We study the supersymmetric partition function of a 2d linear sigma-model whose target space is a torus with a complex structure that varies along one worldsheet direction and a Kähler modulus that varies along the other. This setup is inspired by the dimensional reduction of a Janus configuration of 4d N=4 U(1) Super-Yang-Mills theory compactified on a mapping torus (T^2 fibered over S^1) times a circle with an SL(2,Z) duality wall inserted on S^1, but our setup has minimal supersymmetry. The partition function depends on two independent elements of SL(2,Z), one describing the duality twist, and the other describing the geometry of the mapping torus. It is topological and can be written as a multivariate quadratic Gauss sum. By calculating the partition function in two different ways, we obtain identities relating different quadratic Gauss sums, generalizing the Landsberg-Schaar relation. These identities are a subset of a collection of identities discovered by F. Deloup. Each identity contains a phase which is an eighth root of unity, and we show how it arises as a Berry phase in the supersymmetric Janus-like configuration. Supersymmetry requires the complex structure to vary along a semicircle in the upper half-plane, as shown by Gaiotto and Witten in a related context, and that semicircle plays an important role in reproducing the correct Berry phase.

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

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

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

Least-mean squares (LMS) solvers such as Linear / Ridge / Lasso-Regression, SVD and Elastic-Net not only solve fundamental machine learning problems, but are also the building blocks in a variety of other methods, such as decision trees and matrix factorizations.

We suggest an algorithm that gets a finite set of $n$ $d$-dimensional real vectors and returns a weighted subset of $d + 1$ vectors whose sum is exactly the same. The proof in Caratheodory’s Theorem (1907) computes such a subset in $O(n^2 d^2 )$ time and thus not used in practice. Our algorithm computes this subset in $O(nd)$ time, using $O(logn)$ calls to Caratheodory’s construction on small but “smart” subsets. This is based on a novel paradigm of fusion between different data summarization techniques, known as sketches and coresets.

As an example application, we show how it can be used to boost the performance of existing LMS solvers, such as those in scikit-learn library, up to $x100$. Generalization for streaming and distributed (big) data is trivial. Extensive experimental results and complete open source code are also provided.

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

Determining the computational complexity of matrix multiplication has been one of the central open problems in theoretical computer science ever since in 1969 Strassen presented an algorithm for multiplication of n by n matrices requiring only O(n^2.81) arithmetic operations. I will briefly discuss this problem and its reduction to deciding on which secant variety to the Segre embedding of a product of three projective spaces the matrix multiplication tensor lies. I will explain a recent technique to rule out membership of a fixed tensor in such secant varieties, border apolarity. Border apolarity establishes the existence of certain multigraded ideals implied by membership in a particular secant variety. These ideals may be assumed to be fixed under a Borel subgroup of the group of symmetries of the tensor, and in the simplest case, can consequently be tractably shown not to exist. When ideals exist satisfying the easily checkable properties, one must decide if they are limits of ideals of distinct points on the Segre. This talk discusses joint work with JM Landsberg, Alicia Harper, and Amy Huang.

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

The SSK model was introduced by Kosterlitz, Thouless and Jones as a simplification of the usual SK model with Ising spins. Fluctuations of its observables may be related to quantities from random matrix theory using integral representations.  In this informal talk we discuss some results on fluctuations of this model at critical temperature and with a magnetic field.

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

For any positive integer $n$, we explain why the total number of order $n$ elements in class groups of quadratic fields of discriminant having absolute value at most $X$ is $O_n(X^{5/4})$.

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

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

The increasing dimensionality of data in the modern machine learning age presents new challenges and opportunities. The high-dimensional settings allow one to use powerful asymptotic methods from probability theory and statistical physics to obtain precise characterizations and develop new algorithmic approaches. Statistical mechanics approaches, in particular, are very well suited for such problems. I will give examples of recent works in our group that build on powerful methods of statistical physics of disordered systems to analyze some relevant questions in machine learning and neural networks, including overparameterization, kernel methods, and the behavior gradient descent algorithm in a high dimensional non-convex landscape.

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

I’ll talk about basic ruler and compass constructions, about math as exploration, about making space in mathematics, and about Harvard. Oh, and of course, the election.

Please go to the College Calendar to register.

