I will discuss the recent developments in the mathematical theory of supergravity that lay the mathematical foundations of the universal bosonic sector of four-dimensional ungauged supergravity and its Killing spinor equations in a differential-geometric framework.  I will provide the necessary context and background. explaining the results pedagogically from scratch and highlighting several open mathematical problems which arise in the mathematical theory of supergravity, as well as some of its potential mathematical applications. Work in collaboration with Vicente Cortés and Calin Lazaroiu.

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

Title: Robustness Meets Algorithms

Abstract: Starting from the seminal works of Tukey (1960) and Huber (1964), the field of robust statistics asks: Are there estimators that probably work in the presence of noise? The trouble is that all known provably robust estimators are also hard to compute in high-dimensions.

Here, we study a fundamental problem in robust statistics, posed in various forms in the above works. Given corrupted samples from a high-dimensional Gaussian, are there efficient algorithms to accurately estimate its parameters? We give the first algorithm that is able to tolerate a constant fraction of corruptions that is independent of the dimension. Moreover, we give a general recipe for detecting and correcting corruptions based on tensor-spectral techniques that are applicable to many other problems.

I will also discuss how this work fits into the broader agenda of developing mathematical and algorithmic foundations for modern machine learning.

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A disc potential function plays an important role in studying a symplectic manifold and its Lagrangian submanifolds. In this talk, I will explain how to compute the disc potential function of quadrics. The potential function provides the Landau—Ginzburg mirror, which agrees with Przyjalkowski’s mirror and a cluster chart of Pech—Rietsch—Williams’ mirror

Zoom: https://harvard.zoom.us/j/96709211410?pwd=SHJyUUc4NzU5Y1d0N2FKVzIwcmEzdz09

Polyakov introduced Liouville Conformal Field theory (LCFT) in 1981 as a way to put a naturalmeasure on the set of Riemannian metrics over a two dimensional manifold. Ever since, the work of Polyakov has echoed in various branches of physics and mathematics, ranging from string theory to probability theory and geometry. In the context of 2D quantum gravity models, LCFT is related through the Knizhnik-Polyakov-Zamolodchikov relationsto the scaling limit of Random Planar Maps and through the Alday-Gaiotto-Tachikava correspondence LCFT is conjecturally related to certain 4D Yang-Mills theories. Through the work of Dorn, Otto, Zamolodchikov and Zamolodchikov and Teschner LCFT is believed to be to a certain extent integrable. I will review a probabilistic construction of LCFT and recent proofs concerning the integrability of LCFT developed together with F. David, C. Guillarmou, R. Rhodes and V. Vargas.

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

Random dimensionality reduction is a versatile tool for speeding up algorithms for high-dimensional problems. We study its application to two clustering problems: the facility location problem, and the single-link hierarchical clustering problem, which is equivalent to computing the minimum spanning tree. We show that if we project the input pointset $X$ onto a random $d = O(d_X)$-dimensional subspace (where $d_X$ is the doubling dimension of $X$), then the optimum facility location cost in the projected space approximates the original cost up to a constant factor. We show an analogous statement for minimum spanning tree, but with the dimension $d$ having an extra $\log \log n$ term and the approximation factor being arbitrarily close to $1$. Furthermore, we extend these results to approximating solutions instead of just their costs. Lastly, we provide experimental results to validate the quality of solutions and the speedup due to the dimensionality reduction.

Unlike several previous papers studying this approach in the context of $k$-means and $k$-medians, our dimension bound does not depend on the number of clusters but only on the intrinsic dimensionality of $X$.

Joint work with Shyam Narayanan, Piotr Indyk, Or Zamir.

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

In complex algebraic geometry, the decomposition theorem asserts that semisimple geometric objects remain semisimple after taking direct images under proper algebraic maps. This was conjectured by Kashiwara and is proved by Mochizuki and Sabbah in a series of long papers via harmonic analysis and D-modules. In this talk, I would like to explain a more geometric/topological approach in the case of semisimple local systems adapting de Cataldo-Migliorini. As a byproduct, we can recover a weak form of Saito’s decomposition theorem for variations of Hodge structures. Joint work in progress with Chuanhao Wei.

