Calendar

< 2023 >
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  • CMSA EVENT: CMSA Algebraic Geometry in String Theory: Kähler–Einstein metrics on families of Fano varieties

    Speaker: Chung-Ming Pan – Institut de Mathématiques de Toulouse

    10:00 AM-11:00 AM
    April 3, 2023

    This talk aims to introduce a pluripotential approach to study  uniform a priori estimates of Kähler–Einstein (KE) metrics on families  of Fano varieties. I will first recall basic tools in the pluripotential  theory and the variational approach to complex Monge-Ampère equations. I  will then define a notion of convergence of quasi-plurisubharmonic functions in families of normal varieties and extend several classical properties under this context. Last, I will explain how these elements help to obtain a purely analytic proof of the openness of existing singular KE metrics and a uniform $L^\infty$ estimate of KE potentials.
    This is joint work with Antonio Trusiani.

    **Note special time & location: 10 – 11 AM ET in Room G02**

    https://cmsa.fas.harvard.edu/event/algebraic-geometry-in-string-theory/

  • CMSA EVENT: CMSA Colloquium: Black hole microstate counting from the gravitational path integral

    Speaker: Luca Iliesiu – Stanford University

    11:00 AM-12:00 PM
    April 3, 2023
    20 Garden Street, Cambridge, MA 02138

    Reproducing the integer count of black hole micro-states from the gravitational path integral is an important problem in quantum gravity. In the first part of the talk, I will show that, by using supersymmetric localization, the gravitational path integral for 1/16-BPS black holes in supergravity can reproduce the index obtained in the string theory construction of such black holes. A more refined argument then shows that not only the black hole index but also the total number of black hole microstates within an energy window above extremality that is polynomially suppressed in the charges also matches this string theory index. In the second part of the talk, I will present a second perspective on this state count and show how the BPS Hilbert space can be obtained by directly preparing states using the gravitational path integral. While such a preparation naively gives rise to a Hilbert space of BPS states whose dimension is much larger than expected, I will explain how non-perturbative corrections in the overlap of such states are again responsible for reproducing the correct dimension of the Hilbert space.


     

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  • NUMBER THEORY SEMINAR: Number Theory Seminar: Evaluating the wild Brauer group

    Speaker: Rachel Newton – King’s College London

    3:00 PM-4:00 PM
    April 5, 2023
    1 Oxford Street, Cambridge, MA 02138 USA

    The local global approach to the study of rational points over number fields begins by embedding the set of rational points on a variety X into the set of its adelic points. The Brauer-Manin pairing cuts out a subset of the adelic points, called the Brauer-Manin set, that contains the rational points. If the set of adelic points is non-empty but the Brauer–Manin set is empty then we say there’s a Brauer–Manin obstruction to the existence of rational points on X.  Computing the Brauer–Manin pairing involves evaluating elements of the Brauer group of X at local points.  If an element of the Brauer group has order coprime to p, then its evaluation at a p-adic point factors via reduction of the point modulo p. For elements of order a power of p this is no longer the case: in order to compute the evaluation map one must know the point to a higher p-adic precision. Classifying Brauer group elements according to the precision required to evaluate them at p-adic points gives a filtration which we describe using work of Kato. Applications of our work include addressing Swinnerton-Dyer’s question about which places can play a role in the Brauer–Manin obstruction. This is joint work with Martin Bright.

  • CMSA EVENT: CMSA Probability Seminar: Sampling from the SK and mixed p-spin measures with stochastic localization

    Speaker: Ahmed El Alaoui – Cornell

    3:30 PM-4:30 PM
    April 5, 2023
    20 Garden Street, Cambridge, MA 02138

    I will present an algorithm which efficiently samples from the Sherrington-Kirkpatrick (SK) measure with no external field at high temperature. The approach is based on the stochastic localization process of Eldan, together with a subroutine for computing the mean vectors of a family of measures tilted by an appropriate external field. Conversely, we show that no ‘stable’ algorithm can approximately sample from the SK measure at low temperature. Time permitting, we discuss extensions to the p-spin model.
    This is based on a joint work with Andrea Montanari and Mark Sellke.

