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PhD (Doctor of Philosophy) students
Thesis title: "TBA"
Thesis abstract: TBA.
Thesis title: "TBA"
Thesis abstract: TBA.
Thesis title: "Topics in the geometry of special Riemannian structures"
Thesis abstract: The thesis consists of two chapters. The first chapter is the paper named "Betti numbers of nearly G_{2} and nearly Kähler 6manifolds 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 G_{2} and compact nearly Kähler 6manifolds. First, we establish estimates on two curvaturetype 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 kharmonic maps inducing calibrated fibrations", to appear in the journal Mathematical Research Letters. Here we consider two special classes of kharmonic 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: (L^{k},g) → (M^{n},h) where k ≤ n and α is a calibration kform on M. Away from the critical set, the image is an αcalibrated submanifold of M. These were previously studied by ChengKarigiannisMadnick 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: (M^{n},h) → (L^{k},g) where n ≥ k and α is a calibration (nk)form on M. Away from the critical set, the fibres u^{1}{u(x)} are αcalibrated 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^{n},h) are the BryantSalamon manifolds with exceptional holonomy; we remark on the relevance of this new PDE to the StromingerYauZaslow conjecture for mirror symmetry in terms of special Lagrangian fibrations and to the G_{2} version by GukovYauZaslow in terms of coassociative fibrations; and we present several open questions for future study. PDF version:
Thesis title: "Deformation theory of nearly G_{2}structures and nearly G_{2} instantons"
Thesis abstract: We study two different deformation theory problems on manifolds with a nearly G_{2}structure. The first involves studying the deformation theory of nearly G_{2} manifolds. These are seven dimensional manifolds admitting real Killing spinors. We show that the infinitesimal deformations of nearly G_{2}structures are obstructed in general. Explicitly, we prove that the infinitesimal deformations of the homogeneous nearly G_{2}structure on the AloffWallach space are all obstructed to second order. We also completely describe the de Rham cohomology of nearly G_{2} manifolds. In the second problem we study the deformation theory of G_{2} instantons on nearly G_{2} manifolds. We make use of the onetoone correspondence between nearly parallel G_{2}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 G_{2} manifolds. We also describe the infinitesimal deformation space of the SU(3) instantons on SasakiEinstein 7folds which are nearly G_{2} manifolds with two Killing spinors. A SasakiEinstein structure on a 7dimensional manifold is equivalent to a 1parameter family of nearly G_{2}structures. We show that the deformation space can be described as an eigenspace of a twisted Dirac operator. PDF version:
Thesis title: "Topics in G_{2} geometry and geometric flows"
Thesis abstract: We study three different problems in this thesis, all related to G_{2} structures and geometric flows. In the first problem we study hypersurfaces in a nearly G_{2} manifold. We define various quantities associated to such a hypersurface using the G_{2} 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 G_{2} structure of the ambient manifold to be nearly Kaehler. Then using the nearly G_{2} structure on the round sphere S^{7}, we prove that for a compact minimal hypersurface M^{6} of constant scalar curvature in S^{7} 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 S^{n} of constant scalar curvature. The second problem is related to the study of solitons and almost solitons of the RicciBourguignon flow. We prove some characterization results for compact RicciBourguignon solitons. Taking motivation from Ricci almost solitons, we then introduce the notion of RicciBourguignon almost solitons and prove some results about them which generalize previous results for Ricci almost solitons. We also derive integral formulas for compact gradient RicciBourguignon solitons and compact gradient RicciBourguignon almost solitons. Finally, using the integral formula we show that a compact gradient RicciBourguignon al most 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 G_{2} structures that all induce the same Riemannian metric. This isometric flow is the negative gradient flow of an energy functional. We prove Shitype estimates for the torsion tensor T along the flow. We show that at a finitetime singularity the torsion must blow up, so the flow will exist as long as the torsion remains bounded. We prove a CheegerGromov type compactness theorem for the flow. One possible motivation for studying this isometric flow of G_{2}structures is that it can be coupled with "Ricci flow" of G_{2} structures, which is a flow of G_{2} 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 3form in a way that improves the metric, and then flow the 3form in a way that preserves the metric but still decreases the torsion. More generally, the isometric flow is a particular geometric flow of G_{2}structures distinct from the Laplacian flow, and both fit into a broader class of geometric flows of G_{2}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 scaleinvariant 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 G_{2} structure with divergencefree torsion. We study the singular set of the flow. Finally, we prove that if the singularity is of TypeI then a sequence of blowups of a solution admits a subsequence that converges to a shrinking soliton for the flow. PDF version:
