Landau-Ginzburg orbifold is just another name for a holomorphic function W with its abelian symmetry G. Its Fukaya category can be viewed as a categorification of a homology group of its Milnor fiber. In this introductory talk, we will start with some classical results on the topology of isolated singularities and its Fukaya-Seidel category. Then I will explain a new construction for such category to deal with a non-trivial symmetry group G. The main ingredients are classical variation map and the Reeb dynamics at the contact boundary. If time permits, I will show its application to mirror symmetry of LG orbifolds and its Milnor fiber. This is a joint work with C.-H. Cho and W. Jeong.

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

I will report on a recent joint work with Sorin Popa in which we undertake a systematic study of W*-rigidity paradigms for the embedding relation between II₁ factors and their amplifications. We say that a II₁factor M stably embeds into a II₁ factor N if M may be realized as a subfactor of an amplification of N, not necessarily of finite index. This is a preorder relation and we prove that it is as complicated as it can be: under the appropriate separability assumptions, we concretely realize any partially ordered set inside the preordered class of II₁ factors with embeddability.

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

Cliques are important structures in network science that have been used in numerous applications including spam detection, graph analysis, graph modeling, community detection among others. Obtaining the counts of k-cliques in real-world graphs with millions of nodes and edges is a challenging problem due to combinatorial explosion. Essentially, as k increases, the number of k-cliques goes up exponentially. Existing techniques are (typically) able to count k-cliques for only up to k=5.

In this talk, I will present two algorithms for obtaining k-clique counts that improve the state of the art, both in theory and in practice. The first method is a randomized algorithm that gives a (1+ε)-approximation for the number of k-cliques in a graph. Its running time is proportional to the size of an object called the Turán Shadow, which, for real-world graphs is found to be small. In practice, this algorithm works well for k<=10 and gives orders of magnitude improvement over existing methods. This paper won the Best Paper Award at WWW, 2017.

The second method, a somewhat surprising result, is a simple but powerful algorithm called Pivoter that gives the exact k-clique counts for all k and runs in O(n3^{n/3}) time in the worst case. It uses a classic technique called pivoting that drastically cuts down the search space for cliques and builds a structure called the Succinct Clique Tree from which global and local (per-vertex and per-edge) k-clique counts can be easily read off. In practice, the algorithm is orders of magnitude faster than even other parallel algorithms and makes clique counting feasible for a number of graphs for which clique counting was infeasible before. This paper won the Best Paper Award at WSDM, 2020.

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

High-temperature superconductivity in the cuprates remains an unsolved problem because the cuprates start off their lives as Mott insulators in which no organizing principle such a Fermi surface can be invoked to treat  the electron interactions.  Consequently, it would be advantageous to solve even a toy model that exhibits both Mottness and superconductivity.  Part of the problem is that the basic model for a Mott insulator, namely the Hubbard model is unsolvable in any dimension we really care about.  To address this problem, I will start by focusing on the overlooked Z_2 emergent symmetry of a Fermi surface first noted by Anderson and Haldane.  Mott insulators  break this emergent symmetry.  The simplest model of this type is due to Hatsugai/Kohmoto.  I will argue that this model can be thought of a fixed point for Mottness.  I will then show exactly[1] that this model when appended with a weak pairing interaction exhibits not only the analogue of Cooper’s instability but also a superconducting ground state, thereby demonstrating that a model for a doped Mott insulator can exhibit superconductivity.  The properties of the superconducting state differ drastically from that of the standard BCS theory.  The elementary excitations of this superconductor are not linear combinations of particle and hole states but rather are superpositions of doublons and holons, composite excitations signaling that the superconducting ground state of the doped Mott insulator inherits the non-Fermi liquid character of the normal state. Additional unexpected features of this model are that it exhibits a superconductivity-induced transfer of spectral weight from high to low energies and a suppression of the superfluid density as seen in the cuprates.

[1] PWP, L. Yeo, E. Huang, Nature Physics, 16, 1175-1180 (2020).

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

We will outline a definition of analytic spaces that relates
to complex- or rigid-analytic varieties in the same way that schemes
relate to algebraic varieties over a field. Joint with Dustin Clausen.

Zoom: https://harvard.zoom.us/j/99334398740

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

We will explore the idea of universality in probability theory, as seen in the central limit theorem, in which wide and varied input distributions can all give identical limiting outputs. We will then explore examples of universality in random algebraic structures, such as random vector spaces over finite fields. We will explain how this is connected to recent research on random integral matrices and distributions of class groups of number fields.