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

We introduce the concepts of a symmetry-protected sign problem and symmetry-protected magic, defined by the inability of symmetric finite-depth quantum circuits to transform a state into a nonnegative real wave function and a stabilizer state, respectively. We show that certain symmetry protected topological (SPT) phases have these properties, as a result of their anomalous symmetry action at a boundary. For example, one-dimensional Z2 × Z2 SPT states (e.g. cluster state) have a symmetry-protected sign problem, and two-dimensional Z2 SPT states (e.g. Levin-Gu state) have both a symmetry-protected sign problem and magic. We also comment on the relation of a symmetry-protected sign problem to the computational wire property of one-dimensional SPT states and the greater implications of our results for measurement based quantum computing.

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

Labeled data for imitation learning of theorem proving in large libraries of formalized mathematics is scarce as such libraries require years of concentrated effort by human specialists to be built. This is particularly challenging when applying large Transformer language models to tactic prediction, because the scaling of performance with respect to model size is quickly disrupted in the data-scarce, easily-overfitted regime. We propose PACT (Proof Artifact Co-Training), a general methodology for extracting abundant self-supervised data from kernel-level proof terms for co-training alongside the usual tactic prediction objective. We apply this methodology to Lean, an interactive proof assistant which hosts some of the most sophisticated formalized mathematics to date. We instrument Lean with a neural theorem prover driven by a Transformer language model and show that PACT improves theorem proving success rate on a held-out suite of test theorems from 32\% to 48\%.

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

In the talk, I will present an equidistribution result for families of (non-degenerate) subvarieties in a (general) family of abelian varieties. This extends a result of DeMarco and Mavraki for curves in fibered products of elliptic surfaces. Using this result, one can deduce a uniform version of the classical Bogomolov conjecture for curves embedded in their Jacobians, namely that the number of torsion points lying on them is uniformly bounded in the genus of the curve. This has been previously only known in a few select cases by work of David–Philippon and DeMarco–Krieger–Ye. Finally, one can obtain a rather uniform version of the Mordell-Lang conjecture as well by complementing a result of Dimitrov–Gao–Habegger: The number of rational points on a smooth algebraic curve defined over a number field can be bounded solely in terms of its genus and the Mordell-Weil rank of its Jacobian. Again, this was previously known only under additional assumptions (Stoll, Katz–Rabinoff–Zureick-Brown).

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

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

An evil mathematician has kidnapped the Harvard Pilgrim! To win his freedom, a group of undergrads must each find their name in a row of boxes. The odds look dire—but we’ll use some probability theory and combinatorics to find a strategy that dramatically improves our chances. Can you help save our hapless mascot?

Please go to the College Calendar to register.

‘t Hooft anomalies provide a unique handle to study the nonperturbative infrared dynamics of strongly-coupled theories.  Recently, it has been realized that higher-form global symmetries can also become anomalous, leading to further constraints on the infrared dynamics.  In this talk, I show how one can turn on ‘t Hooft twists in the color, flavor, and baryon number directions in vector-like asymptotically-free gauge theories, which can be used to find new generalized ‘t Hooft anomalies. I give examples of the constraints the generalized anomalies impose on strongly-coupled gauge theories. Then, I argue that the anomaly inflow can explain a non-trivial intertwining that takes place between the light and heavy degrees of freedom on axion domain walls, which leads to the deconfinement of quarks on the walls.  This phenomenon can be explicitly seen in a weakly-coupled model of QCD compactified on a small circle.

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

In the moduli space of one variable complex cubic polynomials with a marked critical point, given any $p \ge 1$, we prove that the loci formed by polynomials with the marked critical point periodic of period $p$ is an irreducible curve.  Our methods rely on techniques used to study one-complex-dimensional parameter spaces.  This is joint work with Matthieu Arfeux.

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

The theory of stable pairs (PT) with descendents, defined on a 3-fold X, is a sheaf theoretical curve counting theory. Conjecturally, it is equivalent to the Gromov-Witten (GW) theory of X via a universal (but intricate) transformation, so we can expect that the Virasoro conjecture on the GW side should have a parallel in the PT world. In joint work with A. Oblomkov, A. Okounkov, and R. Pandharipande, we formulated such a conjecture and proved it for toric 3-folds in the stationary case. The Hilbert scheme of points on a surface S might be regarded as a component of the moduli space of stable pairs on S x P1, and the Virasoro conjecture predicts a new set of relations satisfied by tautological classes on S[n] which can be proven by reduction to the toric case.

