flâneur

Janabel Xia

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on the atlas — 162

highlights — 972

  • On July 19, 2026, Anthropic employee and mathematician Levent Alpöge presented an explicit counterexample in three-dimensional space, discovered by Anthropic's large language model Claude Fable 5, which disproves the conjecture.[6][7]
    Jacobian conjecture
  • The elementary interpolation inequality for Lebesgue spaces, which is a direct consequence of the Hölder's inequality[3]: 707 reads: for exponents 1 ≤ 𝑝 ≤ 𝑟 ≤ 𝑞 ≤ ∞ , every 𝑓 ∈ 𝐿 𝑝 ( 𝑋 , 𝜇 ) ∩ 𝐿 𝑞 ( 𝑋 , 𝜇 ) is also in 𝐿 𝑟 ( 𝑋 , 𝜇 ) , and one has ‖ 𝑓 ‖ 𝐿 𝑟 ≤ ‖ 𝑓 ‖ 𝐿 𝑝 𝑡 ‖ 𝑓 ‖ 𝐿 𝑞 1 − 𝑡 , where, in the case of 𝑝 < 𝑞 < ∞ , 1 𝑟 is written as a convex combination 1 𝑟 = 𝑡 𝑝 + 1 − 𝑡 𝑞 , that is, with 𝑡 := 𝑝 ( 𝑞 − 𝑟 ) 𝑟 ( 𝑞 − 𝑝 ) and 1 − 𝑡 = 𝑞 ( 𝑟 − 𝑝 ) 𝑟 ( 𝑞 − 𝑝 ) ; in the case of 𝑝 < 𝑞 = ∞ , 𝑟 is written as 𝑟 = 𝑝 𝑡 with 𝑡 := 𝑝 𝑟 and 1 − 𝑡 = 𝑟 − 𝑝 𝑟 .
    Interpolation inequality
  • The goal behind our construction will be to ensure that rejection sampling is the only anamorphic technique possibl
    293.pdf
  • ne simple way to do this is to reveal the encryption randomness inside the ciphertext. That is, define
    293.pdf
  • A proper example of a smooth transition function will be:
    Bump function
  • expression x ∈ s \ t expands to x ∈ s ∧ x ∉ t. (The ∉ can be entered as \notin.) It can be rewritten manually using Set.diff_eq and dsimp or Set.mem_diff
    4. Sets and Functions — Mathematics in Lean 0.1 documentation
  • The following examples show their relationship to bounded union and intersection.
    4. Sets and Functions — Mathematics in Lean 0.1 documentation
  • all involve high-bandwidth participation, and so any realistic implementation has to be digital
    The importance of full-stack openness and verifiability
  • obtain ⟨a, ubfa⟩ := ubf
    3. Logic — Mathematics in Lean v4.19.0 documentation
  • The “r” in rcases stands for “recursive,” because it allows us to use arbitrarily complex patterns to unpack nested data. The rintro tactic is a combination of intro and rcases: example : FnHasUb f → FnHasUb g → FnHasUb fun x ↦ f x + g x := by rintro ⟨a, ubfa⟩ ⟨b, ubgb⟩ exact ⟨a + b, fnUb_add ubfa ubgb⟩
    3. Logic — Mathematics in Lean v4.19.0 documentation
  • Focused Research Organization (FRO).
    Lean (proof assistant)
  • Lean was developed primarily by Brazilian computer scientist Leonardo de Moura while employed by Microsoft Research and now Amazon Web Services and has had significant contributions from other coauthors and collaborators during its history.
    Lean (proof assistant)
  • example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x ↦ f x + g x := by rcases ubf with ⟨a, ubfa⟩ rcases ubg with ⟨b, ubgb⟩ use a + b apply fnUb_add ubfa ubgb The rcases tactic unpacks the information in the existential quantifier. The annotations like ⟨a, ubfa⟩, written with the same angle brackets as the anonymous constructors, are known as patterns, and they describe the information that we expect to find when we unpack the main argument
    3. Logic — Mathematics in Lean v4.19.0 documentation
  • emember that you can find theorems like these using Ctrl-space completion (or Cmd-space completion on a Mac). Remember also that you can use .mp and .mpr or .1 and .2 to extract the two directions of an if-and-only-if statement.
    3. Logic — Mathematics in Lean v4.19.0 documentation
  • Notice that we have to introduce the variables even though they are marked implicit:
    3. Logic — Mathematics in Lean v4.19.0 documentation
  • As of 2024, Snowflake proxies are hosted on about 140000 unique IP addresses concurrently
    Snowflake (software)
  • The ease and accessibility of creating proxies increases the difficulty of blocking their IP addresses due to the large number of them in existence.