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4d N=1 super Yang-Mills has multiple gapped vacua arising from the spontaneously broken discrete R-symmetry. Therefore, the theory admits stable domain walls interpolating between any two vacua, but it is a nonperturbative problem to determine the low energy theory on the domain wall. We propose an explicit answer to this question: the domain walls support specific topological quantum field theories. We provide nontrivial evidence for our proposals by exactly matching renormalization group invariant partition functions (twisted by all global symmetries).

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

I will discuss applications of Vojta’s conjecture to some problems in arithmetic dynamics, concerning the growth of sizes of coordinates of orbits, greatest common divisors among coordinates, and prime factors of coordinates.  These problems can be restated and generalized in terms of (local/global) height functions, and I proved estimates on asymptotic behavior of height functions along orbits assuming Vojta’s conjecture.  As corollaries, I showed that Vojta’s conjecture implies the Dynamical Lang-Siegel conjecture for projective spaces (the sizes of coordinates grow in the same speed) and existence of primitive prime divisors in higher dimensional settings.

Go to http://people.math.harvard.edu/~demarco/AlgebraicDynamics/ for Zoom information.

K-theoretic Gromov-Witten invariants of smooth projective varieties have been introduced by YP Lee, using the Euler characteristic of a virtual structure sheaf. In particular, they are integers. In this talk, I look at these invariants for the quintic threefold and I will explain how to compute them modulo 41, using the virtual localization formula under a finite group action, up to genus 19 and degree 40.

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

I will examine the many-body localization (MBL) phase transition in one-dimensional quantum systems with quenched randomness. Having demonstrated the existence of the MBL phase at strong disorder, under a level-statistics assumption, I will focus on the nature of the transition out of this phase, using an approximate strong-disorder renormalization group. In this approach, the phase transition is due to the so-called avalanche instability of the MBL phase. I show that the critical behavior can be determined analytically within this RG. The RG flow near the critical fixed point is qualitatively similar to the Kosterlitz-Thouless (KT) flow, but there are important differences, and so this MBL transition is in a universality class that is distinct from KT. The divergence of the correlation length corresponds to critical exponent ν →∞, but the divergence is weaker than for the KT transition. This is joint work with Alan Morningstar and David Huse.

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

The point location problem is a fundamental question in computational geometry. It asks, given a known partition of R^d into cells by n hyperplanes and an unknown input point x, which cell contains x? More formally, given access to x via linear queries (i.e. for any hyperplane h in R^d we may ask: “is x above or below h?”), our goal is to locate the cell containing x in as few queries as possible.

In its dual formulation, point location is equivalent to a well-studied problem in machine learning: active learning halfspaces. In this version of the problem, we are given a known set of data points X in R^d, and an unknown hyperplane h. The goal is to label all points in X by asking as few linear queries as possible, where in this formulation a linear query simply corresponds to asking for the label of some point in R^d.

It has long been known that solving the point location problem on n hyperplanes in d-dimensions requires at least Omega(dlog(n)) queries. Over the past 40 years, a series of works in the computational geometry literature lowered the corresponding upper bound to O(d^2log(d)log(n)). In this talk, we show how taking a learning theoretic approach to the problem allows us to reach near-linear dependence on d.  The proof combines old margin-based learning and vector-scaling techniques with a novel structural decomposition used for dimensionality reduction.

Based on joint work with Daniel Kane, Shachar Lovett, and Gaurav Mahajan.

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

The problem of determining the birational nature of the moduli space of curves of genus g has received constant attention in the last century and inspired a lot of development in moduli theory. I will discuss progress achieved in the last 12 months. On the one hand, making essential of tropical methods it has been showed that both moduli spaces of curves of genus 22 and 23 are of general type (joint with D. Jensen and S. Payne). On the other hand I will discuss a proof (joint with A. Verra) of the uniruledness of the moduli space of curves of genus 16.

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

The strange metal regime is one of the most  prominent features of the cuprate phase diagram but yet has remained amongst the most mysterious. Seemingly similar metallic behavior is seen in a few other metals. In this talk, I will discuss, in great generality, some properties of  `strange  metals’ in an ideal clean system.  I will discuss general constraints[1] on the emergent low energy symmetries of any such strange metal. These constraints may be viewed as a generalization of the Luttinger theorem of ordinary Fermi liquids.   Many, if not all,   non-Fermi liquids will have the same realization of emergent symmetry as a Fermi liquid (even though they could have very different dynamics). Such phases – dubbed ersatz Fermi liquids –  share some (but not all) universal properties with Fermi liquids. I will discuss the implications for understanding the strange metal physics observed in experiments . Combined with a few experimental observations, I will show that these general model-independent considerations lead to concrete predictions[2] about a class of strange metals. The most striking of these is a divergent susceptibility of an observable that has the same symmetries as the loop current order parameter.