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

Title: Topological Recursion and Enumerative Geometry

Abstract: Given a holomorphic curve in the complex 2-plane together with a suitably normalized symmetric meromorphic bilinear differential, the Chekhov-Eynard-Orantin Topological Recursion defines an infinite sequence of symmetric meromorphic multilinear differentials W_{g,n} on the curve. In many examples, the invariants W_{g,n} provide answers to enumerative problems. I will describe Topological Recursion and present several examples in which the answers are Hodge integrals (which are intersection numbers on moduli of curves) or Gromov-Witten invariants (which are virtual counts of holomorphic maps from Riemann surfaces to a Kahler manifold).

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One of the most fascinating aspects about non-commutative spaces (aka von Neumann algebras), is the way their building data, which is often geometric in nature, impacts on the properties of their quantized symmetries. This is particularly the case for II₁ factors, where symmetries are encoded by their subfactors of finite Jones index. I will discuss some results and open problems that illustrate the unique interplay between analysis and algebra/combinatorics entailed by this interdependence, that’s specific to subfactor theory.

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

Estimating the mean of a distribution from i.i.d samples is a fundamental statistical task. In this talk, we will focus on the high-dimensional setting where we will design estimators achieving optimal recovery guarantees in terms of all relevant parameters. While optimal one-dimensional estimators have been known since the 80s (Nemirovskii and Yudin ’83), optimal estimators in high dimensions have only been discovered recently beginning with the seminal work of Lugosi and Mendelson in 2017 and subsequent work has led to computationally efficient variants of these estimators (Hopkins 2018). We will discuss statistical and computational extensions of these results by developing optimal estimators for settings where the data distribution only obeys a finite fractional moment condition as opposed to the existence of a second moment as assumed previously.

Joint work with Peter Bartlett, Nicolas Flammarion, Michael I. Jordan and Nilesh Tripuraneni.

The talk will be based on the following papers: https://arxiv.org/abs/2011.12433, https://arxiv.org/abs/1902.01998.

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

We will report on a recent joint work with Junyan Cao, cf. arXiv:2012.05063. The main topics we will discuss are revolving around the extension of pluricanonical forms defined on the central fiber of a family of Kaehler manifolds. For our results to hold we need the divisor of zeros of the said forms to be sufficiently “nice”, in a sense that will become clear during the talk.

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

For convex complete intersections, the Gromov-Witten (GW) invariants are often computed using the Quantum Lefshetz Hyperplane theorem, which relates the invariants to those of the ambient space. However, even in the genus 0 theory, the convexity condition often fails when the target is an orbifold, and so Quantum Lefshetz is no longer guaranteed. In this talk, I will showcase a method to compute these invariants, despite the failure of Quantum Lefshetz, for a class of orbifold complete intersections. This talk will be based on joint work with Felix Janda (Notre Dame) and Yang Zhou (Harvard) and upcoming work with Rachel Webb (Berkeley).

Zoom: https://harvard.zoom.us/j/96709211410?pwd=SHJyUUc4NzU5Y1d0N2FKVzIwcmEzdz09

Fracton phases show exotic properties, such as sub-extensive entropy, local particle-like excitation with restricted mobility, and so on. In order to find natural fermionic fracton phases, we explore supersymmetric quantum field theory with exotic symmetry. We use superfield formalism and write down the action of a supersymmetric version of one of the simplest models with exotic symmetry, the φ theory in 3+1 dimensions. It contains a large number of ground states due to the fermionic higher pole subsystem symmetry. Its residual entropy is proportional to the area instead of the volume. This theory has a self-duality similar to that of the φ theory. We also write down the action of a supersymmetric version of a tensor gauge theory, and discuss BPS fractons.

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

I’ll explain a very simple proof of the fact that K3 surfaces of finite height admit (many) CM lifts, a result due independently to Ito-Ito-Koshikawa and Z. Yang, which was used by the former to prove the Tate conjecture for products of K3s. This will be done directly showing that the deformation ring of a polarized K3 surface of finite height admits as a quotient that of its Brauer group. The method applies more generally to many isogeny classes of points on Shimura varieties of abelian type.