     

  • HARVARD-MIT COMBINATORICS SEMINAR: MIT-Harvard-MSR Combinatorics Seminar: Realizable Standard Young Tableaux

    Speaker: Amanda Burcroff – Harvard

    4:15 PM-5:15 PM
    April 5, 2023

    We will discuss enumerative and structural properties of two families of standard Young tableaux, the realizable rectangular tableaux and the realizable staircase tableaux.  The realizable rectangular tableaux come from tropical rank 1 matrices, whereas the realizability condition for staircase tableaux comes from geometric realizability for sorting networks and the Edelman-Greene bijection.  The two notions of realizability are connected via the study of coherent monotone paths on
    the permutahedron, as established by Black and Sanyal in their study of flag polymatroids.  As a consequence of providing tight asymptotic bounds on the size of these families, we make progress on the related studies of random sorting networks, realizable allowable sequences, and sorting algorithms.  Based on joint work with Igor Araujo, Alexander E. Black, Yibo Gao, Robert A. Krueger, and Alex McDonough.
    =======================================================

    For information about the Combinatorics Seminar, please visit…

    http://math.mit.edu/seminars/combin/

    =============================================

  • OPEN NEIGHBORHOOD SEMINAR: Open Neighborhood Seminar: Sums of two cubes

    Speaker: Levent Alpöge – Harvard

    4:30 PM-5:30 PM
    April 5, 2023
    1 Oxford Street, Cambridge, MA 02138 USA

    I will connect the following questions and answers.

    1. 42 = (12602123297335631)^3 + (80435758145817515)^3 + (-80538738812075974)^3.

    2. https://people.math.harvard.edu/~alpoge/fun/fruit%20for%20thought.jpeg , aka:
    Are there positive integers x, y, z such that:
    x / (y + z) + y / (x + z) + z / (x + y) = 4?

    3. The Birch and Swinnerton-Dyer conjecture.

    4. Hilbert’s tenth problem, aka:
    Is there a computer program which “solves all Diophantine equations”?

    5. 0% of integers are a sum of two squares (integral or rational).

    6. A positive proportion of integers are a sum of two rational cubes.


    For more information, please see: https://people.math.harvard.edu/~ana/ons/

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  • CONFERENCE: Current Developments in Mathematics 2023
    All day
    April 7, 2023-April 8, 2023
    1 Oxford Street, Cambridge, MA 02138 USA

    Current Developments in Mathematics 2023

    April 7-8, 2023
    Harvard University Science Center
    Lecture Hall C

    Speakers:

    Amol Aggarwal – Columbia University (Columbia)
    Bhargav Bhatt – Institute for Advanced Study, Princeton University, & University of Michigan (IAS/Princeton/UMichigan)
    Paul Bourgade – New York University, Courant Institute (NYU Courant)
    Vesselin Dimitrov – Institute for Advanced Study & Georgia Institute of Technology (IAS/Georgia Tech)
    Greta Panova – University of Southern California (USC)

    Register Here

    REQUESTS FOR FUNDING ARE CLOSED AS OF MARCH 10TH, 2023.

    Conference Schedule

    Download PDF for a detailed schedule of lectures and events.

    Friday, April 7

    Saturday, April 8

    • 1:30 p.m. – 2:20 p.m. Part 1
    • 2:20 p.m. – 2:30 p.m. Break

    2:30 p.m. – 3:20 p.m. Part 2

    Bhargav Bhatt

    $p$-adic motives

    • 9:05 a.m. – 9:55 a.m. Part 1
    • 9:55 a.m. – 10:05 a.m. Break

    10:05 a.m. – 10:55 a.m. Part 2

    Greta Panova

    Computational complexity in algebraic combinatorics

    3:20 p.m. – 3:35 p.m.

    Break

    10:55 a.m. – 11:10 a.m.

