tensorgami

A great deal of modern mathematics can be read as a wager on compressibility. One camp believes that with the right abstractions the world can be funneled into an additive and exact environment where one can add, subtract, take kernels and cokernels, and push information through functors without losing control. Another camp believes that the essential difficulty lives in nonlinear and metric phenomena that will not dissolve into categories and functorial maps. These are not ideologies so much as priors. They predict what counts as an explanation, where one expects rigor to live, and which skills read as talent.

Linearity in this context does not mean a preference for linear differential equations. It means securing an additive target where superposition and exactness make long arguments commute with finite bookkeeping. Cohomology is the canonical device. A coherent sheaf gives a complex of vector spaces whose cohomology groups are functorial and additive, so exact triangles become long exact sequences. Grothendieck’s constructions turn this into a general method. The Grothendieck group forces the relation whenever is exact, and the Chern character sends theory to cohomology through
,
so index and Riemann Roch statements become identities of linear data. Deformation theory linearizes moduli near a point: tangents appear as , obstructions as , and the Maurer Cartan equation packages higher order structure. Tannakian reconstruction encodes a nonlinear symmetry by its tensor category of linear representations and a fiber functor. The Lefschetz trace formula translates counting into traces of Frobenius on adic cohomology. Hodge theory equips with a decomposition and operators whose algebraic identities capture deep geometry. Even stability is engineered as linear data: one linearizes a group action by a line bundle and reads stability from weights of one parameter subgroups.

None of this is mere linear algebra. The linear invariants are only as good as the geometry that makes them honest. Topoi, descent, representability, deformation and obstruction, purity, monodromy, and Lefschetz type inputs are nonlinear scaffolding that allows the linear target to compute the right thing. Deligne’s proof of the Weil conjectures is emblematic. Bounds on exponential sums become traces of Frobenius acting on adic cohomology with a theory of weights that supplies sharp estimates. The victory is cohomological in spirit and geometric in infrastructure.

Recent foundational work continues the linearization instinct by repackaging analytic geometry so that finiteness, duality, and comparison look algebraic. Proofs replace epsilon and delta with categorical completeness, nuclearity, and derived topological tensor products. The analytic content does not vanish. It is displaced into the axioms of the ambient world and into the definitions of its objects. The burden is redistributed, not erased.

Complex geometry shows where compression leaks. The frontier questions ask for canonical metrics, regularity, and stability for nonlinear equations such as complex Monge Ampere. Yau’s solution of Calabi, the analytic side of the Yau Tian Donaldson correspondence, and methods that drive extension and vanishing all require a priori estimates and compactness. Algebra can predict and package. Stability criteria are refined and tested, birational packages and moduli constructions anticipate where existence should hold, and multiplier ideals carry analytic information into algebraic statements. But when one finally writes the proof, the engine is an energy budget or a convexity argument in a space of potentials. The craft is analytic.

The two grammars of rigor follow from different failure modes. In analysis the detail lives on the page. Referees re run inequalities, track dependence of constants on norms and dimension, check embeddings and interpolations, and verify that each small parameter truly closes a bootstrap. Correctness is legible locally. In algebra the detail is often centralized. Referees audit ambient categories and adjectives, functorialities and base change, and demand canonical sources rather than folklore. Correctness is legible globally. Each side also owns the other’s virtues. Algebra has its line by line grind in the explicit commutative algebra and valuation theory of the minimal model program. Analysis has large black boxes in Calderon Zygmund theory, semiclassical and microlocal calculi, and concentration compactness. The difference is where the culture expects vulnerability to be exposed.

The sociology follows legibility. Engineers and physicists read analysis immediately because their equations are partial differential equations and their metrics are rates, thresholds, and stability regions. A good analyst states a quantitative prediction and hits it. That is falsifiable on the page and it translates into numerics or prototypes. Algebra commands an aura in places where structure is the coin of the realm. Cryptography and coding theory rest on number theoretic and algebraic geometry constructions, and some areas of theoretical physics draw prestige from representation theory and categorical frameworks. Prestige is conditional on what a neighboring field counts as power.

The same prior on compressibility appears in theoretical computer science with other nouns. There is a large linear toolkit that regularizes messy instances into tractable ones. Johnson Lindenstrauss embeddings, CountSketch and second moment estimators, spectral sparsifiers and Laplacian solvers, linear and semidefinite relaxations with bespoke rounding, and the Sum of Squares hierarchy are engines of compression in the convex or spectral sense. There is also designed resistance to compression. Streaming and communication models punish summaries, cell probe lower bounds explain why some data structures cannot be both fast and small, and adaptive data analysis breaks naive sketching. Structure versus randomness and regularity lemmas are themselves compressibility devices. The lower bound machinery explains where that device stalls.

