Calendar

We present recent progress on studies of 2-dimensional topological order in terms of tensor networks and its connections to subfactor theory. We explain how Drinfel’d centers and higher relative commutants naturally appear in this context and use of picture language in this study.

Zoom: https://harvard.zoom.us/j/779283357?pwd=MitXVm1pYUlJVzZqT3lwV2pCT1ZUQT09

Zoom: https://harvard.zoom.us/j/98231541450

Positroid varieties are subvarieties of the Grassmannian obtained by intersecting cyclic rotations of Schubert varieties.  We show that the “top open positroid variety” has mixed Hodge polynomial given by the q,t-rational Catalan numbers (up to a simple factor).  Unlike the Grassmannian, the cohomology of open positroid varieties is not pure.

The q,t-rational Catalan numbers satisfy remarkable symmetry and unimodality properties, and these arise from the Koszul duality phenomenon in the derived category of the flag variety, and from the curious Lefschetz phenomenon for cluster varieties.  Our work is also related to knot homology and to the cohomology of compactified Jacobians.

This talk is based on joint work with Pavel Galashin.

Zoom: https://harvard.zoom.us/j/91794282895?pwd=VFZxRWdDQ0VNT0hsVTllR0JCQytoZz09

TITLE: Is relativity compatible with quantum theory?

ABSTRACT: We review the background, mathematical progress, and open questions in the effort to determine whether one can combine quantum mechanics, special relativity, and interaction together into one mathematical theory. This field of mathematics is known as “constructive quantum field theory.” Physicists believe that such a theory describes experimental measurements made over a 70 year period and now refined to 13-decimal-point precision—the most accurate experiments ever performed.

Talk chair: Zhengwei Liu

Written articles will accompany each lecture in this series and be available as part of the publication “History and Literature of Mathematical Science.”

For more information, please visit the event page.

Register here to attend.

I discuss the interplay between non-Fermi liquid behaviour and pairing near a quantum-critical point (QCP) in a metal. These tendencies are intertwined in the sense that both originate from the same interaction mediated by gapless fluctuations of a critical order parameter. The two tendencies compete because fermionic incoherence destroys the Cooper logarithm, while the pairing eliminates scattering at low energies and restores fermionic coherence. I discuss this physics for a class of models with an effective dynamical interaction V (Ω) ~1/|Ω|^γ (the γ-model). This model describes, in particular, the pairing at a 2D Ising-nematic critical point in (γ=1/3), a 2D antiferromagnetic critical point (γ=1/2) and the pairing by an Einstein phonon with vanishing dressed Debye frequency (γ=2). I argue the pairing wins, unless the pairing component of the interaction is artificially reduced, but because of fermionic incoherence in the normal state, the system develops a pseudogap, preformed pairs behaviour in the temperature range between the onset of the pairing at Tp and the onset of phase coherence at the actual superconducting Tc. The ratio Tc/Tp decreases with γ and vanishes at γ =2. I present two complementary arguments of why this happens. One is the softening of longitudinal gap fluctuations, which become gapless at γ =2. Another is the emergence of a 1D array of dynamical vortices, whose number diverges at γ =2. I argue that once the number of vortices becomes infinite, quasiparticle energies effectively get quantized and do not get re-arranged in the presence of a small phase variation. I show that a new non-superconducting ground state emerges at γ >2.

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

The talk discusses a toy model for phase transitions in mixtures of incompressible droplets. The model consists of non-overlapping hypercubes of side-lengths 2^j, j\in \N_0. Cubes belong to an admissible set such that if two cubes overlap, then one cube is contained in the other, a picture reminiscent of Mandelbrot’s fractal percolation model. I will present exact formulas for the entropy and pressure, discuss phase transitions from a fluid phase with small cubes towards a condensed phase with a macroscopic cube, and briefly sketch some broader questions on renormalization and cluster expansions that motivate the model. Based on arXiv:1909.09546 (J. Stat. Phys. 179 (2020), 309-340).

