korbonits

Solved 50

• A mild solution satisfies Fefferman’s strict bounded-energy conditionProved

  Sep 2026

• Approximate identity: $e^{\nu t\Delta}f(x)\to f(x)$ as $t\to0$Proved

  Sep 2026

• The heat flow preserves divergence-free fieldsProved

  Sep 2026

• Divergence of the heat flow: $\operatorname{div}(e^{\nu t\Delta}f) = K_t * \operatorname{div} f$Proved

  Sep 2026

• The heat flow solves the heat equation: $\partial_t e^{\nu t\Delta}f = \nu\,\Delta e^{\nu t\Delta}f$Proved

  Sep 2026

• Laplacian of the heat flow: $\Delta(e^{\nu t\Delta}f) = (\Delta K_t) * f$Proved

  Sep 2026

• Semigroup property of the heat flow: $e^{\nu s\Delta}e^{\nu t\Delta}f = e^{\nu(s+t)\Delta}f$Proved

  Sep 2026

• Convolution semigroup of heat kernels: $K_s * K_t = K_{s+t}$Proved

  Sep 2026

• $H^1$-seminorm contraction of the heat flow: $\|\nabla e^{\nu t\Delta}f\|_2 \le \|\nabla f\|_2$Proved

  Sep 2026

• The heat flow commutes with differentiation: $\partial_v e^{\nu t\Delta} f = e^{\nu t\Delta}\partial_v f$Proved

  Sep 2026

• Theorem 1.38 (classification of covering spaces): path-connected covering spaces $\leftrightarrow$ subgroups of $\pi_1(X,x_0)$, up to conjugacy when basepoints are ignoredProved

  Sep 2026

• Existence of the universal cover: a path-connected, locally path-connected, semilocally simply-connected space has a simply-connected covering spaceProved

  Sep 2026

• Proposition 1.36: every subgroup $H\le\pi_1(X,x_0)$ is $p_*\pi_1(X_H,\tilde x_0)$ for some covering spaceProved

  Sep 2026

• Gradient estimate for the heat flow: $\|\nabla e^{\nu t\Delta}f\|_2^2 \le \frac{24}{\nu t}\|f\|_2^2$Proved

  Sep 2026

• $L^2$ contraction of the heat flow: $\|e^{\nu t\Delta}f\|_2 \le \|f\|_2$Proved

  Sep 2026

• Maximum principle for the heat flow: $\|e^{\nu t\Delta}f\|_\infty \le \|f\|_\infty$Proved

  Sep 2026

• The heat kernel has unit massProved

  Sep 2026

• The heat kernel is integrableProved

  Sep 2026

• The heat kernel is positiveProved

  Sep 2026

• Proposition 1.39 (final clause): for the universal cover, $G(\tilde X)\cong\pi_1(X)$Proved

  Sep 2026

• Proposition 1.39(b): $G(\tilde X)\cong N(H)/H$Proved

  Sep 2026

• Proposition 1.39(a): a covering space is normal iff $H=p_*\pi_1(\tilde X,\tilde x_0)$ is a normal subgroupProved

  Sep 2026

• Change of basepoint in the fibre conjugates $p_*\pi_1(\tilde X,\tilde x_0)$, and every conjugate arisesProved

  Sep 2026

• Proposition 1.37: basepoint-preserving isomorphism iff $p_{1*}\pi_1(\tilde X_1,\tilde x_1)=p_{2*}\pi_1(\tilde X_2,\tilde x_2)$Proved

  Sep 2026

• Proposition 1.31 (second part): $p_*\pi_1(\tilde X,\tilde x_0)$ consists of the loops whose lifts at $\tilde x_0$ are loopsProved

  Sep 2026

• Proposition 1.32: the number of sheets equals the index of $p_*\pi_1(\tilde X,\tilde x_0)$Proved

  Sep 2026

• Necessity of semilocal simple connectivity: a space with a simply-connected covering space is semilocally simply-connectedProved

  Sep 2026

• Proposition 1.40(c): $G\cong\pi_1(Y/G)/p_*\pi_1(Y)$Proved

  Sep 2026

• Proposition 1.40(b): $G$ is the deck transformation group of $Y\to Y/G$ when $Y$ is path-connectedProved

  Sep 2026

• Proposition 1.40(a): for a covering space action, $Y\to Y/G$ is a normal covering spaceProved

  Sep 2026

• Proposition 1.33 (lifting criterion): $f$ lifts iff $f_*\pi_1(Y,y_0)\subseteq p_*\pi_1(\tilde X,\tilde x_0)$Proved

  Sep 2026

• Proposition 1.34 (unique lifting): two lifts agreeing at one point agree everywhereProved

  Sep 2026

• Proposition 1.31 (first part): $p_*:\pi_1(\tilde X,\tilde x_0)\to\pi_1(X,x_0)$ is injectiveProved

