These notes synthesize material from multiple PMATH 965 instances at the University of Waterloo, drawing only on publicly available textbooks, monographs, and lecture notes. The course is a rotating topics course in geometry and topology, and over the past decade it has covered an unusually wide spectrum: special holonomy, deformation theory, algebraic stacks, rational homotopy theory, harmonic maps, spin geometry, higher stacks, gauge theory, and motivic integration. Sources used below include Joyce’s Compact Manifolds with Special Holonomy (Oxford UP), Besse’s Einstein Manifolds (Springer Classics), Sernesi’s Deformations of Algebraic Schemes (Springer), Hartshorne’s Deformation Theory (Springer GTM), Olsson’s Algebraic Spaces and Stacks (AMS Colloquium), the Stacks Project (stacks.math.columbia.edu), Felix-Halperin-Thomas’s Rational Homotopy Theory (Springer), Eells-Lemaire’s Selected Topics in Harmonic Maps (CBMS), Lawson-Michelsohn’s Spin Geometry (Princeton), Donaldson-Kronheimer’s The Geometry of Four-Manifolds (Oxford), and various publicly available lecture notes (MIT, Stanford, Cambridge).

Chapter 1: Holonomy Groups and Special Riemannian Structures

Taught at UW as PMATH 965 in Winter 2019 by Spiro Karigiannis.

The holonomy group of a Riemannian manifold is one of the most refined geometric invariants attached to its Levi-Civita connection. To define it, fix a point \(p\) in a Riemannian manifold \((M, g)\) and consider all piecewise-smooth loops based at \(p\). Each such loop \(\gamma\) gives rise to a parallel-transport map \(P_\gamma : T_p M \to T_p M\), which is an isometry of the inner product \(g_p\). The set of all such isometries, as \(\gamma\) ranges over loops, forms a subgroup of \(\mathrm{O}(T_p M, g_p) \cong \mathrm{O}(n)\) called the holonomy group \(\mathrm{Hol}_p(g)\). Its identity component, the restricted holonomy group \(\mathrm{Hol}_p^0(g)\), corresponds to null-homotopic loops, and on a simply connected manifold the two coincide. A change of basepoint conjugates the holonomy group, so as an abstract subgroup of \(\mathrm{O}(n)\) it is well-defined up to conjugacy.

The Ambrose-Singer theorem identifies the Lie algebra of \(\mathrm{Hol}_p^0(g)\) with the span of the curvature operators \(R(X,Y)\) parallel-transported to \(p\), so curvature controls holonomy infinitesimally. A parallel tensor field on \(M\) is invariant under holonomy, and conversely every \(\mathrm{Hol}_p\)-invariant tensor at \(p\) extends by parallel transport to a parallel tensor on \(M\). This duality is the principle of holonomy reduction: the existence of special geometric structures (almost complex, Kähler, hyperkähler, \(G_2\), \(\mathrm{Spin}(7)\)) is equivalent to a reduction of the holonomy group to the corresponding stabilizer in \(\mathrm{O}(n)\).

Berger’s classification is striking because it rules out the vast majority of Lie subgroups of \(\mathrm{SO}(n)\) on purely algebraic grounds: the curvature tensor must lie in a Hol-invariant subspace and must satisfy the Bianchi identities, and these constraints, combined with the requirement of acting irreducibly on the tangent space, leave only the listed groups. The de Rham decomposition theorem reduces the locally reducible case to a product, and the Cartan-Ambrose-Hicks theorem covers the locally symmetric case. The remaining cases form the special holonomy groups.

The Calabi conjecture, proved by Yau in 1978, asserts that on a compact Kähler manifold with \(c_1 = 0\) one can find a Ricci-flat Kähler metric in every Kähler class; this produces the first non-trivial examples of compact manifolds with holonomy exactly \(\mathrm{SU}(m)\). Compact \(G_2\) and \(\mathrm{Spin}(7)\) examples were constructed by Joyce in the 1990s by gluing techniques on resolutions of orbifolds, and these remain among the most subtle objects in differential geometry. The relation \(\mathrm{Hol} \subseteq \mathrm{SU}(m)\) implies \(\mathrm{Ric} = 0\) — special holonomy is in this sense a refinement of the Einstein condition, and it provides essentially the only known systematic source of compact Ricci-flat manifolds.

The course in W19 culminated in the proof of the Calabi-Yau theorem via continuity method, the Hard Lefschetz theorem, and the formality of compact Kähler manifolds (a result of Deligne-Griffiths-Morgan-Sullivan), foreshadowing the rational-homotopy techniques developed in Chapter 4.