    Snowflake (software)
  • In 2016, Gowers started Discrete Analysis to demonstrate that a high-quality mathematics journal could be inexpensively produced outside of the traditional academic publishing industry.[33]
    Timothy Gowers
  • AI and machine learning
    Our vision - ESRC Digital Good Network
  • These inequalities are closely related to the Cheeger bound for Markov chains and can be seen as a discrete version of Cheeger's inequality in Riemannian geometry.
    Expander graph
  • For example, the polynomial 𝑧 5 + 3 𝑧 3 + 7 has exactly 5 zeros in the disk | 𝑧 | < 2 since | 3 𝑧 3 + 7 | ≤ 31 < 32 = | 𝑧 5 | for every | 𝑧 | = 2 , and 𝑧 5 , the dominating part, has five zeros in the disk.
    Rouché's theorem
  • It’s never worth forgetting that at the dawn of the Cold War, the US deported Qian Xuesen, the CalTech professor who then built missile delivery systems for Beijing.
    2025 letter | Dan Wang
  • Call me a romantic, but I believe that there will be a future, and indeed a long future, beyond 2027. History will not end. We need to cultivate the skill of exact thinking in demented times.
    2025 letter | Dan Wang
  • But only in San Francisco do people insist that Beijing wants Taiwan for its production of AI chips. In vain do I protest that there are historical and geopolitical reasons motivating the desire, that chip fabs cannot be violently seized, and anyway that Beijing has coveted Taiwan for approximately seven decades before people were talking about AI.
    2025 letter | Dan Wang
  • That contributes to a culture I think of as Silicon Valley’s soft Leninism. When political winds shift, most people fall in line, most prominently this year as many tech voices embraced the right.
    2025 letter | Dan Wang
  • Silicon Valley often speaks in strange tongues, starting podcasts and shows that are popular within the tech world but do not travel far beyond the Bay Area
    2025 letter | Dan Wang
  • Tech has organizations I think of as internal civic institutions that try to build community. They bring people together in San Francisco or retreats north of the city, bringing together young people to learn from older folks.
    2025 letter | Dan Wang
  • For a scalar function f and vector field v, the covariant derivative ∇ 𝑣 𝑓 coincides with the Lie derivative 𝐿 𝑣 ( 𝑓 )
    Covariant derivative
  • d 2 x i dt 2 + n X j,k =1 Γ i jk dx j dt dx k dt = 0
    fundamental_theorem_of_riemannian_geometry.pdf
  • ( ∂ u Γ u uv ) ⃗r u + Γ u uv ∇ u ⃗r u + ( ∂ u Γ v uv ) ⃗r v + Γ v uv ∇ u ⃗r v = ( ∂ u Γ u uv ) ⃗r u + Γ u uv (Γ u uu ⃗r u + Γ v uu ⃗r v ) + ( ∂ u Γ v uv ) ⃗r v + Γ v uv (Γ u uv ⃗r u + Γ v uv ⃗r v )
    gauss_celebrated_theorem.pdf
  • ( ∂ u Γ u uv ) ⃗r u + Γ u uv ∇ u ⃗r u + ( ∂ u Γ v uv ) ⃗r v + Γ v uv ∇ u ⃗r v = ( ∂ u Γ u uv ) ⃗r u + Γ u uv (Γ u uu ⃗r u + Γ v uu ⃗r v ) + ( ∂ u Γ v uv ) ⃗r v + Γ v uv (Γ u uv ⃗r u + Γ v uv ⃗r
    gauss_celebrated_theorem.pdf
  • Gaussian curvature of a surface in the 3-dimensional Euclidean space, which is defined as the product of the two extremal curvature of curves cut out from a plane containing the normal, depends only on the metric of the surface and not on how the surface is embedded inside the 3-dimensional Euclidean space
    gauss_celebrated_theorem.pdf
  • the ring homomorphisms induced by 𝜑 between the stalks of 𝑌 and the stalks of 𝑋 must be local homomorphisms, i.e. for every 𝑥 ∈ 𝑋 the maximal ideal of the local ring (stalk) at 𝑓 ( 𝑥 ) ∈ 𝑌 is mapped into the maximal ideal of the local ring at 𝑥 ∈ 𝑋
    Ringed space - Wikipedia
  • Cartan’s formula ( dω ) ( ξ 1 ∧···∧ ξ k +1 ) = k +1 X j =1 ( − 1) j +1 ξ j  ω  ξ 1 ∧···∧ d ξ j ∧···∧ ξ k +1  + X i ≤ j<ℓ ≤ k +1 ( − 1) j + ℓ ω  [ ξ j , ξ ℓ ] ∧···∧ d ξ j ∧···∧ d ξ k ∧···∧ ξ k +1
    differential_forms.pdf
  • covariant derivative of a type (r, s) tensor field along 𝑒 𝑐 is given by the expression: ( ∇ 𝑒 𝑐 𝑇 ) 𝑎 1 … 𝑎 𝑟 𝑏 1 … 𝑏 𝑠 = ∂ ∂ 𝑥 𝑐 𝑇 𝑎 1 … 𝑎 𝑟 𝑏 1 … 𝑏 𝑠 + Γ 𝑎 1 𝑑 𝑐 𝑇 𝑑 𝑎 2 … 𝑎 𝑟 𝑏 1 … 𝑏 𝑠 + ⋯ + Γ 𝑎 𝑟 𝑑 𝑐 𝑇 𝑎 1 … 𝑎 𝑟 − 1 𝑑 𝑏 1 … 𝑏 𝑠 − Γ 𝑑 𝑏 1 𝑐 𝑇 𝑎 1 … 𝑎 𝑟 𝑑 𝑏 2 … 𝑏 𝑠 − ⋯ − Γ 𝑑 𝑏 𝑠 𝑐 𝑇 𝑎 1 … 𝑎 𝑟 𝑏 1 … 𝑏 𝑠 − 1 𝑑 .
