I will survey the equivariant method of moving frames and how it is used to construct differential invariant signatures for solving equivalence and symmetry problems. I will then present recent applications in image processing, in particular the reassembly of 2D and 3D jigsaw puzzles and the geometric classification and reassembly of broken bones in anthropology.
Moving frames, differential invariants, and the reassembly of broken objects Sponsored by the Meyer Fund
Apr. 02, 2019 1pm (MATH 220)
Agnes Szendrei (CU Boulder)
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This is a continuation of the talk ``Ultrafilters with Model-Theoretically Significant Properties''.
Flexible Ultrafilters and the Theory of the Random Graph
Apr. 02, 2019 2pm (MATH 350)
Lie Theory
John Shareshian (Washington University in St. Louis)
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Given a finite graph , a proper coloring of is a function from to the set of positive integers such that and are distinct whenever is in . Associated to each such is the monomial in variables in which the exponent of is the number of vertices satisfying . The sum of such monomials over all proper colorings is the chromatic symmetric function , introduced by Richard Stanley.
If , one may multiply introduce another variable and multiply each by , where is the number of edges in such that and . The sum of is called the chromatic quasisymmetric function of . These quasisymemtric functions were introduced in joint work with Michelle Wachs. When is a unit interval graph, labeled appropriately, it turns out that this quasisymmetric function is in fact a symmetric function. Moreover, this symmetric function is Schur positive and therefore is the Frobenius characteristic of some representation of the symmetric group .
Patrick Brosnan and Tim Chow proved a conjecture of ours, that the representation in question is, up to a sign twist, the "dot action" (first investigated by Julianna Tymoczko) of on the cohomology of a variety naturally associated to the unit interval graph . This variety is a subvariety of the flag variety that admits a torus action, thus allowing one to study its cohomology by considering a moment graph. (I will explain all of these terms, it is not necessary for the audience to be familiar with them already.)
The Brosnan-Chow Theorem allows a geometric attack on a longstanding conjecture of Stanley and John Stembridge, but this conjecture remains open.
Chromatic quasisymmetric functions and regular semisimple Hessenberg varieties Sponsored by the Meyer Fund
Apr. 02, 2019 4pm (MATH 350)
Topology
Paul Lessard (CU Boulder) Higher categories and the Grothendieck Homotopy Hypothesis
Moving frames, differential invariants, and the reassembly of broken objects