Chapter 2: Deformation Theory and Moduli Spaces

Taught at UW as PMATH 965 in Fall 2020 by Satriano.

Deformation theory studies how a geometric object — a complex structure, a scheme, a coherent sheaf, a representation — varies in families, and it provides the local model for moduli spaces. The basic question is: given an object \(X_0\) over a field \(k\), what are the infinitesimal deformations of \(X_0\), i.e., flat families \(\mathcal{X} \to \mathrm{Spec}\,A\) with central fibre \(X_0\), where \(A\) is an Artin local \(k\)-algebra with residue field \(k\)? The answer is encoded by a deformation functor \(\mathrm{Def}_{X_0} : \mathrm{Art}_k \to \mathrm{Set}\), and the goal is to understand its tangent space, obstructions, and the existence of a hull or a universal formal deformation.

The central technical tool is Schlessinger’s criterion (1968), which gives necessary and sufficient conditions for a functor \(F : \mathrm{Art}_k \to \mathrm{Set}\) to admit a hull — a smooth pro-representing object up to non-canonical isomorphism. The conditions concern the behaviour of \(F\) on fibre products \(A \times_C B\): one needs the natural map \(F(A \times_C B) \to F(A) \times_{F(C)} F(B)\) to be surjective in general and bijective when \(C = k\). When \(F\) is moreover a sheaf for the étale topology on Artinian rings, Schlessinger’s hull is genuinely pro-representable.

Obstructions to extending a deformation from \(A\) to \(A'\) (where \(A' \twoheadrightarrow A\) is a small extension with kernel \(I\)) live in a second cohomology group \(T^2\). For a smooth scheme \(X_0\), \(T^2 = H^2(X_0, T_{X_0})\); for a singular scheme, one must use the cotangent complex \(\mathbb{L}_{X_0/k}\) of Illusie, and \(T^i = \mathrm{Ext}^i(\mathbb{L}_{X_0/k}, \mathcal{O}_{X_0})\). When \(T^2 = 0\), the deformation functor is unobstructed and the formal moduli space is smooth.

A modern viewpoint, due to Deligne, Drinfeld, Hinich, Manetti, Lurie, and Pridham, organizes deformation theory through differential graded Lie algebras (dglas). To each suitable geometric problem one associates a dgla \(L^\bullet\), and the deformation functor is recovered as the Maurer-Cartan functor

where the gauge action is by exponentiated \(L^0\). This is the content of the derived deformation principle: in characteristic zero, every reasonable formal moduli problem is controlled by a dgla (or, more generally, an \(L_\infty\)-algebra), and quasi-isomorphic dglas give equivalent problems. For \(X_0\) a complex manifold, the relevant dgla is the Kodaira-Spencer dgla \((\Omega^{0,\bullet}(X_0, T_{X_0}^{1,0}), \bar\partial, \left[\cdot,\cdot\right])\); for a coherent sheaf \(\mathcal{F}\), it is \(R\mathrm{Hom}(\mathcal{F}, \mathcal{F})\); and so on.

The interplay between obstruction theory and Hodge structures is the subject of Griffiths’ theory of variations of Hodge structures, and it is one of the most powerful tools in algebraic geometry.

Chapter 3: Algebraic Stacks and Equivariant Geometry

Taught at UW as PMATH 965 in Winter 2022 by Satriano.

Many natural moduli problems in algebraic geometry — moduli of curves, of vector bundles, of stable maps — fail to be representable by schemes because the objects they parameterize have non-trivial automorphisms. Stacks are the technology designed to remember those automorphisms. A stack over a site \(\mathcal{C}\) is, informally, a sheaf of groupoids: it assigns to each object \(U \in \mathcal{C}\) a groupoid \(\mathcal{X}(U)\) of “families over \(U\)”, and the assignment satisfies descent both for objects and for morphisms.

The most studied stacks are algebraic: they admit smooth atlases by schemes. Two flavours appear:

The prototypical example is the moduli stack of smooth genus \(g\) curves \(\mathcal{M}_g\), proved by Deligne and Mumford (1969) to be a smooth Deligne-Mumford stack of dimension \(3g - 3\) for \(g \geq 2\). Its compactification \(\overline{\mathcal{M}}_g\) by stable curves is proper. The quotient stack \(\left[X/G\right]\) — for \(G\) acting on a scheme \(X\) — is another fundamental construction: morphisms \(T \to \left[X/G\right]\) are pairs \((P, \varphi)\) where \(P \to T\) is a principal \(G\)-bundle and \(\varphi : P \to X\) is \(G\)-equivariant. When \(G\) acts freely with quotient a scheme, \(\left[X/G\right]\) is just the quotient scheme; in general, it remembers the stabilizers as automorphism groups of points.

