Emmy Noether Takes Center Stage

Just over a year ago on a warm September Saturday, the Harvard Department of Mathematics and the Center of Mathematical Sciences and Applications (CMSA) hosted...
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  • October 2, 2023
    3:00 pm
    Jameel Al-Aidroos Mathematical Pedagogy Lecture Series


    Speaker: Gregory R. Goldsmith - Schmid College of Science and Technology at Chapman University   Title: Jameel Al-Aidroos Mathematical Pedagogy Lecture Series
< 2023 >
  • CMSA EVENT: CMSA Probability Seminar: Correlation decay for finite lattice gauge theories

    Speaker: Arka Adhikari – Stanford University

    1:30 PM-2:30 PM
    September 7, 2023

    In the setting of lattice gauge theories with finite (possibly non-Abelian) gauge groups at weak coupling, we prove exponential decay of correlations for a wide class of gauge invariant functions, which in particular includes arbitrary functions of Wilson loop observables. Based on joint work with Sky Cao.

  • COLLOQUIUMS: Special Colloquium – Systems of points with Coulomb interactions

    Speaker: Sylvia Serfaty – NYU Courant Institute

    3:00 PM-4:00 PM
    September 7, 2023
    1 Oxford Street, Cambridge, MA 02138 USA

    Large ensembles of points with Coulomb interactions arise in various settings of condensed matter physics, classical and quantum mechanics, statistical mechanics, random matrices and even approximation theory, and they give rise to a variety of questions pertaining to analysis, Partial Differential Equations and probability.

    We will first review these motivations, then present the ”mean-field” derivation of effective models and equations describing the system at the macroscopic scale. We then explain how to analyze the next order behavior, giving information on the configurations at the microscopic level and connecting with crystallization questions, and finish with the description of the effect of temperature.

    Talk will be followed by Tea in the Math Common Room – Science Center, 4th Floor

  • COLLOQUIUMS: Special Colloquium: Quantitative homogenization, renormalization and anomalous diffusion

    Speaker: Scott Armstrong – NYU Courant Institute

    3:00 PM-4:30 PM
    September 11, 2023
    1 Oxford Street, Cambridge, MA 02138 USA

    I will begin the talk with an overview of the topic of quantitative homogenization for elliptic and parabolic equations. Homogenization refers to the procedure of replacing a very “noisy” equation– one with rapidly oscillating coefficients– with a nicer, “effective” equation in a large-scale limit. There is a very abstract theory of (qualitative) homogenization, which is classical. We will discuss the more concrete theory of quantitative homogenization, which has been developed recently. A central role in the story concerns certain “coarse-graining” arguments, which can be seen as constituting a rigorous renormalization group-type approach, formulated in the language of analysis. These methods have surprising applications in mathematical physics and probability, which are still emerging. In the second part of the talk, I will discuss one such application (in a recent joint work with V. Vicol) to turbulence theory: namely, a proof of anomalous diffusion for an advection-diffusion equation.

    Talk will be followed by Tea in the Math Common Room – Science Center, 4th Floor

  • CMSA EVENT: CMSA General Relativity Seminar: Pole skipping, quasinormal modes, shockwaves and their connection to chaos

    Speaker: Diandian Wang – Harvard University

    11:00 AM-12:00 PM
    September 12, 2023
    20 Garden Street, Cambridge, MA 02138

    A chaotic quantum system can be studied using the out-of-time-order correlator (OTOC). I will tell you about pole skipping — a recently discovered feature of the retarded Green’s function — that seems to also know things: things like the Lyapunov exponent and the butterfly velocity, which are important quantifiers of the OTOC. Then I will talk about a systematic way of deriving pole-skipping conditions for general holographic CFTs dual to classical bulk theories and how to use this framework to derive a few interesting statements including: (1) theories with higher spins generally violate the chaos bound; (2) the butterfly velocity calculated using pole skipping agrees with that calculated using shockwaves for arbitrary higher-derivative gravity coupled to ordinary matter; (3) shockwaves are related to a special type of quasinormal modes. As we will see, the techniques are entirely classically gravitational, which I will go through with a certain level of details.