    Covariant derivative
  • owever another generalization of directional derivatives which is canonical: the Lie derivative, which evaluates the change of one vector field along the flow of another vector field
    Covariant derivative
  • dω i + n X j =1 ω i j ∧ ω j
    fundamental_theorem_of_riemannian_geometry.pdf
  • Ω ℓ kij + Ω ℓ ijk + Ω ℓ jki
    fundamental_theorem_of_riemannian_geometry.pdf
  • 5. Curvature of Levi-Civita Connection (
    fundamental_theorem_of_riemannian_geometry.pdf
  • γ i ⃗ ξ ( s ) = ⃗ ξ i
    fundamental_theorem_of_riemannian_geometry.pdf
  • The equation for parallel transport of ⃗η along γ ⃗ ξ is d ds ⃗η i + n X j,k =1 Γ i jk ⃗ ξ j ⃗η k = 0
    fundamental_theorem_of_riemannian_geometry.pdf
  • T ( φ ⃗ ξ,ψ⃗η ) = φψT ( ⃗ ξ,⃗η ) and T is a tensor and is called the torsion tensor .
    fundamental_theorem_of_riemannian_geometry.pdf
  • Lie bracket [ ⃗ ξ,⃗η ] yields the same discrepancy terms when ⃗ ξ and ⃗η are respectively multiplied by smooth functions φ and ψ , viz . [ φ ⃗ ξ,ψ⃗η ] = φψ [ ⃗ ξ,⃗η ] + φ ( ⃗ ξψ ) ⃗η − ψ ( ⃗ηφ ) ⃗ ξ. So we conclude that the torsion-free condition is equivalent to the vanishing of ∇ ⃗ ξ ⃗η −∇ ⃗η ⃗ ξ − [ ⃗ ξ,⃗η ] for any pair of tangent vectors ⃗ ξ and ⃗η .
    fundamental_theorem_of_riemannian_geometry.pdf
  • pecialize to the case where the vector bundle V is the tangent bundle T X of X
    fundamental_theorem_of_riemannian_geometry.pdf
  • Γ i jk = 1 2 n X ℓ =1 g iℓ  ∂ ∂x j g kℓ + ∂ ∂x k g jℓ − ∂ ∂x ℓ g jk
    fundamental_theorem_of_riemannian_geometry.pdf
  • with g ij symmetric in
    fundamental_theorem_of_riemannian_geometry.pdf
  • The connection formalizes and generalizes the "rolling without slipping or twisting" method of transporting tangent planes of a smooth surface embedded in 𝑅 3
    Levi-Civita connection - Wikipedia
  • ( ∂ u Γ u uv ) F + Γ u uv (Γ u uu F + Γ v uu G ) + ( ∂ u Γ v uv ) G + Γ v uv (Γ u uv F + Γ v uv G ) − [( ∂ v Γ u uu ) F + Γ u uu (Γ u uv F + Γ v uv G ) + ( ∂ v Γ v uu ) G + Γ v uu (Γ u vv F + Γ v vv G )
    gauss_celebrated_theorem.pdf
  • It permits the calculation of curvature and metric properties of a surface such as length and area in a manner consistent with the ambient space.
    First fundamental form - Wikipedia
  • So the curvature can be defined in an invariant way by R ( X,Y ) s = ∇ X ∇ Y s −∇ Y ∇ X s −∇ [ X,Y ] s
    connection_curvature_chern_classes.pdf