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 Events
IMA, University of Minnesota
Wed 3 Feb 2016, 3:00pm
Department Colloquium
MATH 104
Tiling expression of minors
MATH 104
Wed 3 Feb 2016, 3:00pm-4:00pm

Abstract

The field of enumeration of tilings dates back to the early 1900s when MacMahon proved his classical theorem on plane partitions fitting in a given box. The enumeration of tilings has become a subfield of combinatorics with applications and connections to diverse areas of mathematics. In this talk we will consider a connection between the enumeration of tilings and the theory of electrical networks. 
 
The theory of electrical networks was studied systematically by Colin de Verdiere and Curtis, Ingerman, Moores, and Morrow in the 1990s. Associated with an electrical network is a "response matrix" that measures the response of the network to potential applied at  the nodes. Kenyon and Wilson showed how to test the well-connectivity of  an electrical network with n nodes by checking the positivity of n(n-1)/2 minors of the response matrix. Their test was based on the fact that any "contiguous minor" of a matrix is the generating function of domino tilings of a weighted Aztec diamond.  They conjectured that a larger family of minors, "semicontiguous minors",  can also be expressed in terms of domino tilings of certain regions. We prove this conjecture by describing explicitly the ``tiling expression" of the semicontiguous minors.

Note for Attendees

Refreshments will be served in MATH 125 before the colloquium.
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M.I.T.
Fri 5 Feb 2016, 3:00pm
Department Colloquium
Math Annex 1100
Incidence geometry and low dimensional structure
Math Annex 1100
Fri 5 Feb 2016, 3:00pm-4:00pm

Abstract

Given a collection of points and a collection of lines, circles, or other simple geometric objects, an incidence occurs when a point is contained in one of the objects. Incidence geometry is a branch of extremal combinatorics that studies the maximum number of incidences that can occur amongst all possible arrangements of the objects in question. It turns out that problems from diverse areas of mathematics can be phrased as incidence geometry questions, and often this is an effective way of tackling these problems.

There is a general phenomena in incidence geometry, which is the principle that higher dimensional incidence geometry problems often have fewer incidences than lower dimensional ones unless the objects arrange themselves into a low dimensional structure. For example, any collection of points and lines in three dimensions has relatively few incidences unless many of the points and lines cluster into a plane. I will discuss this phenomena, as well as some of its implications in discrete math and harmonic analysis.

Note for Attendees

Refreshments will be served in MATH 125 before the colloquium.
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