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Current Developments in Mathematics 2021-22
March 18, 2022 - March 19, 2022     
Current Developments in Mathematics 2021-22 March 18-19, 2022 Harvard University Science Center Lecture Hall B   Speakers: Jessica Fintzen (Duke) Ryan O’Donnell (Carnegie Mellon) Jack...
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  • CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics Seminar: Naturalness and muon anomalous magnetic moment
    10:10 AM-11:40 AM
    September 1, 2021
    We study a model for explaining the apparent deviation of the muon anomalous magnetic moment, (g-2), from the Standard Model expectation. There are no new scalars and hence no new hierarchy puzzles beyond those associated with the Standard model Higgs; the only new particles that are relevant for (g-2) are vector-like singlet and doublet leptons. Interestingly, this simple model provides a calculable example violating the Wilsonian notion of naturalness: despite the absence of any symmetries prohibiting its generation, the coefficient of the naively leading dimension-six operator for (g−2) vanishes at one-loop. While effective field theorists interpret this either as a surprising UV cancellation of power divergences, or as a delicate cancellation between matching UV and calculable IR corrections to (g−2) from parametrically separated scales, there is a simple explanation in the full theory: the loop integrand is a total derivative of a function vanishing in both the deep UV and IR. The leading contribution to (g−2) arises from dimension-eight operators, and thus the required masses of new fermions are lower than naively expected, with a sizable portion of parameter space already covered by direct searches at the LHC. All of the viable parameter can be probed by the LHC and planned future colliders.

    https://harvard.zoom.us/j/977347126

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  • NUMBER THEORY SEMINAR
    3:00 PM-4:00 PM
    September 8, 2021
    1 Oxford Street, Cambridge, MA 02138 USA

    The p-torsion in the class group of a number field K is conjectured to be small: of size at most |Disc K|^epsilon, and to have constant average size in families with a given Galois closure group (when p doesn’t divide the order of the group).  In general, the best upper bound we have is |Disc K|^{1/2+epsilon}, and previously the only two cases known with constant average were for 3-torsion in quadratic fields (Davenport and Heilbronn, 1971) and 2-torsion in non-Galois cubic fields (Bhargava, 2005).  We prove that the 3-torsion is constant on average for fields with Galois closure group any 2-group with a transposition, including, e.g. quartic D_4 fields.  We will discuss the main inputs into the proof with an eye towards giving an introduction to the tools in the area.  This is joint work with Robert Lemke Oliver and Jiuya Wang.

  • OPEN NEIGHBORHOOD SEMINAR

    OPEN NEIGHBORHOOD SEMINAR
    Prime numbers and Julia sets

    4:30 PM-5:30 PM
    September 8, 2021
    1 Oxford Street, Cambridge, MA 02138 USA
    Interesting patterns of prime numbers can arise in recursively defined sequences (such as the Fibonacci sequence). For non-linear recursions, there are intriguing connections with complex dynamics and the Mandelbrot set. This is just one of the many ways that Number Theory and Chaotic Dynamical Systems come together. I’ll present a few examples and a glimpse into some of my own research in this direction.
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  • CMSA EVENT: CMSA Quantum Matter in Mathematics & Physics Seminar: The Hilbert Space of large N Chern-Simons matter theories
    10:30 AM-12:00 PM
    September 16, 2021
    We demonstrate that all known formulae for the thermal partition function for large N Chern Simons matter theory admit a simple Hilbert Space interpretation. In each case this quantity equals the partition function of an associated ungauged large $N$ matter theory with a particular local Lagrangian with one additional element: the Fock Space of  this associated theory is projected down to the subspace of its WZW singlets. This projection, in particular,  implies the previously encountered `Bosonic Exclusion Principle’, namely that no single particle state can be occupied by more than $k_B$ particles ($k_B$ is the Chern Simons level). Unlike its Gauss Law counterpart, the WZW constraint does not trivialize in the large volume limit. However thermodynamics does simplify in this limit;  the final partition function reduces to  a product of partition functions associated with each single particle state. These individual single particle state partition functions are a one parameter generalizations of their free boson and free fermion counterparts, and reduce to the later at extreme values of the ‘t Hooft coupling. At generic values of the rank and the level the occupation statistics of each energy level is given by a $q$ deformation of the usual free formulae of Bose and Fermi statistics.