Thesis title: "Weak moment maps in multisymplectic geometry"
Thesis abstract: 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 multiHamiltonian system. We find that a weak moment map interacts with this correspondence in a way analogous to the moment map in symplectic geometry. We define 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 differential 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 torsionfree G_{2} structure. PDF version:
Thesis title: "Moduli space and deformations of special Lagrangian submanifolds with edge singularities"
Thesis abstract: Special Lagrangian submanifolds are submanifolds of a CalabiYau manifold calibrated by the real part of the holomorphic volume form. In this thesis we use elliptic theory for edgedegenerate differential operators on singular manifolds to study general deformations of special Lagrangian submanifolds with edge singularities. We obtain a general theorem describing the local structure of the moduli space. When the obstruction space vanishes the moduli space is a smooth, finite dimensional manifold. PDF version: MMath (Master of Mathematics) students — where (THESIS) denotes Thesis option
Thesis title: "Perspectives on the moduli space of torsionfree G_{2}structures"
Thesis abstract: The moduli space of torsionfree G_{2}structures for a compact 7manifold forms a nonsingular smooth manifold. This was originally proved by Joyce [Joy00]. In this thesis, we present the details of this proof, modifying some of the arguments using techniques in [Kar08] and [DGK23]. Next, we consider the action of gauge transformations of the form e^{tA} where A is a 2tensor, on the space of torsionfree G_{2}structures. This gives us a new framework to study the moduli space. We show that a G_{2}structure φ = P^{*}φ acted upon by a gauge transformation P = e^{tA} is infinitesimal torsionfree condition almost exactly corresponds to A ◇ φ being harmonic if A satisfies a "gaugefixing" condition, where A ◇ φ is a 3form defined using the diamond operator ◇ which features in [DGK23]. 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. PDF version:
Research essay title: "Positive Bisectional Curvature on Compact Kähler surfaces and KählerEinstein Manifolds"
Research essay abstract: This research paper investigates holomorphic bisectional curvature and the Frankel conjecture. The Frankel conjecture states that a connected compact Kähler manifold with positive holomorphic bisectional curvature is biholomorphic to the complex projective space. Following Goldberg and Kobayashi, we present proofs of the conjecture in dimension two and in the case of Kählerâ€“Einstein manifolds. PDF version:
Research essay title: "Elliptic Partial Differential Operators"
Research essay abstract: Elliptic partial differential operators have been become an important class of operators in modern differential geometry, due in part to the AtiyahSinger index theorem, which states that the index of an elliptic operator (defined in terms of the analytic properties of the operator) is equal to its topological index (defined purely in terms of topological data). In this paper, we give an introduction to the theory of elliptic partial differential operators on manifolds, with the main focus being to prove a generalised version of the HodgeDecomposition theorem for elliptic partial differential operators. PDF version:
Thesis title: "The AtiyahHitchinSinger Theorem and an 8dimensional generalization"
Thesis abstract: The AtiyahHitchinSinger theorem states that the twistor almost complex structure on a certain S^{2} bundle over an oriented Riemannian 4manifold (M, g) is integrable if and only if the Weyl curvature tensor of g is selfdual. These ideas were developed by Roger Penrose connecting 4dimensional Riemannian geometry with complex geometry. We present a new approach to the AtiyahHitchinSinger theorem using horizontal lifts and their respective flows, cross products and the quaternions to show that the Nijenhuis tensor vanishes if and only if the Weyl curvature tensor of g is antiselfdual. An eight dimensional generalization is presented when the manifold is R^{8}. PDF version:
Research essay title: "Calculations on Dirac and Spinor Valued Forms"
Research essay abstract: In this paper we review some material on Dirac bundles and spin geometry, and do some calculations on Dirac bundle valued forms and spinor valued forms. PDF version:
Research essay title: "Riemannian Immersions and Submersions"