Zoom: https://harvard.zoom.us/j/98520388668

The Cohen–Lenstra–Martinet conjectures have been verified in
only two cases. Davenport–Heilbronn compute the average size of the
3-torsion subgroups in the class group of quadratic fields and Bhargava
computes the average size of the 2-torsion subgroups in the class groups of cubic fields. The values computed in the above two results are remarkably stable. In particular, work of Bhargava–Varma shows that they do not change if one instead averages over the family of quadratic or cubic fields satisfying any finite set of splitting conditions.

However for certain “thin” families of cubic fields, namely, families of
monogenic and n-monogenic cubic fields, the story is very different. In
this talk, we will determine the average size of the 2-torsion subgroups of
the class groups of fields in these thin families. Surprisingly, these
values differ from the Cohen–Lenstra–Martinet predictions! We will also
provide an explanation for this difference in terms of the Tamagawa numbers of naturally arising reductive groups. This is joint work with Manjul Bhargava and Jon Hanke.

Zoom: https://harvard.zoom.us/j/96767001802

Password: The order of the permutation group on 9 elements.

p-Adic numbers have always been primarily associated with pure Mathematics, and have become especially relevant in algebra and modern number theory. But why did Computer Scientists become interested in them? In this talk we will introduce p-adic numbers and survey their main properties. We will then introduce Dixon’s algorithm, which is the first algorithm that used p-adic numbers to compute the exact rational solution to an integer linear system of equations. We will also explore the latest runtime improvements in p-adic linear algebra algorithms, and discuss whether we can solve linear equation systems faster than matrix multiplication.

Zoom: https://harvard.zoom.us/j/96759150216?pwd=Tk1kZlZ3ZGJOVWdTU3JjN2g4MjdrZz09

Ideas from the early 1990s for regulating chiral fermions in lattice gauge theory led to a number of developments which paralleled roughly concurrent and independent discoveries in condensed matter physics.  I show how the Integer Quantum Hall Effect, Chern Insulators, Topological Insulators, and Majorana edge states all play a role in lattice gauge theories, and how one can also find relativistic versions of the Fractional Quantum Hall Effect, the Quantum Spin Hall Effect and related exotic forms of matter.  How to construct a nonperturbative regulator for chiral gauge theories (like the Standard Model!)  remains an open challenge, however, one that may require new insights from condensed matter physics into exotic states of matter.

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

TITLE: Michael Atiyah: Geometry and Physics

ABSTRACT: In mid career, as an internationally renowned mathematician, Michael Atiyah discovered that some problems in physics responded to current work in algebraic geometry and this set him on a path to develop an active interface between mathematics and physics which was formative in the links which are so active today. The talk will focus, in a fairly basic fashion, on some examples of this interaction, which involved both applying physical ideas to solve mathematical problems and introducing mathematical ideas to physicists.

Talk chair: Peter Kronheimer

Written articles will accompany each lecture in this series and be available as part of the publication “History and Literature of Mathematical Science.”

For more information, please visit the event page.

Register here to attend.

In this talk I describe a holographic perspective to study field spaces that arise in string compactifications. The constructions are motivated by a  general description of the asymptotic, near-boundary regions in complex structure moduli spaces of Calabi-Yau manifolds using asymptotic Hodge theory. For real two-dimensional field spaces, I introduce an auxiliary bulk theory and describe aspects of an associated sl(2) boundary theory. The bulk reconstruction from the boundary data is provided by the sl(2)-orbit theorem of Schmid and Cattani, Kaplan, Schmid, which is a famous and general result in Hodge theory. I then apply this correspondence to the flux landscape of Calabi-Yau fourfold compactifications and discuss how this allows us, in work with C. Schnell, to prove that the number of self-dual flux vacua is finite.

Zoom: https://harvard.zoom.us/j/91780604388?pwd=d3BqazFwbDZLQng0cEREclFqWkN4UT09

In this talk, I will present a new way to look at symmetry. We show that symmetry can be viewed as a non-invertible gravitational anomaly, and a non-invertible gravitational anomaly is classified by topological order in one higher dimension. This leads to a holographic view of symmetry:  symmetry is a shadow of topological order in one higher dimension. This point of view allows us to see the duality (i.e. the equivalence) between symmetries that look very different. It also gives rise to a more general symmetry – algebraic higher symmetry, which is beyond higher group description.