    Break

    • 3:35 p.m. – 4:25 p.m. Part 1
    • 4:25 p.m. – 4:35 p.m. Break

    4:35 p.m. – 5:25 p.m. Part 2

    Amol Aggarwal

    Universality results in random tiling models

    • 11:10 a.m. – 12 p.m. Part 1
    • 12 p.m. – 1:30 p.m. Lunch

    1:30 p.m. – 2:20 p.m. Part 2

    Paul Bourgade

    Random matrices, the Riemann zeta function and branching processes

    2:20 p.m. – 2:35 p.m.

    Break

    • 2:35 p.m. – 3:25 p.m. Part 1
    • 3:25 p.m. – 3:35 p.m. Break

    3:35 p.m. – 4:25 p.m. Part 2

    Vesselin Dimitrov

    Modular forms and arithmetic algebraization methods

     

    Organizers: David Jerison, Paul Seidel, Nike Sun (MIT); Denis Auroux, Mark Kisin, Lauren Williams, Horng-Tzer Yau

    Sponsored by the National Science Foundation, Harvard University Mathematics, Harvard University Center of Mathematical Sciences and Applications, and the Massachusetts Institute of Technology.

    Harvard University is committed to maintaining a safe and healthy educational and work environment in which no member of the University community is, on the basis of sex, sexual orientation, or gender identity, excluded from participation in, denied the benefits of, or subjected to discrimination in any University program or activity. More information can be found here.

  • CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: Enhancing Detection of Topological Order by Local Error Correction

    Speaker: Nishad Maskara – Harvard University

    10:00 AM-11:30 AM
    April 7, 2023
    20 Garden Street, Cambridge, MA 02138

    The exploration of topologically-ordered states of matter is a long-standing goal at the interface of several subfields of the physical sciences. Such states feature intriguing physical properties such as long-range entanglement, emergent gauge fields and non-local correlations, and can aid in realization of  scalable fault-tolerant quantum computation. However, these same features also make creation, detection, and characterization of topologically-ordered states particularly challenging. Motivated by recent experimental demonstrations, we introduce a new paradigm for quantifying topological states—locally error-corrected decoration (LED)—by combining methods of  error correction with ideas of renormalization-group flow. Our approach allows for efficient and robust identification of topological order, and is applicable in the presence of incoherent noise sources, making it particularly  suitable for realistic experiments. We demonstrate the power of LED using numerical simulations of the toric code under a variety of perturbations, and we subsequently apply it to an experimental realization of a quantum spin liquid using a Rydberg-atom quantum simulator.  Finally, we illustrate how LED can be applied to more general phases including non-abelian topological orders.


    This seminar offers the option to attend by Zoom. For information on how to join, please see:
    Quantum Matter in Mathematics and Physics (QMMP) 2023:
    https://cmsa.fas.harvard.edu/event_category/quantum-matter-seminar/

    ——–
    Subscribe to Harvard CMSA Quantum Matter and other seminar videos
    (more to be uploaded):
    https://www.youtube.com/playlist?list=PL0NRmB0fnLJQAnYwkpt9PN2PBKx4rvdup

    Subscribe to Harvard CMSA seminar mailing list:
    https://forms.gle/1ewa7KeP6BxBuBeRA


     

  • SEMINARS: Gauge Theory and Topology Seminar: Instantons on Joyce’s G2-manifolds

    Speaker: Langte Ma – Stony Brook University

    3:30 PM-4:30 PM
    April 7, 2023
    1 Oxford Street, Cambridge, MA 02138 USA

    As 7-manifolds with special holonomy, examples of compact G2-manifolds were first constructed by Joyce as resolutions of flat G2-orbifolds. Later Walpuski constructed non-trivial G2-instantons over Joyce’s manifolds via gluing techniques. In this talk, I will first explain how to define a deformation invariant of G2-orbifolds by counting flat connections, then describe the moduli space of instantons over certain non-compact G2-manifolds that appeared in Joyce’s construction, with the aim to give a complete description of moduli spaces over some examples in Joyce’s list.

