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Demystifying Math 55

By Anastasia Yefremova Few undergraduate level classes have the distinction of nation-wide recognition that Harvard University’s Math 55 has. Officially comprised of Mathematics 55A “Studies...
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< 2022 >
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  • CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics
    10:30 AM-12:30 PM
    July 7, 2022

    Kenichi Konishi will cover both Part I and Part II from 10:30am – 12:30pm.

    Part I: Anomalies, dynamics and phases in strongly-coupled chiral gauge theories: recent developments

    After many years of efforts, still very little is known today about the physics of strongly-coupled chiral gauge theories in four dimensions, in spite of an important role they might play in the physics of fundamental interactions beyond the standard  SU(3)xSU(2)xU(1) model. This is in stark contrast with the vectorlike gauge theories for which we have many solid results, thanks to some exact theorems, to the lattice simulation studies, to the Seiberg-Witten exact solution of N=2 supersymmetric gauge theories, and last, but not the least, to the real-world strong-interaction phenomenology and experimental tests of Quantum Chromodynamics. 

    The purpose of this seminar is to discuss the results of our recent efforts to improve the understanding of the strongly-coupled chiral gauge theories. Among the main tools of analysis are the consideration of anomalies.  We use both the conventional ’t Hooft anomaly-matching ideas, and new, more stringent constraints coming from the generalized anomalies involving some higher-form symmetries.  Also, the so-called strong anomalies, little considered in the context of chiral gage theories, are found to carry significant implications.  

     As the playground we study several classes of SU(N) gauge theories, the so-called Bars-Yankielowicz models, the generalized Georgi-Glashow models, as well as a few other simple theories with the fermions in complex, anomaly-free  representations of the color SU(N).  

     Color-flavor-locked dynamical Higgs phase and dynamical Abelianization, emerge, among others, as two particularly interesting possible phases the system can flow into in the infrared, depending on the matter fermion content of the model.


    Part II: Quantum fluctuations, particles and entanglements: towards the solution of the Quantum Measurement Problem

    The quantum measurement problems are revisited from a new perspective.  One of the main ideas of this work is that the basic entities of our world are various types of particles, elementary or composite. It follows that  each elementary process, hence each measurement process at its core, is a spacetime, pointlike, event. 

    Another key idea is that, when a  microsystem  $\psi$  gets into contact with the experimental device,  factorization of $\psi$ rapidly fails and entangled mixed states appear.

    The wave functions for the microsystem-apparatus coupled systems for different measurement outcomes then lack overlapping spacetime support. It means that  the aftermath of each measurement is a single term in the sum: the fact  sometimes perceived as the “wave-function collapse”. 

    Our discussion leading to a diagonal density matrix shows how the information encoded in the wave function gets transcribed, via entanglement with the experimental device and environment, into the relative frequencies for various experimental results.   

    These results represent new, significant steps towards filling in the logical gaps in the standard interpretation based on Born’s rule, replacing it with a more natural one. Accepting objective reality of quantum fluctuations, independent of any experiments, and independently of human presence, one renounces for good  the idea that in a fundamental, complete theory of Nature the result of each single experiment must necessarily be predictable. 

    A few well-known puzzles such as the Schr\”odinger cat conundrum and the EPR paradox will be briefly revisited: they can all be naturally explained away. 


    For more information on how to join, please see: https://cmsa.fas.harvard.edu/quantum-matter-seminar/

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  • CMSA EVENT: CMSA Interdisciplinary Science Seminar: Topological and geometrical aspects of spinors in insulating crystals
    9:00 AM-10:00 AM
    July 14, 2022

    Introducing internal degrees of freedom in the description of crystalline insulators has led to a myriad of theoretical and experimental advances. Of particular interest are the effects of periodic perturbations, either in time or space, as they considerably enrich the variety of electronic responses. Here, we present a semiclassical approach to transport and accumulation of general spinor degrees of freedom in adiabatically driven, weakly inhomogeneous crystals of dimensions one, two and three under external electromagnetic fields. Our approach shows that spatio-temporal modulations of the system induce a spinor current and density that is related to geometrical and topological objects — the spinor-Chern fluxes and numbers — defined over the higher-dimensional phase-space of the system, i.e., its combined momentum-position-time coordinates.

     

    The results are available here: https://arxiv.org/abs/2203.14902


    Bio: Ioannis Petrides is a postdoctoral fellow at the School of Engineering and Applied Sciences at Harvard University. He received his Ph.D. from the Institute for Theoretical Physics at ETH Zurich. His research focuses on the topological and geometrical aspects of condensed matter system

     

    For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

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  • CMSA EVENT: CMSA Interdisciplinary Science Seminar: Infants’ sensory-motor cortices undergo microstructural tissue growth coupled with myelination
    9:00 AM-10:00 AM
    July 21, 2022

