrere Mengen (mit Abstandsfunktion) entdeckt, in denen diese Eigenschaft nicht gilt. Deshalb nannte man die Geometrien, in denen das Parallelenaxiom nicht gilt nichteuklidische Geometrien. Ein sehr nahe liegendes Beispiele für nichteuklidische Strukturen ist z.B. die Kugel-Oberfläche (damit auch unsere Erdoberfläche) wo die euklidische Geometrie nicht funktioniert. In der Ebene ist der traditionelle Abstand zwischen zwei Punkten die Länge der Strecke, die beide Punkte verbindet. Das lässt sich im Prinzip auf der Kugeloberfläche imitieren, indem man einen Faden zwischen zwei Punkten spannt, dessen Länge dann anschließend am Lineal gemessen wird. Spannt man den Faden aber "falschrum" um die Kugel ist die so beschriebene Strecke aber nicht unbedingt die kürzeste Verbindung zwischen den beiden Punkten. Es gibt aber neben der klassischen Abstandsmessung verschiedene andere sinnvolle Methoden, einen Abstand in der Ebene zu definieren. In unserem Gespräch nennen wir als Beispiel die Pariser Metrik (oder auch SNCF oder Eisenbahnmetrik). Der Name beschreibt, dass man im französischen Schnellzugliniennetz nur mit umsteigen in Paris (sozusagen dem Nullpunkt oder Zentrum des Systems) von Ort A nach Ort B kommt. Für den Abstand von A nach B müssen also zwei Abstände addiert werden, weil man von A nach Paris und dann von Paris nach B fährt. Das verleiht der Ebene eine Baumstruktur. Das ist nicht nur für TGV-Reisende wichtig, sondern gut geeignet, um über Ordnung zu reden. Ebenso sinnvoll ist z.B. auch die sogenannte Bergsteiger-Metrik, die nicht allein die Distanz berücksichtigt, sondern auch den Aufwand (bergauf vs. bergab). Damit ist sie aber in den relevanten Fällen sogar asymmetrisch. D.h. von A nach X ist es "weiter" als von X nach A, wenn X oben auf dem Berg ist und A im Tal. Analog ist es wenn man mit dem Boot oder schwimmend mit bzw. gegen die Strömung oder den Wind unterwegs ist. Dann misst man besser statt der räumlichen Distanz die Kraft bzw. Energie, die man für den jeweiligen Weg braucht. Für Karlsruher interessant ist sicher auch die KVV-Metrik, die wie folgt beschrieben wird: Um den Abstand von einem Punkt A zu einem anderen Punkt B der Ebene zu messen, läuft man von A und B senkrecht zur x-Achse (und trifft diese in Punkten A', bzw B') und addiert zu diesen beiden Abständen den Abstand von A' zu B'. Anschaulich gesprochen muss man also immer erst von A zur Kaiserstrasse, ein Stück die Kaiserstraße entlang und dann zu B. Eben so, wie die KVV ihre Strecken plant. Zwischen einer Ebene und z.B. der Kugeloberfläche gibt es einfach zu verstehende und doch wichtige geometrische Unterschiede. Eine Strecke in der Ebene läßt sich z.B. in zwei Richtungen unendlich weit fortsetzen. Auf der Kugeloberfläche kommt nach einer Umrundung der Kugel die Verlängerung der Strecke an dem Punkt wieder an, wo man die Konstruktion begonnen hat. D.h. insbesondere, dass Punkte auf einer Kugeloberfläche nicht beliebig weit voneinander entfernt sein können. Es gibt außerdem genau einen Punkt, der genau gegenüber liegt und unendlich (!) viele kürzeste Wege dorthin (in jeder Richtung einen). Verblüffend ist dabei auch: So verschieden sich Ebene und Kugeloberfläche verhalten, in einer fußläufigen Umgebung jedes Punktes fühlt sich die Erdoberfläche für uns wie ein Ausschnitt der Ebene an. Mathematisch würde man sagen, dass sich eine Kugel lokal (also in einer sehr kleinen Umgebung) um einen Punkt genauso verhält, wie eine Ebene lokal um einen Punkt. Die Krümmung oder Rundung der Kugel ist dabei nicht spürbar. Versucht man die gesamte Kugel auf einer ebenen Fläche darzustellen, wie zum Beispiel für eine Weltkarte, so kann dies nur gelingen, wenn man Abstände verzerrt. Für unsere ebenen Darstellungen der Erdkugel als Landkarte muss man also immer im Hinterkopf behalten, dass diese (zum Teil stark) verzerrt sind, d.h. Längen, Winkel und Flächen durch die ebene Darstellung verändert werden. Ein wichtiges Konzept zur Unterscheidung von (z.B.) Ebene und Kugeloberfläche ist die eben schon erwähnte Krümmung. Es gibt verschiedene Definitionen - insbesondere, wenn man Flächen eingebettet im dreidimensionalen Raum untersucht. Dabei hat ein flachgestrichenes Blatt Papier keine Krümmung - eine Kugeloberfläche ist gekrümmt. Um das formal zu untersuchen, werden Tangentialflächen an Punkte auf der Oberfläche angelegt. In einer kleinen Umgebung des Berührpunktes wird die Abweichung der Tangentialebene von der Oberfläche betrachtet. Bei der Kugel liegt die Kugeloberfläche immer auf einer Seite von der Tangentialebene. Das muss nicht so sein. Die Tangentialfläche kann z.B. in einem Sattelpunkt die zu untersuchende Fläche durchdringen - d.h. in unterschiedliche Richtungen ist die Krümmung entweder positiv oder negativ. Man braucht aber eigentlich gar keine Tangentialflächen, denn auch Winkelsummen verraten uns etwas über die Krümmung. In der Ebene ergeben die drei Innenwinkel jedes Dreiecks zusammen addiert immer 180 Grad. Auf der Kugel, also auf einer gekrümmten Fläche, sind es immer mehr als 180 Grad. Legt man zum Beispiel einen Punkt in den Nordpol und zwei weitere so auf den Äquator, dass die Verbindungsstrecken zum Nordpol einen Winkel von 90 Grad einschließen, so hat das entstehende Dreieck eine Winkelsumme von 270 Grad. Etwas komplexer ist die Situation bezüglich Krümmung auf einem Torus (der sieht aus wie ein Schwimmreifen oder Donut). Betrachtet man das lokale Krümmungsverhalten in Punkten auf der Donut-/Torusoberfläche ist sie außen so gekrümmt wie eine Kugel, innen sieht sie aber aus wie eine Sattelfläche. Es läßt sich aber auch ein abstraktes Modell des Torus konstruieren, das genauso flach, wie die euklidische Ebene ist. Dazu wähle in der Ebene ein Quadrat mit fester Seitenlänge und klebe gedanklich die gegenüberliegenden Seiten (also oben und unten, sowie links mit rechts) zusammen. Man erhält so ein "periodisches" Quadrat: Wenn man auf einer Seite hinauswandert, kommt man gegenüber an der gleichen Stelle wieder in das Quadrat hinein. Dieses Objekt ist topologisch ebenfalls ein Torus, hat aber, weil das Quadrat Teil der Ebene ist, Krümmung 0. Literatur und weiterführende Informationen D. Hilbert, S. Cohn-Vossen: Anschauliche Geometrie, eine sehr schöne, (in weiten Teilen) auch mit wenig mathematischen Vorkenntnissen gut verständliche Einführung in viele verschiedene Bereiche der Geometrie. D. Burago, Y. Burago, S. Ivanov: A Course in Metric Geometry, eines der Standardlehrbücher über metrische Geometrie. Euklid, Elemente, Digitale Version der 5 Bücher von Euklid. Gromov: Metric Structures for Riemannian and Non-Riemannian Spaces. Das "grüne Buch" - Kursnotizen einer Vorlesung von Gromov, die später in Buchform gebracht wurden.