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  • CONFERENCE: Current Developments in Mathematics 2023
    All day
    April 8, 2023-April 8, 2023
    1 Oxford Street, Cambridge, MA 02138 USA

    Current Developments in Mathematics 2023

    April 7-8, 2023
    Harvard University Science Center
    Lecture Hall C

    Speakers:

    Amol Aggarwal – Columbia University (Columbia)
    Bhargav Bhatt – Institute for Advanced Study, Princeton University, & University of Michigan (IAS/Princeton/UMichigan)
    Paul Bourgade – New York University, Courant Institute (NYU Courant)
    Vesselin Dimitrov – Institute for Advanced Study & Georgia Institute of Technology (IAS/Georgia Tech)
    Greta Panova – University of Southern California (USC)

    Register Here

    REQUESTS FOR FUNDING ARE CLOSED AS OF MARCH 10TH, 2023.

    Conference Schedule

    Download PDF for a detailed schedule of lectures and events.

    Friday, April 7

    Saturday, April 8

    • 1:30 p.m. – 2:20 p.m. Part 1
    • 2:20 p.m. – 2:30 p.m. Break

    2:30 p.m. – 3:20 p.m. Part 2

    Bhargav Bhatt

    $p$-adic motives

    • 9:05 a.m. – 9:55 a.m. Part 1
    • 9:55 a.m. – 10:05 a.m. Break

    10:05 a.m. – 10:55 a.m. Part 2

    Greta Panova

    Computational complexity in algebraic combinatorics

    3:20 p.m. – 3:35 p.m.

    Break

    10:55 a.m. – 11:10 a.m.

    Break

    • 3:35 p.m. – 4:25 p.m. Part 1
    • 4:25 p.m. – 4:35 p.m. Break

    4:35 p.m. – 5:25 p.m. Part 2

    Amol Aggarwal

    Universality results in random tiling models

    • 11:10 a.m. – 12 p.m. Part 1
    • 12 p.m. – 1:30 p.m. Lunch

    1:30 p.m. – 2:20 p.m. Part 2

    Paul Bourgade

    Random matrices, the Riemann zeta function and branching processes

    2:20 p.m. – 2:35 p.m.

    Break

    • 2:35 p.m. – 3:25 p.m. Part 1
    • 3:25 p.m. – 3:35 p.m. Break

    3:35 p.m. – 4:25 p.m. Part 2

    Vesselin Dimitrov

    Modular forms and arithmetic algebraization methods

     

    Organizers: David Jerison, Paul Seidel, Nike Sun (MIT); Denis Auroux, Mark Kisin, Lauren Williams, Horng-Tzer Yau

    Sponsored by the National Science Foundation, Harvard University Mathematics, Harvard University Center of Mathematical Sciences and Applications, and the Massachusetts Institute of Technology.

    Harvard University is committed to maintaining a safe and healthy educational and work environment in which no member of the University community is, on the basis of sex, sexual orientation, or gender identity, excluded from participation in, denied the benefits of, or subjected to discrimination in any University program or activity. More information can be found here.

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  • CMSA EVENT: CMSA Swampland Seminar: Swampland bounds on the abelian gauge sectors

    Speaker: Seung-Joo Lee – IBS Daejeon

    11:00 AM-12:00 PM
    April 10, 2023
    20 Garden Street, Cambridge, MA 02138

    In this talk we will derive various bounds on the 0-form and the 1-form abelian gauge sectors of gravitational effective theories in 6 dimensions with minimal supersymmetry. We will start by considering 6-dimensional F-theory vacua with at least one tensor multiplets, to bound for them the number of the (0-form) U(1) gauge factors as well as the cyclic orders of the 1-form discrete gauge factors. While the two abelian gauge sectors may look rather independent, we will observe that both are heavily constrained by the solitonic heterotic strings present in the spectrum, which provide a common intuition for the derived bounds. Building upon the heterotic intuition, we will also try extending the arena to address analogous bounds for all F-theory vacua in 6 dimensions and even beyond. If time permits, several applications and future directions of research will be discussed at the end of the talk.
  • CMSA EVENT: CMSA Probability Seminar: Localization for random band matrices

    Speaker: Ron Peled – Tel Aviv University

    3:00 PM-4:00 PM
    April 10, 2023
    20 Garden Street, Cambridge, MA 02138
    **Note unusual day and time**

     I will explain an approach via “an adaptive Mermin-Wagner style shift” which proves localization of N x N Gaussian random band matrices with band width W satisfying W << N^{1/4}.
    Joint work with Giorgio Cipolloni, Jeffrey Schenker and Jacob Shapiro.