Research essay abstract: We discuss the Gauss, Codazzi, and Ricci equations for a Riemannian immersion. The O'Neill tensors for a Riemannian submersion are introduced and applied to the FubiniStudy metric on CP^{n}. Moreover, we discuss the case of a vector bundle equipped with a connection and fibre metric over a Riemannian manifold and show how to equip the vector bundle with an induced Riemannian metric. This construction is analyzed as a Riemannian immersion and a Riemannian submersion, and the precise conditions under which the O'Neill tensors vanish is determined. PDF version:
Thesis title: "Differential Operators on Manifolds with G_{2}Structure"
Thesis abstract: In this thesis, we study differential operators on manifolds with torsionfree G_{2}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 D_{T} 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 spinorvalued 1forms to examine the properties of the twisted Dirac operator D_{T}. 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 D_{T}) = b^{2} + b^{3}. PDF version:
Research essay title: "The moduli space of ASD connections on compact 4manifolds"
Research essay abstract: Some analytic aspects of the theory of antiselfdual connections on compact oriented Riemannian 4manifolds is discussed, including a local version of Uhlenbeck's sequential compactness theorem and a description of the irreducible part of the ASD moduli space as the zero set of a smooth map between finitedimensional vector spaces. PDF version:
Thesis title: "Derived geometry and the integrability problem for Gstructures"
Thesis abstract: In this thesis, we study the integrability problem for Gstructures. Broadly speaking, this is the problem of determining topological obstructions to the existence of principal Gsubbundles 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 a differentialgeometric analogue of a wellknown result from derived algebraic geometry, as well as the identification of the infinitesimal generator of the natural S^{1}action corresponding to loop rotation with the de Rham differential. As a short corollary we obtain a natural isomorphism identifying the standard Gerstenhaber bracket with the Schouten bracket. These two results are wellknown in derived algebraic geometry and are folklore 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 Gstructures. PDF version:
Thesis title: "KTheory for C^{*}algebras and for topological spaces"
Thesis abstract: This thesis is an introduction to both the Ktheory of C^{*}algebras and the Ktheory of compact Hausdorff spaces, including a proof of the equivalence of the two theories in a certain special case. PDF version:
Research essay title: "Nöether's theorem under the Legendre transform"
Research essay abstract: In this paper we demonstrate how the Legendre transform connects the statements of Nöether's theorem in Hamiltonian and Lagrangian mechanics. We give precise definitions of symmetries and conserved quantities in both the Hamiltonian and Lagrangian frameworks and discuss why these notions in the Hamiltonian framework are somewhat less rigid. We explore conditions which, when put on these definitions, allow the Legendre transform to set up a onetoone correspondence between them. We also discuss how to preserve this correspondence when the definitions of symmetries and conserved quantities are less restrictive. PDF version:
Research essay title: "Characterizations of the Chern characteristic class"
Research essay abstract: Chern's characteristic class may be defined by several different ways, including algebraic topology, differential geometry, and sheaf theory. All these approaches are presented, with the main goal being to show that even though the definitions lie in different spaces, they all satisfy the Chern class axioms and are isomorphic by various theorems. To reach this goal, strong background machinery is constructed, including the complex Grassmannian as a CWcomplex, a detailed setup for the splitting principle, and a thorough proof of the ChernWeil theorem. PDF version:
Research essay title: "A Review of Whitehead's Asphericity Conjecture"
Research essay abstract: This paper summarizes some of the work done to date on Whitehead's question about the asphericity of subcomplexes of an aspherical 2complex. We start with a review of the theory of higher homotopy groups. Next, we study some of their particular properties for 2complexes; including their translation into an algebraic structure called crossed modules. The next section includes a translation of Whitehead's conjecture using properties of crossed modules. We also review a different approach using homotopy of finite spaces; we include a short summary of the main definitions and results of that theory, and the implications for Whitehead's conjecture. We finish the paper by considering some interesting questions that arise from the above mentioned translations. PDF version:
Research essay title: "Milnor's Exotic Spheres"
