Demystifying Math 55

By Anastasia Yefremova Few undergraduate level classes have the distinction of nation-wide recognition that Harvard University’s Math 55 has. Officially comprised of Mathematics 55A “Studies...
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upcoming events

< 2021 >
    3:00 AM-4:00 AM
    December 1, 2021

    will speak on:

    Lagrangians and mirror symmetry in the Higgs bundle moduli space

    The talk concerns recent work with Tamas Hausel in asking how SYZ mirror symmetry works for the moduli space of Higgs bundles. Focusing on C^*-invariant Lagrangian submanifolds, we use the notion of virtual multiplicity as a tool firstly to examine if the Lagrangian is closed, but  also to open up new features involving finite-dimensional algebras which are deformations of cohomology algebras. Answering some of the questions raised  requires revisiting basic constructions of stable bundles on curves.


    For details visit:


    Zoom Link:

    Meeting ID: 992 8582 5827
    Passcode: 20211201

  • CMSA EVENT: CMSA New Technologies in Mathematics Seminar: The Principles of Deep Learning Theory
    2:00 PM-3:00 PM
    December 1, 2021

    Deep learning is an exciting approach to modern artificial intelligence based on artificial neural networks. The goal of this talk is to provide a blueprint — using tools from physics — for theoretically analyzing deep neural networks of practical relevance. This task will encompass both understanding the statistics of initialized deep networks and determining the training dynamics of such an ensemble when learning from data.

    In terms of their “microscopic” definition, deep neural networks are a flexible set of functions built out of many basic computational blocks called neurons, with many neurons in parallel organized into sequential layers. Borrowing from the effective theory framework, we will develop a perturbative 1/n expansion around the limit of an infinite number of neurons per layer and systematically integrate out the parameters of the network. We will explain how the network simplifies at large width and how the propagation of signals from layer to layer can be understood in terms of a Wilsonian renormalization group flow. This will make manifest that deep networks have a tuning problem, analogous to criticality, that needs to be solved in order to make them useful. Ultimately we will find a “macroscopic” description for wide and deep networks in terms of weakly-interacting statistical models, with the strength of the interactions between the neurons growing with depth-to-width aspect ratio of the network. Time permitting, we will explain how the interactions induce representation learning.

    This talk is based on a book, “The Principles of Deep Learning Theory,” co-authored with Sho Yaida and based on research also in collaboration with Boris Hanin. It will be published next year by Cambridge University Press.


    Mod p points on Shimura varieties

    3:00 PM-4:00 PM
    December 1, 2021
    1 Oxford Street, Cambridge, MA 02138 USA

    The study of mod p points on Shimura varieties was originally motivated by the study of the Hasse-Weil zeta function for Shimura varieties. It involves some rather subtle problems which test just how much we know about motives over finite fields. In this talk I will explain some recent results, and applications, and what still remains conjectural.

    Mod p points on Shimura varieties

    3:00 PM-4:00 PM
    December 1, 2021
    1 Oxford Street, Cambridge, MA 02138 USA

    The study of mod p points on Shimura varieties was originally motivated by the study of the Hasse-Weil zeta function for Shimura varieties. It involves some rather subtle problems which test just how much we know about motives over finite fields. In this talk I will explain some recent results, and applications, and what still remains conjectural.
    3:30 PM-4:30 PM
    December 3, 2021
    1 Oxford Street, Cambridge, MA 02138 USA


    Rietsch and Williams relate cluster structures and mirror symmetryfor type A Grassmannians Gr(k, n), and use this interaction to construct Newton-Okounkovbodies and associated toric degenerations. In this talk, we define a cluster seed for theLagrangian Grassmannian, construct the associated Newton-Okounkov body, and prove that it agrees (up to unimodular equivalence) with a polytope obtained from the superpotential defined by Pech and Rietsch on the mirror Orthogonal Grassmannian.


  • CMSA EVENT: CMSA Colloquium : Induced subgraphs and tree decompositions
    9:30 AM-10:30 AM
    December 8, 2021

    Tree decompositions are a powerful tool in both structural graph theory and graph algorithms. Many hard problems become tractable if the input graph is known to have a tree decomposition of bounded “width”. Exhibiting a particular kind of a tree decomposition is also a useful way to describe the structure of a graph. Tree decompositions have traditionally been used in the context of forbidden graph minors; bringing them into the realm of forbidden induced subgraphs has until recently remained out of reach. Over the last couple of years we have made significant progress in this direction, exploring both the classical notion of bounded tree-width, and concepts of more structural flavor. This talk will survey some of these ideas and results.