Zoom: https://harvard.zoom.us/j/779283357?pwd=MitXVm1pYUlJVzZqT3lwV2pCT1ZUQT09

Moduli spaces of curves admit finite covers by moduli spaces which parametrize curves together with so-called level structures. In my talk, I will discuss how the cohomology of these spaces at infinite level is related to a profinite property of the mapping class group. I will then explain why tools from p-adic geometry yield vanishing statements for these cohomologies in high degree.

Zoom: https://harvard.zoom.us/j/91794282895?pwd=VFZxRWdDQ0VNT0hsVTllR0JCQytoZz09

The temperature versus doping phase diagram of the cuprate high-T_c superconductors features an enigmatic pseudogap region whose microscopic origin remains a subject of intensive study. Experimentally resolving its symmetry properties is imperative for narrowing down the list of possible explanations. In this talk I will give an overview of how optical second harmonic generation (SHG) can be used as a sensitive probe of symmetry breaking, and recap the ways it has been used to solve outstanding problems in condensed matter physics. I will then describe how we have been applying SHG polarimetry and spectroscopy to interrogate the cuprate pseudogap. In particular, I will discuss our data on YBa₂Cu₃O_y [1], which show an order parameter-like increase in SHG intensity below the pseudogap temperature T* across a broad range of doping levels. I will then focus on our more recent results on a model parent cuprate Sr₂CuO₂Cl₂ [2], where evidence of anomalous broken symmetries surprisingly also exists. Possible connections between these observations will be speculated upon.

[1] L. Zhao, C. A. Belvin, R. Liang, D. A. Bonn, W. N. Hardy, N. P. Armitage and D. Hsieh, “A global inversion-symmetry-broken phase inside the pseudogap region of YBa₂Cu₃O_y,” Nature Phys. 13, 250 (2017).

[2] A. de la Torre, K. L. Seyler, L. Zhao, S. Di Matteo, M. S. Scheurer, Y. Li, B. Yu, M. Greven, S. Sachdev, M. R. Norman and D. Hsieh. “Anomalous mirror symmetry breaking in a model insulating cuprate Sr₂CuO₂Cl₂,” Preprint at https://arxiv.org/abs/2008.06516.

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

There are several ways to specify a number field. One can provide the minimal polynomial of a primitive element, the multiplication table of a $\bf Q$-basis, the traces of a large enough family of elements, etc. From any way of specifying a number field one can hope to deduce a bound on the number $N_n(H)$ of number fields of given degree $n$ and discriminant bounded by $H$. Experimental data suggest that the number of isomorphism classes of number fields of degree $n$ and discriminant bounded by $H$ is equivalent to $c(n)H$ when $n\geqslant 2$ is fixed and $H$ tends to infinity. Such an estimate has been proved for $n=3$ by Davenport and Heilbronn and for $n=4$, $5$ by Bhargava. For an arbitrary $n$ Schmidt proved a bound of the form $c(n)H^{(n+2)/4}$ using Minkowski’s theorem. Ellenberg et Venkatesh have proved that the exponent of $H$ in $N_n(H)$ is less than sub-exponential in $\log (n)$. I will explain how Hermite interpolation (a theorem of Alexander and Hirschowitz) and geometry of numbers combine to produce short models for number fields and sharper bounds for $N_n(H)$.

Zoom: https://harvard.zoom.us/j/96767001802

Password: The order of the permutation group on 9 elements.

In this talk we will discuss the application of Machine Learning techniques to obtain numerical approximations to various metrics of SU(3) structure on six manifolds. More precisely, we will be interested in SU(3) structures whose torsion classes make them suitable backgrounds for various string compactifications. A variety of aspects of this topic will be covered. These will include learning moduli dependent Ricci-Flat metrics on Calabi-Yau threefolds and obtaining numerical approximations to torsional SU(3) structures.

