Pure Mathematics

Permanent URI for this collectionhttps://uwspace.uwaterloo.ca/handle/10012/9932

This is the collection for the University of Waterloo's Department of Pure Mathematics.

Research outputs are organized by type (eg. Master Thesis, Article, Conference Paper).

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Now showing 1 - 7 of 7

Deformation theory of nearly G₂-structures and nearly G₂ instantons
(University of Waterloo, 2021-08-30) Singhal, Ragini; Charbonneau, Benoit; Karigiannis, Spiro
We study two different deformation theory problems on manifolds with a nearly G₂-structure. The first involves studying the deformation theory of nearly G₂ manifolds. These are seven dimensional manifolds admitting real Killing spinors. We show that the infinitesimal deformations of nearly G₂-structures are obstructed in general. Explicitly, we prove that the infinitesimal deformations of the homogeneous nearly G₂-structure on the Aloff–Wallach space are all obstructed to second order. We also completely describe the de Rham cohomology of nearly G₂ manifolds. In the second problem we study the deformation theory of G₂ instantons on nearly G₂ manifolds. We make use of the one-to-one correspondence between nearly parallel G₂-structures and real Killing spinors to formulate the deformation theory in terms of spinors and Dirac operators. We prove that the space of infinitesimal deformations of an instanton is isomorphic to the kernel of an elliptic operator. Using this formulation we prove that abelian instantons are rigid. Then we apply our results to explicitly describe the deformation space of the canonical connection on the four normal homogeneous nearly G₂ manifolds. We also describe the infinitesimal deformation space of the SU(3) instantons on Sasaki–Einstein 7-folds which are nearly G₂ manifolds with two Killing spinors. A Sasaki–Einstein structure on a 7-dimensional manifold is equivalent to a 1-parameter family of nearly G₂-structures. We show that the deformation space can be described as an eigenspace of a twisted Dirac operator.
Derived Geometry and the Integrability Problem for G-Structures
(University of Waterloo, 2017-08-24) McCormick, Anthony; Karigiannis, Spiro
In this thesis, we study the integrability problem for G-structures. Broadly speaking, this is the problem of determining topological obstructions to the existence of principal G-subbundles of the frame bundle of a manifold, subject to certain differential equations. We begin this investigation by introducing general methods from homological algebra used to obtain cohomological obstructions to the existence of solutions to certain geometric problems. This leads us to a precise analogy between deformation theory and the formal integrability properties of partial differential equations. Along the way, we prove the following differential-geometric analogue of a well-known result from derived algebraic geometry. Namely, the cohomology of the normalized complex associated to the simplicial object obtained by tensoring the simplicial set correpsonding to the circle with the smooth algebra of smooth functions on a smooth manifold is precisely the algebra of differential forms. We also identify the infinitesimal generator of the natural circle-action with the de Rham differential. As a short corollary we obtain a natural isomorphism between the dual of this algebra and the algebra of poly-vector-fields, leading to a comparison between the Gerstenhaber bracket of Hochshild cohomology and the Schouten bracket. These two results are well-known in derived algebraic geometry and are folk-lore in differential geometry, where we were unable to find an explicit proof in the literature. In the end, this machinery is used to provide what the author believes is a new perspective on the integrability problem for G-structures.
Differential Operators on Manifolds with G2-Structure
(University of Waterloo, 2020-12-16) Suan, Caleb; Karigiannis, Spiro
