One of the first algorithm any mathematician learns about is the Euclidean division algorithm for the rational integer ring Z. When asking whether other integer rings have similar such division algorithms, we are then led naturally to a geometric interpretation of this algorithm which concerns lattices and their (multiplicative) covering radius.
Since the seminal work of Arnold on the Euler equations (1966), many equations in hydrodynamics were shown to be geodesic equations of diffeomorphism groups of manifolds, with respect to various Sobolev norms. This led to new ways to study these PDEs, and also initiated the study of of the geometry ofthose groups as (infinite dimensional) Riemannian manifolds.
Speaker: Zur Luria, ETH
Title: New bounds for the n-queen's problem
The famous n-queens problem asks: In how many ways can n nonattacking queens be placed on an n by n chessboard? This question also makes sense on the toroidal chessboard, in which opposite sides of the board are identified. In this setting, the n-queens problem counts the number of perfect matchings in a certain regular hypergraph. We give an extremely general bound for such counting problems, which include Sudoku squares and designs.
Speaker: Joel Friedman, UBC
Title: Open Problems Related to the Zeta Functions
We express some open problems in graph theory in terms of Ihara graph zeta
functions, or, equivalently, non-backtracking matrices of graphs. We focus
on "expanders" and random regular graphs, but touch on some seemingly
unrelated problems encoded in zeta functions.
We suggest that zeta functions of sheaves on graphs may have relevance to
complexity theory and to questions of Stark and Terras regarding whether
coverings of a fixed graph can ramify like number field extensions.
Salmon and Cayley proved the celebrated 19th century result that a smooth cubic surface over the complex numbers contains exactly 27 lines. By contrast, the count over the real numbers depends on the surface, and these possible counts were classified by Segre. A number of researchers have recently made the striking observation that Segre’s work shows a certain signed count is always 3. In my talk, I will explain how to extend this result to an arbitrary field.