Cohomology on stacks generalizes both étale cohomology of schemes and group cohomology of finite groups: for the classifying stack \(BG = \left[\mathrm{pt}/G\right]\) of a finite group \(G\), one has \(H^*_{\acute{e}t}(BG, \mathbb{F}_\ell) = H^*(G, \mathbb{F}_\ell)\). For \(G\) reductive acting on \(X\), \(H^*(\left[X/G\right]) = H^*_G(X)\) is equivariant cohomology, recovering the Borel construction \(H^*(EG \times_G X)\).

The descent theory underlying stacks is the étale or fppf version of Grothendieck’s faithfully flat descent, and the technical bedrock is provided by the Stacks Project (stacks.math.columbia.edu), which is fully public and the canonical modern reference. Olsson’s Algebraic Spaces and Stacks gives a more compact textbook account.

Chapter 4: Rational Homotopy Theory in Geometry

Taught at UW as PMATH 965 in Fall 2024 by Milivojević.

Rational homotopy theory is the study of topological spaces up to maps that induce isomorphisms on rational homotopy and rational cohomology. Its central insight, due to Sullivan and Quillen in the late 1960s, is that the rational homotopy type of a simply connected space \(X\) is captured by an entirely algebraic object: a commutative differential graded algebra (cdga) over \(\mathbb{Q}\), which can be chosen to be free as a graded commutative algebra and minimal in a precise sense. The cohomology of \(X\) is then the cohomology of this minimal model, and the rational homotopy groups \(\pi_*(X) \otimes \mathbb{Q}\) are read off from its generators.

To a simply connected space \(X\) of finite type one associates Sullivan’s cdga \(A_{PL}(X)\) of polynomial differential forms, and Sullivan’s theorem produces a minimal Sullivan algebra \((\Lambda V, d)\) with a quasi-isomorphism \((\Lambda V, d) \xrightarrow{\sim} A_{PL}(X)\); this minimal model is unique up to isomorphism. For a smooth manifold one may replace \(A_{PL}(X)\) with the de Rham algebra \(\Omega^*(X)\) tensored with \(\mathbb{R}\), which gives the real homotopy type.

A space is formal if its minimal model is quasi-isomorphic to \((H^*(X; \mathbb{Q}), 0)\) — i.e., the rational homotopy type is determined by the cohomology ring alone. The formality theorem of Deligne-Griffiths-Morgan-Sullivan asserts that compact Kähler manifolds are formal; the proof uses the \(\partial\bar\partial\)-lemma to construct a chain of quasi-isomorphisms. Symmetric spaces are formal; \(H\)-spaces are formal; but generic compact symplectic manifolds need not be (counterexamples by McDuff and others).

Quillen’s parallel approach uses differential graded Lie algebras instead of cdgas: to a simply connected space he associates a dgla \(\lambda(X)\) whose homology is \(\pi_*(\Omega X) \otimes \mathbb{Q}\) with the Whitehead product as bracket. The Quillen and Sullivan models are Koszul-dual, and the equivalence between them is one of the original instances of Koszul duality.

The standard public reference is Felix-Halperin-Thomas, Rational Homotopy Theory (Springer GTM 205); a leaner introduction is Felix-Oprea-Tanré, Algebraic Models in Geometry (Oxford GTM).

Chapter 5: Harmonic Maps

Taught at UW as PMATH 965 in Winter 2025 by Spiro Karigiannis.

A harmonic map between Riemannian manifolds simultaneously generalizes harmonic functions (the case where the target is \(\mathbb{R}\)) and geodesics (the case where the source is an interval). Given Riemannian manifolds \((M,g)\) and \((N,h)\) and a smooth map \(\varphi : M \to N\), the energy density is \(e(\varphi) = \tfrac{1}{2} |d\varphi|^2_{g \otimes \varphi^*h}\), and the energy is \(E(\varphi) = \int_M e(\varphi) \, \mathrm{vol}_g\). The map \(\varphi\) is harmonic if it is a critical point of \(E\) with respect to compactly supported variations.

When \(N = \mathbb{R}\) the tension field reduces to \(\Delta_g \varphi\), so harmonic functions are recovered. When \(M\) is an interval, the equation reduces to \(\nabla_t \dot\varphi = 0\), i.e., geodesic. When \(M = N\) and \(\varphi = \mathrm{id}\), the map is harmonic iff \(g\) is a critical point of an action that turns out to be related to scalar curvature. The harmonic-map equation is a quasilinear elliptic system on \(\varphi\), but it is generally non-linear in a serious way because of the curvature terms involving \(\Gamma^N\).