The analysis of complex systems often hinges on our ability to extract the relevant degrees of freedom from among the many others. Recently the information bottleneck (IB), a signal processing tool, was proposed as an unbiased means for such order parameter extraction. While IB optimization was considered intractable for many years, new deep-learning-based techniques seem to solve it quite efficiently. In this talk, I’ll introduce IB in the real-space renormalization context (a.k.a. RSMI), along with two recent theoretical results. One links IB optimization to the short-rangeness of coarse-grained Hamiltonians. The other provides a dictionary between the quantities extracted in IB, understood only qualitatively thus far, and relevant operators in the underlying field theory (or eigenvectors of the transfer matrix). Apart from relating field-theory and information, these results suggest that deep learning in conjunction with IB can provide useful and interpretable tools for studying complex systems.

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

Lacunary trigonometric sums are known to exhibit several properties that are typical of sums of iid random variables such as the central limit theorem, established by Salem and Zygmund, and the law of the iterated logarithm, due to Erdos and Gal.  We initiate an investigation of large deviation principles for such sums, and show that they display several interesting features, including sensitivity to the arithmetic properties of the corresponding lacunary sequence.  This is joint work with C. Aistleitner, N. Gantert, Z. Kabluchko and J. Prochno.

Zoom: https://brandeis.zoom.us/j/93794552542

High‐temperature superconductivity in cupper oxides, with critical temperature well above what was anticipated by the BCS theory, remains a major unsolved physics problem. The problem is fascinating because it is simultaneously simple ‐ being a single band and 1⁄2 spin system, yet extremely rich ‐ boasting d‐wave superconductivity, pseudogap, spin and charge orders, and strange metal phenomenology. For this reason, cuprates emerge as the most important model system for correlated electrons – stimulating conversations on the physics of Hubbard model, quantum critical point, Planckian metal and beyond. Central to this debate is whether the Hubbard model, which is the natural starting point for the undoped magnetic insulator, contains the essential ingredients for key physics in cuprates. In this talk, I will discuss our photoemission evidence for a multifaceted answer to this question [1‐3]. First, we show results that naturally points to the importance of Coulomb and magnetic interactions, including d‐wave superconducting gap structure [4], exchange energy (J) control of bandwidth in single‐hole dynamics [5]. Second, we evidence effects beyond the Hubbard model, including band dispersion anomalies at known phonon frequencies [6, 7], polaronic spectral lineshape and the emergence of quasiparticle with doping [8]. Third, we show properties likely of hybrid electronic and phononic origin, including the pseudogap [9‐11], and the almost vertical phase boundary near the critical 19% doping [12]. Fourth, we show examples of small q phononic coupling that cooperates with d‐wave superconductivity [13‐15]. Finally, we discuss recent experimental advance in synthesizing and investigating doped one‐dimensional (1D) cuprates [16]. As theoretical calculations of the 1D Hubbard model are reliable, a robust comparison can be carried out. The experiment reveals a near‐neighbor attractive interaction that is an order of magnitude larger than the attraction generated by spin‐superexchange in the Hubbard model. Addition of such an attractive term, likely of phononic origin, into the Hubbard model with canonical parameters provides a quantitative explanation for all important experimental observable: spinon and holon dispersions, and holon‐ holon attraction. Given the structural similarity of the materials, It is likely that an extended two‐dimensional (2D) Hubbard model with such an attractive term, will connect the dots of the above four classes of experimental observables and provide a holistic understanding of cuprates, including the elusive d‐wave superconductivity in 2D Hubbard model.

[1] A. Damascelli, Z. Hussain, and Z.‐X. Shen, Review of Modern Physics, 75, 473 (2003)
[2] M. Hashimoto et al., Nature Physics 10, 483 (2014)
[3] JA Sobota, Y He, ZX Shen ‐ arXiv preprint arXiv:2008.02378, 2020; submitted to Rev. of Mod. Phys.
[4] Z.‐X. Shen et al., Phys. Rev. Lett. 70, 1553 (1993)
[5] B.O. Wells et al., Phys. Rev. Lett. 74, 964 (1995)
[6] A. Lanzara et al., Nature 412, 510 (2001)
[7] T. Cuk et al., Phys. Rev. Lett., 93, 117003 (2004)
[8] K.M. Shen et al., Phys. Rev. Lett., 93, 267002 (2004)
[9] D.M. King et al., J. of Phys. & Chem of Solids 56, 1865 (1995)
[10] D.S. Marshall et al., Phy. Rev. Lett. 76, 484 (1996)
[11] A.G. Loeser et al., Science 273, 325 (1996)
[12] S. Chen et al., Science, 366, 6469 (2019)
[13] T.P. Devereaux, T. Cuk, Z.X. Shen, N. Nagaosa, Phys. Rev. Lett., 93, 117004 (2004)
[14] S. Johnston et al., Phys. Rev. Lett. 108, 166404 (2012)
[15] Yu He et al., Science, 362, 62 (Oct. 2018)
[16] Z. Chen, Y. Wang et al., preprint, 2020

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

I shall discuss a recent work on how p-adic strings can produce perturbative quantum gravity, and an adelic physics interpretation of Tate’s thesis.

