Calendar

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.”

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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