Negativity of the curvature of \(N\) makes \(J_\varphi\) positive-definite, which underlies many existence and rigidity results. The single most influential theorem in the subject is:

The proof uses a Bochner-type identity expressing \(\Delta e(\varphi) = |\nabla d\varphi|^2 + \mathrm{Ric}^M(d\varphi, d\varphi) - \langle R^N(d\varphi)d\varphi, d\varphi\rangle\), combined with parabolic theory for the heat flow. Non-positive curvature of \(N\) makes the curvature term favourable, giving uniform energy bounds and convergence.

The standard references used in W25 are Eells-Lemaire’s CBMS Selected Topics in Harmonic Maps, Urakawa’s Calculus of Variations and Harmonic Maps, and the original Eells-Sampson paper Harmonic mappings of Riemannian manifolds (Amer. J. Math. 86, 1964).

Chapter 6: Spin Geometry

Taught at UW as PMATH 965 in Fall 2026.

Spin geometry concerns geometric structures on Riemannian manifolds beyond the orthogonal frame bundle: it asks for a lift of the structure group \(\mathrm{SO}(n)\) to its double cover \(\mathrm{Spin}(n)\), and on such a lift it constructs the natural first-order elliptic operator now called the Dirac operator. The algebraic foundation is the Clifford algebra \(\mathrm{Cl}(V, q)\) of a quadratic vector space, which has many equivalent definitions: as the universal associative algebra generated by \(V\) subject to \(v^2 = -q(v)\cdot 1\); as the quotient of the tensor algebra by this relation; or, in low dimensions, as a matrix algebra over \(\mathbb{R}\), \(\mathbb{C}\), or \(\mathbb{H}\) according to Bott periodicity (mod 8 in the real case, mod 2 in the complex case).

A spin structure exists iff the second Stiefel-Whitney class \(w_2(M) \in H^2(M, \mathbb{Z}/2)\) vanishes; when it exists, the set of spin structures is a torsor over \(H^1(M, \mathbb{Z}/2)\). A spin\(^c\) structure exists iff \(w_2(M)\) lifts to an integral class, which is automatic on every almost complex manifold (the canonical \(\mathrm{Spin}^c\) structure has determinant line \(K_M^{-1}\)).

\(D\) is a first-order elliptic differential operator, formally self-adjoint on a closed manifold, and its square satisfies the Lichnerowicz formula

where \(R\) is the scalar curvature. This identity is the source of the most celebrated obstructions in the field: if \(R > 0\) everywhere on a closed spin manifold, then \(D^2 \geq \nabla^*\nabla > 0\) on harmonic spinors, so \(\ker D = 0\), hence \(\mathrm{ind}\,D = 0\). Combined with the Atiyah-Singer index theorem one obtains:

This was the first obstruction to positive scalar curvature beyond elementary topology, and it ruled out positive scalar curvature on, for example, the K3 surface. The Atiyah-Singer index theorem, applied to twisted Dirac operators \(D_E = D \otimes \nabla^E\) on \(\mathcal{S} \otimes E\), computes

recovering Hirzebruch-Riemann-Roch (twist by a holomorphic line bundle on a Kähler manifold), the Hirzebruch signature theorem (twist by \(\Lambda^*T^*M\) and pair appropriately), and the Gauss-Bonnet-Chern theorem (Euler-characteristic Dirac operator).

The standard reference is Lawson-Michelsohn, Spin Geometry (Princeton Mathematical Series 38), supplemented by Berline-Getzler-Vergne’s Heat Kernels and Dirac Operators (Springer Grundlehren) and the public lecture notes of Dan Freed (UT Austin) and Mike Hopkins (Harvard).

Chapter 7: Higher and Derived Stacks

Taught at UW as PMATH 965 in Fall 2026 (offered as a second instance under the title Algebraic Stacks; the framing here follows the modern higher-categorical tradition).

The stacks of Chapter 3 take values in groupoids — that is, in 1-truncated spaces. Many natural geometric problems demand higher truncation: deformations of objects in a derived category, mapping stacks between schemes, intersections of subschemes that are not transverse. Higher stacks take values in \(\infty\)-groupoids (Kan complexes); derived stacks take values in simplicial commutative rings or connective \(E_\infty\)-rings. Together they form the foundation of derived algebraic geometry, developed by Toën-Vezzosi and Lurie in the 2000s.