[1]. Dominic Else, Ryan Thorngren, T. Senthil, https://arxiv.org/abs/2007.07896
[2]. Dominic Else, T. Senthil, https://arxiv.org/abs/2010.10523

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

Zoom: https://harvard.zoom.us/j/99334398740

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

Autoregressive language models pretrained on large corpora have been successful at solving downstream tasks, even with zero-shot usage. However, there is little theoretical justification for their success. This paper considers the following questions: (1) Why should learning the distribution of natural language help with downstream classification tasks? (2) Why do features learned using language modeling help solve downstream tasks with linear classifiers? For (1), we hypothesize, and verify empirically, that classification tasks of interest can be reformulated as next word prediction tasks, thus making language modeling a meaningful pretraining task. For (2), we analyze properties of the cross-entropy objective to show that eps-optimal language models in cross-entropy (log-perplexity) learn features that are O(sqrt{eps}) good on such linear classification tasks, thus demonstrating mathematically that doing well on language modeling can be beneficial for downstream tasks. We perform experiments to verify assumptions and validate our theoretical results. Our theoretical insights motivate a simple alternative to the cross-entropy objective that performs well on some linear classification tasks.

Zoom: https://harvard.zoom.us/j/99018808011?pwd=SjRlbWFwVms5YVcwWURVN3R3S2tCUT09

Gapped phases of matter, including topological and fracton phases, are deformation classes of gapped quantum systems, and exhibit a rich array of phenomena. An interesting generalization is to consider parametrized families of gapped systems, and the deformation classes of such families. This talk will describe examples of such parametrized families and their physical properties in the bulk and at spatial boundaries. In particular, we will describe a family of one-dimensional systems that realizes a Chern number pump, which can change the quantized Chern number of a zero-dimensional family placed at its boundary.

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

It is natural to study automorphisms of hypersurfaces in projective spaces. In this talk, I will discuss a new approach to determine all possible orders of automorphisms of smooth hypersurfaces with fixed degree and dimension. Then we consider the specific case of cubic fourfolds, and discuss the relation with Hodge theory.

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

Confusingly for the uninitiated, experts in infinite-dimensional category theory make use of different definitions of an ∞-category, and theorems in the ∞-categorical literature are often proven “analytically”, in reference to the combinatorial specifications of a particular model. In this talk, we present a new point of view on the foundations of ∞-category theory, which allows us to develop the basic theory of ∞-categories — adjunctions, limits and colimits, co/cartesian fibrations, and pointwise Kan extensions — “synthetically” starting from axioms that describe an ∞-cosmos, the infinite-dimensional category in which ∞-categories live as objects. We demonstrate that the theorems proven in this manner are “model-independent”, i.e., invariant under change of model. Moreover, there is a formal language with the feature that any statement about ∞-categories that is expressible in that language is also invariant under change of model, regardless of whether it is proven through synthetic or analytic techniques. This is joint work with Dominic Verity.

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

Prediction algorithms assign numbers to individuals that are popularly understood as individual “probabilities” — e.g., what is the probability of 5-year survival after cancer diagnosis? — and which increasingly form the basis for life-altering decisions. The understanding of individual probabilities in the context of such unrepeatable events has been the focus of intense study for decades within probability theory, statistics, and philosophy. Building off of notions developed in complexity theory and cryptography, we introduce and study Outcome Indistinguishability (OI). OI predictors yield a model of probabilities that cannot be efficiently refuted on the basis of the real-life observations produced by Nature.

We investigate a hierarchy of OI definitions, whose stringency increases with the degree to which distinguishers may access the predictor in question.  Our findings reveal that OI behaves qualitatively differently than previously studied notions of indistinguishability.  First, we provide constructions at all levels of the hierarchy.  Then, leveraging recently-developed machinery for proving average-case fine-grained hardness, we obtain lower bounds on the complexity of the more stringent forms of OI.  The hardness result provides scientific grounds for the political argument that, when inspecting algorithmic risk prediction instruments, auditors should be granted oracle access to the algorithm, not simply historical predictions.

Joint work with Cynthia Dwork, Omer Reingold, Guy N. Rothblum, Gal Yona; to appear at STOC 2021.

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

Given a smooth plane curve C defined over an arbitrary field k, we say that C is transverse-free if it has no transverse lines defined over k. If k is an infinite field, then Bertini’s theorem guarantees the existence of a transverse line defined over k, and so the transverse-free condition is interesting only in the case when k is a finite field F_q. After fixing a finite field F_q, we can ask the following question: For each degree d, what is the fraction of degree d transverse-free curves among all the degree d curves? In this talk, we will investigate an asymptotic answer to the question as d tends to infinity. This is joint work with Brian Freidin.

