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One of the most exotic phenomena in condensed matter systems is the emergence of fractionalized particles. However, until now, only a few experimental systems are known to realize fractionalized excitations. This calls for more systematic ways to find and understand systems with fractionalization. One natural starting point is to look for an exotic quantum criticality, where the fundamental degrees of freedom become insufficient to describe the system accurately. Furthermore, understandings in exotic quantum critical phenomena would provide a unified perspective on nearby gapped phases, i.e. a guiding principle to engineer the system in a desirable direction that may host anyons. In this talk, I would present my works on two different types of quantum criticality: (1) Deconfined quantum critical point (DQCP) between plaquette valence-bond solids and Neel ordered state in Shastry-Sutherland lattice models [PRX 9, 041037 (2019)], where two distinct symmetry breaking order parameters become unified by the fractionalized degree of freedom. (2) Transitions between fractional Chern/Quantum Hall insulators tuned by the strength of lattice potential [PRX 8, 031015 (2018)]. Here, the low-lying excitations are already fractionalized; therefore, the deconfined fractional excitations follows more naturally, which is described by Chern-Simons quantum electrodynamics. The numerical results using iDMRG as well as theoretical analysis of their emergent critical properties would be presented. In the end, I would discuss their spectroscopic signatures, providing a full analysis of experimental verification.

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A Simplified Approach to Interacting Bose Gases

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I will discuss some new results about an effective theory introduced by Lieb in 1963 to approximate the ground state energy of interacting Bosons at low density. In this regime, it agrees with the predictions of Bogolyubov. At high densities, Hartree theory provides a good approximation. In this talk, I will show that the ’63 effective theory is actually exact at both low and high densities, and numerically accurate to within a few percents in between, thus providing a new approach to the quantum many body problem that bridges the gap between low and high density.

via Zoom Video Conferencing: https://harvard.zoom.us/j/136830668

via Zoom Video Conferencing:  https://harvard.zoom.us/j/972495373

We will describe how the problem of finding periodic trajectories in a regular pentagon can be solved using a new height on P^1 coming from real multiplication.

Online via Zoom Video Conferencing: https://harvard.zoom.us/meeting/977347126

via Zoom Video Conferencing: https://harvard.zoom.us/j/140789806

We finish the computation of the automorphisms of rationalized E_n-operads when n is at least 3, by verifying that the conditions of the Goldman-Millson theorem are satisfied for the map from the (dual) graph complex to the deformation complex of maps from the graphs cooperad to the cooperadic W-construction of the Poisson cooperad.

via Zoom Video Conferencing: https://harvard.zoom.us/j/635180669

In this talk, we will discuss the rigidity of positive mass theorem for asymptotically hyperbolic manifolds. That is, if the mass equality holds, then the manifold is isometric to hyperbolic space. The proof used a variational approach with the scalar curvature constraint. It also involves an investigation on a type of Obata’s equations, which leads to recent splitting results with Galloway. This talk is based on the joint works with L.-H. Huang and D. Martin, and with G. J. Galloway.

*note unusual time*

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Title and Abstract TBA

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Circularly polarized light (i.e. helicity) is a concept defined in terms of
plane wave expansions of solutions to Maxwell’s equations.  We wish to find  an analogous concept for classical and quantized Yang-Mills fields. Since the classical (hyperbolic) Yang-Mills equation is a non-linear equation, a gauge invariant  plane wave expansion does not exist.  We will first
show, in electromagnetism,  an equivalence between the usual plane wave characterization  of helicity and a characterization in terms of (anti-)self  duality of a gauge potential on a half space of Euclidean R^4. The transition from Minkowski space to Euclidean space is implemented by the
Maxwell-Poisson equation. We will then replace the Maxwell- Poisson equation by the Yang-Mills-Poisson equation to find a decomposition of the Yang-Mills configuration space into submanifolds arguably corresponding to positive and negative helicity. This is a report on the paper [1].
References
[1] https://doi.org/10.1016/j.nuclphysb.2019.114685

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I will report on some recent progress on the problem of understanding the collapsing behavior of Ricci-flat Kahler metrics on Calabi-Yau manifolds that admit a fibration structure, when the volume of the fibers shrinks to zero. Based on joint works with Gross-Zhang and with Hein.

