Calendar

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.

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