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

How does a child learn to speak, without prior direct communication, nor with having dictionary to translate words from another language? How do we learn to play chess, with no prior intuition about a myriad of different positions on the board nor with tactics to achieve those positions? How do scientists manage to move into the unknown, with no one guiding them through the right steps? And, how do they discover the previously unknown “right steps,” tools, and techniques in the first place? Curiously, there are many questions like these, which we face on a day-to-day basis and to which we have no good answers. Yet, we all find ways to make progress. How is it possible? We will take a look at this magic process by putting the smooth 4-dimensional Poincaré conjecture into the framework of Natural Language Processing (NLP).

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

Given a variation of Hodge structures $V$ on a smooth complex quasi-projective variety $S$, its Hodge locus is the set of points $s$ in $S$ where the Hodge structure $V_s$ admits exceptional Hodge tensors. A famous result of Cattani, Deligne and Kaplan shows that this Hodge locus is a countable union of irreducible algebraic subvarieties of $S$, called the special subvarieties of $(S, V)$. In this talk I will discuss the geometry of the Zariski closure of the union of the positive dimensional special subvarieties. This is joint work with Ania Otwinowska.

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

How does a child learn to speak, without prior direct communication, nor with having dictionary to translate words from another language? How do we learn to play chess, with no prior intuition about a myriad of different positions on the board nor with tactics to achieve those positions? How do scientists manage to move into the unknown, with no one guiding them through the right steps? And, how do they discover the previously unknown “right steps,” tools, and techniques in the first place? Curiously, there are many questions like these, which we face on a day-to-day basis and to which we have no good answers. Yet, we all find ways to make progress. How is it possible? We will take a look at this magic process by putting the smooth 4-dimensional Poincaré conjecture into the framework of Natural Language Processing (NLP).

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

Recent advances in deep learning exploit the structure in data and architectures. Graph Neural Network (GNN) is a powerful framework for learning with graph-structured objects, and for learning the interaction of objects on a graph. Applications include recommender systems, drug discovery, physical and visual reasoning, program synthesis, and natural language processing.

In this talk, we study GNNs from the following aspects: expressive power, generalization, and extrapolation. We characterize the expressive power of GNNs from the perspective of graph isomorphism tests. We show an upper bound that GNNs are at most as powerful as a Weisfeiler-Lehman test. We then show conditions to achieve this upper bound, and present a maximally powerful GNN. Next, we analyze the generalization of GNNs. The optimization trajectories of over-parameterized GNNs trained by gradient descent correspond to those of kernel regression using a specific graph neural tangent kernel. Using this relation, we show GNNs provably learn a class of functions on graphs. More generally, we study how the architectural inductive biases influence generalization in a task. We introduce an algorithmic alignment measure, and show better alignment implies better generalization. Our framework suggests GNNs can sample-efficiently learn dynamic programming algorithms. Finally, we study how neural networks trained by gradient descent extrapolate, i.e., what they learn outside the support of the training distribution (e.g., on larger graphs or edge weights). We prove a linear extrapolation behavior of ReLU multilayer perceptrons (MLPs), and identify conditions under which MLPs and GNNs extrapolate well. Our results suggest how a good representation or architecture can help extrapolation.

Talk based on:
https://arxiv.org/abs/1810.00826 
ICLR’19 (oral)
https://arxiv.org/abs/1905.13192 
NeurIPS’19
https://arxiv.org/abs/1905.13211 
ICLR’20 (spotlight)
https://arxiv.org/abs/2009.11848 

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

The 2D Toda system consists of a complicated set of infinitely many coupled PDEs in infinitely many variables that is known to assemble into an infinite-dimensional integrable system. Krichever and Zabrodin made the remarkable observation that the poles of some special meromorphic solutions to the 2D Toda system are known to evolve in time according to the Ruijsenaars-Schneider many particle integrable system. In this talk I will describe work in progress to establish this 2D Toda-RS correspondence via a Fourier-Mukai equivalence of derived categories: a category of “RS spectral sheaves” on one side, and a category of “Toda micro-difference operators” on another. This description of the 2D Toda-RS correspondence mirrors that of the KP-CM corrspondence previously established by two of the authors and suggests the existence of a conjectural elliptic integrable hierarchy.