Research essay abstract: In 1956, John Milnor surprised the mathematical community by exhibiting examples of smooth manifolds that were homeomorphic to the 7sphere but not diffeomorphic to it with its standard smooth structure; this was the first example of socalled "exotic" manifolds. This paper concerns itself with John Milnor's exotic spheres. After establishing some familiar terminology and notation, we will use Morse theoretical methods to provide a means of determining whether a given manifold is homeomorphic to the nsphere. We shall then use tools from the theory of characteristic classes to define a quantity (Milnor's invariant) that distinguishes smooth structures on manifolds. We will give Milnor's original construction of his exotic spheres and show that they are all homeomorphic to the 7sphere but that they are not all diffeomorphic to the 7sphere with its standard smooth structure by means of computing Milnor's invariant for these spaces. This paper assumes familiarity with elementary smooth manifold theory and Riemannian geometry, including differential forms and integration thereof, familiarity with vector bundles, elements of algebraic topology and quaternion arithmetic. Facts pertaining to these topics are freely used throughout, though many definitions are repeated to establish terminology and notation. PDF version:
Thesis title: "Mean Curvature Flow in Euclidean spaces, Lagrangian Mean Curvature Flow, and Conormal Bundles"
Thesis abstract: We present the mean curvature flow in Euclidean spaces and the Lagrangian mean curvature flow. We will first study the mean curvature evolution of submanifolds in Euclidean spaces, with an emphasis on the case of hypersurfaces. Along the way we will demonstrate the basic techniques in the study of geometric flows in general (for example, various maximum principles and the treatment of singularities). After that we will move on to the study of Lagrangian mean curvature flows. We will make the relevant definitions and prove the fundamental result that the Lagrangian condition is preserved along the mean curvature flow in KählerEinstein manifolds, which started the extensive, and still ongoing, research on Lagrangian mean curvature flows. We will also define special Lagrangian submanifolds as calibrated submanifolds in CalabiYau manifolds. Finally, we will study the mean curvature flow of conormal bundles as submanifolds of C^{n}. Using some tools developed recently, we will show that if a surface has strictly negative curvatures, then away from the zero section, the Lagrangian mean curvature flow starting from a conormal bundle does not develop Type I singularities. PDF version:
Research essay title: "The Kähler and Special Lagrangian Calibrations"
Research essay abstract: Definition and basic properties of calibrations and calibrated submanifolds, including the fundamental theorem of calibrated geometry. Two important examples: Kähler and special Lagrangian calibrations. Explicit constructions of examples involving high degrees of symmetry. Relations between them in the hyperKäher case. PDF version: Undergraduate Research students (listed by project) — where (NSERC) denotes NSERC USRA
Project title: "Explicit symmetric examples of a special class of second order U(n)structures on complex Euclidean nspace"
Project abstract: Let M be a 2ndimensional smooth manifold. A U(n)structure on M is a pair consisting of a Riemannian metric g and an orthogonal almost complex structure J. When J is parallel with respect to the LeviCivita connection, this is the celebrated case of a Kahler manifold. The authors de la Ossa, Karigiannis, and Svanes have introduced a new class of U(n)structures on 2ndimensional real manifolds that satisfy a second order system of nonlinear PDEs. This class generalizes the Kahler class in that some analogous identities and Hodge theoretic relations continue to hold in a particular sense. It is of interest to seek explicit nonKahler solutions to this system of PDEs. A natural starting point is to consider the cohomogeneity one case, where one imposes enough symmetry to reduce the system to a (still nonlinear) ODEs which one could hope to solve explicitly. Such an approach has had much success in other related geometric constructions.
Project title: "Aspects of symplectic theory which generalize to G_{2} geometry"
Project abstract: The theory of symplectic manifolds has many interesting aspects, including MarsdenWeinstein reduction, and a kind of "reverse" procedure, whereby one constructs a new symplectic structure on the total space of a complex line bundle over a symplectic manifold satisfying certain conditions, which admits an S^{1} action by symplectomorphisms. A G_{2}structure on a 7manifold is a special kind of 3form which has many properties in common with a symplectic structure. Indeed, it is a particular restrictive type of multisymplectic structure. We intend to investigate to what extent some of these aspects of symplectic theory can generalize to this setting. Since G_{2}structures only exist in dimension 7, we must necessarily broaden our scope. For example, given a quaternionic line bundle over a manifold with G_{2}structure, can we construct a special type of 3form on the total space, which is invariant under an Sp(1) action? Or, given a quaternionic line bundle over an oriented Riemannian 3manifold, what kind of Sp(1)invariant G_{2}structures can we construct on the total space?