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

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

In the past, new conjectures about fundamental constants were discovered sporadically by famous mathematicians such as Newton, Euler, Gauss, and Ramanujan. The talk will present a different approach – a systematic algorithmic approach that discovers new mathematical conjectures on fundamental constants. We call this approach “the Ramanujan Machine”. The algorithms found dozens of well-known formulas as well as previously unknown ones, such as continued fraction representations of π, e, Catalan’s constant, and values of the Riemann zeta function. Some of the conjectures are in retrospect simple to prove, whereas others remain so far unproven. We will discuss these puzzles and wider open questions that arose from this algorithmic investigation – specifically, a newly-discovered algebraic structure that seems to generalize all the known formulas and connect between fundamental constants. We will also discuss two algorithms that proved useful in finding conjectures: a variant of the meet-in-the-middle algorithm and a gradient descent algorithm tailored to the recurrent structure of continued fractions. Both algorithms are based on matching numerical values; consequently, they conjecture formulas without providing proofs or requiring prior knowledge of the underlying mathematical structure. This way, our approach reverses the conventional usage of sequential logic in formal proofs; instead, using numerical data to unveil mathematical structures and provide leads to further mathematical research.

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

Quantum systems out of equilibrium can exhibit different dynamical phases that are fundamentally characterized by their entanglement dynamics and entanglement scaling. Random quantum circuits with non-unitarity induced by measurement or other sources provide a large class of systems for us to investigate the nature of these different entanglement phases and associated criticality. While numerical studies have provided a lot of insight into the behavior of such quantum circuit models, analytical understanding of the entanglement criticality in these models has remained challenging in many cases. In this talk, I will focus on the random non-unitary fermionic Gaussian circuits, namely non-unitary circuits for non-interacting fermions. I will first present a numerical study of an entanglement critical phase in this type of circuit. Then, I will discuss the analytical understanding of general entanglement phases in this type of circuit via a general correspondence among (1) non-unitary fermionic Gaussian circuits, (2) fermionic Gaussian tensor network, and (3) unitary non-interacting fermions subject to quenched disorder. In particular, we show that the critical entanglement phase numerically found in the non-unitary Gaussian circuit without any symmetry can be described by the theory of (unitary) disordered metal in the symmetry class DIII. I will comment on the entanglement critical phases that correspond to unitary disordered fermion critical points or unitary disordered metals in other symmetry classes.

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https://www.youtube.com/channel/UCBmPO-OK1sa8T1oX_9aVhAg/playlists (all in playlist)

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

I will discuss recent developments at a new scientific interface between quantum optics, quantum many-body physics, information science and engineering. Specifically, I will focus on two examples at this interface involving realization of programmable quantum systems and their first scientific applications. In the first example, I will describe the recent advances involving programmable, coherent manipulation of quantum many-body systems using atom arrays excited into Rydberg states. Recent progress involving programmable quantum simulations with over 200 qubits in two-dimensional arrays, the exploration of exotic many-body phenomena, as well as realization and testing of quantum optimization algorithms will be discussed. In the second example, I will discuss progress towards realization of quantum repeaters for long-distance quantum communication. Specifically, I will describe experimental realization of memory-enhanced quantum communication, which utilizes a solid-state spin memory integrated in a nanophotonic diamond resonator to implement asynchronous Bell-state measurements. Prospects for scaling up these techniques, including realization of larger quantum processors and quantum networks, as well as their novel applications will be discussed.

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

Title: Recent progress on random field Ising model

Abstract: Random field Ising model is a canonical example to study the effect of disorder on long range order. In 70’s, Imry-Ma predicted that in the presence of weak disorder, the long-range order persists at low temperatures in three dimensions and above but disappears in two dimensions. In this talk, I will review mathematical development surrounding this prediction, and I will focus on recent progress on exponential decay and on correlation length in two dimensions. The talk is based on a joint work with Jiaming Xia and a joint work with Mateo Wirth.

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We introduce a derived version of Lie algebras in characteric p and describe two recent applications: first, we use them to classify infinitesimal deformations, generalising the Lurie-Pridham theorem in characteristic zero; second, we prove a Galois correspondence for purely inseparable field extension, extending work of Jacobson at height one. This talk is based on joint works with Mathew and Waldron.