    Zoom Link:
    Password: cmsa

  • HARVARD-MIT ALGEBRAIC GEOMETRY SEMINAR: Harvard–MIT Algebraic Geometry Seminar: Enumerative geometry, wall-crossing and Virasoro constraints

    Speaker: Miguel Moreira – MIT

    3:00 PM-4:00 PM
    September 12, 2023

    Given a moduli space of either sheaves on a smooth projective variety or a moduli space of representations of a quiver, there are several invariants that we can extract. One of the ways to get numbers out of a moduli space is to integrate (possibly against a virtual fundamental class) certain tautological classes. Such numbers often have interesting structures behind, and I will talk about two: how they change when one changes a stability condition (wall-crossing formulas) and some universal and explicit linear relations that those invariants always seem to satisfy (Virasoro constraints). Both of these phenomena are related to a vertex algebra found by D. Joyce. For simplicity I will mostly focus on the case of representations of a quiver. The talk is based on joint work with A. Bojko and W. Lim.

  • CMSA EVENT: CMSA Topological Quantum Matter Seminar: Homotopy classes of loops of Clifford unitaries

    Speaker: Roman Geiko – UCLA

    4:00 PM-5:00 PM
    September 12, 2023
    20 Garden Street, Cambridge, MA 02138

    We study Clifford locality-preserving unitaries and stabilizer
    Hamiltonians by means of Hermitian K-theory. We demonstrate how
    the notion of algebraic homotopy of modules over Laurent polynomial
    rings translates into the connectedness of two short-range entangled
    stabilizer Hamiltonians by a shallow Clifford circuit. We apply this
    observation to a classification of homotopy classes of loops of Clifford
    unitaries. The talk is based on a work in collaboration with Yichen Hu

    This seminar will be held in person and on Zoom:

    Password: 353114

  • CMSA EVENT: CMSA Colloquium: An invitation to strong-field scattering

    Speaker: Tim Adamo – University of Edinburgh

    12:30 PM-1:30 PM
    September 14, 2023
    20 Garden Street, Cambridge, MA 02138

    Scattering amplitudes in strong background fields provide an arena where perturbative and non-perturbative physics meet, with important applications ranging from laser physics to black holes, but their study is hampered by the cumbersome nature of QFT in the background field formalism. In this talk, I will try to convince you that strong-field scattering amplitudes contain a wealth of physical information which cannot be obtained with standard perturbative techniques, ranging from all-order classical observables to constraints on exact solutions. Furthermore, I will discuss how amplitudes in certain chiral strong fields can be obtained to all-multiplicity twistor and string methods.


  • CMSA EVENT: CMSA Active Matter Seminar: Frustration-free states of cell fate networks: the case of the epithelial-mesenchymal transition

    Speaker: Herbert Levine – Northeastern University

    1:00 PM-2:00 PM
    September 14, 2023
    20 Garden Street, Cambridge, MA 02138

    Cell fate decisions are made by allowing external signals to govern the steady-state pattern adopted by networks of interacting regulatory factors governing transcription and translation. One of these decisions, of importance for both developmental processes and for cancer metastasis, is the epithelial-mesenchymal transition (EMT). In this talk, we will argue that these biological networks have highly non-generic interaction structures such that they allow for phenotypic states with very low frustration, i.e. where most interactions are satisfied. This property has important consequences for the allowed dynamics of these systems.

    This seminar will be held in person and on Zoom. For more information on how to join, please see:

  • CMSA EVENT: CMSA General Relativity Seminar: Quantization of causal diamonds in 2+1 dimensional gravity

    Speaker: Rodrigo Silva – University of Maryland

    11:00 AM-12:00 PM
    September 19, 2023

    We develop the reduced phase space quantization of causal diamonds in $2+1$ dimensional gravity with a nonpositive cosmological constant. The system is defined as the domain of dependence of a spacelike topological disk with a fixed boundary metric. By solving the constraints in a constant-mean-curvature time gauge and removing all the spatial gauge redundancy, we find that the phase space is the cotangent bundle of $Diff^+(S^1)/PSL(2, \mathbb{R})$, i.e., the group of orientation-preserving diffeomorphisms of the circle modulo the projective special linear subgroup. Classically, the states correspond to causal diamonds embedded in $AdS_3$ (or $Mink_3$ if $\Lambda = 0$), with a fixed corner length, that has the topological disk as a Cauchy surface. Because this phase space does not admit a global system of coordinates, a generalization of the standard canonical (coordinate) quantization is required — in particular, since the configuration space is a homogeneous space for a Lie group, we apply Isham’s group-theoretic quantization scheme. The Hilbert space of the associated quantum theory carries an irreducible unitary representation of the $BMS_3$ group and can be realized by wavefunctions on a coadjoint orbit of Virasoro with labels in irreducible unitary representations of the corresponding little group. A surprising result is that the twist of the diamond boundary loop is quantized in terms of the ratio of the Planck length to the corner length.


