Yang–Mills existence and mass gap
Construct four-dimensional quantum Yang–Mills theory for every compact simple gauge group and prove that it has a positive mass gap.
问题
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符合的问题
Bell violation by the vacuum at arbitrarily large distances
Decide whether the vacuum of a free massive scalar field violates the CHSH inequality between bounded regions at arbitrarily large distances.
The mean of the Tracy–Widom distribution
Find a closed form for the mean μ ≈ −1.7710868074 of the Tracy–Widom distribution of the largest GUE eigenvalue.
The critical mass of the Townes soliton
Find a closed form for the critical mass N_c ≈ 11.7008965246 of the Townes soliton, the collapse threshold of the 2D cubic NLS.
Smallest output dimension for a minimum-output-entropy violation
Find the smallest output dimension in which two quantum channels violate additivity of minimum output entropy; it lies between 2 and 182.
Additivity of minimum output entropy for qubit channels
Prove or refute that minimum output entropy is additive when one of two channels is an arbitrary, possibly non-unital, qubit channel.
Almost-sure global existence in the Newtonian N-body problem
Prove that for Newtonian point masses, the initial conditions whose motion breaks down in finite time form a set of measure zero.
The multidimensional L² conjecture for Schrödinger operators
Show that in two or more dimensions, −Δ + V keeps a.c. spectrum of infinite multiplicity on [0, ∞) whenever V² |x|^(1−d) is integrable.
The Madelung constant of rock salt
Find a closed form for the Madelung constant of rock salt, M ≈ 1.7475645946, the alternating lattice sum of unit charges on Z^3.
The Ising-class integral C5
Find a closed form for the Ising-class integral C5 = (4/15)∫ t K0(t)^5 dt ≈ 0.6657598002 of Bailey, Borwein and Crandall.
Feigenbaum's period-doubling constant
Find a closed form for Feigenbaum's period-doubling constant δ ≈ 4.6692016091, known to 10,000 digits but not proven irrational.
The ionization conjecture
Prove that a nucleus of charge Z can bind at most Z + C electrons, with a constant C independent of Z.