Speaker: Tingran Gao (U. Chicago)
Title: Manifold Learning on Fibre Bundles
Abstract: Spectral geometry has played an important role in modern
geometric data analysis, where the technique is widely known as
Laplacian eigenmaps or diffusion maps. In this talk, we present a
geometric framework that studies graph representations of complex
datasets, where each edge of the graph is equipped with a non-scalar
transformation or correspondence. This new framework models such a
dataset as a fibre bundle with a connection, and interprets the
collection of pairwise functional relations as defining a horizontal
diffusion process on the bundle driven by its projection on the base.
The eigenstates of this horizontal diffusion process encode the
“consistency” among objects in the dataset, and provide a lens through
which the geometry of the dataset can be revealed. We demonstrate an
application of this geometric framework on evolutionary anthropology.
SPEAKER: Max Alekseyev (George Washington U)
Transfer-Matrix Method as a Combinatorial Hammer:
Enumeration of Silent Circles, Graph Cycles, and Seating Arrangements
I will discuss application of the transfer-matrix method to a variety
of enumeration problems concerning the party game “silent circles”,
Hamiltonian cycles in the antiprism graphs, simple paths/cycles in
arbitrary graphs, and generalized menage problem. While this method
does not always lead to nice formulas, it often provides an efficient
way of computing the corresponding quantities.
Max Alekseyev is an Associate Professor of Mathematics and
Computational Biology at the George Washington University. He holds
M.S. in mathematics (1999) and Ph.D. in computer science (2007), and
is a recipient of the NSF CAREER award (2013) and the John Riordan
prize (2015). Dr. Alekseyev’s research interests range from discrete
mathematics (particularly, combinatorics and graph theory) to
computational biology (particularly, comparative genomics and genome
assembly). He is an Editor-in-Chief of the Online Encyclopedia of
Integer Sequences (http://oeis.org).