    https://harvard.zoom.us/j/977347126

  • CMSA EVENT: CMSA Active Matter Seminar: The many phases of a cell
    1:00 PM-2:00 PM
    September 16, 2021
    I will begin by introducing an emerging paradigm of cellular organization – the dynamic compartmentalization of biochemical pathways and molecules by phase separation into distinct and multi-phase condensates. Motivated by this, I will discuss two largely orthogonal problems, united by the theme of phase separation in multi-component and chemically active fluid mixtures.
    1. I will propose a theoretical model based on Random-Matrix Theory, validated by phase-field simulations, to characterizes the rich emergent dynamics, compositions, and steady-state properties that underlie multi-phase coexistence in fluid mixtures with many randomly interacting components.
    2. Motivated by puzzles in gene-regulation and nuclear organization, I will propose a role for how liquid-like nuclear condensates can be organized and regulated by the active process of RNA synthesis (transcription) and RNA-protein coacervation. Here, I will describe theory and simulations based on a Landau formalism and recent experimental results from collaborators.
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  • CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics Seminar: Symmetry types in QFT and the CRT theorem
    10:30 AM-12:00 PM
    September 22, 2021

    I will discuss ideas around symmetry and Wick rotation contained in joint work with Mike Hopkins (https://arxiv.org/abs/1604.06527). This includes general symmetry types for relativistic field
    theories and their Wick rotation.  I will then indicate how thebasic CRT theorem works for general symmetry types, focusing on the case of the pin groups.  In particular, I expand on a subtlety first flagged by Greaves-Thomas.

    https://harvard.zoom.us/j/977347126

  • NUMBER THEORY SEMINAR
    3:00 PM-4:00 PM
    September 22, 2021
    1 Oxford Street, Cambridge, MA 02138 USA

    Given a variety over a number field, its geometric etale fundamental group comes equipped with an action of the Galois group. This induces a Galois action on the pro-algebraic completion of the etale fundamental group and hence the ring of functions on that pro-algebraic
    completion provides a supply of Galois representations.

    It turns out that any finite-dimensional p-adic Galois representation contained in the ring of functions on the pro-algebraic completion of the fundamental group of a smooth variety satisfies the assumptions of the Fontaine-Mazur conjecture: it is de Rham at places above p and is a. e. unramified.

    Conversely, we will show that every semi-simple representation of the Galois group of a number field coming from algebraic geometry (that is, appearing as a subquotient of the etale cohomology of an algebraic variety) can be established as a subquotient of the ring of functions on the pro-algebraic completion of the fundamental group of the projective line with 3 punctures.

  • OPEN NEIGHBORHOOD SEMINAR

    OPEN NEIGHBORHOOD SEMINAR
    Projective vs. abelian geometry

    4:30 PM-5:30 PM
    September 22, 2021
    1 Oxford Street, Cambridge, MA 02138 USA
    Projective space and complex tori are two of the simplest types of manifolds we encounter, and in many ways they seem very different from each other. I will try to convince you however that, at least if we consider a special (but at the same time very common) class of tori called principally polarized abelian varieties, then the geometry of their subvarieties exhibits surprising, and to date mostly unexplained, similarities to the geometry of subvarieties in projective space.
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  • CMSA EVENT: CMSA Interdisciplinary Science Seminar: The number of n-queens configurations
    9:00 AM-10:00 AM
    September 23, 2021
    The n-queens problem is to determine Q(n), the number of ways to place n mutually non-threatening queens on an n x n board. The problem has a storied history and was studied by such eminent mathematicians as Gauss and Polya. The problem has also found applications in fields such as algorithm design and circuit development.
    Despite much study, until recently very little was known regarding the asymptotics of Q(n). We apply modern methods from probabilistic combinatorics to reduce understanding Q(n) to the study of a particular infinite-dimensional convex optimization problem. The chief implication is that (in an appropriate sense) for a~1.94, Q(n) is approximately (ne^(-a))^n. Furthermore, our methods allow us to study the typical “shape” of n-queens configurations.
    Zoom ID: 950 2372 5230 (Password: cmsa)
  • CMSA EVENT: CMSA Quantum Matter in Mathematics & Physics Seminar: Applications of instantons, sphalerons and instanton-dyons in QCD
    10:30 AM-12:00 PM
    September 23, 2021