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

Symmetry protected topological (SPT) phases are inevitable phases of quantum matter that are distinct from trivial phases only in the presence of unbroken global symmetries. These are characterized by anomalous boundaries which host emergent symmetries and protected degeneracies and gaplessness. I will present results from an ongoing series of works with Juven Wang on boundary symmetries of fermionic SPT phases, generalizing a previous work: arxiv:1804.11236. In 1+1 d, I will argue that the boundary of all intrinsically fermionic SPT phases can be recast as supersymmetric (SUSY) quantum mechanical systems and show that by extending the boundary symmetry to that of the bulk, all fermionic SPT phases can be unwound to the trivial phase. I will also present evidence that boundary SUSY seems to be present in various higher dimensional examples also and might even be a general feature of all intrinsically fermionic SPT phases.

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

For a smooth proper (formal) scheme X defined over a valuation ring of mixed characteristic, the crystalline cohomology H of its special fiber has the structure of an F-crystal, to which one can attach a Newton polygon and a Hodge polygon that describe the ”slopes of the Frobenius action on H”. The shape of these polygons are constrained by the geometry of X — in particular by the Hodge numbers of both the special fiber and the generic fiber of X. One instance of such constraints is given by a beautiful conjecture of Katz (now a theorem of Mazur, Ogus, Nygaard etc.), another constraint comes from the notion of “weakly admissible” Galois representations.

In this talk, I will discuss some results regarding the shape of the Frobenius action on the F-crystal H and the Hodge numbers of the generic fiber of X, along with generalizations in several directions. In particular, we give a new proof of the fact that the Newton polygon of the special fiber of X lies on or above the Hodge polygon of its generic fiber, without appealing to Galois representations. A new ingredient that appears is (a generalized version of) the Nygaard filtration of the prismatic/Ainf cohomology, developed by Bhatt, Morrow and Scholze.

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

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

The complexity of biological systems (among others) makes demands on the complexity of the mathematical modeling enterprise that could be satisfied with mathematical artificial intelligence of both symbolic and numerical flavors. Technologies that I think will be fruitful in this regard include (1) the use of machine learning to bridge spatiotemporal scales, which I will illustrate with the “Dynamic Boltzmann Distribution” method for learning model reduction of stochastic spatial biochemical networks and the “Graph Prolongation Convolutional Network” approach to course-graining the biophysics of microtubules; (2) a meta-language for stochastic spatial graph dynamics, “Dynamical Graph Grammars”, that can represent structure-changing processes including microtubule dynamics and that has an underlying combinatorial theory related to operator algebras; and (3) an integrative conceptual architecture of typed symbolic modeling languages and structure-preserving maps between them, including model reduction and implementation maps.

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

We consider eigenvector statistics of large symmetric random matrices. When the matrix entries are sampled from independent Gaussian random variables, eigenvectors are uniformly distributed on the sphere and numerous properties can be computed exactly. In particular, we can bound their extremal coordinates with high probability. There has been an extensive amount of work on generalizing such a result, known as delocalization, to more general entry distributions. After giving a brief overview of the previous results going in this direction, we present an optimal delocalization result for matrices with sub-exponential entries for all eigenvectors. The proof is based on the dynamical method introduced by Erdos-Yau, an analysis of high moments of eigenvectors as well as new level repulsion estimates which will be presented during the talk. This is based on a joint work with P. Lopatto.

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

This talk is organized as follows:
1. Physical Principles leading to Loop-current order and quantum criticality as the central feature in the physics of Cuprates.
2. Summary of the essentially exact solution of the dissipative xy model for Loop-current fluctuations.
3. Quantitative comparison of theory for the quantum-criticality with a variety of experiments.
4. Topological decoration of loop-current order to understand ”Fermi-arcs” and small Fermi-surface magneto-oscillations.

Time permitting,
(i) Quantitative theory and experiment for fluctuations leading to d-wave superconductivity.
(ii) Extensions to understand AFM quantum-criticality in heavy-fermions and Fe-based superconductors.
(iii) Problems.

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

A multidimensional polynomial system is said to be sparse when the monomials present in the polynomials are fixed a priori. I will present classic and recent upper and lower bounds for the number of positive solutions of systems of n sparse real polynomials in n variables. I will also discuss basic open questions.

