My MS thesis

notes §

• Homogeneous and isotropic Riemannian manifolds
• Boundary of CAT(0) metric spaces
  • Boundary of simply connected Hadamard manifolds
  • Visual metric on the boundary of CAT(-1) spaces
• Symmetric spaces of Euclidean/non-compact type
  • $\mathsf{Pd}(n,\mathbb{R})$
    • $GL(n,\mathbb{R})\curvearrowright \mathsf{Pd}(n,\mathbb{R})$
    • $\mathsf{Pd}(2,\mathbb{R})$ (coming soon!)
    • $\mathsf{SPd}(2,\mathbb{R})$ (coming soon!)
  • $\mathsf{T}$-closed, closed subgroups of $GL(n,\mathbb{R})$ (coming soon!)
  • $\mathsf{T}$-closed, $\exp$-closed closed subgroups of $GL(n,\mathbb{R})$ $\leftrightarrow$ completely geodesic submanifolds of $\mathsf{Pd}(n,\mathbb{R})$ (coming soon!)
  • $\mathsf{T}$-closed Lie subalgebras are reductive (coming soon!)
• Quasi-conformal maps on $S^n$
• $\mathbb{R}{H}^n$ and hyperbolic manifolds
  • $SL(2,\mathbb{R})\curvearrowright H_{\mathrm{U}}^2$
  • Fuchsian groups and hyperbolic surfaces
    • Geodesic flow of Fuchsian groups and hyperbolic surfaces (coming soon!)
  • Discrete subgroups of $O^{+}(1,n)$ and hyperbolic manifolds/orbifolds
    • Counting orbits of discrete subgroups of $O^{+}(1,n)(\mathbb{R})$ in $\mathbb{R}{H}^n$
    • Exponential sum of a discrete subgroup of $\mathrm{Isom}(\mathbb{R}{H}^n)$
• Riemannian manifolds with negative curvature
  • Geodesic flow on Riemannian manifolds of negative curvature
  • Discrete subgroup of isometry group of a simply connected Riemannian manifold of negative curvature
• Harmonic functions composed with conformal maps