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

In this talk, I will discuss some structural results for the cohomology of the moduli of semi-stable SL_n Higgs bundles on a curve. One consequence is a new proof of the Hausel-Thaddeus conjecture proven previously by Groechenig-Wyss-Ziegler via p-adic integration. If time permits, we will also discuss the case where the rank of the Higgs bundle is not coprime to the degree, so that the moduli spaces are singular due to the presence of the strictly semi-stable loci. We will explain that how intersection cohomology comes into play naturally. Based on joint work with Davesh Maulik.

Zoom: https://harvard.zoom.us/j/96709211410?pwd=SHJyUUc4NzU5Y1d0N2FKVzIwcmEzdz09

Two-dimensional QCD models form an interesting playground for studying phenomena such as confinement and screening.  In this talk I will describe one such model, namely a 2d SU(N) gauge theory with an adjoint and a fundamental fermion, and explain how to compute the spectrum of bound states using discretized light-cone quantization at large N.  Surprisingly, the spectrum of the discretized theory exhibits a large number of exact degeneracies, for which I will provide two different explanations.  I will also discuss how these degeneracies provide a physical picture of confinement in 2d QCD with just a massless adjoint fermion.  This talk is based on joint work with R. Dempsey and I. Klebanov.

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

Following work of Gross-Wallach, Gan-Gross-Savin defined what are called “modular forms” on the split exceptional group G_2.  These are a special class of automorphic forms on G_2.   I’ll review their definition, and give an update about what is known about them.  Results include a construction of cuspidal modular forms with all algebraic Fourier coefficients, and the exact functional equation of the completed standard L-function of certain cusp forms.  The results on L-functions are joint with Fatma Cicek, Giuliana Davidoff, Sarah Dijols, Trajan Hammonds, and Manami Roy.

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

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

The periodicity of certain functions has been a topic of study throughout the long history of mathematics. It was a topic of study for you in high school trigonometry, and perhaps again in college Fourier analysis!  In this talk, I will revisit some of the ideas related to periodicity and Fourier analysis, and explain some of their relationships to other topics in mathematics, such as group theory and number theory.   In particular, I hope to say something about the important role of *non-abelian* groups of periods in contemporary mathematics.

Please go to the College Calendar to register.

Standard lore suggests that four-dimensional SU(N) gauge theory with 2 massless adjoint Weyl fermions (“adjoint QCD”) flows to a phase with confinement and chiral symmetry breaking. In this two-part talk, we will test and present new evidence for this lore. Our strategy involves realizing adjoint QCD in the deep IR of an RG flow descending from SU(N) Seiberg-Witten theory, deformed by a soft supersymmetry (SUSY) breaking mass for its adjoint scalars. We review what is known about the simplest case N=2, before presenting results for higher values of N. A crucial role in the analysis is played by a dual Lagrangian that originates from the multi-monopole points of Seiberg-Witten theory, and which can be used to explore the phase diagram as a function of the SUSY-breaking mass. The semi-classical phases of this dual Lagrangian suggest that the softly broken SU(N) theory traverses a sequence of phases, separated by first-order transitions, that interpolate between the Coulomb phase of Seiberg-Witten theory and the confining, chiral symmetry breaking phase expected for adjoint QCD.

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

Let K be a complete and algebraically closed field, such as C or the p-adic field C_p, and let f\in K(z) be a rational function of degree d\geq 2. The map f is said to be hyperbolic if there is some metric on its Julia set with respect to which it is expanding. A celebrated 1983 theorem of Mane, Sad, and Sullivan shows that for K=C, hyperbolic maps are J-stable, meaning that nearby maps in moduli space have topologically conjugate dynamics on their Julia sets. In this talk, we show that if K is non-archimedean, an a priori weaker bounded-contraction condition also yields J-stability. This project is joint work with Junghun Lee.

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

Flip is a fundamental surgery operation for constructing minimal models in higher-dimensional birational geometry. In this talk, I will introduce a series of flips from Lie theory and investigate their derived categories. This is a joint program with Conan Leung.

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

Title: Metric representations: Algorithms and Geometry

Abstract: Given a data set or a set of distances amongst data points, determining what combinatorial representation is most “consistent” with the input distances or the metric that best captures the relevant geometric features of the data is a key step in many machine learning algorithms. We seek such representations to gain new insights into the data generation process or to uncover fundamental structures in the data (especially for scientific discovery). We may also seek representations that improve computational efficiency.