    I start with a general map of gauge topology, including monopoles, instantons and instanton-dyons. Then comes reminder of the “topological landscape”, the minimal energy gauge field configurations, as a function of Chern-Simons number Ncs and r.m.s. size. It includes “valleys” at integer Ncs separated by mountain ridges. The meaning of instantons,
    instanton-antiinstanton “streamlines” or thimbles, and sphalerons are reminded, together with some proposal to produce sphalerons at LHC and RHIC.

    Applications of instanton ensembles, as a model of QCD vacuum, are mostly related to their fermionic zero modes and t’Hooft effective Lagrangian, which explains explicit and spontaneous breaking of chiral symmetries. Recent applications are related with hadronic wave
    functions, at rest and in the light front (LFWFs). Two application would be spin-dependent forces and  the so called “flavor asymmetry of antiquark sea” of the nucleons. At temperatures comparable to deconfinement transition, instantons get split into constituents called instanton-dyons. Studies of their ensemble explains both deconfinement and chiral transitions, in ordinary and deformed QCD.

    https://harvard.zoom.us/j/977347126
    Password: cmsa

  • CMSA EVENT: CMSA Active Matter Seminar: The many phases of a cell
    1:00 PM-2:00 PM
    September 23, 2021
    I will begin by introducing an emerging paradigm of cellular organization – the dynamic compartmentalization of biochemical pathways and molecules by phase separation into distinct and multi-phase condensates. Motivated by this, I will discuss two largely orthogonal problems, united by the theme of phase separation in multi-component and chemically active fluid mixtures.
    1. I will propose a theoretical model based on Random-Matrix Theory, validated by phase-field simulations, to characterizes the rich emergent dynamics, compositions, and steady-state properties that underlie multi-phase coexistence in fluid mixtures with many randomly interacting components.
    2. Motivated by puzzles in gene-regulation and nuclear organization, I will propose a role for how liquid-like nuclear condensates can be organized and regulated by the active process of RNA synthesis (transcription) and RNA-protein coacervation. Here, I will describe theory and simulations based on a Landau formalism and recent experimental results from collaborators.
    *rescheduled from 9/16/21
  • ALGEBRAIC DYNAMICS SEMINAR
    4:00 PM-6:00 PM
    September 23, 2021

    Given a rational map defined over a number field, the Galois orbits of points with canonical height tending to zero will equidistribute to a measure supported on the Julia set. If one is able to extend the space of test functions to include those with certain logarithmic poles, then it is possible to obtain finiteness results on S-integral points. In this talk, we will study quantitative versions of logarithmic equidistribution in some special situations and their implications.

    http://people.math.harvard.edu/~demarco/AlgebraicDynamics/

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  • CMSA EVENT: CMSA Colloquium: Langlands duality for 3 manifolds
    9:30 AM-10:30 AM
    September 29, 2021

    Langlands duality began as a deep and still mysterious conjecture in number theory, before branching into a similarly deep and mysterious conjecture of Beilinson and Drinfeld concerning the algebraic geometry of Riemann surfaces. In this guise it was given a physical explanation in the framework of 4-dimensional super symmetric quantum field theory by Kapustin and Witten.  However to this day the Hilbert space attached to 3-manifolds, and hence the precise form of Langlands duality for them, remains a mystery.

    In this talk I will propose that so-called “skein modules” of 3-manifolds give natural candidates for these Hilbert spaces at generic twisting parameter Psi , and I will explain a Langlands duality in this setting, which we have conjectured with Ben-Zvi, Gunningham and Safronov.

