Kai @kaichung.bsky.social · 3hSorry so the male wasps are born, impregnate their unborn sisters, then dig their way out and die. Wtf 010
Reposted by KaiRev. Dr. Everett Howe @ewhowe.com · 07/10/2026Human mathematicians are left to sort out this mess. So it’s like someone dropping a burning bag of shit on your doorstep, telling you there’s a silver dollar inside, and wondering why you’re not happy. 1205
Kai @kaichung.bsky.social · 30/09/2026more vacuous leaves greater room to fill it with whatever vibes appeal to you: profundity, divinity, awe-some-ness, etc 120
Kai @kaichung.bsky.social · 30/09/2026presumably the answer is "profundity"? What is more profound than (a) God and (b) Reality? Of course why these concepts should be considered profound is almost tautalogical. (and also, what is more mundane than reality?) 120
Reposted by KaiJay Willis @jaywillis.net · 23/09/2026Sometimes you only need to read the caption to know that a Supreme Court case is going to be an absolute bloodbath 510714
Kai @kaichung.bsky.social · 24/09/2026I would call it a "guitar pick" personally, which is hopefully unambitious! 021
Reposted by KaiRyan Boyd @ryanboyd.bsky.social · 21/09/2026NEW BRAIN DROPPEDmed.stanford.eduHuman brain is two separate organs, Stanford Medicine-led research findsStanford Health Care delivers the highest levels of care and compassion. SHC treats cancer, heart disease, brain disorders, primary care issues, and many more. 232602780
Kai @kaichung.bsky.social · 22/09/2026"The most important thing about mathematics is how it resides in the human brain. Mathematics is not something we sense directly: it lives in our imagination and we sense it only indirectly." 000
Kai @kaichung.bsky.social · 22/09/2026"Logic, formalization, and symbols can be very powerful tools for humans to use, but we are actually very poor at purely formal reasoning; computers are far better at formal computation and formal reasoning, but humans are far better mathematicians..." 100
Kai @kaichung.bsky.social · 22/09/2026"If a mathematical topic seems rich, beautiful and interesting, that signals that it fills a significant mental role. If we allow ourselves to drink it in, it's highly likely to become useful, even if we don't have applications in mind." 110
Kai @kaichung.bsky.social · 22/09/2026"My view is that in mathematics, an internal test is the best gauge for the significance of a subject. If it is rich and interconnected and if it grabs your interest, then it is very likely to be become significant to you, even though in many cases you can't foresee how..."matrixeditions.comTeichmuller Theory: foreword by William Thurstonforeword by William Thurston to John Hubbard's book Teichmuller Theory and Applications to Geometry, Topology, and Dynamics 110
Kai @kaichung.bsky.social · 22/09/2026to add, a wall in a continuously varying field would be like a wall where the density of a fluid discontinuously changes. The system would rapidly relax into a smooth configuration by distributing the gradient over a large region of space 030
Kai @kaichung.bsky.social · 21/09/2026here is the arxiv version: arxiv.org/abs/2506.053...arxiv.orgA 2D-CFT Factory: Critical Lattice Models from Competing Anyon Condensation Processes in SymTO/SymTFTIn this paper, we introduce a ``CFT factory'' : a novel algorithm of methodically generating 2D lattice models that would flow to 2D conformal fixed points in the infrared. These 2D models are realise... 010
Kai @kaichung.bsky.social · 20/09/2026The overall point here is that when you have a (generalized) symmetry, that organizes the spectrum of your theory in precise ways, into symmetry charges and symmetry defects (kinks), depending on how the symmetry is realized on the vacuum. Condensing/proliferating those objects leads to other phases 010
Kai @kaichung.bsky.social · 20/09/2026then decreasing h/J will reduce the mass of these excitations until then condense into the vacuum. This means that in L = K (df/dx)^2 + m^2 f^2 the m^2 becomes negative and we go to the SSB phase with two ground states. In the lattice when h=0 they are the all-up and all-down states exchanged by U 000
Kai @kaichung.bsky.social · 20/09/2026The symmetry charges are created by acting with a single Z, creating a state Z_i |0> = ... |+> |+> |-> |+> |+> ... where X|-> = -1 |->. The J term gives these charges kinetic energy and h controls their mass, we can loosely think of this like a scalar field theory with L = J(df/dx)^2 + h f^2 100
Kai @kaichung.bsky.social · 20/09/2026Take the 1+1D transverse field Ising chain, with Hamiltonian H = - J \sum_i Z_i Z_{i+1} - h \sum_i X_i where XZ=-ZX are Pauli operators. The Z2 symmetry is generated by U = \prod_i X_i The symmetric phase is h/J>>1, with ground state |0> = \otimes_i \ket{+}_i with U |0> = |0> 100
Kai @kaichung.bsky.social · 20/09/2026The SSB phases are also gapped. The excitations of a symmetry broken phase come in two varieties: domain walls or "kinks" between two symmetry broken vacua and charges of the unbroken H symmetry labeled by irreps of H. They can also combine together to form composite bound "dyonic" objects 100
Kai @kaichung.bsky.social · 20/09/2026The unbroken phase is generally a gapped phase with a unique ground state. Symmetry broken phases are characterized by a subgroup H<G which is the residual symmetry. The SSB phase has multiple degenerate ground states (think all-up and all-down in the Ising model) labeled by cosets G/H 200
Kai @kaichung.bsky.social · 20/09/2026I mean, if I have a U(1) symmetry charges labeled by irreps of U(1) which are just integers. That integer is just the charge of the particle. It works the same for any group G. 100