background §

• 2506 Summer School on Rigidity of Discrete Groups, June 30 – July 4, 2025 at IISER Mohali
• topological
  • Types of topological spaces
  • vector spaces
    • Topological vector space over $\mathbb{R}$ or $\mathbb{C}$
    • Dual space of a topological vector space
    • Weak topology on a topological vector space
    • Weak${}^{★}$ topology on dual of a topological vector space
  • Locally compact Hausdorff spaces
  • Discrete subsets of topological spaces
  • Closed map between topological spaces
  • Covering space of a topological space
  • Local homeomorphisms between topological spaces
  • Proper maps between topological spaces
  • When is a local homeomorphism between topological spaces a covering map
  • When is a local homeomorphism between topological spaces a closed map
  • When does a map between topological spaces send discrete sets to discrete sets
  • $Y^X$ (coming soon!)
  • groups
    • Locally compact Hausdorff topological group
  • group actions
    • Discontinuous group actions on topological spaces
    • Covering space action
• smooth
  • Counting compact topological manifolds upto homeomorphism (coming soon!)
  • group actions
    • Submersions, immersions and embeddings of smooth manifolds (coming soon!)
    • Submanifolds of smooth manifolds (coming soon!)
    • Proper Lie group actions (coming soon!)
    • Orbits of smooth Lie group actions
    • Quotient manifolds (coming soon!)
    • Homogeneous manifold (coming soon!)
  • groups
    • Lie algebras (coming soon!)
    • Virtual subgroups of Lie groups (coming soon!)
    • Connected Lie groups with non-surjective exponential map
    • Discrete normal subgroups of Lie groups (coming soon!)
    • Connected Lie groups with discrete center (coming soon!)
    • Classification of Lie groups
    • Semi-simple $\mathbb{R}$-Lie groups (coming soon!)
      • Compact Lie groups
      • Almost simple groups
      • Semi-simple $\mathbb{R}$-Lie groups of dimension 6 (coming soon!)
    • $SL(2,\mathbb{R})$
      • Fuchsian groups and hyperbolic surfaces
    • Subgroups of $SL(n,\mathbb{C})$ that are semi-simple (coming soon!)
    • Zariski closure of subgroups of $GL(n,\mathbb{C})$ (coming soon!)
    • Subgroups of $GL(n,\mathbb{R})$ (coming soon!)
    • Compact subgroups of $GL(n)(\mathbb{R})$ (coming soon!)
  • group FinVec-representations
    • $\mathfrak{sl}(2,\mathbb{C})$ (coming soon!)
      • ${\mathsf{FinVec}}_{\mathbb{C}}$ irreducible representations of $\mathfrak{sl}(2,\mathbb{C})$ (coming soon!)
      • $2\otimes 3=5\oplus 3\oplus 1$ for ${\mathsf{FinVec}}_{\mathbb{C}}$ irreps of $\mathfrak{sl}(2,\mathbb{C})$ (coming soon!)
      • ${Sym}^2{Sym}^2({\mathbb{C}}^2){\cong }_{{SL}_{\mathbb{C}}(2)}{Sym}^0({\mathbb{C}}^2)\oplus {Sym}^4({\mathbb{C}}^2)$ (coming soon!)
  • $\mathbb{R}$-manifold semi-Riemannian
    • Riemannian connection
    • Curvature of a Riemannian manifold
    • Exponential map from tangent bundle
    • Quotients of Riemannian manifolds
    • Moduli space of quotients
    • Totally geodesic submanifolds of semi-Riemannian manifolds
    • Isometries of a semi-Riemannian manifold
    • Complete Riemannian manifolds
    • Invariant Riemannian metrics on homogeneous manifolds
    • Homogeneous manifold with compact stabilizer (coming soon!)
    • Riemannian homogeneous manifolds where ${\exp }_p$ is a diffeomorphism
    • Laplacian on a Riemannian manifold
    • Uniquely geodesic complete Riemannian manifolds
    • Riemannian manifolds with non-positive sectional curvature
  • low dimensions and geometrization
    • smooth $\mathbb{R}$-manifolds of dimension 2
    • smooth $\mathbb{R}$-manifolds of dimension 3
  • transformation groups
    • invariant Riemannian metric in a proper Lie group action (coming soon!)
    • Smooth action by a compact Lie group (coming soon!)
    • The only compact connected Lie group that can act faithfully on a torus is itself (coming soon!)
    • Transformation Groups in Differential Geometry (coming soon!)
• dynamical
  • Iterative dynamics of endomorphism of a set
  • Measure preserving endomorphisms
  • Recurrence in iterations of a measure preserving endomorphism
  • Interpretation of measure preserving endomorphisms
• Metric spaces (coming soon!)
  • continuous maps
    • $\mathrm{End}\mathsf{Top}(X,Y)$ (coming soon!)
    • Pre-compact subsets of $(\mathrm{End}\mathsf{Top}(X,Y),{\mathfrak{T}}_{\text{cpt}})$
  • metric geometry
    • Geodesics in a metric space (coming soon!)
    • Metric space isometries (coming soon!)
    • undefined (coming soon!)
    • CAT($\kappa$) spaces
      • Isometries of CAT(0) spaces (coming soon!)
  • Isometric action on a metric space
  • quasi-metric geometry
    • Quasi-isometries between metric spaces
  • Mobius geometry
    • Cross ratios and Mobius maps between metric spaces
  • conformal geometry
  • quasi-conformal geometry
  • Metric simplicial complexes (coming soon!)
  • Buildings
• Finitely generated groups
  • When a group plays ping pong
  • Lamplighter group on ${\mathbb{Z}}_2$ (coming soon!)
  • Subgroups of finite index of a finitely generated group
• discrete subgroups of semi-simple/reductive Lie groups

references §

• References for GGD GTG GGT AGT
• Rigidity theorems by Mostow, Sullivan, Tukia, Yue, Ivanov

presentations §

• The slides for my PRJ501 presentation (after half of my MS thesis) is here
• The slides for my PRJ502 presentation will be here in a few months
• The poster on the MS thesis will be here too

memes §

Made this “thesis mood sticker” in the summer:

which loosely says (in Hindi) “What do I know? I am just doing my thesis…”