    The establishment of neural circuitry during early infancy is critical for developing visual, auditory, and motor functions. However, how cortical tissue develops postnatally is largely unknown. By combining T1 relaxation time from quantitative MRI and mean diffusivity (MD) from diffusion MRI, we tracked cortical tissue development in infants across three timepoints (newborn, 3 months, and 6 months). Lower T1 and MD indicate higher microstructural tissue density and more developed cortex. Our data reveal three main findings: First, primary sensory-motor areas (V1: visual, A1: auditory, S1: somatosensory, M1: motor) have lower T1 and MD at birth than higher-level cortical areas. However, all primary areas show significant reductions in T1 and MD in the first six months of life, illustrating profound tissue growth after birth. Second, significant reductions in T1 and MD from newborns to 6-month-olds occur in all visual areas of the ventral and dorsal visual streams. Strikingly, this development was heterogenous across the visual hierarchies: Earlier areas are more developed with denser tissue at birth than higher-order areas, but higher-order areas had faster rates of development. Finally, analysis of transcriptomic gene data that compares gene expression in postnatal vs. prenatal tissue samples showed strong postnatal expression of genes associated with myelination, synaptic signaling, and dendritic processes. Our results indicate that these cellular processes may contribute to profound postnatal tissue growth in sensory cortices observed in our in-vivo measurements. We propose a novel principle of postnatal maturation of sensory systems: development of cortical tissue proceeds in a hierarchical manner, enabling the lower-level areas to develop first to provide scaffolding for higher-order areas, which begin to develop more rapidly following birth to perform complex computations for vision and audition.

     

    This work is published here: https://www.nature.com/articles/s42003-021-02706-w


    For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

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  • CONFERENCE: Advances in Mathematical Physics: A Conference in Honor of Elliott H. Lieb on his 90th Birthday.
    All day
    July 30, 2022-August 1, 2022

    Advances in Mathematical Physics

    A Conference in Honor of Elliott H. Lieb on his 90th Birthday

    Dates: July 30-August 1, 2022

    Harvard University

    July 30 – 31, 2022: Hall B, Science Center, 1 Oxford Street, Cambridge, MA, 02138
    August 1, 2022: Gymnasium and the Coolidge Room, Knafel Center, Radcliffe, 18 Mason Street, Cambridge, 02138

    Register Here

    Organizers:
    Michael Aizenman, Princeton University
    Joel Lebowitz, Rutgers University
    Ruedi Seiler, Technische Universität Berlin
    Herbert Spohn, Technical University of Munich
    Horng-Tzer Yau, Harvard University
    Shing-Tung Yau, Harvard University
    Jakob Yngvason, University of Vienna

    Speakers:
    Rafael Benguria, Pontificia Universidad Catolica de Chile
    Eric Carlen, Rutgers University
    Philippe Di Francesco, University of Illinois
    Hugo Duminil-Copin, IHES
    László Erdös, Institute of Science and Technology Austria
    Rupert Frank, The California Institute of Technology
    Jürg Fröhlich, ETH Zurich
    Alessandro Giuliani, Università degli Studi Roma Tre
    Bertrand Halperin, Harvard University
    Klaus Hepp, Institute for Theoretical Physics, ETH Zurich
    Sabine Jansen, Ludwig Maximilian University of Munich
    Mathieu Lewin, Université Paris-Dauphine
    Bruno Nachtergaele, The University of California, Davis
    Yoshiko Ogata, University of Tokyo
    Ron Peled, Tel Aviv University
    Benjamin Schlein, University of Zurich
    Robert Seiringer, Institute of Science and Technology Austria
    Jan Philip Solovej, University of Copenhagen
    Hal Tasaki, Gakushuin University
    Simone Warzel, Technical University of Munich
    Jun Yin, The University of California, Los Angeles

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  • CONFERENCE: Advances in Mathematical Physics: A Conference in Honor of Elliott H. Lieb on his 90th Birthday.
    All day
    July 31, 2022-August 1, 2022

    Advances in Mathematical Physics

    A Conference in Honor of Elliott H. Lieb on his 90th Birthday

    Dates: July 30-August 1, 2022

    Harvard University

    July 30 – 31, 2022: Hall B, Science Center, 1 Oxford Street, Cambridge, MA, 02138
    August 1, 2022: Gymnasium and the Coolidge Room, Knafel Center, Radcliffe, 18 Mason Street, Cambridge, 02138

    Register Here

    Organizers:
    Michael Aizenman, Princeton University
    Joel Lebowitz, Rutgers University
    Ruedi Seiler, Technische Universität Berlin
    Herbert Spohn, Technical University of Munich
    Horng-Tzer Yau, Harvard University
    Shing-Tung Yau, Harvard University
    Jakob Yngvason, University of Vienna

    Speakers:
    Rafael Benguria, Pontificia Universidad Catolica de Chile
    Eric Carlen, Rutgers University
    Philippe Di Francesco, University of Illinois
    Hugo Duminil-Copin, IHES
    László Erdös, Institute of Science and Technology Austria
    Rupert Frank, The California Institute of Technology
    Jürg Fröhlich, ETH Zurich
    Alessandro Giuliani, Università degli Studi Roma Tre
    Bertrand Halperin, Harvard University
    Klaus Hepp, Institute for Theoretical Physics, ETH Zurich
    Sabine Jansen, Ludwig Maximilian University of Munich
    Mathieu Lewin, Université Paris-Dauphine
    Bruno Nachtergaele, The University of California, Davis
    Yoshiko Ogata, University of Tokyo
    Ron Peled, Tel Aviv University
    Benjamin Schlein, University of Zurich
    Robert Seiringer, Institute of Science and Technology Austria
    Jan Philip Solovej, University of Copenhagen
    Hal Tasaki, Gakushuin University
    Simone Warzel, Technical University of Munich
    Jun Yin, The University of California, Los Angeles

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