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

We will discuss an elementary way of detecting some global anomalies from the way the symmetry algebra is realized on the torus Hilbert space of the anomalous theory, give a physical description of the imprint of the “layers” that enter in the cobordism classification of anomalies and discuss applications, including how anomalies can imply a supersymmetric spectrum in strongly coupled (nonsupersymmetric) gauge theories.

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

There is a long standing connection between the Tate conjecture in codimension 1 and finiteness properties, which first appeared in Tate’s seminal work on the endomorphisms of abelian varieties. I will explain how one can possibly extend this connection to codimension 2 cycles, using the theory of Brauer groups, moduli of twisted sheaves, and twisted derived equivalences, and prove the Tate conjecture for K3 squares. This recovers an earlier result of Ito-Ito-Kashikawa, which was established via a CM lifting theory, and moreover provides a recipe of constructing all the cycles on these varieties by purely geometric methods.

Zoom: https://harvard.zoom.us/j/99334398740

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

We will begin by discussing hyperbolic geometry, and how it can be used to build “evenly curved” metrics on a donut with one point removed. We will then discuss Maryam Mirzakhani’s computation of the “size” of the universe of all such metrics (the Weil-Petersson volume of the moduli space of complete hyperbolic metrics on a punctured torus)

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Motivated by the relation between anomaly and topological/SPT order in one higher dimension, we propose a solution to the chiral fermion problem. In particular, we find several sufficient conditions, such that a chiral fermion field theory can be regularized by an
interacting lattice model in the same dimension. We also discuss some related issues, such as mass without mass term, and why ‘topological’ phase transitions are usually not “topological” phase transitions.

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

Strominger–Yau–Zaslow conjecture predicts the existence of special Lagrangian fibrations on Calabi–Yau manifolds. The conjecture inspires the development of mirror symmetry while the original conjecture has little progress. In this talk, I will confirm the conjecture for the complement of a smooth anti-canonical divisor in del Pezzo surfaces. Moreover, I will also construct the dual torus fibration on its mirror. As a consequence, the special Lagrangian fibrations detect a non-standard semi-flat metric and some Ricci-flat metrics that don’t obviously appear in the literature. This is based on a joint work with T. Collins and A. Jacob.

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

TITLE: Homological (homotopical) algebra and moduli spaces in Topological Field theories

ABSTRACT: Moduli spaces of various gauge theory equations and of various versions of (pseudo) holomorphic curve equations have played important role in geometry in these 40 years. Started with Floer’s work people start to obtain more sophisticated object such as groups, rings, or categories from (system of) moduli spaces. I would like to survey some of those works and the methods to study family of moduli spaces systematically.

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.

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By developing high-performance quantum light sources, the multi-photon interference has been scaled up to implement Boson sampling with up to 76 photons out of a 100-mode interferometer, which yields a Hilbert state space dimension of 10³⁰ and a rate that is 10¹⁴ faster than using the state-of-the-art simulation strategy on supercomputers. Such a demonstration of quantum computational advantage is a much-anticipated milestone for quantum computing.

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

We consider the problem of estimating the number of distinct elements in a large data set from a random sample of its elements. The problem occurs in many applications, including biology, genomics, computer systems and linguistics. A line of research spanning the last decade resulted in algorithms that estimate the support up to $\pm \epsilon n$ from a sample of size $O(\log^2(1/\epsilon) \cdot n/\log n)$, where $n$ is the data set size. Unfortunately, this bound is known to be tight, limiting further improvements to the complexity of this problem. In this paper we consider estimation algorithms augmented with a machine-learning-based predictor that, given any element, returns an estimation of its frequency. We show that if the predictor is correct up to a constant approximation factor, then the sample complexity can be reduced significantly, to $\log (1/\eps) \cdot n^{1-\Theta(1/\log(1/\eps))}.$ We evaluate the proposed algorithms on a collection of data sets, using the neural-network based estimators from {Hsu et al, ICLR’19} as predictors. Our experiments demonstrate substantial (up to 3x) improvements in the estimation accuracy compared to the state of the art algorithm.

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

Title: Replica Symmetry Breaking for Random Regular NAESAT

Abstract: Ideas from physics have predicted a number of important properties of random constraint satisfaction problems such as the satisfiability threshold and the free energy (the exponential growth rate of the number of solutions). Another prediction is the condensation regime where most of the solutions are contained in a small number of clusters and the overlap of two random solutions is concentrated on two points. We establish this phenomena for the first time in sparse CSPs in the random regular NAESAT model.

