Embedded Files
PhD Projects
PhD Projects
Some Embedding Calculus Generalizations
Some Embedding Calculus Generalizations
The Goodwillie-Weiss embedding calculus is a machinery to access embedding spaces between manifolds by taking successive approximations. This has found applications in many things -- for example, determining the weak homotopy types of certain diffeomorphism groups. Much is known when the underlying manifolds are compact (much remains unknown as well), but it is far less clear what one should do when non-compact manifolds are allowed. This ongoing project is to explore to what extent can one say something (if anything) about compactly-supported embedding spaces and diffeomorphism groups.
Undergraduate Projects
Undergraduate Projects
VC-dimension of generalized progressions in some nonabelian groups
VC-dimension of generalized progressions in some nonabelian groups
The Vapnik–Chervonenkis-dimension (VC-dimension) measures a certain complexity in a set family, i.e. a subset of a power set. If the set family consist of certain arithmetic progressions in an abelian group, then the VC-dimension is well known.
One may generalize the notion of a progression into an arbitrary finitely generated group with a choice of a generating set, but now very little is known of its VC-dimension.
In this paper, we studied some special cases in free groups and the discrete Heisenberg group, showing finite (or better) bounds of VC-dimension of progressions.
A certain embedding of a progression in the discrete Heisenberg group into Euclidean space
Box-ball systems and RSK recording tableaux
Box-ball systems and RSK recording tableaux
Evolution of a box-ball system into steady state
A box-ball system (BBS) is a certain discrete dynamical system consisting of balls and boxes. A BBS eventually must stabilize into a certain steady state. Every permutation is associated with a BBS, whose steady state configuration, in the form of a standard Young tableau, provides information about the permutation.
A permutation is also associated with a pair of standard Young tableau via the RSK correspondence. There are some very surprising relationships between tableaux generated via the two different methods.
In this paper, we studied BBS in general, and certain aspects of the aforementioned relationships.
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