    Intriguingly, the precise formulation of such a conjecture in the classical limit Psi=0 is still an open question, beyond the scope of the talk.

    Zoom link: https://harvard.zoom.us/j/95767170359 (Password: cmsa)

  • CMSA EVENT: CMSA JOINT QUANTUM MATTER IN MATH & PHYSICS and STRONGLY CORRELATED QUANTUM MATERIALS & HIGH-TEMPERATURE SUPERCONDUCTORS SEMINAR: Oscillations in the thermal conductivity of a spin liquid*
    11:30 AM-1:00 PM
    September 29, 2021

    The layered honeycomb magnet alpha-RuCl3 orders below 7 K in a zigzag phase in zero field. An in-plane magnetic field H||a suppresses the zigzag order at 7 Tesla, leaving a spin-disordered phase widely believed to be a quantum spin liquid (QSL) that extends to ~12 T. We have observed oscillations in the longitudinal thermal conductivity Kxx vs. H from 0.4 to 4 K. The oscillations are periodic in 1/H (with a break-in-slope at 7 T). The amplitude function is maximal in the QSL phase (7 –11.5 T). I will describe a benchmark for crystalline disorder, the reproducibility and intrinsic nature of the oscillations, and discuss implications for the QSL state. I will also show detailed data on the thermal Hall conductivity Kxy measured from 0.4 K to 10 K and comment on recent half-quantization results.

    *Czajka et al., Nature Physics 17, 915 (2021).

    Collaborators: Czajka, Gao, Hirschberger, Lampen Kelley, Banerjee, Yan, Mandrus and Nagler.

     


    https://harvard.zoom.us/j/977347126
     Password: cmsa

  • NUMBER THEORY SEMINAR

    NUMBER THEORY SEMINAR
    Density of arithmetic Hodge loci

    3:00 PM-4:00 PM
    September 29, 2021
    1 Oxford Street, Cambridge, MA 02138 USA

    I will explain a conjecture on density of arithmetic Hodge loci which includes and generalizes several recent density results of these loci in arithmetic geometry. This conjecture has also analogues over functions fields that I will survey. As a particular instance, I will outline the proof of the following result: a K3 surface over a number field admits infinitely many specializations where its Picard rank jumps. This last result is joint work with Ananth Shankar, Arul Shankar and Yunqing Tang.

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  • CMSA EVENT: CMSA Active Matter Seminar: Cytoskeletal Energetics and Energy Metabolism
    1:00 PM-2:00 PM
    September 30, 2021
    Life is a nonequilibrium phenomenon. Metabolism provides a continuous flux of energy that dictates the form and function of many subcellular structures. These subcellular structures are active materials, composed of molecules which use chemical energy to perform mechanical work and locally violate detailed balance. One of the most dramatic examples of such a self-organizing structure is the spindle, the cytoskeletal based assembly which segregates chromosomes during cell division. Despite its central role, very little is known about the nonequilibrium thermodynamics of active subcellular matter, such as the spindle. In this talk, I will describe ongoing work from my lab aimed at understanding the flows of energy which drive the nonequilibrium behaviors of the cytoskeleton in vitro and in vivo.
    *rescheduled from 9/16/21
  • COLLOQUIUMS
    4:30 PM-5:30 PM
    September 30, 2021
    1 Oxford Street, Cambridge, MA 02138

    The field of Geometric Set Theory studies structures on sets of countable objects (typically Polish spaces) by considering virtual objects, typically uncountable sets representing members of the space under consideration in some larger model of set theory. This approach can be used to study analytic equivalence relations on Polish spaces, where the virtual objects represent equivalence classes. The representatives of the virtual classes can be used for instance to prove non-reducibility results between such equivalence relations. Another set of applications involves separating forms of the Axiom of Choice, specifically forms asserting the existence of a set of reals with certain first order properties. Typical examples include Vitali sets, Hamel bases, discontinuous homomorphisms on the real line or countable colorings of various graphs on Euclidean space. We will give a brief tour of some of the landmarks in the area, and discuss some directions for further research.

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