13 January 2025

  • [SAL] Transformation representations of diagram monoids | James East (Western Sydney University)

    13 January 2025 - 11:00 - 12:00

    Zoom link: https://videoconf-colibri.zoom.us/j/92565611443?pwd=diOblm3Pu1aC5vYRMoN1BKd0DyktmB.1

    Cayley's Theorem states that any finite monoid can be faithfully represented as a semigroup of transformations (self-maps) of a finite set. The minimum size of such a set is the (minimum transformation) degree of the monoid.

    We obtain formulae for the degrees of the most well-studied families of finite diagram monoids, including the partition, Brauer, Temperley-Lieb and Motzkin monoids. For example, the partition monoid Pn has degree 1 + ( B(n + 2) - B(n + 1) + B(n) ) / 2 for n ≥ 2, where these are Bell numbers. The proofs involve constructing explicit faithful representations of the minimum degree, many of which can be realised as (partial) actions on projections.

    This is joint work with Reinis Cirpons and James Mitchell, both at Univ St Andrews.

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14 January 2025

  • [SAn] On the DiPerna-Majda gap problem for 2D Euler equations | Óscar Domínguez Bonilla (UNEF University-Madrid, Spain)

    14 January 2025 - 14:15 - 15:15
    Room 1.6, building VII

    A famous result ofDelort (1991) establishes the concentration-

    cancellationphenomenon for approximating solutions of 2D Euler equations

    with a vortex sheetwhose vorticity maximal function has a log-decay of order

    1/2 . On the otherhand, DiPerna and Majda (1987) showed that if the log-

    decay assumption isstrictly larger than 1 then the lack of concentration

    (and hence energyconservation) holds. Then the so-called DiPerna-Majda

    gap problem asks:concentration-cancellation vs. energy conservation in the

    remaining log-range(1/2,1]?

    In this talk, afterreviewing earlier contributions to the DiPerna-Majda

    gap problem, I willpresent a new approach to this question based on sparse-

    ness. This is based onjoint projects with Mario Milman and Daniel Spector.

    The talk will beself-contained, and no additional prerequisites are needed.

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