When \(\mathcal{X}\) is \(n\)-truncated (i.e., takes values in \(n\)-types), this recovers the classical notion of an \(n\)-stack: \(0\)-stacks are sheaves of sets, \(1\)-stacks are stacks of groupoids (Chapter 3), \(2\)-stacks involve 2-categories of objects, morphisms, and 2-morphisms, and so on. The classifying stack \(K(G, n)\) of an abelian group \(G\) is an \(n\)-stack; \(K(\mathbb{G}_m, 2)\) classifies \(\mathbb{G}_m\)-gerbes; \(K(\mathbb{G}_m, n)\) classifies higher \(\mathbb{G}_m\)-gerbes used in twisted K-theory and elliptic cohomology.

Derived stacks make sense of intersections “with multiplicity” by computing \(\mathrm{Tor}\): if \(Y\) and \(Z\) are subschemes of \(X\), the derived intersection is the derived tensor product \(\mathcal{O}_Y \otimes^L_{\mathcal{O}_X} \mathcal{O}_Z\), and the resulting derived scheme has cotangent complex with cohomology in negative degrees recording the failure of transversality.

Two of the most striking applications are the construction of shifted symplectic structures by Pantev-Toën-Vaquié-Vezzosi (PTVV, 2013) — for instance the moduli stack of \(G\)-bundles on a Calabi-Yau \(d\)-fold carries a canonical \((2-d)\)-shifted symplectic form — and the systematic treatment of derived loop spaces: \(LX = \mathrm{Map}(S^1, X)\) computed in derived algebraic geometry produces \(\mathcal{O}_{LX} \simeq \mathrm{HH}_*(X)\), the Hochschild homology, recovering the HKR isomorphism for smooth schemes in characteristic zero.

Chapter 8: Geometric Aspects of Gauge Theory

Taught at UW as PMATH 965 in past iterations (companion to the Winter 2020 Moraru gauge-theory course treated separately in the promoted pmath965-gauge file).

The smooth topology of a compact 4-manifold — classified topologically by Freedman — is not determined by its intersection form. The smooth category contains strictly more information, and the tool that first revealed this extra information was gauge theory: the study of connections on principal bundles modulo the infinite-dimensional gauge group. Chapter 8 develops the language and gives a road map to the Donaldson–Seiberg–Witten revolution that transformed 4-manifold topology between 1982 and the present.

Donaldson’s proof uses the moduli space \(\mathcal{M}\) of anti-self-dual (\(F_A^+ = 0\)) connections on an \(\mathrm{SU}(2)\)-bundle with \(c_2 = 1\). For generic metrics this moduli space is a smooth oriented 5-manifold whose boundary consists of the manifold itself (from flat connections) together with finitely many isolated reducible connections (from \(\mathrm{U}(1)\)-subbundles indexed by characteristic classes). Cobordism-theoretic invariants of \(\mathcal M\) then force the intersection form to have the claimed shape.

The Donaldson–Uhlenbeck–Yau theorem (1985) closes a remarkable circle: a holomorphic vector bundle on a compact Kähler manifold is stable (in Mumford’s sense) iff it admits a Hermitian–Einstein connection — a gauge-theoretic condition. Algebra and differential geometry converge, and the moduli of stable bundles acquires a metric structure. This is the Kähler–Calabi–Yau branch of the gauge-theoretic tree; Seiberg–Witten is the differential-topological branch; \(G_2\)-instantons are the special-holonomy branch. All three converge on the same algebraic objects when the underlying manifold has additional structure, and Karigiannis’s Waterloo research program lives at exactly this triple intersection.

Chapter 9: Motivic Integration

Mentioned as a possible topic in past PMATH 965 outlines (W22); chapter framed from the standard public literature, which is extensive and well-developed.

Motivic integration, introduced by Maxim Kontsevich in a 1995 Orsay talk, is a form of integration whose values live not in a field but in the Grothendieck ring of varieties — a ring built from isomorphism classes of algebraic varieties under scissor and product relations. The striking consequence of Kontsevich’s construction is that two birationally equivalent Calabi–Yau manifolds have the same Hodge numbers, a fact that was not previously amenable to any direct geometric argument. Since then motivic integration has become a central tool in singularity theory, \(p\)-adic analysis, and the arithmetic theory of varieties.

The standard public references are Eduard Looijenga’s Bourbaki seminar (1999/2000, no. 874) for the original Kontsevich construction; Denef–Loeser’s survey articles on arxiv for the zeta-function applications; and the monograph of Chambert-Loir, Nicaise, and Sebag, Motivic Integration (Birkhäuser Progress in Math, 2018) for a modern textbook treatment.