    Zoom link: Password: cmsa

  • CMSA EVENT: CMSA : Defects, link invariants and exact WKB
    10:30 AM-12:00 PM
    December 8, 2021

    I will describe some of my recent work on defects in supersymmetric field theories. The first part of my talk is focused on line defects in certain large classes of 4d N=2 theories and 3d N=2 theories. I will describe geometric methods to compute the ground states spectrum of the bulk-defect system, as well as implications on the construction of link invariants. In the second part I will talk about some perspectives of surface defects in 4d N=2 theories and related applications on the exact WKB method for ordinary differential equations. This talk is based on past joint work with A. Neitzke, various work in progress with D. Gaiotto, S. Jeong, A. Khan, G. Moore, as well as work by myself.
    Password: cmsa

  • CMSA EVENT: CMSA New Technologies in Mathematics Seminar: Hierarchical Transformers Are More Efficient Language Models
    2:00 PM-3:00 PM
    December 8, 2021

    Transformer models yield impressive results on many NLP and sequence modeling tasks. Remarkably, Transformers can handle long sequences which allows them to produce long coherent outputs: full paragraphs produced by GPT-3 or well-structured images produced by DALL-E. These large language models are impressive but also very inefficient and costly, which limits their applications and accessibility. We postulate that having an explicit hierarchical architecture is the key to Transformers that efficiently handle long sequences. To verify this claim, we first study different ways to upsample and downsample activations in Transformers so as to make them hierarchical. We use the best performing upsampling and downsampling layers to create Hourglass – a hierarchical Transformer language model. Hourglass improves upon the Transformer baseline given the same amount of computation and can yield the same results as Transformers more efficiently. In particular, Hourglass sets new state-of-the-art for Transformer models on the ImageNet32 generation task and improves language modeling efficiency on the widely studied enwik8 benchmark.

  • CMSA EVENT: CMSA Math-Science Literature Lecture Series
    9:30 AM-11:00 AM
    December 9, 2021

    will speak on:

    The Noether Theorems in Geometry: Then and Now

    The 1918 Noether theorems were a product of the general search for energy and momentum conservation in Einstein’s newly formulated theory of general relativity. Although widely referred to as the connection between symmetry and conservation laws, the theorems themselves are often not understood properly and hence have not been as widely used as they might be. In the first part of the talk, I outline a brief history of the theorems, explain a bit of the language, translate the first theorem into coordinate invariant language and give a few examples. I will mention only briefly their importance in physics and integrable systems. In the second part of the talk, I describe why they are still relevant in geometric analysis: how they underlie standard techniques and why George Daskalopoulos and I came to be interested in them for our investigation into the best Lipschitz maps of Bill Thurston. Some applications to integrals on a domain a hyperbolic surface leave open possibilities for applications to integrals on domains which are locally symmetric spaces of higher dimension. The talk finishes with an example or two from the literature.

    Register Online

    4:30 PM-5:30 PM
    December 9, 2021
    1 Oxford Street, Cambridge, MA 02138 USA

    The dynamical degree of an invertible self-map of projective space is an asymptotic measure of the algebraic complexity of the iterates of the map. This numerical invariant controls many aspects of the dynamics of the map, and in this talk I will survey the significance of the dynamical degree and discuss some important examples. In these examples, the dynamical degree is an integer or an eigenvalue of an integer matrix, so an algebraic number, as was conjecturally the case for all such maps. I will discuss joint work with Bell, Diller, and Jonsson in which we refute this conjecture by constructing invertible maps of projective 3-space which have transcendental dynamical degree.

  • 2:30 PM-4:00 PM
    December 10, 2021

    Quasiparticle excitations in 3 + 1 dimensions can be either bosons or fermions. In this work, we introduce the notion of fermionic loop excitations in 3 + 1 dimensional topological phases. Specifically, we construct a new many-body lattice invariant of gapped Hamiltonians, the loop self-statistics μ = ±1, that distinguishes two bosonic topological orders that both superficially resemble 3 + 1d Z2 gauge theory coupled to fermionic charged matter. The first has fermionic charges and bosonic Z2 gauge flux loops (FcBl) and is just the ordinary fermionic toric code. The second has fermionic charges and fermionic loops (FcFl) and, as we argue, can only exist at the boundary of a non-trivial 4 + 1d invertible phase, stable without any symmetries i.e., it possesses a gravitational anomaly. We substantiate these claims by constructing an explicit exactly solvable 4 + 1d Walker–Wang model and computing the loop self-statistics in the fermionic Z2 gauge theory hosted at its boundary. We also show that the FcFl phase has the same gravitational anomaly as all-fermion quantum electrodynamics. Our results are in agreement with the recent classification of nondegenerate braided fusion 2- categories, and with the cobordism prediction of a non-trivial Z2-classified 4+1d invertible phase with action S = (1/2) w2 w3.
    Password: cmsa