Zoom: https://northeastern.zoom.us/j/93522278073?pwd=REdwenR0Z0RWSHJNeWJDYW8wREErUT09

For password email Andrew McGuinness

TITLE: Knot Invariants From Gauge Theory in Three, Four, and Five Dimensions

ABSTRACT: I will explain connections between a sequence of theories in two, three, four, and five dimensions and describe how these theories are related to the Jones polynomial of a knot and its categorification.

Talk chair: Cliff Taubes

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.

TITLE: Classical and quantum integrable systems in enumerative geometry

ABSTRACT: For more than a quarter of a century, thanks to the ideas and questions originating in modern high energy physics, there has been a very fruitful interplay between enumerative geometry and integrable system, both classical and quantum. While it impossible to summarize even the most important aspects of this interplay in one talk, I will try to highlight a few logical points with the goal to explain the place and the role of certain more recent developments.

Talk chair: Cumrun Vafa

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.

We report on a new development in asymptotic Hodge theory, arising from work of Golyshev–Zagier and Bloch–Vlasenko, and connected to the Gamma Conjectures in Fano/LG-model mirror symmetry. The talk will focus exclusively on the Hodge/period-theoretic aspects through two main examples. Given a variation of Hodge structure M on a Zariski open in P^1, the periods of the limiting mixed Hodge structures at the punctures are interesting invariants of M.  More generally, one can try to compute these asymptotic invariants for iterated extensions of M by “Tate objects”, which may arise for example from normal functions associated to algebraic cycles. The main point of the talk will be that (with suitable assumptions on M) these invariants are encoded in an entire function called the motivic Gamma function, which is determined by the Picard-Fuchs operator L underlying M. In particular, when L is hypergeometric, this is easy to compute and we get a closed-form answer (and a limiting motive).  In the non-hypergeometric setting, it yields predictions for special values of normal functions; this part of the story is joint with V. Golyshev and T. Sasaki.

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

Consider a free group and its group von Neumann algebra A. Finding
criteria on the boundedness or complete boundedness of multipliers on the Lp(A) is a major subject of analysis on free groups. A remarkable result of U↵e Haagerup and his co-authors characterizes the completely bounded radial Fourier multipliers on A (i.e., for p = 1). However, the case of finite p 6= 2 is a considerably more delicate matter, as it is for abelian groups. One of very few existing significant results is that on the free Hilbert transform recently proved by Tao Mei and Eric Ricard. In this talk I will present some new work, joint with these authors. A more-detailed abstract can be found in the seminar announcement at https://mathpicture.fas.harvard.edu/seminar.

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

Economics and social science research often require analyzing datasets of sensitive personal information at fine granularity, with models fit to small subsets of the data. Unfortunately, such fine-grained analysis can easily reveal sensitive individual information. We study algorithms for simple linear regression that satisfy differential privacy, a constraint which guarantees that an algorithm’s output reveals little about any individual input data record, even to an attacker with arbitrary side information about the dataset. We consider the design of differentially private algorithms for simple linear regression for small datasets, with tens to hundreds of datapoints, which is a particularly challenging regime for differential privacy. Focusing on a particular application to small-area analysis in economics research, we study the performance of a spectrum of algorithms we adapt to the setting. We identify key factors that affect their performance, showing through a range of experiments that algorithms based on robust estimators (in particular, the Theil-Sen estimator) perform well on the smallest datasets, but that other more standard algorithms do better as the dataset size increases. See https://arxiv.org/abs/2007.05157 for more details.

Joint work with Audra McMillan, Jayshree Sarathy, Adam Smith, and Salil Vadhan.

If time permits, I will chronicle past work on differentially private linear regression, discussing previous works on distributed linear regression and hypothesis testing in the general linear model.

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

I will discuss recent work on computing the top weight cohomology of A_g for g up to 7. We use combinatorial methods coming from the relationship between the top weight cohomology of A_g and the homology of the link of the moduli space of tropical abelian varieties to carry out the computation. This is joint work with Madeline Brandt, Juliette Bruce, Melody Chan, Margarida Melo, and Corey Wolfe.

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

I will discuss the problem of understanding the collapsing behavior of Ricci-flat Kahler metrics on a Calabi-Yau manifold that admits a holomorphic fibration structure, when the Kahler class degenerates to the pullback of a Kahler class from the base. I will present recent work with Hans-Joachim Hein where we obtain a priori estimates of all orders for the Ricci-flat metrics away from the singular fibers, as a corollary of a complete asymptotic expansion.