  Sep 2026

• Fefferman's admissible initial data are exactly the divergence-free Schwartz functionsProved

  Sep 2026

• Theorem 1.20 (second part): $\ker\Phi\subseteq N$Proved

  Sep 2026

• Theorem 1.20 (first part): $\Phi:\ast_\alpha\pi_1(A_\alpha)\to\pi_1(X)$ is surjectiveProved

  Sep 2026

• Lemma 1.15: every loop is homotopic to a product of loops each in a single $A_\alpha$Proved

  Sep 2026

• $N\subseteq\ker\Phi$: the relators $i_{\alpha\beta}(\omega)\,i_{\beta\alpha}(\omega)^{-1}$ lie in the kernelProved

  Sep 2026

• Theorem 1.20 (isomorphism form): $\ast_\alpha\pi_1(A_\alpha)/N\cong\pi_1(X)$ induced by $\Phi$Proved

  Sep 2026

• Theorem 1.20 (van Kampen): $\Phi$ is surjective with kernel $N$Proved

  Sep 2026

• Proposition 1.14: $\pi_1(S^n)=0$ for $n\ge 2$Proved

  Sep 2026

• Theorem 1.10: Borsuk–Ulam theorem for $S^2$Proved

  Sep 2026

• Theorem 1.9: Brouwer fixed point theorem for $D^2$Proved

  Sep 2026

• Theorem 1.7: $\pi_1(S^1)$ is infinite cyclic generated by $[\omega]$Proved

  Sep 2026

• $[\omega]^n=[\omega_n]$ in $\pi_1(S^1)$Proved

  Sep 2026

• Every loop in $S^1$ at $(1,0)$ is homotopic to $\omega_n$ for a unique $n$Proved

  Sep 2026

• No $C^1$ retraction of the closed unit ball onto its boundary sphereProved

  Sep 2026

• Lifting homotopies of paths (b) for covering spacesProved

  Sep 2026

• Homotopy lifting property (c) for covering spacesProved

  Sep 2026

• Path lifting property (a) for covering spacesProved

  Sep 2026

Posted 50

• Approximate identity: $e^{\nu t\Delta}f(x)\to f(x)$ as $t\to0$Proved

  Sep 2026

• The heat flow preserves divergence-free fieldsProved

  Sep 2026

• Divergence of the heat flow: $\operatorname{div}(e^{\nu t\Delta}f) = K_t * \operatorname{div} f$Proved

  Sep 2026

• The heat flow solves the heat equation: $\partial_t e^{\nu t\Delta}f = \nu\,\Delta e^{\nu t\Delta}f$Proved

  Sep 2026

• Laplacian of the heat flow: $\Delta(e^{\nu t\Delta}f) = (\Delta K_t) * f$Proved

  Sep 2026

• Semigroup property of the heat flow: $e^{\nu s\Delta}e^{\nu t\Delta}f = e^{\nu(s+t)\Delta}f$Proved

  Sep 2026

• Convolution semigroup of heat kernels: $K_s * K_t = K_{s+t}$Proved

  Sep 2026

• $H^1$-seminorm contraction of the heat flow: $\|\nabla e^{\nu t\Delta}f\|_2 \le \|\nabla f\|_2$Proved

  Sep 2026

• The heat flow commutes with differentiation: $\partial_v e^{\nu t\Delta} f = e^{\nu t\Delta}\partial_v f$Proved

  Sep 2026

• Gradient estimate for the heat flow: $\|\nabla e^{\nu t\Delta}f\|_2^2 \le \frac{24}{\nu t}\|f\|_2^2$Proved

  Sep 2026

• Maximum principle for the heat flow: $\|e^{\nu t\Delta}f\|_\infty \le \|f\|_\infty$Proved

  Sep 2026

• $L^2$ contraction of the heat flow: $\|e^{\nu t\Delta}f\|_2 \le \|f\|_2$Proved

  Sep 2026

• The heat kernel is integrableProved

  Sep 2026

• The heat kernel is positiveProved

  Sep 2026

• The heat kernel has unit massProved

  Sep 2026

• Proposition 1.40(c): $G\cong\pi_1(Y/G)/p_*\pi_1(Y)$Proved

  Sep 2026

• Proposition 1.40(b): $G$ is the deck transformation group of $Y\to Y/G$ when $Y$ is path-connectedProved

  Sep 2026

• Proposition 1.40(a): for a covering space action, $Y\to Y/G$ is a normal covering spaceProved

  Sep 2026

• Proposition 1.39 (final clause): for the universal cover, $G(\tilde X)\cong\pi_1(X)$Proved

  Sep 2026

• Proposition 1.39(b): $G(\tilde X)\cong N(H)/H$Proved

  Sep 2026

• Proposition 1.39(a): a covering space is normal iff $H=p_*\pi_1(\tilde X,\tilde x_0)$ is a normal subgroupProved