via Zoom Video Conferencing: https://harvard.zoom.us/j/977347126

Every physicist knows that the classical electromagnetism is described by Maxwell’s equations and that it is invariant under the electromagnetic duality S: (E, B) → (B, −E). However, the properties of the electromagnetic duality in the quantum theory might not be as well known to physicists in general, and in fact are not very well understood in the literature. This is particularly true when going around a nontrivial path in the spacetime results in a duality transformation. In our recent work, we uncovered a feature of the Maxwell theory and its duality symmetry in such a situation, namely that it has a quantum anomaly. We found that the anomaly of this system in a particular formulation is 56 times that of a Weyl fermion. Our result reproduces, as a special case, the known anomaly of the all-fermion electrodynamics—a version of the Maxwell theory where particles of odd (electric or magnetic) charge are fermions—discovered in the last few years.

via Zoom Video Conferencing: https://harvard.zoom.us/j/136830668

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The Oppenheim Conjecture, proved by Margulis, states that any irrational quadratic form, has values (at integer coordinates) that are dense on the real line. However, obtaining effective estimates for any given form is a very difficult problem. In this talk I will discuss recent results, where such effective estimates are obtained for generic forms using a combination of methods from dynamics and analytic number theory. I will also discuss some results on analogous problems for inhomogenous forms and more general higher degree polynomials.

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The goal of this talk is to precisely describe how certain operator properties of continuum QFT (e.g. operator product expansions, current algebras, vertex operator algebras) emerge from an underlying lattice theory.  The main lesson will be that a “continuum limit” must always involve two or more cutoffs being taken to zero in a specific order.  In other words, the naive statement that continuum theories are obtained from lattice ones by letting a “lattice spacing” go to zero is never sufficient to describe the lattice-continuum correspondence.  Using these insights, I will show in detail how the Kac-Moody algebra arises from a nonperturbatively well defined, fully regularized model of free fermions, and I will comment on generalizations and applications to bosonization.  Time permitting, I will describe more intricate examples involving scalar fields, and I will discuss several open questions.

via Zoom Video Conferencing:  https://harvard.zoom.us/j/837429475

via Zoom Video Conferencing:  https://harvard.zoom.us/j/779283357

We study the nonlinear Schroedinger equation (NLS) with bounded initial data which
does not vanish at infinity. Examples include periodic, quasi-periodic and random initial data.
On the lattice we prove that solutions are polynomially bounded in time for any bounded data.
In the continuum, local existence is proved for real analytic data by a Newton iteration scheme.
Global existence for NLS with a regularized nonlinearity follows by analyzing a local energy norm.

This is joint work with B. Dodson and A. Soffer.

via Zoom Video Conferencing: link TBA

The anisotropic Calderon inverse problem consists in recovering the metric of a compact connected Riemannian manifold with boundary from the knowledge of the Dirichlet-to-Neumann map at fixed energy. A fundamental result due to Lee and Uhlmann states that there is uniqueness in the analytic case. We shall present counterexamples to uniqueness in cases when: 1) The metric smooth in the interior of the manifold, but only Holder continuous on one connected component of the boundary, with the Dirichlet and Neumann data being measured on the same proper subset of the boundary. 2) The metric is smooth everywhere and Dirichlet and Neumann data are measured on disjoint subsets of the boundary. This is joint work with Thierry Daude (Cergy-Pontoise) and Francois Nicoleau (Nantes).

I will talk about some ideas that are essential to build a general framework for topological phases in arbitrary dimensions. I will also discuss how these ideas are applied when global symmetry or higher symmetry is present, and how to understand and classify such higher SET/SPT orders.

via Zoom Video Conferencing:  https://harvard.zoom.us/j/977347126

via Zoom Video Conferencing: https://harvard.zoom.us/j/972495373

Strata of abelian differentials have long been of interest for their dynamical and algebro-geometric properties, but relatively little is understood about their topology. I will describe a project aimed at understanding the (orbifold) fundamental groups of non-hyperelliptic stratum components. The centerpiece of this is the monodromy representation valued in the mapping class group of the surface relative to the zeroes of the differential. For g \ge 5, we give a complete description of this as the stabilizer of the framing of the (punctured) surface arising from the flat structure associated to the differential. This is joint work with Aaron Calderon.