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

TITLE: Log Calabi-Yau fibrations

ABSTRACT: Fano and Calabi-Yau varieties play a fundamental role in algebraic geometry, differential geometry, arithmetic geometry, mathematical physics, etc. The notion of log Calabi-Yau fibration unifies Fano and Calabi-Yau varieties, their fibrations, as well as their local birational counterparts such as flips and singularities. Such fibrations can be examined from many different perspectives. The purpose of this talk is to introduce the theory of log Calabi-Yau fibrations, to remind some known results, and to state some open problems.

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.

Simplified as it is, the Hubbard model embodies much of the complexity of the `strong correlation problem’ and has established itself as a paradigmatic model in the field. In this talk, I will argue that several key aspects of its physics in two dimensions can now be established beyond doubt, thanks to the development of controlled and accurate computational methods. These methods implement different and complementary points of view on the quantum many-body problem. Along with pushing forward each method, the community has recently embarked into a major effort to combine and critically compare these approaches, and in several instances a consistent picture of the physics has emerged as a result. I will review in this perspective our current understanding of the emergence of a pseudogap in both the weak and strong coupling regimes. I will present recent progress in understanding how the pseudogap phase may evolve into a stripe-dominated regime at low temperature, and briefly address the delicate question of the competition between stripes and superconductivity. I will also emphasize outstanding questions which are still open, such as the possibility of a Fermi surface reconstruction without symmetry breaking. Whenever possible, connections to the physics of cuprate superconductors will be made. If time permits, I may also address the question of Planckian transport and bad metallic transport at high temperature.

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

Hierarchical spin glasses such as the generalised random energy model are known to faithfully model typical energy landscapes in the classical theory of mean-field spin glasses. Their built-in hierarchical structure is known to emerge spontaneously in the spin-glass phase of, e.g., the Sherrington-Kirkpatrick model. In this talk, I will review recent results on the effects of a transversal magnetic field on such hierarchical quantum spin glasses.
In particular, I will present a formula of Parisi-type for their free energy which allows to make predictions about the phase diagram.

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

We review how historically the problem of string phenomenology lead theoretical physics first to algebraic/diffenretial geometry, and then to computational geometry, and now to data science and AI.
With the concrete playground of the Calabi-Yau landscape, accumulated by the collaboration of physicists, mathematicians and computer scientists over the last 4 decades, we show how the latest techniques in machine-learning can help explore problems of physical and mathematical interest, from geometry, to group theory, to combinatorics and number theory.

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

The well-known classical Eichler-Shimura relation for modular curves asserts that the Hecke operator $T_p$ is equal, as an algebraic correspondence over the special fiber, to the sum of Frobenius and Verschebung. Blasius and Rogawski proposed a generalization of this result for general Shimura varieties with good reduction at $p$, and conjectured that the Frobenius satisfies a certain Hecke polynomial. I will talk about a recent proof of this conjecture for Shimura varieties of Hodge type, assuming a technical condition on the unramified sigma-conjugacy classes in the associated Kottwitz set.

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

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

I will argue that the orders that are present in high temperature superconductors naturally arise with the same strength and are better regarded as intertwined rather than competing. I illustrate this concept in the context of the orders that are present in the pair-density-wave state and the phase diagrams that result from this analysis.

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

TITLE: Homotopy spectra and Diophantine equations

ABSTRACT: For a long stretch of time in the history of mathematics, Number Theory and Topology formed vast, but disjoint domains of mathematical knowledge.

Origins of number theory can be traced back to the Babylonian clay tablet Plimpton 322 (about 1800 BC)  that contained a list of integer solutions of the “Diophantine” equation $a^2+b^2=c^2$: archetypal theme of number theory, named after Diophantus of Alexandria (about 250 BC).

Topology was born much later, but arguably, its cousin — modern measure theory, — goes back  to Archimedes, author of Psammites (“Sand Reckoner”), who was approximately a contemporary of Diophantus.

In modern language, Archimedes measures the volume of observable universe by counting the number of small grains of sand necessary to fill this volume. Of course, many qualitative geometric models and quantitative estimates of the relevant distances precede his calculations. Moreover, since the estimated numbers of grains of sands are quite large (about $10^{64}$), Archimedes had to invent and describe a system of notation for large numbers going far outside the possibilities of any of the standard ancient systems.

The construction of the first bridge between number theory and topology  was accomplished only about fifty years ago: it is the theory of spectra in stable homotopy theory.

In particular, it connects $Z$, the initial object in the theory of commutative rings, with the sphere spectrum $S$.

This connection poses the challenge: discover a new information in number theory using the developed independently machinery of homotopy theory.