     

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  • CMSA EVENT: CMSA Colloquium: Unexpected Uses of Neural Networks: Field Theory and Metric Flows

    Speaker: James Halverson – Northeastern University

    12:30 PM-1:30 PM
    April 12, 2023
    20 Garden Street, Cambridge, MA 02138

    We are now quite used to the idea that deep neural networks may be trained in a variety of ways to tackle cutting-edge problems in physics and mathematics, sometimes leading to rigorous results. In this talk, however, I will argue that breakthroughs in deep learning theory are also useful for making progress, focusing on applications to field theory and metric flows. Specifically, I will introduce a neural network approach to field theory with a different statistical origin, that exhibits generalized free field behavior at infinite width and interactions at finite width, and that allows for the study of symmetries via the study of correlation functions in a different duality frame. Then, I will review recent progress in approximating Calabi-Yau metrics as neural networks and cast that story into the language of neural tangent kernel theory, yielding a theoretical understanding of neural network metric flows induced by gradient descent and recovering famous metric flows, such as Perelman’s formulation of Ricci flow, in particular limits.


     

  • NUMBER THEORY SEMINAR: Number Theory Seminar: Square root p-adic L-functions

    Speaker: Michael Harris – Columbia

    3:00 PM-4:00 PM
    April 12, 2023
    1 Oxford Street, Cambridge, MA 02138 USA

    The Ichino–Ikeda conjecture, and its generalization to unitary groups by N. Harris, gives explicit formulas for central critical values of a large class of Rankin–Selberg tensor products. The version for unitary groups is now a theorem, and expresses the central critical value of -functions of the form L(s, Π × Π′) in terms of squares of automorphic periods on unitary groups. Here Π×Π′ is an automorphic representation of GL(n, F) × GL(n − 1, F) that descends to an automorphic representation of U(V) × U(V′), where and are hermitian spaces over , with respect to a Galois involution c of F, of dimension and n − 1, respectively.

    I will report on the construction of a -adic interpolation of the automorphic period — in other words, of the square root of the central values of the -functions — when Π′ varies in a Hida family. The construction is based on the theory of -adic differential operators due to Eischen, Fintzen, Mantovan, and Varma. Most aspects of the construction should generalize to higher Hida theory. I will explain the archimedean theory of the expected generalization, which is the subject of work in progress with Speh and Kobayashi.

     

  • CMSA EVENT: CMSA Probability Seminar: Large deviations of Selberg’s central limit theorem

    Speaker: Emma Bailey – CUNY

    3:30 PM-4:30 PM
    April 12, 2023
    20 Garden Street, Cambridge, MA 02138

    Selberg’s CLT concerns the typical behaviour of the Riemann zeta function and shows that the random variable $\Re \log \zeta(1/2 + i t)$, for a uniformly drawn $t$, behaves as a Gaussian random variable with a particular variance.  It is natural to investigate how far into the tails this Gaussianity persists, which is the topic of this work. There are also very close connections to similar problems in circular unitary ensemble characteristic polynomials.  It transpires that a `multiscale scheme’ can be applied to both situations to understand these questions of large deviations, as well as certain maxima and moments. In this talk I will focus more on the techniques we apply to approach this problem and I will assume no number theoretic knowledge. This is joint work with Louis-Pierre Arguin.