    Password: cmsa

  • HARVARD-MIT ALGEBRAIC GEOMETRY SEMINAR: Harvard–MIT Algebraic Geometry Seminar: Cycling in Cambridge

    Speaker: Elden Elmanto – University of Toronto

    3:00 PM-4:00 PM
    September 19, 2023

    I spent most of my time here cycling (or is it biking?) and thinking about algebraic cycles from a homotopical viewpoint. I will speak about the latter. In joint work with Matthew Morrow, we developed a theory of motivic cohomology of schemes beyond the case of smooth schemes over a field. I will explain the cycle-theoretic aspects of this construction, focusing on the case of surfaces, revisiting older results of Krishna and Srinivas.

  • CMSA EVENT: CMSA Topological Quantum Matter Seminar: Exact Results in Flat Band Hubbard Models

    Speaker: Jonah Herzog-Arbeitman – Princeton University

    10:30 AM-11:30 AM
    September 20, 2023
    20 Garden Street, Cambridge, MA 02138

    Flat bands, like those in the kagome lattice or twisted bilayer graphene, are a
    natural setting for studying strongly coupled physics since the interaction
    strength is the only energy scale in the problem. They can exhibit
    unconventional behavior in the multi-orbital case: the mean-field theory of
    flat band attractive Hubbard models shows the possibility of
    superconductivity even though the Fermi velocity of the bands is strictly zero.
    However, it is not necessary to resort to this approximation. We demonstrate
    that the groundstates and low-energy excitations of a large class of attractive
    Hubbard models are exactly solvable, offering a rare, microscopic view of
    their physics. The solution reveals the importance of quantum geometry in
    escaping (some of) BCS phenomenology within a tractable and nontrivial
    strong coupling theory.

    This seminar will be held in person and on Zoom:

    Password: 353114

  • CMSA EVENT: CMSA New Technologies in Mathematics Seminar: The TinyStories Dataset: How Small Can Language Models Be And Still Speak Coherent English?

    Speaker: Ronan Eldan – Microsoft Research

    2:00 PM-3:00 PM
    September 20, 2023
    20 Garden Street, Cambridge, MA 02138

    While generative language models exhibit powerful capabilities at large scale, when either the model
    or the number of training steps is too small, they struggle to produce coherent and fluent text:
    Existing models whose size is below a few billion parameters often do not generate coherent text
    beyond a few sentences. Hypothesizing that one of the main reasons for the strong reliance on size is
    the vast breadth and abundance of patterns in the datasets used to train those models, this
    motivates the following question: Can we design a dataset that preserves the essential elements of
    natural language, such as grammar, vocabulary, facts, and reasoning, but that is much smaller and
    more refined in terms of its breadth and diversity?
    In this talk, we introduce TinyStories, a synthetic dataset of short stories that only contain words
    that 3 to 4-year-olds typically understand, generated by GPT-3.5/4. We show that TinyStories can
    be used to train and analyze language models that are much smaller than the state-of-the-art models
    (below 10 million parameters), or have much simpler architectures (with only one transformer
    block), yet still produce fluent and consistent stories with several paragraphs that are diverse and
    have almost perfect grammar, and demonstrate certain reasoning capabilities. We also show that
    the trained models are substantially more interpretable than larger ones, as we can visualize and
    analyze the attention and activation patterns of the models, and show how they relate to the
    generation process and the story content. We hope that TinyStories can facilitate the development,
    analysis and research of language models, especially for low-resource or specialized domains, and
    shed light on the emergence of language capabilities in LMs.

    This seminar will be held in person and on Zoom:

  • NUMBER THEORY SEMINAR: Number Theory Seminar: Harris–Venkatesh plus Stark

    Speaker: Robin Zhang – MIT

    3:00 PM-4:00 PM
    September 20, 2023
    1 Oxford Street, Cambridge, MA 02138 USA

    The class number formula describes the behavior of the Dedekind zeta function at $s=0$ and $s=1$. The Stark conjecture extends the class number formula, describing the behavior of Artin $L$-functions and $p$-adic $L$-functions at $s=0$ and $s=1$ in terms of units. The Harris–Venkatesh conjecture describes the residue of Stark units modulo $p$, giving a modular analogue to the Stark and Gross conjectures while also serving as the first verifiable part of the broader conjectures of Venkatesh, Prasanna, and Galatius. In this talk, I will draw an introductory picture, formulate a unified conjecture combining Harris–Venkatesh and Stark for weight one modular forms, and describe the proof of this in the imaginary dihedral case.