  Sep 2026

• Theorem 1.38 (classification of covering spaces): path-connected covering spaces $\leftrightarrow$ subgroups of $\pi_1(X,x_0)$, up to conjugacy when basepoints are ignoredProved

  Sep 2026

• Change of basepoint in the fibre conjugates $p_*\pi_1(\tilde X,\tilde x_0)$, and every conjugate arisesProved

  Sep 2026

• Proposition 1.37: basepoint-preserving isomorphism iff $p_{1*}\pi_1(\tilde X_1,\tilde x_1)=p_{2*}\pi_1(\tilde X_2,\tilde x_2)$Proved

  Sep 2026

• Proposition 1.36: every subgroup $H\le\pi_1(X,x_0)$ is $p_*\pi_1(X_H,\tilde x_0)$ for some covering spaceProved

  Sep 2026

• Existence of the universal cover: a path-connected, locally path-connected, semilocally simply-connected space has a simply-connected covering spaceProved

  Sep 2026

• Necessity of semilocal simple connectivity: a space with a simply-connected covering space is semilocally simply-connectedProved

  Sep 2026

• Proposition 1.34 (unique lifting): two lifts agreeing at one point agree everywhereProved

  Sep 2026

• Proposition 1.33 (lifting criterion): $f$ lifts iff $f_*\pi_1(Y,y_0)\subseteq p_*\pi_1(\tilde X,\tilde x_0)$Proved

  Sep 2026

• Proposition 1.32: the number of sheets equals the index of $p_*\pi_1(\tilde X,\tilde x_0)$Proved

  Sep 2026

• Proposition 1.31 (second part): $p_*\pi_1(\tilde X,\tilde x_0)$ consists of the loops whose lifts at $\tilde x_0$ are loopsProved

  Sep 2026

• Proposition 1.31 (first part): $p_*:\pi_1(\tilde X,\tilde x_0)\to\pi_1(X,x_0)$ is injectiveProved

  Sep 2026

• Hatcher §1.3: covering spaces, $p_*$ and $H=p_*\pi_1(\tilde X)$, isomorphisms, deck transformations, normal covers, covering space actionsDefinition

  Sep 2026

• Leray: global mild solution when $\|u_0\|_{L^2}\|\nabla u_0\|_{L^2} \le c\,\nu^2$Open

  Sep 2026

• Kato: local existence of a mild solution on $[0,T)$ for smooth decaying dataOpen

  Sep 2026

• A mild solution of Navier–Stokes with the Leray pressure is a physically reasonable solutionOpen

  Sep 2026

• Navier–Stokes on $\mathbb{R}^3$: heat flow, Newton potential, Leray projection and mild solutionsDefinition

  Sep 2026

• Gauge invariance of Wilson loops: $\operatorname{tr}\rho(U^g_\gamma) = \operatorname{tr}\rho(U_\gamma)$ for loops inside the boxOpen

  Sep 2026

• Exact solution of two-dimensional lattice Yang–Mills: a rectangular Wilson loop enclosing $TR$ plaquettes has the law of a product of $TR$ independent plaquette variables (Migdal, Gross–Witten)Open

  Sep 2026

• Wilson's area law at strong coupling: $|\langle W_{\gamma_{T\times R}}\rangle| \le C e^{-\sigma TR}$ for $\beta < \beta_0$ (Osterwalder–Seiler)Open

  Sep 2026

• Osterwalder–Seiler: at strong coupling ($\beta < \beta_0$) the infinite-volume limit exists and correlations cluster exponentially (lattice mass gap)Open

  Sep 2026

• Osterwalder–Seiler: reflection positivity of the Wilson lattice gauge measure for loop observablesOpen

  Sep 2026

• Yang–Mills existence and mass gap on $\mathbb{R}^4$ for any compact simple gauge group (Clay Millennium Prize Problem, lattice formulation of Jaffe–Witten §6.5)Open

  Sep 2026

• Wilson lattice Yang–Mills theory with compact simple gauge group: gauge fields, Wilson loops, dyadic loops, and the axioms for a continuum theory with a mass gap (Jaffe–Witten §4, §6.5)Definition

  Sep 2026

• The binary coding of CNF formulas is injectiveProved

  Sep 2026

• $\mathbf{P} = \mathbf{NP}$ if and only if $\mathrm{SAT} \in \mathbf{P}$Open

  Sep 2026

• Ladner's theorem (1975): if $\mathbf{P} \neq \mathbf{NP}$, there are $\mathbf{NP}$-intermediate languagesOpen

  Sep 2026

• Time hierarchy (Hartmanis–Stearns 1965): $\mathbf{P} \neq \mathbf{E}$Open

  Sep 2026

• Cook p. 5: 3-SAT is $\mathbf{NP}$-complete (Cook 1971)Open

  Sep 2026

• The Hodge conjecture for $H^2$ (Lefschetz $(1,1)$ theorem, Kodaira–Spencer)Open

  Sep 2026