Zoom: https://harvard.zoom.us/j/96047767096?pwd=M2djQW5wck9pY25TYmZ1T1RSVk5MZz09

In this talk, I will introduce an analytic bootstrap approach for two-point correlation functions in CFTs on real projective space, and CFTs with a conformal boundary. We will use holography as a kinematical tool to derive universal results. By examining the conformal block decomposition properties of exchange diagrams in AdS space, we identify a useful new basis for decomposing correlators. The dual basis gives rise to a basis of functionals, whose actions we can compute explicitly via holography. Applying these functionals to the crossing equations, we can systematically extract constraints on the CFT data in the form of sum rules. I will demonstrate this analytic method in the canonical example of \phi^4 theory in d=4-\epsilon, fixing the CFT data to \epsilon^2.

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

A two-dimensional rotating shallow-water model describes a layer of water, in guise of oceans covering the Earth. It is formally analogue to a Schrödinger equation where the tools from topological insulators are relevant. Once regularized at small scale by an odd-viscous term, such a model has a well-defined bulk topological index. However, in presence of a sharp boundary, the number of edge modes depends on the boundary condition, showing an explicit violation of the bulk-edge correspondence. We study a continuous family of boundary conditions with a rich phase diagram, and explain the origin of this mismatch. Our approach relies on scattering theory and Levinson’s theorem. The latter does not apply at infinite momentum because of the analytic structure of the scattering amplitude there, which is ultimately the reason for the violation. (Joint work with H. Jud and C. Tauber.)

Zoom: https://harvard.zoom.us/j/779283357?pwd=MitXVm1pYUlJVzZqT3lwV2pCT1ZUQT09

The Lightning Network is a second-layer solution built above Bitcoin, aimed to solve Bitcoin’s scalability and immediacy problems. A channel in the Lightning Network allows two parties to secure bitcoin payments and escrow holdings between them. Designed to increase transaction immediacy and reduce blockchain congestion, this has the potential to solve many issues associated with Bitcoin.

In this talk, we study the economics of the Lightning Network. We present conditions under which two parties optimally establish a channel and give explicit formulas for channels’ costs. Using these, we derive implications for the network’s structure under cooperation assumptions among small sets of users. We show both local implications, such as the wastefulness of certain structures, and global implications, such as a (low) upper bound on the Lightning Network’s average degree.

Zoom: https://harvard.zoom.us/j/98231541450

The complex phenomenon in the high-Tc cuprate calls for a microscopic understanding based on general principles. In this Lecture, an exact organizing principle for a typical doped Mott insulator will be presented, in which the fermion sign structure is drastically reduced to a mutual statistics. Its nature as a long-range spin-charge entanglement of many-body quantum mechanics will be exemplified by exact numerical calculations. The phase diagram of the cuprate may be unified in a “bottom-up” fashion by a “parent” ground state ansatz with hidden orders constructed based on the organizing principle. Here the pairing mechanism will go beyond the “RVB” picture and the superconducting state is of non-BCS nature with modified London equation and novel elementary excitations. In particular, the Bogoliubov/Landau quasiparticle excitation are emerging with a two-gap structure in the superconducting state and the Fermi arc in a pseudogap regime. A mathematic framework of fractionalization and duality transformation guided by the organizing principle will be introduced to describe the above emergent phenomenon.

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Despite the fact that papers submitted to glossy journals universally start by bemoaning the absence of theoretical understanding, I will argue that the answer to the title question is “quite a lot.” To focus the discussion, I will take the late P.W. Anderson’s “Last Words on the Cuprates” (arXiv:1612.03919) as a point of departure, although from a perspective that differs from his in many key points.

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

Imagine that we are in Arthur’s office, and I will give an informal lecture. This lecture has no main theme; instead I will talk about some insightful examples and fresh ideas inspired by math and physics—as well as some potential connections between different areas.

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I’ll describe some quantum field theories that gap fermions without breaking chiral symmetries.

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