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Abstract of Adam's MMath project: In 1956, John Milnor surprised the mathematical community by exhibiting examples of smooth manifolds that were homeomorphic to the 7-sphere but not diffeomorphic to it with its standard smooth structure; this was the first example of so-called "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 n-sphere. 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 7-sphere but that they are not all diffeomorphic to the 7-sphere 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:
Milnor's Exotic Spheres
© Adam Bognat, 2011.

Description of Li's research project: There is a deep relationship between minimal surfaces in Rⁿ 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 R³. 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 non-linear differential equation. One type of calibrated submanifolds are the so-called "special Lagrangian" submanifolds, which are half-dimensional minimal submanifolds of a certain type in R²ⁿ. In this case, too, there is a circle family of such submanifolds. In the special case of surfaces in R⁴, it would be of interest to see how these two circle families of minimal (special Lagrangian) surfaces interact.

Abstract of Nat's MMath thesis: 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 Kähler-Einstein 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 Calabi-Yau manifolds. Finally, we will study the mean curvature flow of conormal bundles as submanifolds of Cⁿ. 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:
Mean Curvature Flow in Euclidean spaces, Lagrangian Mean Curvature Flow, and Conormal Bundles
© Nat Chun Ho Leung, 2011.

Description of Thea's MMath project: 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 hyper-Käher case.

PDF version:
The Kähler and Special Lagrangian Calibrations
© Thea Gegenberg, 2010.

Description of Zach's research project: There exists a skew-symmetric 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 Frenet-Serret 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 non-associative eight dimensional division algebra known as the octonions or Cayley numbers.

Description of Ho Yeung's and Nat's joint research project: 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 n-dimensional special Lagrangian submanifold of R2n = Cn. Another interesting case occurs only in seven dimensional Euclidean space, and consists of 3-dimensional associative submanifolds and 4-dimensional coassociative submanifolds. There is also a case of 4-dimensional 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 8-dimensional non-associative 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 well-known 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.

Information for potential students or postdoctoral fellows

I am available to advise students at the PhD, Master's, and undergraduate levels. Some idea of my research interests can be discerned by looking at my papers, available under the "Research" link on the menu. I plan to write an explanation of my research interests aimed at current and future graduate students, in the near future. Keep watching this space.

If you are a student interested in differential geometry, feel free to contact me for more information.

Information about graduate studies in the Department of Pure Mathematics at the University of Waterloo can be found here: http://www.math.uwaterloo.ca/PM_Dept/Grad/grad.shtml

Other links for potential graduate students at UW (including information about available graduate scholarships) can be found here: http://www.math.uwaterloo.ca/PM_Dept/Grad/Info/links.shtml

Undergraduates at UW interested in working on a research project in differential geometry under my supervision should apply for an NSERC Undergraduate Student Research Award (USRA). Details can be found here: http://www.math.uwaterloo.ca/PM_Dept/Undergrad/NSERC-USRA.shtml

In Canada, postdoctoral fellows are usually financially supported through NSERC grants of supervising faculty members. As such, it usually requires the sharing of grant money from 2 or 3 faculty members to support a postdoctoral fellow. Anyone interested in a postdoctoral position in geometry at UW should contact one or more of the geometry/topology faculty at UW, who can be found here: http://www.math.uwaterloo.ca/PM_Dept/Research/interests.shtml

Alternatively, Canadian citizens and permanent residents can also apply for an NSERC Postdoctoral Fellowship. Currently these pay $40,000 per year and the possibility exists that successful applicants could have their stipend 'topped-up' by an additional $5,000 from NSERC grants of the supervising faculty members. Information about applying for NSERC Postdoctoral Fellowships can be found here: http://www.nserc-crsng.gc.ca/Students-Etudiants/PD-NP/PDF-BP_eng.asp

If anyone knows of other funding opportunities for graduate students, undergraduate students, or postdoctoral fellows in Canada, please pass along this information to me so I can post it here.