Project title: "Mean curvature flow of small sections of the tangent bundle"
Project abstract: Let (M,g) be a Riemannian manifold. The LeviCivita connection of g induces a Riemannian metric G on the total space of the tangent bundle TM, sometimes called the Sasaki metric. A small section s of TM is diffeomorphic to the zero section M. Applying mean curvature flow to this section for small time and then projecting back to M gives a flow of Riemannian metrics on M. It is of interest to determine to what extent this flow of metrics is or is not the classical Ricci flow. A natural starting point for this investigation is the setting of parallelizable manifolds such as Lie groups, where the computations should be completely tractable. In particular, for a compact Lie group M equipped with a leftinvariant metric, one can consider "leftequivariant" sections. It is very likely that this condition would be preserved under the flow and thus the flow would reduce to a fully nonlinear ODE, which may be explicitly solvable in some cases. If this flow turns out to differ from the Ricci flow, it is of interest to determine if it still admits nice properties such as preservations of positive scalar curvature or positive curvature operator.
Project title: "Partial factorization of exceptional Lie group G_{2}"
Project abstract: The exceptional Lie group G_{2} plays a very important role in modern theoretical physics, being the holonomy group of a Ricciflat Riemannian manifold in 7 dimensions that admits a parallel spinor, and thus is a candidate for a supersymmetric compactification space in Mtheory. A better understanding the the Lie theoretic properties of the group G_{2} could lead to useful applications in physics, algebra, and geometry. The group G_{2} embeds naturally inside SO(7), and the associated Lie algebra decomposition is so(7) = g_{2} ⊕ V, where V is isomorphic to the standard representation of so(7). Note that V is a subspace but is *not* a subalgebra. Using the relations between G_{2}, the octonion algebra, and the cross product in 7 dimensions, we plan to look for a "partial" factorization of an element A of SO(7) into A = BP where B lies in G_{2} and P is in some sense 'as compatible as possible' with V. Such a factorization would be similar in spirit to the polar factorization of matrices, but is necessarily much more complicated. It is possible that the correct notion of "as compatible as possible" will have to do with killing off as many terms as we can in the CampellBakerHausdorff series expansion.
Project title: "Explicit cohomogeneity one solutions to a new second order class of U(m)structures"
Project abstract: Let M be a 2mdimensional smooth manifold. A U(m)structure on M is a pair consisting of a Riemannian metric g and an orthogonal almost complex structure J. When J is parallel with respect to the LeviCivita connection, this is the celebrated case of a Kahler manifold. The authors de la Ossa, Karigiannis, and Svanes have introduced a new class of U(m)structures on 2mdimensional real manifolds that satisfy a second order system of nonlinear PDEs. This class generalizes the Kahler class in that some analogous identities and Hodge theoretic relations continue to hold in a particular sense. It is of interest to seek explicit nonKahler solutions to this system of PDEs. A natural starting point is to consider the cohomogeneity one case, where one imposes enough symmetry to reduce the system to a (still nonlinear) ODEs which one could hope to solve explicitly. Such an approach has had much success in other related geometric constructions.
Project title: "Reformulations of certain natural equations in Riemannian geometry"
Project abstract: A Riemannian metric g on a manifold M can be described locally as g = A^{*}A where A is a tangent bundle isomorphism, modulo certain redundancies. It is of interest to derive an expression for the Ricci curvature of g in terms of A and its first two covariant derivatives. Such a description may shed new insight on the Ricci flow equation. Similarly, a G_{2} structure on a Riemannian 7manifold is a special kind of 3form φ, and it can be described locally in terms of a tangent bundle endomorphism A, modulo certain redundancies. The most interesting G_{2} structures are those which are covariantly constant. These are also called torsionfree. It is of interest to derive an expression for the torsion of φ in terms of A and its first covariant derivative. Such a description may shed new light on the structure of the space of torsionfree G_{2} structures on M.