In this talk, we focus on a variety of metric representation problems. The first three are specific metric constrained problems, a class of optimization problems with metric constraints: metric nearness, weighted correlation clustering on general graphs, and metric learning. The fourth problem is sparse metric repair, a non-convex version of the metric nearness problem. The final problem seeks a combinatorial representation of data sets in the form of trees or sparse graphs (which we then embed into hyperbolic space).

Because of the large number of constraints in the metric constrained problems, however, these and other researchers have been forced to restrict either the kinds of metrics learned or the size of the problem that can be solved. We provide an algorithm, PROJECT AND FORGET, that uses Bregman projections with cutting planes, to solve metric constrained problems with many (possibly exponentially) inequality constraints.

We discuss the surprising features of the fourth problem, sparse metric repair problem; in one setting it has a simple polynomial time solution and in the other settings, it is fiendishly difficult. Finally, we end with something different. We learn combinatorial structures rather than metrics or data geometry. We then use those to embed the data sets into metric spaces.

This is joint work with a number of collaborators and students: Rishi Sonthalia (Univ. of Michigan), Lalit Jain (Univ. of Washington), Benjamin Raichel (Univ. of Texas-Dallas), and Greg van Buskirk (Univ. of Texas-Dallas).

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The last two decades produced a substantial noncommutative (in the free algebra) real and complex algebraic geometry. The aim of this subject is to develop a systematic theory of equations and inequalities for noncommutative polynomials of operator variables. The talk will focus on a few topics which bear on quantum games, then shift attention to quantum strategies for XOR games.

Two and three player XOR games historically played a major role, with the Bell inequalities an instance of 2XOR. A family of 3XOR games was the first to illustrate unbounded advantage of quantum strategies. Recent results proved with Adam Bene Watts show that one can decide in polynomial time, whether or not a (perfect) solution exists to 3XOR. We do this with a constructive proof: if a perfect quantum strategy exists, it is achievable in 8 dimensions; but the quantum advantage over a classical strategy is bounded.

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

Most work to date on mitigating the COVID-19 pandemic is focused urgently on biomedicine and epidemiology. However, pandemic-related policy decisions cannot be made on health information alone but need to consider the broader impacts on people and their needs. In addition, understanding the disparate impacts of the pandemic and its policies on a full spectrum of human needs, especially for vulnerable populations, is critical for designing response and recovery efforts for major disruptions. Quantifying human needs across the population is challenging as it requires high geo-temporal granularity, high coverage across the population, and appropriate adjustment for seasonal and other external effects. Quantifying disparities across population groups require careful disentanglement of key factors that are engrained in our societal structure. In this talk, I will present computational approaches to leveraging web search interactions as a unique lens through which to examine changes in human needs as well as disparities in the expression of those needs during the COVID-19 pandemic. Grounding our analyses on well-established frameworks of human needs and social determinants of health, I will demonstrate how web search interactions can be used to enhance and complement our understanding of human behaviors during global crises.

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

TITLE: Indistinguishability Obfuscation: How to Hide Secrets within Software

ABSTRACT: At least since the initial public proposal of public-key cryptography based on computational hardness conjectures (Diffie and Hellman, 1976), cryptographers have contemplated the possibility of a “one-way compiler” that translates computer programs into “incomprehensible” but equivalent forms. And yet, the search for such a “one-way compiler” remained elusive for decades.

In this talk, we look back at our community’s attempts to formalize the notion of such a compiler, culminating in our 2001 work with Barak, Goldreich, Impagliazzo, Rudich, Vadhan, and Yang, which proposed the notion of indistinguishability obfuscation (iO). Roughly speaking, iO requires that the compiled versions of any two equivalent programs (with the same size and running time) be indistinguishable to any efficient adversary. Leveraging the notion of punctured programming, introduced in our work with Waters in 2013, well over a hundred papers have explored the remarkable power of iO.

We’ll then discuss the intense effort that recently culminated in our 2020 work with Jain and Lin, finally showing how to construct iO in such a way that, for the first time, we can prove the security of our iO scheme based on well-studied computational hardness conjectures in cryptography.