In this this talk based upon the authors’ (Yu. Manin and M. Marcolli) joint research project, I suggest to apply homotopy spectra to the problem of distribution of rational points upon algebraic manifolds.

Talk chair: Michael Hopkins

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.

TITLE: Noncommutative Geometry, the Spectral Aspect

ABSTRACT: This talk will be a survey of the spectral side of noncommutative geometry, presenting the new paradigm of spectral triples and showing its relevance for the fine structure of space-time, its large scale structure and also in number theory in connection with the zeros of the Riemann zeta function.

Talk chair: Peter Kronheimer

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

For more information, please visit the event page.

Register here to attend.

TITLE: Subfactors–in Memory of Vaughan Jones

ABSTRACT: Jones initiated modern subfactor theory in early 1980s and investigated this area for his whole academic life. Subfactor theory has both deep and broad connections with various areas in mathematics and physics. One well-known peak in the development of subfactor theory is the discovery of the Jones polynomial, for which Jones won the Fields Metal in 1990. Let us travel back to the dark room at the beginning of the story, to appreciate how radically our viewpoint has changed.

Talk chair: Arthur Jaffe

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.

Derived categories and motives are important invariants of algebraic varieties invented by Grothendieck and his collaborators around 1960s. In 2005, Orlov conjectured that they will be closely related and now there are several evidences supporting his conjecture. On the other hand, moduli spaces of vector bundles on curves provide attractive and important examples of algebraic varieties and there have been intensive works studying them. In this talk, I will discuss derived categories and motives of moduli spaces of vector bundles on curves. This talk is based on joint works with I. Biswas and T. Gomez.

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

Multi-robot systems are already playing a crucial role in manufacturing, warehouse automation, and natural resource monitoring, and in the future they will be employed in even broader domains from space exploration to search-and-rescue. Moreover, these systems will likely be incorporated in our daily lives through drone delivery services and smart mobility systems that comprise of thousands of autonomous vehicles. The anticipated benefits of multi-robot systems are numerous, ranging from automating dangerous jobs, to broader societal facets such  as easing traffic congestion and sustainability. However, to reap those rewards we must develop control mechanisms for such systems that can adapt rapidly to unexpected changes on a massive scale. Importantly, these mechanisms must capture: (i) dynamical and collision-avoidance constraints of individual robots; (ii) interactions between multiple robots; and (iii) more broadly, the  interaction of those systems with the environment. All these considerations give rise to extremely complex and high-dimensional optimization problems that need to be solved in real-time.

In this talk I will present recent progress on the design of algorithms for  control and decision-making to allow the safe, effective, and societally-equitable deployment of multi-robot systems. I will highlight both results on fundamental capabilities for multi-robot systems (e.g., motion planning and task allocation), as well as applications in smart mobility, including multi-drone delivery and autonomous mobility-on-demand systems. Along the way, I will mention a few related open problems in mathematics and algorithm design.

BIO:
Kiril Solovey is roboticist specializing in multi-robot systems and their applications to smart mobility. He is currently a Postdoctoral Scholar at the Department of Aeronautics and Astronautics, Stanford University, working with Marco Pavone, where he is supported by the Center for Automotive Research (CARS). He obtained a PhD in Computer Science from Tel Aviv University, where he was advised by Dan Halperin.

Kiril’s research focuses on the design of effective control and decision-making mechanisms to allow multi-robot systems to tackle complex problems for the benefit of the society. His work draws upon ideas that span across the disciplines of engineering, computer science, and transportation science, to develop scalable optimization approaches with substantial guarantees regarding quality and robustness of the solution. For his work he received multiple awards, including the Clore Scholars and Fulbright Postdoctoral Fellowships, best paper awards and nominations (at Robotics: Science and Systems, International Conference on Robotics and Automation, International Symposium on Multi-Robot and Multi-Agent System, and European Control Conference), and teaching awards.

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

Multi-robot systems are already playing a crucial role in manufacturing, warehouse automation, and natural resource monitoring, and in the future they will be employed in even broader domains from space exploration to search-and-rescue. Moreover, these systems will likely be incorporated in our daily lives through drone delivery services and smart mobility systems that comprise of thousands of autonomous vehicles. The anticipated benefits of multi-robot systems are numerous, ranging from automating dangerous jobs, to broader societal facets such  as easing traffic congestion and sustainability. However, to reap those rewards we must develop control mechanisms for such systems that can adapt rapidly to unexpected changes on a massive scale. Importantly, these mechanisms must capture: (i) dynamical and collision-avoidance constraints of individual robots; (ii) interactions between multiple robots; and (iii) more broadly, the  interaction of those systems with the environment. All these considerations give rise to extremely complex and high-dimensional optimization problems that need to be solved in real-time.