Project title: "Flows of metrics induced from mean curvature flow"
Project abstract: The mean curvature flow is a flow of an isometrically immersed submanifold M inside an ambient Riemannian manifold X in the direction of the mean curvature vector field. The Ricci flow is an intrinsic flow of a metric on a Riemannian manifold M in the direction of (minus) the Ricci tensor. Let M be a Riemannian manifold and consider it as the zero section inside one of its tensor bundles X, such as the cotangent bundle X = T^{*} M or the real canonical bundle X = Λ^{n} (T^{*} M) where n = dim(M). Using the LeviCivita connection of M, there is a canonical Riemannian metric induced on X, which makes the zero section isometrically immersed. Therefore one can evolve M inside X by the mean curvature flow. For small time, this evolution M_{t} will be canonically diffeomorphic to M using the exponential map on the mean curvature vector field. Pulling back via this diffeomorphism, the mean curvature flow of M in X thus induces an intrinsic flow of metrics on M. The natural question is: what is this induced flow of metrics? If it is the Ricci flow, this gives a new interesting characterization of Ricci flow. If it is not the Ricci flow, this could be a new interesting canonical flow of metrics. Moreover, in the case when X = T^{*} M, the zero section is Lagrangian, and mean curvature flow preserves the Lagrangian condition. The Lagrangian mean curvature flow is wellbehaved. Similarly, when X = Λ^{n} (T^{*} M), then M is a hypersurface in X. The hypersurface mean curvature flow is also wellbehaved.
Project title: "A Modern Characterization of the Walker Torsion Derivation"
Project abstract: An almost complex manifold is a real 2mdimensional smooth manifold together with a smooth endomorphism J of the tangent bundle that squares to minus the identity, allowing one to identity each tangent space with a complex mdimensional vector space. In classical literature of the 1950's and 1960's, there exists a notion called the "torsional derivation" on an almost complex manifold, introduced by Walker and later expanded on by Willmore. This notion seems to have disappeared from the literature. Most likely, it is equivalent to an algebraic or Lie derivation on the bundle of forms as discussed by Michor et al in their text on "Natural Operations in Differential Geometry". The goal of this project is to understand what almost complex manifolds are, what algebraic and Lie derivations are, and to deduce whether or not the torsional derivation of Walker/Willmore is really a special case of one of these modern derivations. In the process, the students will also consider related questions for other vectorvalued forms that are generalizations of almost complex structures, as arise, for example, in G_{2} geometry.
Project title: "Differential Invariants of Vector Cross Products"
Project abstract: Vector cross products on manifolds were classified in the 1960's by Brown and Gray and fall into 4 types: the Hodge star operator, almost complex structures, and the 2fold and 3fold cross products associated to G_{2} and Spin(7) structures, respectively. The covariant derivatives of the associated calibration forms measure the "torsion" of the geometric structure, and are obvious fundamental differential invariants. However, the cross products themselves are vectorvalued differential forms, and as such one can compute their exterior covariant derivatives with respect to either the LeviCivita connection or any other canonical connection in this context. These are additional differential invariants associated to such structures. It would be interesting to relate these to the classical torsion. If they are equal, this gives a new geometric characterization of torsion in this context. If they are not equal, it would be useful to interpret these new invariants in terms of classical invariants. In a very closely related vein, for the case of almost complex structures one can define the notion of the "Nijenhuis tensor" which is a first order differential invariant that is defined using the FrolicherNijenhuis bracket of vector valued forms. There is an obvious generalization available here to the setting of the other vector cross products, and it is unclear how such generalized Nijenhuis tensors are related to the aforementioned classical torsion. Moreover, in analogy with the almost complex case, an investigation of these Nijenhuis tensors may provide a new geometric interpretation of torsionfree vector cross product structures in terms of integrable distributions of the tangent bundle.
Project title: "Generalization of the MarsdenWeinstein Reduction from symplectic geometry to G_{2} geometry"
Project abstract: The MarsdenWeinstein symplectic reduction theorem constructs a new symplectic manifold from an existing symplectic manifold M that admits a symplectic action by a Lie group G. The principal object involved in such a construction is the moment map, which is a map from M to g^{*}, the dual of the Lie algebra of G. The purpose of this project is to study to what extent such a reduction theory is possible in the context of manifolds with closed G_{2} structures, which are generalizations of symplectic manifolds, admitting a closed nondegenerate 3form rather than a 2form. It appears that the moment map should be generalized to a g^{*} valued 1form rather than a 0form. Moreover, the correct reduced space should just be an oriented Riemannian 3manifold. It would be interesting to see if we can construct special 3manifolds using such a reduction procedure.