Talk chair: Sergiy Verstyuk

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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I will present work on establishing the correspondence between the (appropriately defined) Hodge numbers of the moduli spaces of parabolic Higgs bundles for the structure groups SL_n and PGL_n, building on previous results of Groechenig-Wyss-Ziegler on the non-parabolic case. I will first describe the strategy used by Groechenig-Wyss-Ziegler, which combines p-adic integration with the generic duality between the Hitchin systems. Then I will talk about the new ingredients that come into play in the parabolic setting.

Zoom: https://harvard.zoom.us/j/96709211410?pwd=SHJyUUc4NzU5Y1d0N2FKVzIwcmEzdz09

Standard lore suggests that four-dimensional SU(N) gauge theory with 2 massless adjoint Weyl fermions (“adjoint QCD”) flows to a phase with confinement and chiral symmetry breaking. In this two-part talk, we will test and present new evidence for this lore. Our strategy involves realizing adjoint QCD in the deep IR of an RG flow descending from SU(N) Seiberg-Witten theory, deformed by a soft supersymmetry (SUSY) breaking mass for its adjoint scalars. We review what is known about the simplest case N=2, before presenting results for higher values of N. A crucial role in the analysis is played by a dual Lagrangian that originates from the multi-monopole points of Seiberg-Witten theory, and which can be used to explore the phase diagram as a function of the SUSY-breaking mass. The semi-classical phases of this dual Lagrangian suggest that the softly broken SU(N) theory traverses a sequence of phases, separated by first-order transitions, that interpolate between the Coulomb phase of Seiberg-Witten theory and the confining, chiral symmetry breaking phase expected for adjoint QCD.

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

DeepWalk is an approach we have developed to construct vertex embeddings: vector representations of vertices which be applied to a very general class of problems in data mining and information retrieval.  DeepWalk exploits an appealing analogy between sentences as sequences of words and random walks as sequences of vertices to transfer deep learning (unsupervised feature learning) techniques from natural language processing to network analysis.  It has become extremely popular, having been cited by over 4600 research papers since its publication at KDD 2014.   In this talk, I will introduce the notion of graph embeddings, and demonstrate why they make such powerful features for machine learning applications.  I will focus on more recent efforts concerning (1) fast embedding methods for very large networks, (2) techniques for embedding dynamic graphs, and (3) embedding spaces as models for knowledge generation.

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

I’ll discuss 4 different types of integration — one in the complex setting, one in the tropical setting, and two in the p-adic setting, and the relationships between them. In particular, we explain how to compute Vologodsky’s “single-valued” iterated integrals on curves of bad reduction in terms of Berkovich integrals, and how to give a single-valued integration theory on complex varieties. Time permitting, I’ll explain some potential arithmetic applications. This is a report on joint work in progress with Sasha Shmakov (in the complex setting) and Eric Katz (in the p-adic setting).

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

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

Differential topology is the study of smooth manifolds. I hope to tell you where the frontier lies between knowledge and ignorance with regards to smooth 4-dimensional manifolds (which is by far the hardest dimension to understand).

Zoom: https://harvard.zoom.us/j/98248914765?pwd=Q01tRTVWTVBGT0lXek40VzdxdVVPQT09

String-net models are exactly solvable lattice models that can realize a large class of (2+1)D topological phases. I will review basic aspects of these models, including their Hamiltonians, ground-state wave functions, and anyon excitations. I will also discuss the relationship between the original string-net models, proposed in 2004, and the more recent, “generalized’’, string-net models.

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

Quantum cohomology is a deformation of the cohomology of a projective variety governed by counts of stable maps from a curve into this variety. Quantum K-theory is in a similar way a deformation of K-theory but also of quantum cohomology, It has recently attracted attention in physics since a realization in a physical theory has been found. Currently, both the structure and examples in quantum K-theory are far less understood than in quantum cohomology.
We will explain the properties of quantum K-theory in comparison with quantum cohomology, and we will discuss the examples of projective space and the quintic hypersurface in P^4.

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

In the first part of the talk, I will establish the rationality of generating series formed from Euler characteristics of tautological bundles over Hilbert schemes of points on surfaces. In the second part, I will present results on virtual invariants of Quot schemes parameterizing rank zero quotients of trivial bundles on surfaces. The second part of the talk is based on work with Y. Kononov and work with D. Johnson, W. Lim, D. Oprea and R. Pandharipande.