  • CMSA EVENT: CMSA Probability Seminar: Solving spin systems, the Babylonian way

    Speaker: Nicola Kistler – Johann Wolfgang Goethe-Universität Frankfurt am Main

    3:30 PM-4:30 PM
    September 20, 2023
    20 Garden Street, Cambridge, MA 02138

    The replica method, together with Parisi’s symmetry breaking mechanism, is an extremely powerful tool to compute the limiting free energy of virtually any mean field disordered system. Unfortunately, the tool is dramatically flawed from a mathematical point of view. I will discuss a truly elementary procedure which allows to rigorously implement two (out of three) steps of the replica method, and conclude with some remarks on the relation between this new point of view and old work by Mezard and Virasoro on the microstructure of ultrametricity, the latter being the fundamental yet unjustified Ansatz in the celebrated Parisi solution. We are still far from a clear understanding of the issues, but quite astonishingly, evidence is mounting that Parisi’s ultrametricity assumption, the onset of scales and the universal hierarchical self-organisation of random systems in the infinite volume limit, is intimately linked to hidden geometrical properties of large random matrices which satisfy rules reminiscent of the popular SUDOKU game.

  • HARVARD-MIT COMBINATORICS SEMINAR: Richard P. Stanley Seminar in Combinatorics: An SL(4) web basis from hourglass plabic graphs

    Speaker: Christian Gaetz – Cornell

    4:15 PM-5:15 PM
    September 20, 2023

    The SL(3) web basis is a special basis of certain spaces of tensor invariants developed in the late 90’s by Khovanov and Kuperberg as a tool for computing quantum link invariants. Since then this basis has found connections and applications to cluster algebras, canonical bases, dimer models, quantum topology, and tableau combinatorics. A main open problem has remained: how to find a basis replicating the desirable properties of this basis for SL(4) and beyond? I will describe joint work with Oliver Pechenik, Stephan Pfannerer, Jessica Striker, and Josh Swanson in which we construct such a basis for SL(4). Modified versions of plabic graphs and the six-vertex model and new tableau combinatorics will appear along the way.


    For more info, see

  • OPEN NEIGHBORHOOD SEMINAR: Open Neighborhood Seminar: A moduli space problem in condensed matter physics

    Speaker: Dan Freed – Harvard University

    4:30 PM-5:30 PM
    September 20, 2023
    1 Oxford Street, Cambridge, MA 02138 USA

    This talk is an elementary account of what is at first a surprising application of stable homotopy theory to theoretical physics. No specialized background is required; the storyline is the focus, not technical details. The main result is a classification of invertible gapped phases in quantum mechanical systems. We will take a journey to get there, along the way meeting moduli problems in geometry and topological field theory. This is joint work with Mike Hopkins.



  • CMSA EVENT: CMSA Quantum Matter in Mathematic and Physics Seminar: Floquet codes, automorphisms, and quantum computation

    Speaker: Margarita Davydova – MIT

    10:00 AM-11:30 AM
    September 22, 2023
    20 Garden Street, Cambridge, MA 02138

    In this talk, I will introduce a new kind of measurement-based quantum computation inspired by Floquet codes. In this model, the quantum logical gates are implemented by short sequences of low-weight measurements which simultaneously encode logical information and enable error correction. We introduce a new class of quantum error-correcting codes generalizing Floquet codes that achieve this, which we call dynamic automorphism (DA) codes.
    As in Floquet codes, the instantaneous codespace of a DA code at any fixed point in time is that of a topological code. In this case, the quantum computation can be viewed as a sequence of time-like domain walls implementing automorphisms of the topological order, which can be understood in terms of reversible anyon condensation paths in a particular parent model. This talk will introduce all of these concepts as well as provide a new perspective for thinking about Floquet codes.
    The explicit examples that we construct, which we call DA color codes, can implement the full Clifford group of logical gates in 2+1d by two- and, rarely three-body measurements. Using adaptive two-body measurements, we can achieve a non-Clifford gate in 3+1d, making the first step towards universal quantum computation in this model.
    The talk is based on recent work with Nathanan Tantivasadakarn, Shankar Balasubramanian, and David Aasen [arxiv: 2307.10353].

    This seminar will be held in person and on Zoom:

    Password: cmsa

  • HARVARD-MIT COMBINATORICS SEMINAR: Richard P. Stanley Seminar in Combinatorics: Pair constructions for hypergraph Ramsey numbers

    Speaker: Xiaoyu He – Princeton University

    3:00 PM-4:00 PM
    September 22, 2023

    The Ramsey number r_k(G, H) of two k-uniform hypergraphs G and H is the smallest n such that any edge-coloring of the complete k-uniform hypergraph on n vertices contains either a red copy of G or a blue copy of H. Hypergraph Ramsey theory is concerned with the growth rates of such Ramsey numbers, particularly when one or both of {G, H} is a clique of size tending to infinity. Classical arguments of Erdős-Rado and Erdős-Hajnal reduce most major problems in this area from all higher uniformities to uniformity k=3, but fail to bridge the gap from the graph case k=2 to the hypergraph case k=3.

