MathDB

Miscellaneous (problem 6)

\begin{enumerate}[label=\alph*)] \item Relate tr(T(m,n))tr(T(m,n)) to rings of quasi-invariants and to A. Wilson's conjecture on ∇p1n\nabla_{p_1}^n. \item Describe H∗(Hilb(xnd=yn))H^*(\text{Hilb}(x^{nd}=y^n)) as a module over rational Double Affine Hecke algebra (DAHA) and relate it to the above. \end{enumerate}

Equivalent formulations 2Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Miscellaneous (problem 6)

    Evaluate

    ∑∫tjtj+T7/8∣ζ(12+it)∣4 dt,\sum \int_{t_j}^{t_j+T^{7/8}} |\zeta(\tfrac{1}{2}+it)|^4 \,dt,

    where the sum is over a set of tjt_j's with modulus ≤T\leq T and spaced more than T7/8T^{7/8} apart from each other (related to work of Heath-Brown and of Zavorotnyi)

    source: AimPL: Moments of zeta and correlations of divisor sums (posed by H. Iwaniec)

  2. Miscellaneous (problem 6)

    If XX and YY are i.i.d.\ real random variables such that Var(X)=Var(Y)=1Var(X) = Var(Y) = 1. The question is, for Z∼N(0,1)Z \sim \mathcal{N}(0,1) and ZZ independent of X,YX,Y, is it true that

    h(X+Y)≤h(X+Z),h(X + Y) \leq h(X + Z),

    and

    I(X+Y)≥I(X+Z).I(X+Y) \geq I(X+Z).

    source: AimPL: Entropy power inequalities (posed by Piotr Nayar)

References

Primary source

AimPL: Algebra, geometry, and combinatorics of link homology

number-theoryprobability
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