Zoom: https://harvard.zoom.us/j/96709211410?pwd=SHJyUUc4NzU5Y1d0N2FKVzIwcmEzdz09

TITLE: On the History of quantum cohomology and homological mirror symmetry

ABSTRACT: About 30 years ago, string theorists made remarkable discoveries of hidden structures in algebraic geometry.  First, the usual cup-product on the cohomology of a complex projective variety admits a canonical multi-parameter deformation to so-called quantum product, satisfying a nice system of differential equations (WDVV equations).  The second discovery, even more striking,  is Mirror Symmetry, a duality between families of Calabi-Yau varieties acting as a mirror reflection on the Hodge diamond.

Later it was realized that the quantum product belongs to the realm of symplectic geometry, and a half of mirror symmetry (called Homological Mirror Symmetry) is a duality between complex algebraic and symplectic varieties. The search of correct definitions and possible generalizations lead to great advances in many domains, giving mathematicians new glasses, through which they can see familiar objects in a completely new way.

I will review the history of major mathematical advances in the subject of HMS, and the swirl of ideas around it.

Talk chair: Paul Seidel

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.

The hyperfinite II1 factor contains a wealth of subfactors, many of which give rise to new and fascinating mathematical structures. For instance, the standard representation of a subfactor generates a certain unitary tensor category that Jones described as (what he called) a “planar algebra.” It is a complete invariant for amenable, hyperfinite subfactors due to a deep result of Popa. However, generic subfactors are not amenable, and one typically does not know how to distinguish them. I will discuss a notion of “noncommutativity” for a subfactor that provides an invariant that is complementary to the planar algebra. Bare hand constructions of hyperfinite subfactors generally lead to “commutative” examples, and I will explain a theorem that allows us to produce “very noncommutative” ones as well. It involves actions of suitable groups on the hyperfinite II1 factor.

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

This describes joint work in progress with M. Mustata, in which we study the filtration on local cohomology sheaves induced by their natural mixed Hodge module structure. Special properties of this filtration, for instance strictness, lead to a number of different applications, including an injectivity theorem for dualizing complexes, local vanishing for forms with log poles, and especially a characterization of the local cohomological dimension of a closed subscheme in terms of  data arising from a log resolution of singularities.

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

The magnetoroton is the neutral excitation of a gapped fractional quantum Hall state. We argue that at zero momentum the magnetoroton has spin ±2, and show how the spin of the magnetoroton can be determined by polarized Raman scattering. We suggest that polarized Raman scattering may help to determine the nature of the ν=5/2 state. Ref: D.X. Nguyen and D.T. Son, arXiv:2101.02213.

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

The O’Connell-Yor polymer is a fundamental model of a polymer in a random environment. It corresponds to the positive temperature version of Brownian Last Passage percolation. Although much is known about this model thanks to remarkable algebraic structure uncovered by O’Connell, Yor and others, basic estimates for the behavior of the tails of the centered partition function for finite N that are available for zero temperature models are missing. I will present an iterative estimate to obtain strong concentration and localization bounds for the O’Connell-Yor polymer on an almost optimal scale N^{1/3+\epsilon}.

In the second part of the talk, I will introduce a system of interacting diffusions describing the successive increments of partition functions of different sizes. For this system, the N^{2/3} variance upper bound known for the OY polymer can be proved for a general class of interactions which are not expected to correspond to integrable models.

Joint work with Christian Noack and Benjamin Landon.

Zoom: https://harvard.zoom.us/j/99333938108?pwd=eklLTS9qaGVrWWx5elJWb2IrS284Zz09

Are proof assistants relevant to mathematics? One approach to this question is to explore the breadth of mathematical topics that can be formalised. The partition calculus was introduced by Erdös and R. Rado in 1956 as the study of “analogues and extensions of Ramsey’s theorem”. Highly technical results were obtained by Erdös-Milner, Specker and Larson (among many others) for the particular case of ordinal partition relations, which is concerned with countable ordinals and order types. Much of this material was formalised last year (with the assistance of Džamonja and Koutsoukou-Argyraki). Some highlights of this work will be presented along with general observations about the formalisation of mathematics, including ZFC, in simple type theory.

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