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Spacetimes with compact directions, which have special holonomy such as Calabi-Yau spaces, play an important role in supergravity and string theory. In this talk I will discuss the global, non-linear stability for the vacuum Einstein equations on a spacetime which is a cartesian product of a high dimensional Minkowski space with a compact Ricci flat internal space with special holonomy. I will start by giving a brief overview of related stability problems which have received a lot of attention recently, including the black hole stability problem. This is based on joint work with Pieter Blue, Zoe Wyatt and Shing-Tung Yau.

*via Zoom Video Conferencing: https://harvard.zoom.us/j/952543678

via Zoom Video Conferencing: https://harvard.zoom.us/j/977347126

Quantum spin systems are many-body models which are of wide interest in modern physics and at the same time amenable to rigorous mathematical analysis. A central question about a quantum spin system is whether its Hamiltonian exhibits a spectral gap above the ground state. The existence of such a spectral gap has far-reaching consequences, e.g., for the ground state complexity. In this talk, we survey recent progress regarding spectral gaps for frustration-free quantum spin systems in dimensions greater than 1 such as the antiferromagnetic models of Affleck-Kennedy-Lieb-Tasaki (AKLT).

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The derived equivalences of K3 surfaces and the K3 categories of certain cubic fourfolds are known to be realizable as Hodge isometries, i.e. lattice isometries preserving Hodge structures. On the other hand, Hodge isometries are also known to appear when one factorizes a birational map between varieties and tracks the actions on the middle cohomologies. When does a Hodge isometry induced from the derived equivalence of K3 surfaces/categories arise from a birational map? This is the first of two related talks discussing this question. In this talk, I will exhibit such examples for general K3 surfaces of degree 12. As a corollary, I will introduce how the construction gives an interesting relation in the Grothendieck ring of algebraic varieties. This is joint work with Brendan Hassett.

via Zoom Video Conferencing: https://harvard.zoom.us/j/837429475

Work of Ernst Specker (1960), Simon Kochen and Ernst Specker (1967), and John Bell (1964) concerning the non-existence of a hidden-variables theory reproducing the predictions of Quantum Mechanics is reviewed.
Subsequently, a novel approach to Quantum Mechanics yielding – among other things – a solution of the measurement problem is outlined.

via Zoom Video Conferencing: https://harvard.zoom.us/j/779283357

In this talk I’ll discuss work in preparation on the unknottedness of asymptotically conical self shrinkers in R^3.

via Zoom Video Conferencing: if you would like to attend, please email spicard@math.harvard.edu

Global anomalies in gauge theories are detected by the exponentiated eta-invariant, which becomes a cobordism invariant when perturbative anomalies vanish. We analyse global anomalies in four distinct (but equally valid) versions of the Standard Model of particle physics by computing the appropriate cobordism groups. In two cases we find that there are no global anomalies beyond the Witten anomaly associated with the SU(2) factor, while in the other cases we show that there are no global anomalies at all. This uncovers a subtle interplay between local and global anomalies in closely related gauge theories. We will then discuss a more subtle version of this `anomaly interplay’ occurring in a U(2) gauge theory defined without a spin structure.

via Zoom Video Conferencing: https://harvard.zoom.us/j/977347126

We introduce a class of UV-regularized two-body interactions for fermions in $\R^d$ and prove a Lieb-Robinson estimate for the dynamics of this class of many-body systems. As a step towards this result, we
also prove a propagation bound of Lieb-Robinson type for continuum one-particle Schr\“odinger operators. We apply the propagation bound to prove the existence of a strongly continuous infinite-volume dynamics on the CAR algebra.

via Zoom Video Conferencing: https://harvard.zoom.us/j/147308224

We give a construction of a pseudo-Anosov map of a surface starting from (and almost isomorphic to) a hyperbolic automorphism of an n-torus. The construction arises from a peano curve based on an invariant space-filling tree. This construction allows to confirm (for degree 3) a conjecture of Fried regarding stretch factors of pseudo-Anosov maps.