It is tempting to describe this split with the slogan of versus . A better description is a spectrum of priors on compressibility. A temperament expects uniform procedures that compress difficulty into reusable structure. An temperament expects stubborn local hardness and invests in bespoke methods under constraints. Neither temperament is a proof about the world. Each is a prediction about which explanations will persuade you and which kinds of rigor feel like truth.

A literary image captures both the ambition and the risk. In Ted Chiang’s Tower of Babylon the builders ascend a colossal tower, pierce the vault of heaven, and emerge back where they began. The cosmos closes on itself, so the pilgrimage up the edifice reveals a loop. Mathematics has its own loops. There are equivalence loops that are victories by design. GAGA, nonabelian Hodge, and Yau Tian Donaldson are deliberate loops that translate between worlds and return with isomorphisms in hand. There are abstraction loops that reframe analytic proofs in a richer category where theorems read as formal consequences. The epsilon and delta move off the page, but they reappear as completeness conditions and exactness properties. There are social loops in which the climb becomes its own justification and progress is measured by conformity to a canon rather than by contact with problems that push back.

One wants a spiral rather than a loop. A spiral returns near the start with strictly more reach. There are practical tests. A conservation test asks whether the new framework proves only what could already be proved with comparable cost. A prediction test asks whether it forecasts new theorems or thresholds that later survive contact with proofs. A transfer test asks whether results move between previously separate areas without hiding hard parts in a fog of definitions. A cost test asks whether the number of irreducible estimates one must pay actually goes down. Positive answers mark a spiral.

Nonlinear problems that resist linear packaging have recognizable signatures. They sit at scaling thresholds where compactness fails or concentrates. The nonlinear Schrödinger equation illustrates the point. For

with scaling
,
the critical Sobolev index is . At or above criticality the natural embeddings are scale invariant, concentration appears, and one needs a choreography of concentration compactness, profile decompositions, Morawetz type monotonicity, and rigidity arguments. General relativity offers a different signature. Diffeomorphism invariance and null geometry erase naive coercivity. The stability of Minkowski and the analysis of Kerr require vector field methods, robust energy multipliers, and mode analysis that extract decay from the geometry of propagation. Free boundary problems and geometric measure theory develop singular sets whose structure must be controlled by monotonicity formulas, blow up analysis, and improvement of flatness. Kinetic theory mixes transport and collisions in a way that defeats additive bookkeeping until one crafts a weighted energy that couples derivatives and moments. Singular stochastic dynamics rebuild multiplication at the correct scales and renormalize divergent terms through regularity structures or paracontrolled calculus. Convex integration shows that even existence and uniqueness can fail for fluid equations in a way that no linear invariant foresees. In each case the decisive move is an estimate that a category cannot supply.

There is a reason this divide provokes strong feeling. Large edifices are coordination technologies. They amortize effort, standardize interfaces, move facts across contexts, and let a field scale beyond a handful of masters. They also create canons, dialects, and hierarchies. Refereeing can drift from checking arguments to checking allegiance. The analytic craft looks opposite. It is legible on the page and falsifiable by recomputation. Both grammars have their dogmas. Any well liked toolkit tempts its users to read the world through it. The right corrective is not cynicism but diagnostics. Ask what the framework actually makes cheap that was expensive before. Ask what would count as a counterexample to its ambition. Ask where the argument still needs an estimate that the category does not provide.

A pragmatic heuristic falls out. Before choosing tools, ask whether your problem compresses to linear data under a known functor or relaxation. If the answer is yes, build the ambient category carefully and push formal machinery as far as it will go. If the answer is no, budget for a nonlinear existence argument with and , an invariant that survives iteration, and a compactness scheme that does not lose control of constants. When you borrow across the aisle, make the dictionary explicit. If you import a black box, state the hypotheses and where they are checked. If you export an estimate into algebra, package it so that others can invoke it without reconstructing the analysis.

The honest use of Chiang’s tower is as a vantage point, not an eschatology. Build just high enough to see the correspondences you can prove and the statements you can transport. Then climb down and do the work that only lives on the ground. The promised land in mathematics is rarely beyond a vault. It is more often at the point where a linear dictionary and a nonlinear estimate meet and make each other cheaper.