Chern–Simons Action

damn I can write Chern-Simons (Simons of Renaissance Technologies fame). that's awesome. The Chern–Simons action for a gauge field $A$ on a 3-dimensional manifold $M$ is given by:

Definitions

• $A$: Lie algebra–valued connection 1-form
• $M$: 3-dimensional manifold
• $k$: Level (an integer in quantum theory)
• $\mathrm{Tr}$: Invariant bilinear form on the Lie algebra
• $\wedge$: Wedge product of differential forms

Notes

• The theory is topological (independent of the metric on $M$).
• Under gauge transformations, the action is invariant up to a multiple of $2\pi k$, which leads to quantization of $k$.
• Plays a key role in:
  • Topological quantum field theory (TQFT)
  • Knot invariants (e.g. Jones polynomial)
  • 3D quantum gravity (in certain formulations)

Abelian Case (U(1))

For an abelian gauge field:

Alain Connes' stuff.

Connes' Distance Formula

One of the foundational results in noncommutative geometry is that the notion of distance on a space can be recovered purely from operator-algebraic data.

Let $(\mathcal{A}, \mathcal{H}, D)$ be a spectral triple, where:

• $\mathcal{A}$ is an involutive algebra acting on a Hilbert space $\mathcal{H}$,
• $D$ is a self-adjoint (Dirac) operator with compact resolvent,
• $[D,a]$ is bounded for all $a \in \mathcal{A}$.

Then the distance between two states $\varphi, \psi$ on $\mathcal{A}$ is given by:

Key Insight

• This formula generalizes the usual geodesic distance on a manifold.

• When $\mathcal{A} = C^\infty(M)$, $\mathcal{H}$ is the space of spinors, and $D$ is the Dirac operator, the formula reproduces the classical Riemannian distance:

  $$d(x,y) = \inf_{\gamma} \text{length}(\gamma)$$

Importance

• Replaces points with states on an algebra.
• Encodes geometry entirely in spectral data.
• Serves as a bridge between:
  • differential geometry
  • operator algebras
  • quantum physics

a big Grothendieck Theorem.

Grothendieck–Riemann–Roch Theorem

Let $f : X \to Y$ be a proper morphism between smooth quasi-projective varieties. Then the following diagram commutes after applying characteristic classes:

Equivalently,

Definitions

• $E \in K^0(X)$: a vector bundle (or class in K-theory)
• $f_!$: pushforward in K-theory (derived direct image)
• $f_*$: pushforward in cohomology (integration along the fiber)
• $\mathrm{ch}$: Chern character
• $\mathrm{Td}$: Todd class
• $T_f$: relative tangent bundle

Significance

• Vast generalization of the classical Riemann–Roch theorem.
• Unifies:
  • topology (K-theory),
  • algebraic geometry (sheaves, morphisms),
  • differential geometry (characteristic classes).
• Foundation for modern intersection theory and index theory.

Insight

The theorem expresses a deep compatibility between:

• algebraic data (K-theory),
• and geometric/topological invariants (cohomology),

showing that “pushforward” operations commute up to universal correction factors given by characteristic classes.

Ed Witten result.

Witten’s Interpretation of the Jones Polynomial (1989)

Witten showed that the Jones polynomial of a knot can be understood as an expectation value in a 3-dimensional quantum field theory—Chern–Simons theory.

Let $K \subset S^3$ be a knot and $A$ a gauge connection. Then:

Statement

• The expectation value of the Wilson loop observable $W_R(K)$ in Chern–Simons theory:

  • is a topological invariant of the knot $K$,
  • and reproduces the Jones polynomial (and its generalizations).

Ingredients

• $S_{CS}(A)$: Chern–Simons action
• $\mathcal{P} \exp$: path-ordered exponential
• $\mathrm{Tr}_R$: trace in representation $R$
• $\mathcal{D}A$: path integral over gauge fields

Why This Is Awesome

• Connects quantum field theory ↔ knot theory.
• Turns knot invariants into physical observables.
• Launched entire fields:
  • Topological quantum field theory (TQFT)
  • Quantum topology
  • Relations to quantum computing (anyon braiding)

Deep Insight

Topology of knots is encoded in quantum phases of gauge fields:

This was one of the first times physics predicted and explained deep structures in pure mathematics.

Something from Gromov.

Gromov’s Compactness Theorem

One of Mikhail Gromov’s huge theorems is the compactness theorem for pseudoholomorphic curves.

Let $(M,\omega)$ be a compact symplectic manifold with an almost complex structure $J$ compatible with $\omega$.
Let

be a sequence of $J$-holomorphic curves with uniformly bounded energy:

Then, after passing to a subsequence, the curves $u_n$ converge to a stable $J$-holomorphic curve.