    **location in Room G-10 is tentative. Posting will be updated with new location if necessary**

  • HARVARD-MIT COMBINATORICS SEMINAR: MIT-Harvard-MSR Combinatorics Seminar: Vector bundles, valuations, tropical linear spaces and matriods

    Speaker: Kiumars Kaveh – University of Pittsburgh

    4:15 PM-5:15 PM
    April 12, 2023

    Torus equivariant rank r vector bundles on a toric variety (toric vector bundles) were famously classified by Klyachko (1989) using certain combinatorial data of compatible filtrations in an r-dimensional vector space E. This data can be thought of as a higher rank generalization of an (integer-valued) piecewise linear function. In this talk, we give interpretations of Klyachko data in terms of valuations with values in a certain meet-join lattice as well as points on a tropical linear space. Since tropical linear spaces correspond to linear matroids, this point of view leads us to introduce the notion of a “matroidal vector bundle”, a generalization of toric vector bundles to general matroids (possibly non-representable). The talk focuses on the combinatorial side of the story and I will give a brief review of toric varieties at the beginning. This is a work in progress with Chris Manon (Kentucky).
    =======================================================

    For information about the Combinatorics Seminar, please visit…

    http://math.mit.edu/seminars/combin/

    =============================================

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  • CMSA EVENT: CMSA General Relativity Seminar: Resolving the photon ring

    Speaker: Shahar Hadar – University of Haifa

    9:30 AM-10:30 AM
    April 13, 2023

    In the past few years, the Event Horizon Telescope has released the first close-up interferometric images of two supermassive black holes, M87* and SgrA*. It is believed that within these images is embedded a fine, yet-unresolved brightness enhancement called the photon ring. The ring is a universal consequence of strong lensing by the black hole and thereby conveys information on its spacetime geometry, potentially providing a new independent avenue for tests of general relativity in the strong-field regime. In the talk I will briefly review the theory of the photon ring and its corresponding spacetime region, the photon shell, which governs the universal lensing structure. I will then describe some current efforts and future prospects for resolving the ring, which include both the construction of transformative new instruments and the development of novel analysis methods. Focusing on the latter, I will present an upcoming proposal to use spectro-temporal autocorrelations in signals emitted from black hole environs as a probe of strong lensing effects.

    This seminar will be broadcast over Zoom: https://harvard.zoom.us/j/98794872462

  • CMSA EVENT: CMSA Active Matter: Control of actin cable length by decelerated growth and network geometry

    Speaker: Shane McInally – Brandeis University

    1:00 PM-2:00 PM
    April 13, 2023
    20 Garden Street, Cambridge, MA 02138

    The sizes of many subcellular structures are coordinated with cell size to ensure that these structures meet the functional demands of the cell. In eukaryotic cells, these subcellular structures are often membrane-bound organelles, whose volume is the physiologically important aspect of their size. Scaling organelle volume with cell volume can be explained by limiting pool mechanisms, wherein a constant concentration of molecular building blocks enables subcellular structures to increase in size proportionally with cell volume. However, limiting pool mechanisms cannot explain how the size of linear subcellular structures, such as cytoskeletal filaments, scale with the linear dimensions of the cell. Recently, we discovered that the length of actin cables in budding yeast (used for intracellular transport) precisely matches the length of the cell in which they are assembled. Using mathematical modeling and quantitative
    imaging of actin cable growth dynamics, we found that as the actin cables grow longer, their extension rates slow (or decelerate), enabling cable length to match cell length. Importantly, this deceleration behavior is cell-length dependent, allowing cables in longer cells to grow faster, and therefore reach a longer length before growth stops at the back of the cell. In addition, we have unexpectedly found that cable length is specified by cable shape. Our imaging analysis reveals that cables progressively taper as they extend from the bud neck into the mother cell, and further, this tapering scales with cell length. Integrating observations made for tapering actin networks in other systems, we have developed a novel mathematical model for cable length control that recapitulates our quantitative experimental observations. Unlike other models of size control, this model does not require length-dependent rates of assembly
    or disassembly. Instead, feedback control over the length of the cable is an emergent property due to the cross-linked and bundled architecture of the actin filaments within the cable. This work reveals a new strategy that cells use to coordinate the size of their internal parts with their linear dimensions. Similar design principles may control the size and scaling of other subcellular structures whose physiologically important dimension is their length.