via Zoom Video Conferencing: https://harvard.zoom.us/j/972495373

The most puzzling aspect of the `strange metal’ behavior of correlated electron compounds is that the linear in temperature resistivity often extends down to low temperatures, lower than natural microscopic energy scales. We consider recently proposed deconfined critical points (or phases) in models of
electrons in large dimension lattices with random nearest-neighbor exchange interactions. The criticality is in the class of Sachdev-Ye-Kitaev models, and exhibits a time reparameterization soft mode representing quantum gravity in dual holographic theories. We compute the low temperature resistivity in a large $M$ limit of models with SU($M$) spin symmetry, and find that the dominant temperature dependence arises from this soft mode. The resistivity is linear in temperature down to zero temperature at the critical point, with a co-efficient universally proportional to the product of the residual resistivity and the co-efficient of the linear in temperature specific heat. We argue that the time reparameterization soft mode offers a promising and generic mechanism for resolving the strange metal puzzle.

via Zoom Video Conferencing: https://harvard.zoom.us/j/977347126

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It is conjectured that two cubic fourfolds are birational if their associated K3 categories are equivalent. We prove this conjecture for very general cubic fourfolds of discriminant 20, where the birational maps are produced via certain Cremona transformations defined by Veronese surfaces. Using these birational maps, we find new rational cubic fourfolds. Joint work with Kuan-Wen Lai.

via Zoom Video Conferencing: https://harvard.zoom.us/j/837429475

Experimentalists are getting better and better at building qubits, but no matter how hard they try, their qubits will never be perfect. In order to build a large quantum computer, we will almost certainly need to encode the qubits using quantum error-correcting codes and encode the quantum circuits using fault-tolerant protocols. The central result of the theory of fault tolerance is the threshold theorem, which states that arbitrarily long and reliable quantum computations are possible if the error rate per gate or time step is below some constant threshold value. Fault tolerance can be nicely defined using graphical techniques, allowing for a relatively straightforward proof of the threshold theorem.

via Zoom: https://harvard.zoom.us/j/779283357

The SYZ conjecture predicts that for polarised Calabi-Yau manifolds undergoing the large complex structure limit, there should be a special Lagrangian torus fibration. A weak version asks if this
fibration can be found in the generic region. I will discuss my recent work proving this weak SYZ conjecture for the degenerating hypersurfaces in the Fermat family. Although these examples are quite
special, this is the first construction of generic SYZ fibrations that works uniformly in all complex dimensions.

If you would like to attend, please email spicard@math.harvard.edu

I will give an account of the work ArXiv:1907.08221 where we explore
the phase structure of the Standard Model as the relative strengths of
the SU(2) weak force and SU(3) strong force are varied. With a single
generation of fermions, the structure of chiral symmetry breaking
suggests that there is no phase transition as we interpolate between the
SU(3)-confining phase and the SU(2)-confining phase. Remarkably, the
massless left-handed neutrino, familiar in our world, morphs smoothly
into a massless right-handed down-quark. With multiple generations, a
similar metamorphosis occurs, but now proceeding via a phase
transition.

via Zoom Video Conferencing: https://harvard.zoom.us/j/977347126

Spin wave theory suggests that low temperature properties of the Heisenberg model can be described in terms of noninteracting quasiparticles called magnons. In my talk I will review the basic concepts and predictions of spin wave approximation and report on recent rigorous results in that direction. Based on joint work with Robert Seiringer.

via Zoom Video Conferencing: https://harvard.zoom.us/j/147308224

Triangulated surfaces are compact (hyperbolic) Riemann surfaces that admit a conformal triangulation by equilateral triangles. Brooks and Makover started the study of the geometry of random large genus triangulated surfaces. Mirzakhani later proved analogous results for random hyperbolic surfaces. These results, along with many others, suggest that the geometry of triangulated surfaces mirrors the geometry of arbitrary hyperbolic surfaces especially in the case of large genus asymptotics. In this talk, I will describe an approach to show that triangulated surfaces are asymptotically well-distributed in moduli space.

via Zoom: https://harvard.zoom.us/j/972495373