    In this talk, we survey the known approaches for lifting Ramsey graphs to 3-uniform hypergraphs, most famously the stepping-up construction of Erdős-Hajnal. We collect these constructions under the umbrella term “pair constructions” and present new variations proving new lower bounds. In particular, we describe an explicit family of 3-uniform hypergraphs H (including links of odd cycles and tight cycles of length not divisible by 3) which satisfy r_3(H, K_n) > 2^{c n log n}. We also prove the existence of a linear 3-uniform hypergraph H for which r_3(H, K_n) grows superpolynomially.

    Based on joint work with David Conlon, Benjamin Gunby, Jacob Fox, Dhruv Mubayi, Andrew Suk, and Jacques Verstraete.


    For more info, see

  • HARVARD-MIT COMBINATORICS SEMINAR: Richard P. Stanley Seminar in Combinatorics: Coefficientwise Hankel-total positivity

    Speaker: Alan Sokal – University College London

    3:00 PM-4:00 PM
    September 22, 2023

    A matrix M of real numbers is called totally positiveif every minor of $M$ is nonnegative. Gantmakher and Krein showedin 1937 that a Hankel matrix $H = (a_{i+j})_{i,j \ge 0}$of real numbers is totally positive if and only if the underlyingsequence $(a_n)_{n \ge 0}$ is a Stieltjes moment sequence,i.e.\ the moments of a positive measure on $[0,\infty)$.Here I will introduce a generalization: a matrix $M$ of polynomials(in some set of indeterminates) will be called{\em coefficientwise totally positive}\/ if every minor of $M$is a polynomial with nonnegative coefficients. And a sequence$(a_n)_{n \ge 0}$ of polynomials will be called{\em coefficientwise Hankel-totally positive}\/ if the Hankel matrix$H = (a_{i+j})_{i,j \ge 0}$ associated to $(a_n)$ is coefficientwisetotally positive.It turns out that many sequences of polynomials arising naturallyin enumerative combinatorics are (empirically) coefficientwiseHankel-totally positive. I will discuss some methods for proving this.But these proofs fall far short of what appears to be true.


    For more info, see

  • CMSA EVENT: Algebraic Geometry in String Theory Seminar: Species Scale across String Moduli Spaces

    Speaker: Damian van de Heisteeg – CMSA

    10:00 AM-11:30 AM
    September 25, 2023

    String theories feature a wide array of moduli spaces. We propose that the energy cutoff scale of these theories – the so-called species scale – can be determined through higher-curvature corrections. This species scale varies with the moduli; we use it both asymptotically to bound the diameter of the field space, as well as in the interior to determine a “desert point” where it is maximized.


    Please note that there will be a pre-talk aimed at graduate students by David Wu from 10:00-10:30am

  • CMSA EVENT: CMSA Colloquium: Predicting non-continuous functions

    Speaker: Sean Cox – Virginia Commonwealth University

    4:30 PM-5:30 PM
    September 25, 2023
    20 Garden Street, Cambridge, MA 02138

    One of the strangest consequences of the Axiom of Choice is the following Hardin-Taylor 2008 result:  there is a “predictor” such that for every function $f$ from the reals to the reals—even nowhere continuous $f$—the predictor applied to $f \restriction (-\infty,t)$ correctly predicts $f(t)$ for *almost every* $t \in R$.  They asked how robust such a predictor could be, with respect to distortions in the time (input) axis; more precisely, for which subgroups $H$ of Homeo^+(R) do there exist $H$-invariant predictors?  Bajpai-Velleman proved an affirmative answer when H=Affine^+(R), and a negative answer when H is (the subgroup generated by) C^\infty(R).  They asked about the intermediate region; in particular, do there exist analytic-invariant predictors?  We have partially answered that question:  assuming the Continuum Hypothesis (CH), the answer is “no”. Regarding other subgroups of Homeo^+(R), we have affirmative answers that rely solely on topological group-theoretic properties of the subgroup.  But these properties are very restrictive; e.g., all known positive examples are metabelian.  So there remain many open questions. This is joint work with Aldi, Buffkin, Cline, Cody, Elpers, and Lee.