Project title: "First order elliptic equations and calibrated geometry"
Project abstract: A system of first order partial differential equations with the same number of unknown functions as equations is called elliptic if a certain property is satisfied, having to do with the invertibility of the prinicpal symbol of the associated linear differential operator. Elliptic systems enjoy good regularity properties. The classical example is the CauchyRiemann equations of complex analysis in any number of complex variables. In the geometry of isometrically immersed submanifolds of R^{n} there is a distinguished class of submanifolds known as calibrated submanifolds. Examples include complex submanifolds and special Lagrangian submanifolds of C^{n}, and associative and coassociative submanifolds of R^{7}. Such submanifolds satisfy first order nonlinear equations, which are in some sense elliptic. The aim of this project is to understand the notion of ellipticity both for linear an nonlinear systems, and to verify explicitly the ellipticity of the CauchyRiemann, special Lagrangian, associative, and coassociative systems of equations.
Project title: "Hopf fibrations and the NavierStokes equations"
Project abstract: The Hopf fibrations in geometry are fibrations of spheres over projective spaces with spheres as the fibres. All the Hopf fibrations are related to the four finite dimensional normed real division algebras: the real numbers R, the complex numbers C, the quaternions H, and the octonions O. In particular the "classical" Hopf fibration of S^{3} over S^{2} = CP^{1} with fibre S^{1} is related to the quaternions. Closely related to these algebras is the notion of a "cross product", which exists nontrivially only in dimensions 3 and 7, being determined by the imaginary part of quaternion or octonion multiplication, respectively. This cross product allows us to define the notion of the "curl" of a vector field in R^{3} and R^{7}. The NavierStokes equations of fluid dynamics in R^{3} are a very complex nonlinear system of partial differential equations that have been studied for centuries. These equations involve the cross product and curl operations on R^{3} in an essential way. Remarkably, an explicit exact solution in a certain special case is given by the classical Hopf fibration. The aim of this project is to look for an analogue of the NavierStokes equations in 7 dimensional space, exploiting the existence of curl and cross product only in dimensions 3 and 7. Furthermore, the correct equations could possibly be found by demanding that one of the other Hopf fibrations, related to the octonions, provide an explicit exact solution. A potential byproduct may be a method to reduce solutions to the 7d "NavierStokes" equations to obtain new exact solutions to the classical 3d NavierStokes equations.
Project title: "Octonionic surfaces in sevendimensional space"
Project abstract: If a smooth twodimensional oriented surface L is immersed into ndimensional Euclidean space, one can define a "second fundamental form" which is a symmetric bilinear form on M whose values are normal vector fields to L. Then the trace of this bilinear form, with respect to the induced metric, gives a distinguished normal vector field H on L called the "mean curvature vector field". When this vector field vanishes, the surface is called "minimal" and one can show that it is a critical point of the area functional with respect to nearby variations. In sevendimensional space, there is a notion of a "cross product", with similar but slightly different properties to the usual cross product in threedimensional space. This operation is intimately connected with the nonassociative algebra of the octonions. This cross product allows one to pick out a distinguished normal vector field N to an oriented smooth surface L, immersed in sevendimensional space. One can then consider the class of such surfaces for which only the component of the mean curvature vector field H in the direction of N vanishes. It may be possible to give a variational characterization of such surfaces. One can also attempt to explore other aspects of the extrinsic geometry of surfaces in sevendimensional space that are adapted to this octonionic structure. For example, every oriented surface L in sevendimensional space can be (in some sense uniquely) partially "thickened" to a socalled "associative submanifold", which is a class of calibrated submanifolds first defined by Reese Harvey and Blaine Lawson in 1982. These are of interest in Mtheory and supergravity in modern physics. It would be interesting to see what consequences on the "thickened associative submanifold" arise from a priori conditions on the extrinsic geometry of the immersed surface L.
Project title: "Generalized symmetries of the equations of calibrated geometry"
Project abstract: There exists a general method for determining the (usually compact) Lie group G that is the "symmetry group" of a system of partial differential equations in the sense that it transforms solutions to solutions, and is like a Galois group for PDE's. For example, the symmetry group of Maxwell's equations of electromagnetism is the Poincaré group of Minkowski space. If a system of PDE's is kth order, then this symmetry group G naturally extends to act on the kth jet bundle of the domain on which the equations are defined. Conversely, it sometimes happens that a group acts on the kth jet bundle, taking solutions to solutions, but it does not arise by extension of a group action on the domain. These are called "generalized symmetries" and usually are present in equations that exhibit qualities of "integrable systems." One natural class of PDE's that occurs in geometry (inspired from physics) are the equations for calibrated submanifolds. In particular, the equations of special Lagrangian geometry do admit integrable systems interpretations, and therefore are particularly well suited to study by these techniques. We propose to look for generalized symmetries (of second order) for the special Lagrangian differential equations. This is a problem that can easily be solved (in the sense that we can determine unambiguously if such symmetries exist, and if they do, exactly what they are) during the summer term. This would be an interesting addition to the literature, at the interface of the fields of (i) symmetry groups of differential equations and (ii) calibrated geometry. If successful, we can use similar methods to study other calibrated geometries, which although they are only first order, tend to be much more nonlinear.