The limiting object may contain bubbling: spheres or disks can appear when energy concentrates.

where $u_\infty$ is a stable map, possibly with bubble components.

Why This Is Huge

Gromov compactness is foundational in symplectic geometry.

It makes possible:

• Gromov–Witten invariants
• Floer homology
• modern symplectic topology
• mirror symmetry
• enumerative geometry via pseudoholomorphic curves

Key Insight

Sequences of holomorphic curves do not simply disappear or diverge.

Instead, their limits are controlled by a compactified moduli space:

whose boundary corresponds to bubbling and nodal degenerations.

In slogan form:

something from William Thurston, eh this didn't render so well. will work on it

Thurston’s Geometrization Conjecture (Theorem)

Every compact, orientable 3-manifold can be decomposed into pieces that each admit one of eight model geometries.

More precisely, let $M$ be a compact, orientable 3-manifold. Then there exists a canonical decomposition (along spheres and incompressible tori) such that each component admits a complete locally homogeneous Riemannian metric modeled on one of the following eight geometries:

Structure

The theorem combines two major decompositions:

1. Prime decomposition (Kneser–Milnor):

   $$M = M_1 \# M_2 \# \cdots \# M_k$$

2. JSJ decomposition (along incompressible tori): splits each $M_i$ into geometric pieces.

Significance

• Vast generalization of the Poincaré conjecture
• Reduces topology of 3-manifolds to geometry
• Introduced hyperbolic geometry as the generic case

Resolution

• Proved by Grigori Perelman (2002–2003) using Ricci flow with surgery
• One of the greatest achievements in modern mathematics

Insight

Topology in dimension 3 is governed by geometry:

Thurston’s vision turned classification into a problem of understanding curvature and geometric models.

Atiyah Singer.

Atiyah–Singer Index Theorem

Let $D$ be an elliptic differential operator on a compact manifold $M$. Then:

Meaning

• The analytic index:

  $$\mathrm{index}(D) = \dim \ker D - \dim \ker D^*$$

  counts solutions of a differential equation.

• The topological index:

  $$\int_M \mathrm{ch}(\sigma(D)) \cdot \mathrm{Td}(TM)$$

  depends only on global topological data.

Ingredients

• $D$: elliptic operator (e.g. Dirac operator)
• $\sigma(D)$: symbol of $D$
• $\mathrm{ch}$: Chern character
• $\mathrm{Td}$: Todd class
• $TM$: tangent bundle of $M$

Why This Is Huge

• Bridges analysis ↔ topology ↔ geometry
• Generalizes:
  • Gauss–Bonnet theorem
  • Riemann–Roch theorem
• Fundamental in:
  • quantum field theory
  • anomaly calculations
  • noncommutative geometry (Connes builds on this)

Insight

Local differential equations encode global topology:

This theorem is one of the deepest unifications in all of mathematics.

Bott

Bott Periodicity Theorem

Bott periodicity is a fundamental result describing a deep periodic structure in the homotopy groups of classical groups.

Complex Case

For the infinite unitary group $U$, there is a periodicity of period 2:

Equivalently, in topological K-theory:

Real Case

For the infinite orthogonal group $O$, there is periodicity of period 8:

Why This Is Huge

• Completely determines stable homotopy groups of classical groups
• Foundation of topological K-theory
• Explains recurring algebraic patterns across topology
• Used throughout:
  • index theory (Atiyah–Singer)
  • homotopy theory
  • mathematical physics

Geometric Insight

Topology repeats itself in a stable regime:

Deeper Interpretation

Bott’s original proof used Morse theory on loop spaces, revealing that:

• critical points correspond to geodesics
• topology of loop spaces encodes group structure

This connects:

• geometry (geodesics),
• analysis (Morse theory),
• and topology (homotopy groups)

Slogan

Perleman.

Perelman’s Proof of the Poincaré Conjecture

Poincaré Conjecture (1904)

Let $M$ be a closed, simply connected 3-manifold. Then:

That is, every such manifold is homeomorphic to the 3-sphere.

Perelman’s Theorem (2002–2003)

Using Ricci flow with surgery, Perelman proved the Poincaré Conjecture as a special case of Thurston’s Geometrization Conjecture.

Key Equation: Ricci Flow

Ideas

• Introduced entropy functionals controlling the flow
• Proved no local collapsing theorem
• Analyzed singularities via surgery
• Showed the flow evolves manifolds into canonical geometric pieces

Outcome

• Complete proof of Poincaré Conjecture
• Completion of Thurston’s Geometrization Program

Fun Facts

• Declined the Fields Medal (2006)
• Declined the $1 million Clay Prize
• Withdrew from the mathematical community

Insight

Geometry evolves to reveal topology:

seems to work ok so far