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Math Community Web Page

Following up on the #shutdownSTEM discussions, the Department of Mathematics has launched a community web page, with evolving content to be created through community effort.
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#shutdownSTEM–10 June 2020

In partnership with the Harvard Division of Science, the Mathematics Department used the call to #shutdownSTEM on 10 June to initiate a discussion of actions...
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  • Non-random behaviour in sums of modular symbols
    3:00 PM-4:00 PM
    March 4, 2020
    1 Oxford Street, Cambridge, MA 02138 USA

    We give explicit expressions for the Fourier coefficients of Eisenstein series twisted by Dirichlet characters and modular symbols on $\Gamma_0(N)$ in the case where $N$ is prime and equal to the conductor
    of the Dirichlet character. We obtain these expressions by computing the spectral decomposition of automorphic functions closely related to these Eisenstein series. As an application, we then evaluate certain sums of modular symbols in a way which parallels past work of Goldfeld, O’Sullivan, Petridis, and Risager. In one case we find less cancellation in this sum than would be predicted by the common phenomenon of “square root cancellation”, while in another case we find more cancellation.

  • **CANCELED** Informal Geometry & Dynamics Seminar: Pseudo-Anosov mapping classes with large dilatation
    4:00 PM-6:00 PM
    March 4, 2020
    I’ll talk about some subclasses of pseudo-Anosov mapping classes whose dilatations are bounded away from 1.
  • Well-ordering principles in Proof theory and Reverse Mathematics
    4:30 PM-5:30 PM
    March 4, 2020
    There are several familiar theories of reverse mathematics that can be characterized by well-ordering principles of the form
    (*) “if X is well ordered then f(X) is well ordered”,
    where f is a standard proof theoretic function from ordinals to ordinals (such f’s are always dilators). Some of these equivalences have been obtained by recursion-theoretic and combinatorial methods. They (and many more) can also be shown by proof-theoretic methods, employing search trees and cut elimination theorems in infinitary logic with ordinal bounds. One could perhaps generalize and say that every cut elimination theorem in ordinal-theoretic proof theory encapsulates a theorem of this type.
    One aim of the talk is to present a general methodology underlying these results that enables one to construct omega-models of particular theories from (*).
    As (*) is of complexity $\Pi^1_2$ such a principle cannot characterize stronger comprehensions at the level of $\Pi^1_1$-comprehension. This requires a higher order version of (*) that employs ideas from ordinal representation systems with collapsing functions used in impredicative proof theory.  The simplest one is the Bachmann construction. Relativizing the latter construction to any dilator f and asserting that this always yields a well-ordering turns out to be equivalent to $\Pi^1_1$-comprehension. This result has been conjectured more than 10 years ago, but its proof has only been worked out by Anton Freund in recent years
  • CMSA Colloquium: Derandomizing Algorithms via Spectral Graph Theory
    4:45 PM-5:45 PM
    March 4, 2020
    20 Garden Street, Cambridge, MA 02138
    Randomization is a powerful tool for algorithms; it is often easier to design efficient algorithms if we allow the algorithms to “toss coins” and output a correct answer with high probability.  However, a longstanding conjecture in theoretical computer science is that every randomized algorithm can be efficiently “derandomized” — converted into a deterministic algorithm (which always outputs the correct answer) with only a polynomial increase in running time and only a constant-factor increase in space (i.e. memory usage).

    In this talk, I will describe an approach to proving the space (as opposed to time) version of this conjecture via spectral graph theory.  Specifically, I will explain how randomized space-bounded algorithms are described by random walks on directed graphs, and techniques in algorithmic spectral graph theory (e.g. solving Laplacian systems) have yielded deterministic space-efficient algorithms for approximating the behavior of such random walks on undirected graphs and Eulerian directed graphs (where every vertex has the same in-degree as out-degree).  If these algorithms can be extended to general directed graphs, then the aforementioned conjecture about derandomizing space-efficient algorithms will be resolved.

    Joint works with Jack Murtagh, Omer Reingold, Aaron Sidford,  AmirMadhi Ahmadinejad, Jon Kelner, and John Peebles.

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  • **CANCELED** A scale-critical trapped surface formation criterion for the Einstein-Maxwell system
    10:30 AM-11:30 AM
    March 6, 2020
    Few notions within the realm of mathematical physics succeed in capturing the imagination and inspiring awe as well as that of a black hole. First encountered in the Schwarzschild solution, discovered a few months after the presentation of the Field Equations of General Relativity at the Prussian Academy of Sciences, the black hole as a mathematical phenomenon accompanies and prominently features within the history of General Relativity since its inception. In this talk we will lay out a brief history of the question of dynamical black hole formation in General Relativity and discuss a recent result, in collaboration with Xinliang An, on a scale-critical trapped surface formation criterion for the Einstein-Maxwell system.
  • A topological approach to convexity in complex surfaces
    3:30 PM-4:30 PM
    March 6, 2020
    1 Oxford Street, Cambridge, MA 02138 USA

    We will discuss the classical notion of J-convexity of subsets of complex manifolds, and the closely related notion of Stein manifolds. The theory is particularly subtle in complex dimension 2. Surprisingly, progress can be made using topological 4-manifold theory. Every tame CW 2-complex topologically embedded in a complex surface can be perturbed so that it becomes J-convex in the sense of being a nested intersection of Stein open subsets. These Stein neighborhoods are all topologically equivalent to each other, but can be very different when viewed in the smooth category. As applications, we obtain uncountable families of distinct smoothings of R^4 admitting convex or concave holomorphic structures. We can also generalize the notion of J-convex hypersurfaces to the topological category. The resulting topological embeddings behave like their smooth counterparts, but are much more common.

