Lower bounds, circuit complexity and the landscape around P vs NP. Most progress is careful partial results and documented barriers, reviewed with stated reasons.
Construct explicit n×n matrices that stay high-rank even after many entry changes, with parameters strong enough for Valiant's circuit lower bounds. Random matrices are highly rigid, but no explicit matrix is known to meet the required parameters.
Is the deterministic communication complexity of every Boolean matrix M bounded by a polynomial in log rank(M)? The best upper bound is O(√rank) (Sudakov–Tomon), and the largest known separation is quadratic in log rank.
Khot's conjecture (2002) that approximating the value of unique games is NP-hard. Its imperfect-completeness 2-to-2 variant was proven in 2018, but the full conjecture remains open.
Prove larger circuit-size lower bounds for explicit Boolean functions. The best bound for general fan-in-2 circuits is still only about 3.1n, and a superpolynomial bound for a function in NP would separate P from NP.
Decide whether every problem whose solutions can be verified in polynomial time can also be solved in polynomial time (Clay Millennium Prize Problem). A full solution is not expected here; the goal is mapped barriers and verifiable partial results.
Let f:\-1,1\^n→\-1,1\ be a Boolean function with Fourier expansion f(x)=Σ_S⊆[n]hat f(S)χ_S(x). Its spectral entropy is H(hat f^2) := Σ_S⊆[n]hat f(S)^2log_21/hat f(S)^2, and its total influence is Inf(f) := Σ_S⊆[n]hat f(S)^2 lvert Srvert.
Let f:0,1^n→0,1 be a Boolean function. Let deg(f) denote the degree of the unique multilinear polynomial over ℝ that agrees with f on 0,1^n. A variable x_i is relevant if f depends on it (equivalently: x_i appears in some monomial with nonzero coefficient in the multilinear representation of f).
Let m,n be positive integers and let V be a set of n Boolean variables. By a random formula of density r = m/n, we mean a collection of m clauses selected u.a.r. with replacement from the set of 8C(n, 3) clauses on three distinct variables from V.
Let f be a Boolean function on n bits, i.e. f:0,1^n → 0,1 with n≥ 2. For x∈ 0,1^n and 1≤ i≤ n, let x^(i) be x with the i-th bit flipped. The (pointwise) sensitivity of f at x is s(f)(x):=Σ_i=1^n |f(x)-f(x^(i))|, and the (max) sensitivity is s(f):=max_x∈0,1^n s(f)(x).
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