Project title: "Relations between two circle families of minimal surfaces in 4dimensional Euclidean space"
Project abstract: There is a deep relationship between minimal surfaces in R^{n} and the theory of holomorphic functions, which is encoded by the classical Weierstrass representation of minimal surfaces in terms of holomorphic data. Understanding this relationship involves a mixture of complex analysis and the differential geometry of surfaces in Euclidean space. One interesting aspect of the Weierstrass representation is that it reveals the existence of a continuous family of "associated" minimal surfaces, parametrized by a circle. For example, this family continuously deforms the catenoid to the helicoid through minimal surfaces in R^{3}. More recently, there has been intense interest in the study of "calibrated submanifolds" of Euclidean space, which are a special class of absolutely volume minimizing submanifolds defined by a first order nonlinear differential equation. One type of calibrated submanifolds are the socalled "special Lagrangian" submanifolds, which are halfdimensional minimal submanifolds of a certain type in R^{2n}. In this case, too, there is a circle family of such submanifolds. In the special case of surfaces in R^{4}, it would be of interest to see how these two circle families of minimal (special Lagrangian) surfaces interact.
Project title: "Special submanifolds of seven dimensional Euclidean space"
Project abstract: There exists a skewsymmetric multiplication of vectors in R7, which is analogous to the standard cross product in R3. However, this seven dimensional cross product does not satisfy all the same identities as its three dimensional counterpart, but rather it satisfies more complicated relations. It is well known that in this setting one can define special classes of three and four dimensional submanifolds (called associative and coassociative, respectively) which are examples of minimal submanifolds: they have vanishing mean curvature, and are critical points of the volume functional. An interesting question which has not yet been satisfactorily addressed is the following: are there natural classes of submanifolds of other dimensions (specifically curves, surfaces, and five and six dimensional submanifolds) which are somehow nicely compatible with the cross product structure on R7? If so, what kind of curvature properties do such submanifolds possess? For example, one can attempt to study the analogue in seven dimensional space of the FrenetSerret formulas for curves in R3 (where the cross product plays an important role). Such a project involves an interplay of the differential geometry of submanifolds of Euclidean space with the exceptional algebraic structures arising from a nonassociative eight dimensional division algebra known as the octonions or Cayley numbers.
Project title: "Constructions of calibrated submanifolds in seven and eight dimensions"
Project abstract: Minimal submanifolds of Euclidean space are critical points of the volume functional, and have zero mean curvature. They are solutions to a second order differential equation. In certain specific dimensions, some exceptional algebraic structures lead to the existence of special minimal submanifolds called 'calibrated submanifolds.' These are solutions to certain first order differential equations which are not only critical points of volume, but are actually global minimizers. One example of a calibrated submanifold is an ndimensional special Lagrangian submanifold of R2n = Cn. Another interesting case occurs only in seven dimensional Euclidean space, and consists of 3dimensional associative submanifolds and 4dimensional coassociative submanifolds. There is also a case of 4dimensional Cayley submanifolds which exist only in eight dimensional Euclidean space. All three of these types of submanifolds are intimately related to the algebra of the octonions, an exceptional real 8dimensional nonassociative division algebra. Many explicit examples have been found of calibrated submanifolds by assuming certain symmetries and reducing the problem to more tractable differential equations (sometimes enough symmetry actually leads to ordinary differential equations.) In this research project, the students will study a particular wellknown construction of special Lagrangian submanifolds in Cn, that of the twisted normal cone construction of Harvey and Lawson, and attempt to generalize this construction to the case of associative or coassociative submanifolds of R7 and to Cayley submanifolds of R8. 
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