In this thesis various aspects of target-space duality in closed bosonic string theory are studied. It begins by introducing generalized geometry as the main mathematical framework. In analogy to general relativity with the Riemannian metric as dynamical quantity, a unified description for string backgrounds – Riemannian metrics together with Kalb-Ramond two-form fields – is approached via Courant algebroids on the generalized tangent bundle equipped with a generalized metric. The dual background configuration, i.e. a metric and a bivector field, is described by the generalized cotangent bundle. The absence of a conventional curvature tensor and consequently the problem of defining generalized gravity theories on Courant algebroids is investigated in detail. This leads to the introduction of Lie algebroids whose differential geometry is suitable for the formulation of gravity theories. Different such theories are shown to be interrelated by appropriate homomorphisms. This proves to be useful for describing non-geometric backgrounds. Target-space duality is introduced in terms of O(d,d)-duality which identifies two-dimensional non-linear sigma models for different string backgrounds as physically equivalent under certain conditions: The backgrounds and coordinates of the dual theories have to be related by certain O(d, d) transformations. In particular, integrability conditions of the dual coordinates are formulated in terms of Courant algebroids. Apart from (non-abelian) T-duality, O(d,d)-duality contains the novel Poisson-duality induced by Poisson structures. T- and Poisson-duality are applied to the three-torus with constant H-flux which shows the existence of non-geometric backgrounds. The latter exceed conventional conceptions of geometry as they cannot be described globally. The problem of describing non-geometric backgrounds is approached with generalizes geometry. A unified description of T-dual backgrounds is given in terms of proto-Lie bialgebroids – one for the geometric sector and another for the non-geometric one. They combine into a Courant algebroid whose anomalous Jacobi identity provides conditions for the concurrent appearance of dual fluxes. The absence of a gravity theory leads to the restriction to Lie algebroids. Their gravity theories allow for a global description of non-geometric backgrounds by an exact prescription for the patching of these backgrounds. The description extends to all possible supergravity theories. The question whether a unified description of dual backgrounds is possible is reconsidered in a manifestly T-duality invariant conformal field theory approach. Dual coordinates are treated on equal footing. Modular invariance of the one-loop partition function together with the premise of physical intermediate states in four-tachyon scattering inevitably leads to the appearance of the strong constraint of double field theory on non-compact spaces. Toroidally compactified directions do not require a constraint. This explains the appearance of the strong constraint and justifies possible attenuations.