    This seminar will be held in person and on Zoom. For more information on how to join, please see: https://cmsa.fas.harvard.edu/event_category/active-matter-seminar/

  • THURSDAY SEMINAR SEMINAR: Thursday Seminar: Real-etale localization

    Speaker: Lucy Yang – Harvard

    3:30 PM-5:30 PM
    April 13, 2023
    1 Oxford Street, Cambridge, MA 02138 USA

    No additional detail for this event.

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  • CMSA EVENT: CMSA Swampland Seminar: The Tameness of Quantum Field Theories

    Speaker: Thomas Grimm – Utrecht University

    11:00 AM-12:00 PM
    April 24, 2023
    17 Oxford St, Cambridge, MA 02138

    Tameness is a generalized notion of finiteness that is restricting the geometric complexity of sets and functions. The underlying mathematical foundation lies in tame geometry, which is built from o-minimal structures introduced in mathematical logic. In this talk I formalize the connection between quantum field theories and logical structures and argue that the tameness of a quantum field theory relies on its UV definitionI quantify our expectations on the tameness of effective theories that can be coupled to quantum gravity and on CFTs. In particular, I present tameness conjectures about CFT observables and propose universal constraints that render spaces of CFTs to be tame sets. I then highlight the relation of these conjectures to other swampland conjectures, e.g., by arguing that the tameness of CFT observables restricts having parametrical gaps in the operator spectrum.
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  • OTHER MATHEMATICS DEPARTMENT EVENTS: Special Lecture: Finding Balance in Chaos: Approximating orthogonal polynomials on Julia sets

    Speaker: Madison Shirazi – Harvard Class of 2023

    9:45 AM-10:15 AM
    April 28, 2023
    1 Oxford Street, Cambridge, MA 02138 USA

    Equilibrium measures of compact sets in two dimensions are a generalization of equilibrium charge distributions on perfect conductors. Using equilibrium measures, we can compute the capacity of a compact set: the ability of the set to hold energy. We will compare two methods of approximating these equilibrium measures and thus the capacity of sets using finite collections of points, and then study the convergence behavior and properties of these approximations.

  • CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: Fracton Self-Statistics

    Speaker: Hao Song – ITP-CAS

    10:00 AM-11:30 AM
    April 28, 2023

    Fracton order describes novel quantum phases of matter that host quasiparticles with restricted mobility, and thus lies beyond the existing paradigm of topological order. In particular, excitations that cannot move without creating other excitations are called fractons. Here we address a fundamental open question — can the notion of self-exchange statistics be naturally defined for fractons, given their complete immobility as isolated excitations? Surprisingly, we demonstrate how fractons can be exchanged, and show their self-statistics is a key part of the characterization of fracton orders. We derive general constraints satisfied by the fracton self-statistics in a large class of abelian fracton orders. Finally, we show the existence of semionic or fermionic fracton self-statistics in some twisted variants of the checkerboard model and Haah’s code, establishing that these models are in distinct quantum phases as compared to their untwisted cousins.

    References: H Song, N Tantivasadakarn, W Shirley, M Hermele, arXiv:2304.00028.


    This seminar will be virtual.

    Password: cmsa
    For more information, please see:
    Quantum Matter in Mathematics and Physics (QMMP) 2023:
    https://cmsa.fas.harvard.edu/event_category/quantum-matter-seminar/

    ——–
    Subscribe to Harvard CMSA Quantum Matter and other seminar videos
    (more to be uploaded):
    https://www.youtube.com/playlist?list=PL0NRmB0fnLJQAnYwkpt9PN2PBKx4rvdup

    Subscribe to Harvard CMSA seminar mailing list:
    https://forms.gle/1ewa7KeP6BxBuBeRA


     

  • OTHER MATHEMATICS DEPARTMENT EVENTS: Special Lecture: Mostow rigidity and hyperbolic 3-manifolds