In this talk I will present recent progress on the design of algorithms for  control and decision-making to allow the safe, effective, and societally-equitable deployment of multi-robot systems. I will highlight both results on fundamental capabilities for multi-robot systems (e.g., motion planning and task allocation), as well as applications in smart mobility, including multi-drone delivery and autonomous mobility-on-demand systems. Along the way, I will mention a few related open problems in mathematics and algorithm design.

BIO:
Kiril Solovey is roboticist specializing in multi-robot systems and their applications to smart mobility. He is currently a Postdoctoral Scholar at the Department of Aeronautics and Astronautics, Stanford University, working with Marco Pavone, where he is supported by the Center for Automotive Research (CARS). He obtained a PhD in Computer Science from Tel Aviv University, where he was advised by Dan Halperin.

Kiril’s research focuses on the design of effective control and decision-making mechanisms to allow multi-robot systems to tackle complex problems for the benefit of the society. His work draws upon ideas that span across the disciplines of engineering, computer science, and transportation science, to develop scalable optimization approaches with substantial guarantees regarding quality and robustness of the solution. For his work he received multiple awards, including the Clore Scholars and Fulbright Postdoctoral Fellowships, best paper awards and nominations (at Robotics: Science and Systems, International Conference on Robotics and Automation, International Symposium on Multi-Robot and Multi-Agent System, and European Control Conference), and teaching awards.

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

One possible interpretation of the SYZ conjecture is that for a polarized family of CY manifolds near the large complex structure limit, there is a special Lagrangian fibration on the generic region of the CY manifold. Generic here means a set with a large percentage of the CY measure, and the percentage tends to 100% in the limit. I will discuss my recent progress on this version of the SYZ conjecture, with emphasis on how differential geometers think about this problem, and give some hint about where nonarchimedean geometry comes in.

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

TITLE: Theorems of Torelli type

ABSTRACT: Given a closed manifold of even dimension 2n, then Hodge
showed around 1950 that a kählerian complex structure on
that manifold determines a decomposition of its complex
cohomology. This decomposition, which can potentially vary
continuously with the complex structure, extracts from a non-linear
given, linear data. It can contain a lot of information. When there
is essentially no loss of data in this process, we say that the Torelli
theorem holds. We review the underlying theory and then survey
some cases where this is the case. This will include the classical
case n=1, but the emphasis will be on K3 manifolds (n=2) and
more generally, on hyperkählerian manifolds. These cases stand
out, since one can then also tell which decompositions occur.

Talk chair: Shing-Tung Yau

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.

Strongly correlated electron systems often show bad-metal behavior, as operationally specified in terms of a resistivity at room temperature that reaches or exceeds the Mott-Ioffe-Regel limit. They display a rich landscape of electronic orders, which provide clues to the underlying microscopic physics. Iron-based superconductors present a striking case study, and have been the subject of extensive efforts during the past decade or so. They are well established to be bad metals, and their phase diagrams prominently feature various types of electronic orders that are essentially always accompanied by nematicity. In this talk, I will summarize these characteristic features and discuss our own efforts towards understanding the normal state through the lens of the electronic orders and their fluctuations. Implications for superconductivity will be briefly discussed. In the second part of the talk, I will consider the nematic correlations that have been observed in the graphene-based moiré narrow-band systems. I will present a theoretical study which demonstrates nematicity in a “fragile insulator”, predicts its persistence in the bad metal regime and provides an overall perspective on the phase diagram of these correlated systems.

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

In this talk, I will introduce a new construction for the K-theoretic mirror symmetry of toric varieties/stacks, based on the 3d N=2 mirror symmetry introduced by Dorey-Tong. Given the toric datum, i.e. a  short exact sequence 0 -> Z^k -> Z^n -> Z^{n-k} -> 0, we consider the toric Artin stack of the form [C^n / (C^*)^k]. Its mirror is constructed by taking the Gale dual of the defining short exact sequence. As an analog of the 3d N=4 case, we consider the K-theoretic I-function, with a suitable level structure, defined by counting parameterized quasimaps from P^1. Under mirror symmetry, the I-functions of a mirror pair are related to each other under the mirror map, which exchanges the K\”ahler and equivariant parameters and maps q to q^{-1}. This is joint work with Yongbin Ruan and Yaoxiong Wen.

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