    Future schedule is found here: https://scholar.harvard.edu/gerig/seminar

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  • CMSA Quantum Matter/Quantum Field Theory Seminar: Quantized Graphs and Quantum Error Correction
    10:30 AM-12:00 PM
    March 11, 2020
    20 Garden Street, Cambridge, MA 02138

    Graph theory is important in information theory. We introduce a quantization process on graphs and apply the quantized graphs in quantum information. The quon language provides a mathematical theory to study such quantized graphs in a general framework. We give a new method to construct graphical quantum error correcting codes on quantized graphs and characterize all optimal ones. We establish a further connection to geometric group theory and construct quantum low-density parity-check stabilizer codes on the Cayley graphs of groups. Their logical qubits can be encoded by the ground states of newly constructed exactly solvable models with translation-invariant local Hamiltonians. Moreover, the Hamiltonian is gapped in the large limit when the underlying group is infinite.

  • **CANCELED** Schinzel-Zassenhaus, height gap and overconvergence
    3:00 PM-4:00 PM
    March 11, 2020
    1 Oxford Street, Cambridge, MA 02138 USA

    We explain a new height gap result on holonomic power series
    with rational coefficients, and prove the Schinzel-Zassenhaus conjecture
    as its consequence: a monic irreducible non-cyclotomic integer polynomial
    of degree $n > 1$ has at least one complex root of modulus exceeding
    $2^{1/(4n)}$. The method, an arithmetic algebraization argument with input
    from two fixed places of the global field $\mathbb{Q}$, takes a
    combination of a $p$-adic overconvergence for a certain algebraic
    function, at a fixed prime $p$ on the one hand (this is the place where
    the cyclotomic examples get recognized and excluded); and, on the other
    hand, of the automatic Archimedean analytic continuation for solutions of
    linear ODE. The input at the Archimedean place has a potential-theoretic
    flavor and comes out of Dubinin’s solution of an extremal problem of
    Gonchar about harmonic measure. A possible connection to the $p$-curvature
    conjecture is indicated, as an analogous height gap hypothesis for
    G-operators with infinite monodromy.

    Repeating the same pattern, we shall follow up by a sharp arithmetic
    criterion on a formal power series to satisfy a linear differential
    equation with polynomial coefficients. It upgrades the classical
    Polya-Bertrandias rationality criterion, and we shall conclude by
    explaining how this new criterion serves to amplify Calegari’s p-adic
    counterpart of Apery’s theorem, yielding thus a proof of irrationality of
    the Kubota-Leopoldt 2-adic zeta value $\zeta_2(5)$. The latter is joint
    work with Frank Calegari and Yunqing Tang.

  • Informal Geometry & Dynamics Seminar: Billiards, heights and non-arithmetic groups
    4:00 PM-6:00 PM
    March 11, 2020

    No additional detail for this event.

  • CMSA Colloquium: Menu Costs and the Volatility of Inflation
    4:30 PM-5:30 PM
    March 11, 2020
    20 Garden Street, Cambridge, MA 02138
    We present a state-dependent equilibrium pricing model that generates inflation rate fluctuations from idiosyncratic shocks to the cost of price changes of individual firms.  A firm’s nominal price increase lowers other firms’ relative prices, thereby inducing further nominal price increases. We first study a mean-field limit where the equilibrium is characterized by a variational inequality and exhibits a constant rate of inflation. We use the limit model to show that in the presence of a large but finite number n of firms the snowball effect of repricing causes fluctuations to the aggregate price level  and these fluctuations converge to zero slowly as n grows. The fluctuations caused by this mechanism are larger when the density of firms at the repricing threshold is high, and the density at the threshold is high when the trend inflation level is high. However a calibration to US data shows that this mechanism is quantitatively important even at modest levels of trend inflation and  can account for the positive relationship between inflation level and volatility that has been observed empirically.

     

    (Joint with Makoto Nirei, University of Tokyo.)