    Speaker: Benjy Firester – Harvard Class of 2023

    10:20 AM-10:50 AM
    April 28, 2023
    1 Oxford Street, Cambridge, MA 02138 USA

    My thesis develops the theory of hyperbolic manifolds and explains two proofs of the foundational Mostow rigidity theorem. Great math theorems bridge fields, enabling us to transfer tools from one to another, exhibiting overarching themes that unify mathematics. Mostow rigidity is such a theorem. It proves the uniqueness of geometric structures on spaces of dimension at least three, demonstrating a deep connection between topology and geometry. Most 3-dimensional spaces are hyperbolic, meaning “negatively curved” like a hyperboloid. Understanding hyperbolic spaces is important to topology, algebra, dynamics, and more. This theory is critical to the geometrization theorem, perhaps the most celebrated result in geometry. Mostow rigidity shows that any homotopy equivalence between two hyperbolic manifolds, a topological relationship, can be uniquely deformed to an isometry, a geometric one. Mostow’s own proof uses quasi-conformal theory to improve an initial function only preserves the topological structure into one that preserves the geometric structure, linking the two notions of equivalence. The second proof defines a topological quantity capturing a manifold’s complexity, and computes the volume for hyperbolic manifolds. The algebraic realization of a hyperbolic manifold encodes the geometric data of which simplices (higher-dimensional triangles) have maximal volume, which are rigid above dimension two.

  • OTHER MATHEMATICS DEPARTMENT EVENTS: Special Lecture: From triangles to algebraic geometry

    OTHER MATHEMATICS DEPARTMENT EVENTS
    Special Lecture: From triangles to algebraic geometry

    Speaker: Dori Bejleri – Harvard University

    2:00 PM-2:45 PM
    April 28, 2023
    1 Oxford Street, Cambridge, MA 02138 USA

    The classification of shapes is one of the central pursuits of geometry. In this talk, we will begin with the problem of classifying triangles and see how it leads naturally to the notion of moduli spaces in algebraic geometry.

  • HARVARD-MIT COMBINATORICS SEMINAR: MIT-Harvard-MSR Combinatorics Seminar: Equiangular lines and large multiplicity of fixed second eigenvalue

    Speaker: Carl Schildkraut – MIT

    3:00 PM-4:00 PM
    April 28, 2023

    Given a fixed angle alpha and growing dimension n, what is the maximum number of lines in n dimensions, all pairs of which meet at the same angle alpha? In 2019, Jiang, Tidor, Yao, Zhang, and Zhao determined this to be n + o(n) for “most” angles alpha, and determined the answer within O(1) for the others; the main technical portion was a sublinear upper bound on the multiplicity of the second-largest eigenvalue of bounded degree graphs. We present two constructions of bounded degree graphs with second-largest eigenvalue of large multiplicity. The first gives multiplicity about n^(1/2) using group-theoretic techniques. The second gives multiplicity only log log n, but allows precise control on the value of the second eigenvalue. This corresponds to families of n + log log n equiangular lines with the same fixed angle alpha. For some values of alpha, this answers a question of Jiang and Polyanskii, as well as Jiang, Tidor, Yao, Zhang, and Zhao, in the negative. Partially based on joint work with Milan Haiman, Shengtong Zhang, and Yufei Zhao.

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    For information about the Combinatorics Seminar, please visit:

    http://math.mit.edu/seminars/combin/

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  • OTHER MATHEMATICS DEPARTMENT EVENTS: Special Lecture: When do varieties map to each other?

    OTHER MATHEMATICS DEPARTMENT EVENTS
    Special Lecture: When do varieties map to each other?

    Speaker: Mihnea Popa – Harvard University

    3:15 PM-4:00 PM
    April 28, 2023
    1 Oxford Street, Cambridge, MA 02138 USA

    Algebraic geometers study algebraic varieties, and the maps between them. A basic question is whether there can be any non-constant maps between two (smooth, projective) varieties of different type. I will explain some basic and some more sophisticated obstructions to the existence of such maps.

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