Kai @kaichung.bsky.social · 20/09/2026Generally, given a G symmetry the SSB phases are labeled by subgroups H, and are associated to an order parameter algebra on the set of cosets, Fun(G/H), which decomposes into irreps under the G action. Condensing a charge C breaks G to an H such that Fun(G/H) contains C 120
Kai @kaichung.bsky.social · 20/09/2026Proliferation is the inverse process (in my usage or the terminology, which I'm not certain is standard). In the SSB phase you have gapped defects (domain walls for discrete G, homotopy defects for continuous G). Making a defect soft proliferates it and restores part of the broken symmetry 130
Kai @kaichung.bsky.social · 20/09/2026but that view generalizes to any symmetric phase with G symmetry with charges labeled by irreps of G. Condensing some charge species drives one into some G SSB phase 130
Kai @kaichung.bsky.social · 20/09/2026I view it as a phase transition driven by making a particular charge species "soft". The most well known example I guess is a superfluid, where we go from a U(1) symmetric phase with charge conservation to a SSB phase by condensing the charge into the vacuum. 130
Kai @kaichung.bsky.social · 20/09/2026Yeah indirectly, generally been learning about symmetry breaking and phase transitions through the lens of SymTFT, particularly with an eye towards non-Abelian anyon condensation/proliferation bsky.app/profile/kaic... 120
Kai @kaichung.bsky.social · 20/09/2026alright so Ive been meaning to reply to this for a while but wasn't sure what papers to mention, but I'm reading one right now that is really fantastic: "An algorithm to generate two-dimensional critical lattice models using competing anyon condensation" www.nature.com/articles/s41...nature.comAn algorithm to generate two-dimensional critical lattice models using competing anyon condensation - Nature PhysicsConformal field theories have a central role in statistical and high-energy theoretical physics. An algorithm to systematically generate critical lattice models described by different conformal field ... 140
Kai @kaichung.bsky.social · 20/09/2026reading about Frobenius algebras and unitary fusion categories 1100
Reposted by KaiPeter @notalawyer.bsky.social · 20/09/2026a lot of people act like there's a trade being offered where we give up trans athletes in sports and in exchange fascism is defeated forever 187137432917
Reposted by KaiJoshua Grochow @joshuagrochow.bsky.social · 16/09/2026Same but said a little differently: Associativity is a kind of commutativity: the operations of “left mult by A” and “right mult by C” commute 071
Kai @kaichung.bsky.social · 14/09/2026geometrically frustrated systems are by far the most interesting! 010
Kai @kaichung.bsky.social · 14/09/2026In sum: (S=1) coherent states are disk-like uniaxial (S=3/2) ALL states saturate the quadrupole moment, uniform on the 4-sphere (S=2) coherent states are uniform on the 4-sphere (S>2) rod-like uniaxial states dominate 000
Kai @kaichung.bsky.social · 14/09/2026This means that for a Hamiltonian with interacting quadruple moments, the phase diagram will depend sensitively on the angular moment/spin quantum number. Here we demonstrate mean-field phase diagrams for S=3/2 and S=1 110
Kai @kaichung.bsky.social · 14/09/2026For S=2 the spectra are not the same, but the maximum eigenvalues are degenerate, so at low energies also they exhibit the uniform distribution on the 4-sphere. For S>2 the uniaxial operator has a larger maximum eigenvalue, so rod-like uniaxial states dominate 100
Kai @kaichung.bsky.social · 14/09/2026It turns out that S=3/2 very special: all the quadrupole operators magically have the same spectrum for S=3/2, and there is no distinction between uniaxial and biaxial. Thus S=3/2 quadrupoles behave like a spin on S^4 with an effective time-reversal symmetry Q \to -Q. 100
Kai @kaichung.bsky.social · 14/09/2026Unlike dipole operators, it turns out that quadrupole operators are not all unitarily equivalent. You can see this visually by looking at the five L=2 spherical harmonic orbitals. Four are equivalent by rotation, while the fifth is clearly not. Indeed, four are biaxial and one is uniaxial 100
Kai @kaichung.bsky.social · 14/09/2026these states are related not just by an SO(3) rotation but also by a sort of "shape parameter" that interpolates between uniaxial and biaxial configurations 100
Kai @kaichung.bsky.social · 14/09/2026We define similarly quadrupolar coherent states, which are max weight eigenstates of a quadrupole operator projected along a unit-norm symmetric trace-free tensor. The space of such unit tensors is a 4-sphere 100
Kai @kaichung.bsky.social · 14/09/2026Consider a dipolar coherent state: a max-weight eigenstate of a spin operator along an axis. All dipolar coherent states are related to each other by a simple rotation, and so are parameterized by a point on a 2-sphere 100
![As the fig develops, the wasp eggs hatch and develop into larvae. After going through the pupal stage, the mature male's first act is to mate with a female - before the female hatches.
Consequently, the female will emerge pregnant.
The males of many species lack wings and cannot survive outside the fig for a sustained period of time. After mating, a male wasp begins to dig out of the fig, creating a tunnel through which the females escape. 16]
Once out of the fig, the male wasps quickly die.
The females find their way out, picking up pollen as they do. They then fly to another tree of the same species, where they deposit their eggs and allow the cycle to begin again.](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:dwi2chdew4aglyfnypmvyxzl/bafkreibpgej7u2vzkljjnqkhpkdkgdklfbgn4j72rhnquf3jber4dmnhlm)