In this thesis, we study differential operators on manifolds with torsion-free G2-structure. In particular, we use an identification of the spinor bundle S of such a manifold M with the bundle R ⊕ T*M to reframe statements regarding the Dirac operator in terms of three other first order differential operators: the divergence, the gradient, and the curl operators. We extend these three operators to act on tensors of one degree higher and study the properties of the extended operators. We use the extended operators to describe a Dirac bundle structure on the bundle T*M ⊕ (T*M ⊗ T*M) = T*M ⊗ (R ⊕ T*M) as well as its Dirac operator. We show that this Dirac operator is equivalent to the twisted Dirac operator DT defined using the original identification of S with R ⊕ T*M. As the two Dirac operators are equivalent, we use the T*M ⊕ (T*M ⊗ T*M) = T*M ⊗ (R ⊕ T*M) description of the bundle of spinor-valued 1-forms to examine the properties of the twisted Dirac operator DT. Using the extended divergence, gradient, and curl operators, we study the kernel of the twisted Dirac operator when M is compact and provide a proof that dim (ker DT) = b2 + b3.
Perspectives on the moduli space of torsion-free G2-structures
(University of Waterloo, 2024-08-27) Romshoo, Faisal; Karigiannis, Spiro
The moduli space of torsion-free G₂-structures for a compact 7-manifold forms a non-singular smooth manifold. This was originally proved by Joyce. In this thesis, we present the details of this proof, modifying some of the arguments using new techniques. Next, we consider the action of gauge transformations on the space of torsion-free G₂-structures. This gives us a new framework to study the moduli space. We show that the torsion-free condition under the action of gauge transformations almost exactly corresponds to a particular 3-form, which arises naturally from the G₂-structure and the gauge transformation, being harmonic when we add a "gauge-fixing" condition. This may be the first step in giving an alternate proof of the fact that the moduli space forms a manifold in our framework of gauge transformations.
Topics in G₂ geometry and geometric flows
(University of Waterloo, 2020-05-15) Dwivedi, Shubham; Karigiannis, Spiro
We study three different problems in this thesis, all related to G₂ structures and geometric flows. In the first problem we study hypersurfaces in a nearly G₂ manifold. We define various quantities associated to such a hypersurface using the G₂ structure of the ambient manifold and establish several relations between them. In particular, we give a necessary and sufficient condition for a hypersurface with an almost complex structure induced from the G₂ structure of the ambient manifold to be nearly Kähler. Then using the nearly G₂ structure on the round sphere S⁷, we prove that for a compact minimal hypersurface M⁶ of constant scalar curvature in S⁷ with the shape operator A satisfying |A|² > 6, there exists an eigenvalue λ > 12 of the Laplace operator on M such that |A|² = λ − 6, thus giving the next discrete value of |A|² greater than 0 and 6 in terms of the spectrum of the Laplace operator on M. The latter is related to a question of Chern on the values of the scalar curvature of compact minimal hypersurfaces in Sⁿ of constant scalar curvature. The second problem is related to the study of solitons and almost solitons of the Ricci- Bourguignon flow. We prove some characterization results for compact Ricci-Bourguignon solitons. Taking motivation from Ricci almost solitons, we then introduce the notion of Ricci-Bourguignon almost solitons and prove some results about them which generalize previous results for Ricci almost solitons. We also derive integral formulas for compact gradient Ricci-Bourguignon solitons and compact gradient Ricci-Bourguignon almost solitons. Finally, using the integral formula we show that a compact gradient Ricci-Bourguignon almost soliton is isometric to an Euclidean sphere if it has constant scalar curvature or if its associated vector field is conformal. In the third problem we study a flow of G₂ structures that all induce the same Riemannian metric. This isometric flow is the negative gradient flow of an energy functional. We prove Shi-type estimates for the torsion tensor T along the flow. We show that at a finite-time singularity the torsion must blow up, so the flow will exist as long as the torsion remains bounded. We prove a Cheeger–Gromov type compactness theorem for the flow. One possible motivation for studying this isometric flow of G₂-structures is that it can be coupled with “Ricci flow” of G₂ structures, which is a flow of G₂ structures that induces precisely the Ricci flow on metrics, in contrast to the Laplacian flow which induces Ricci flow plus lower order terms involving the torsion. In effect, one may hope to first flow the 3-form in a way that improves the metric, and then flow the 3-form in a way that preserves the metric but still decreases the torsion. More generally, the isometric flow is a