In this thesis, we investigate the question when a non-compact manifold can be quasi-isometric to a leaf in a foliation of a compact manifold. The point of departure is the result of Paul Schweitzer's that every non-compact manifold carries a Riemannian metric so that the resulting Riemannian manifold is not quasi-isometric to a leaf in a codimension one foliation of a compact manifold. We show that the coarse homology of these non-leaves is not finitely generated. This observation motivates the main question of this thesis: Does every leaf in a foliation of a compact manifold have finitely generated coarse homology? The answer to this question is a double negative: Firstly, we show that there exists a large class of two-dimensional leaves in codimension one foliations that have non-finitely generated coarse homology. Moreover, we improve Schweitzer's construction by showing that every Riemannian metric can be deformed to a codimension one non-leaf without affecting the coarse homology. In particular, we find non-leaves with trivial coarse homology. In order to answer these questions we develop computational tools for the coarse homology. Furthermore, we show that certain known criteria for manifolds to be a leaf are independent of one another and of the coarse homology.

Leonard Susskind continues his discussion of Riemannian geometry and uses it as a foundation for general relativity. (October 8, 2012)

Micheli, M (University of California, Los Angeles) Friday 26 August 2011, 11:45-12:30

In this thesis we study the geometry and topology of Riemannian 3-orbifolds which are locally volume collapsed with respect to a curvature scale. Our main result is that a sufficiently collapsed closed 3-orbifold without bad 2-suborbifolds satisfies Thurston’s Geometrization Conjecture. We also prove a version of this result with boundary. Kleiner and Lott indepedently and simultanously proved similar results ([KL11]). The main step of our proof is to construct a graph decomposition of sufficiently collapsed (closed) 3-orbifolds. We describe a coarse stratification of roughly 2-dimensional Alexandrov spaces which we then promote to a decomposition into suborbifolds for collapsed 3-orbifolds; this decomposition can then be reduced to a graph decomposition. We complete our proof by showing that graph orbifolds without bad 2-suborbifolds satisfy the Geometrization Conjecture.