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  • CMSA Colloquium: Math, Music and the Mind; Mathematical analysis of the performed Trio Sonatas of J.S. Bach
    4:00 PM-5:00 PM
    March 12, 2020
    20 Garden Street, Cambridge, MA 02138

    The works by J.S. Bach discussed in this talk will be performed in a free recital by Prof. Forger at Harvard’s Memorial Church at 7:30pm that evening (March 12th)

    I will describe a collaborative project with the University of Michigan Organ Department to perfectly digitize many performances of difficult organ works (the Trio Sonatas by J.S. Bach) by students and faculty at many skill levels. We use these digitizations, and direct representations of the score to ask how music should encoded in the mind. Our results challenge the modern mathematical theory of music encoding, e.g., based on orbifolds, and reveal surprising new mathematical patterns in Bach’s music. We also discover ways in which biophysical limits of neuronal computation may limit performance.

    Daniel Forger is the Robert W. and Lynn H. Browne Professor of Science, Professor of Mathematics and Research Professor of Computational Medicine and Bioinformatics at the University of Michigan. He is also a visiting scholar at Harvard’s NSF-Simons Center and an Associate of the American Guild of Organists.

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  • **CANCELED** Lifting cobordisms and Kontsevich-type recursions for counts of real curves **CANCELED**
    11:00 AM-12:00 PM
    March 17, 2020

    Kontsevich’s recursion, proved in the early 90s, is a recursion formula for the counts of rational holomorphic curves in complex manifolds. For complex fourfolds and sixfolds with a real structure (i.e. a conjugation), signed invariant counts of real rational holomorphic curves were defined by Welschinger in 2003. Solomon interpreted Welschinger’s invariants as holomorphic disk counts in 2006 and proposed Kontsevich-type recursions for them in 2007, along with an outline of a potential approach of proving them. For many symplectic fourfolds and sixfolds, these recursions determine all invariants from basic inputs. We establish Solomon’s recursions by re-interpreting his disk counts as degrees of relatively oriented pseudocycles from moduli spaces of stable real maps and lifting cobordisms from Deligne–Mumford moduli spaces of stable real curves (this is very different from Solomon’s approach).

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  • CMSA Mathematical Physics Seminar: Bit Threads: Understanding Gravitation from Quantum Entanglement
    12:00 PM-1:00 PM
    March 23, 2020

    via Zoom Video Conferencing:  https://harvard.zoom.us/j/738333299

    The AdS/CFT correspondence stipulates that gravitational evolution in a bulk spacetime is dual to a boundary description that has no gravity. In the AdS/CFT picture the bulk spacetime evolves gravitationally against an anti-de Sitter space background, and the boundary dual theory is a conformal gauge theory in a spacetime of one dimension less. Recent insights by Ryu and Takayanagi have conjectured that quantum entangled boundary states quantitatively give rise to geometry in the bulk. They do so by explicitly referring to “minimal surfaces” in the bulk, connecting them to the entropy of a related area in the boundary. I will present a conceptually and technically powerful complementary holographic entanglement picture, reformulating Ryu–Takayanagi to no longer refer to minimal surfaces, and suggesting a new way to think about the holographic principle and the connection between spacetime gravitation and information. I will introduce the idea of bit threads, and show how they can be used for fun and profit.

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  • The spacetime positive mass theorem and path connectedness of initial data sets
    10:30 AM-11:30 AM
    March 27, 2020

    via Zoom Video Conferencing: https://harvard.zoom.us/j/635180669

    The purpose of this talk is twofold: First we present a new proof of the spacetime positive mass theorem (joint with Demetre Kazaras and Marcus Khuri); second we discuss some new results about the topology of initial data sets (joint with Martin Lesourd).
    The spacetime positive mass theorem that the mass of an initial data set is non-negative with equality if and only if the initial data set arises as subset of Minkowski space. This result has first been proven by Schoen and Yau using Jang’s equation. There are further proofs by Witten using spinors and by Eichmair, Huang, Lee and Schoen using MOTS. Our proof uses Stern’s integral formula technique and also leads to a new explicit lower bound of the mass which is even valid when the dominant energy condition is not satisfied.
    A central conjecture in mathematical relativity is the final state conjecture which states that initial data sets will eventually approach Kerr black holes. In particular, this would imply that the space of initial data sets is path connected. Building upon the work of Marques and using deep and beautiful results of Carlotto and Li, we show that indeed the space of initial data set with compact trapped interior boundary is path connected.

     

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  • Liquid Crystals and the Heilmann-Lieb Model
    10:00 AM-11:00 AM
    March 31, 2020

    via Zoom Video Conferencing:  https://harvard.zoom.us/j/779283357

    A liquid crystal is a phase of matter in which order and disorder coexist: for some degrees of freedom, there is order, whereas for others, disorder. Such materials were discovered in the late XIXth century, but it took over a century to understand, from microscopic models, how such phases form. In 1979, O. Heilmann and E.H. Lieb introduced an interacting dimer model with the goal of proving the emergence of such a liquid crystal phase. In this setting, this amounts to showing that dimers spontaneously align, but do not fully crystallize: there is no translational order. Heilmann and Lieb proved that dimers do, indeed, align, and conjectured that there is no translational order. In this talk, I will discuss a recent proof of this conjecture, that is, a proof of the emergence of a liquid crystal phase in the Heilmann-Lieb model. This is joint work with E.H. Lieb.

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