particular geometric flow of G₂-structures distinct from the Laplacian flow, and both fit into a broader class of geometric flows of G₂-structures with good analytic properties. In the final section, we summarize the rest of the results on the isometric flow which include an Uhlenbeck type trick and the definition of a scale-invariant quantity ϴ for any solution of the flow and the proof that it is almost monotonic along the flow. We also introduce an entropy functional and prove that low entropy initial data lead to solutions of the flow that exist for all time and converge smoothly to a G₂ structure with divergence-free torsion. We study the singular set of the flow. Finally, we prove that if the singularity is of Type-I then a sequence of blow-ups of a solution admits a subsequence that converges to a shrinking soliton for the flow.
Topics in the Geometry of Special Riemannian Structures
(University of Waterloo, 2024-07-26) Iliashenko, Anton; Karigiannis, Spiro
The thesis consists of two chapters. The first chapter is the paper named “Betti numbers of nearly G₂ and nearly Kähler 6-manifolds with Weyl curvature bounds” which is now in the journal Geometriae Dedicata. Here we use the Weitzenböck formulas to get information about the Betti numbers of compact nearly G₂ and compact nearly Kähler 6-manifolds. First, we establish estimates on two curvature-type self adjoint operators on particular spaces assuming bounds on the sectional curvature. Then using the Weitzenböck formulas on harmonic forms, we get results of the form: if certain lower bounds hold for these curvature operators then certain Betti numbers are zero. Finally, we combine both steps above to get sufficient conditions of vanishing of certain Betti numbers based on the bounds on the sectional curvature. The second chapter is the paper written with my supervisor Spiro Karigiannis named “A special class of k-harmonic maps inducing calibrated fibrations”, to appear in the journal Mathematical Research Letters. Here we consider two special classes of k-harmonic maps between Riemannian manifolds which are related to calibrated geometry, satisfying a first order fully nonlinear PDE. The first is a special type of weakly conformal map u:(Lᵏ,g)→(Mⁿ,h) where k≤n and α is a calibration k-form on M. Away from the critical set, the image is an α-calibrated submanifold of M. These were previously studied by Cheng–Karigiannis–Madnick when α was associated to a vector cross product, but we clarify that such a restriction is unnecessary. The second, which is new, is a special type of weakly horizontally conformal map u:(Mⁿ,h)→(Lᵏ,g) where n≥k and α is a calibration (n-k)-form on M. Away from the critical set, the fibres u⁻¹{u(x)} are α-calibrated submanifolds of M. We also review some previously established analytic results for the first class; we exhibit some explicit noncompact examples of the second class, where (M,h) are the Bryant–Salamon manifolds with exceptional holonomy; we remark on the relevance of this new PDE to the Strominger–Yau–Zaslow conjecture for mirror symmetry in terms of special Lagrangian fibrations and to the G₂ version by Gukov–Yau–Zaslow in terms of coassociative fibrations; and we present several open questions for future study.
Weak Moment Maps in Multisymplectic Geometry
(University of Waterloo, 2018-07-04) Herman, Jonathan; Karigiannis, Spiro; Hu, Shengda
We introduce the notion of a weak (homotopy) moment map associated to a Lie group action on a multisymplectic manifold. We show that the existence/uniqueness theory governing these maps is a direct generalization from symplectic geometry. We use weak moment maps to extend Noether's theorem from Hamiltonian mechanics by exhibiting a correspondence between multisymplectic conserved quantities and continuous symmetries on a multi-Hamiltonian system. We nd that a weak moment map interacts with this correspondence in a way analogous to the moment map in symplectic geometry. We de ne a multisymplectic analog of the classical momentum and position functions on the phase space of a physical system by introducing momentum and position forms. We show that these di erential forms satisfy generalized Poisson bracket relations extending the classical bracket relations from Hamiltonian mechanics. We also apply our theory to derive some identities on manifolds with a torsion-free G2 structure.

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Browsing Pure Mathematics by Author "Karigiannis, Spiro"