There are competing schools of thought about the question of whether spacetime is fundamentally continuous or discrete. Here, we consider the possibility that spacetime could be simultaneously continuous and discrete, in the same mathematical way that information can be simultaneously continuous and discrete. The equivalence of continuous information and discrete information, which is of key importance in signal processing, is established by the Shannon sampling theory: for any band-limited signal, it suffices to record discrete samples to be able to perfectly reconstruct it everywhere, if the samples are taken at a rate of at least twice the band limit. Physical fields on generic curved spaces obey a sampling theorem if they possess an ultraviolet cutoff. Recently, methods of spectral geometry have been employed to show that also the very shape of a curved space (i.e. of a Riemannian manifold) can be discretely sampled and then reconstructed up to the cutoff scale.

Calogero, S (Granada) Friday 10 September 2010, 14:00-14:40

On classical Lie groups, which act by means of a unitary representation on finite dimensional Hilbert spaces H, we identify two classes of tensor field constructions. First, as pull-back tensor fields of order two from modified Hermitian tensor fields, constructed on Hilbert spaces by means of the property of having the vertical distributions of the C_0-principal bundle H_0 over the projective Hilbert space P(H) in the kernel. And second, directly constructed on the Lie group, as left-invariant representation-dependent operator-valued tensor fields (LIROVTs) of arbitrary order being evaluated on a quantum state. Within the NP-hard problem of deciding whether a given state in a n-level bi-partite quantum system is entangled or separable (Gurvits, 2003), we show that both tensor field constructions admit a geometric approach to this problem, which evades the traditional ambiguity on defining metrical structures on the convex set of mixed states. In particular by considering manifolds associated to orbits passing through a selected state when acted upon by the local unitary group U(n)xU(n) of Schmidt coefficient decomposition inducing transformations, we find the following results: In the case of pure states we show that Schmidt-equivalence classes which are Lagrangian submanifolds define maximal entangled states. This implies a stronger statement as the one proposed by Bengtsson (2007). Moreover, Riemannian pull-back tensor fields split on orbits of separable states and provide a quantitative characterization of entanglement which recover the entanglement measure proposed by Schlienz and Mahler (1995). In the case of mixed states we highlight a relation between LIROVTs of order two and a class of computable separability criteria based on the Bloch-representation (de Vicente, 2007).

Due to the singularities arising in quantum field theory and the difficulties in quantizing gravity it is often believed that the description of spacetime by a smooth manifold should be given up at small length scales or high energies. In this work we will replace spacetime by noncommutative structures arising within the framework of deformation quantization. The ordinary product between functions will be replaced by a *-product, an associative product for the space of functions on a manifold. We develop a formalism to realize algebras defined by relations on function spaces. For this porpose we construct the Weyl-ordered *-product and present a method how to calculate *-products with the help of commuting vector fields. Concepts developed in noncommutative differential geometry will be applied to this type of algebras and we construct actions for noncommutative field theories. In the classical limit these noncommutative theories become field theories on manifolds with nonvanishing curvature. It becomes clear that the application of *-products is very fruitful to the solution of noncommutative problems. In the semiclassical limit every *-product is related to a Poisson structure, every derivation of the algebra to a vector field on the manifold. Since in this limit many problems are reduced to a couple of differential equations the *-product representation makes it possible to construct noncommutative spaces corresponding to interesting Riemannian manifolds. Derivations of *-products makes it further possible to extend noncommutative gauge theory in the Seiberg-Witten formalism with covariant derivatives. The resulting noncommutative gauge fields may be interpreted as one forms of a generalization of the exterior algebra of a manifold. For the Formality *-product we prove the existence of the abelian Seiberg-Witten map for derivations of these *-products. We calculate the enveloping algebra valued non abelian Seiberg-Witten map pertubatively up to second order for the Weyl-ordered *-product. A general method to construct actions invariant under noncommutative gauge transformations is developed. In the commutative limit these theories are becoming gauge theories on curved backgrounds. We study observables of noncommutative gauge theories and extend the concept of so called open Wilson lines to general noncommutative gauge theories. With help of this construction we give a formula for the inverse abelian Seiberg-Witten map on noncommutative spaces with nondegenerate *-products.