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In this talk, we study certain moduli spaces of semistable objects in the Kuznetsov component of a cubic fourfold. We show that they admit a symplectic resolution \tilde{M} which is a smooth projective hyperkaehler manifold deformation equivalent to the 10-dimensional example constructed by O’Grady. As a first application, we construct a birational model of \tilde{M} which is a compactification of the twisted intermediate Jacobian fiberation of the cubic fourfold. Secondly, we show that \tilde{M} is the MRC quotient of the main component of the Hilbert scheme of elliptic quintic curves in the cubic fourfold, as conjectured by Castravet. This is a joint work with Chunyi Li and Laura Pertusi.

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

I will describe joint work with Yiannis Sakellaridis and Akshay Venkatesh, in which ideas originating in quantum field theory are applied to a problem in number theory.
A fundamental aspect of the Langlands correspondence — the relative Langlands program — studies the representation of L-functions of Galois representations as integrals of automorphic forms. However, the data that naturally index the period integrals (spherical varieties for G) and the L-functions (representations of the dual group G^) don’t seem to line up.
We present an approach to this problem via the Kapustin-Witten interpretation of the [geometric] Langlands correspondence as electric-magnetic duality for 4-dimensional supersymmetric Yang-Mills theory. Namely, we rewrite the relative Langlands program as duality in the presence of supersymmetric boundary conditions. As a result, the partial correspondence between periods and L-functions is embedded in a natural duality between Hamiltonian actions of the dual groups.

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Zoom: https://harvard.zoom.us/j/95505022117

Walker and I wrote down a lattice model schema to realize the (3+1)-Crane-Yetter TQFTs based on unitary pre-modular categories many years ago, and application of the model is found in a variety of places such as quantum cellular automata and fracton physics.  I will start with the conceptual origin of this model as requested by the organizer.  Then I will discuss a general idea for writing down lattice realizations of state-sum TQFTs based on gluing formulas of TQFTs and explain the model for Crane-Yetter TQFTs on general three manifolds.  In the end, I will mention lattice models that generalize the Haah codes in two directions:  general three manifolds and more than two qubits per site.

If the path integral of a quantum field theory is regarded as a generalization of the ordinary definite integral, then a lattice model of a quantum field theory could be regarded as an analogue of a Riemann sum.  New lattice models in fracton physics raise an interesting question:  what kinds of quantum field theories are they approximating if their continuous limits exist?  Their continuous limits would be rather unusual as the local degrees of freedom of such lattice models increase under entanglement renormalization flow.

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

A controlled natural language for mathematics is an artificial language that is designed in an explicit way with precise computer-readable syntax and semantics. It is based on a single natural language (which for us is English) and can be broadly understood by mathematically literate English speakers. This talk will describe the design of a controlled natural language for mathematics that has been influenced by the Lean theorem prover, by TeX, and by earlier controlled natural languages. The semantics are provided by dependent type theory.

Zoom: https://harvard.zoom.us/j/99018808011?pwd=SjRlbWFwVms5YVcwWURVN3R3S2tCUT09

Emerton’s completed cohomology gives, at present, the most general notion of a space of p-adic automorphic forms. Important properties of completed cohomology, such as its ‘size’, is predicted by a conjecture of Calegari and Emerton, which may be viewed as a non-abelian generalization of the Leopoldt conjecture. I will discuss the proof of many new cases of this conjecture, using a mixture of techniques from p-adic and real geometry. This is joint work with David Hansen.

Zoom: https://harvard.zoom.us/j/99334398740

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

Walker and I wrote down a lattice model schema to realize the (3+1)-Crane-Yetter TQFTs based on unitary pre-modular categories many years ago, and application of the model is found in a variety of places such as quantum cellular automata and fracton physics.  I will start with the conceptual origin of this model as requested by the organizer.  Then I will discuss a general idea for writing down lattice realizations of state-sum TQFTs based on gluing formulas of TQFTs and explain the model for Crane-Yetter TQFTs on general three manifolds.  In the end, I will mention lattice models that generalize the Haah codes in two directions:  general three manifolds and more than two qubits per site.

If the path integral of a quantum field theory is regarded as a generalization of the ordinary definite integral, then a lattice model of a quantum field theory could be regarded as an analogue of a Riemann sum.  New lattice models in fracton physics raise an interesting question:  what kinds of quantum field theories are they approximating if their continuous limits exist?  Their continuous limits would be rather unusual as the local degrees of freedom of such lattice models increase under entanglement renormalization flow.

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