Preface The selections in this volume are mainly from the 2013 Midwest Geometry Con- ference held at Oklahoma State University on October 19, 2013 and partly from the 2012 Midwest Geometry Conference held at The University of Oklahoma, May 12– 13, 2012. The Midwest Geometry Conference was an annual event in the Midwest region from 1991 until 2007 and was revived in 2012 at The University of Oklahoma (cf.e.g. The 2013 Midwest Geome- try Conference focused on Plateau problems, equivariant motivic cohomology, ideal theory and classification of isoparametric hypersurfaces, stable submanifolds, cali- brated geometry, p-harmonic geometry, and Dolbeault cohomology groups. The first article in this volume gives a nice, brief introduction to Plateau prob- lems in general and describes the recent result of Hardt with his collaborators T. De Pauw and W. Pfeffer. Hardt discusses the fundamental compactness and rectifiabil- ity and the applications to the Plateau problem and optimal transport paths. Dos Santos, Lima-Filho, and Hardt show possibly the best motivation for Voevodsky’s definition of motivic cohomology and apply it to singular and ordinary equivariant cohomology theories. Li shows that the Reidemeister number of a smooth map on the representation variety induced from the braid action provides a knot invariant of the corresponding braid, and the Reidemeister zeta function from the dynamic system point of view is a rational function for certain classes of braids. Chen and Li present a survey on systolic inequalities and systolic freedom in two and three dimensions. They investigate the optimal systolic ratio and the real- ized metric for surfaces and discuss systolic inequalities and freedoms of homotopy, homology and stable and conformal systoles for 3-manifolds. Chi applies regular sequences, Cohen-Macaulayness, Serre’s criterion on reducedness, Serre’s criterion of primeness, Serre’s criterion of normality for homogeneous polynomials for his purpose of classifying isoparametric hypersurfaces of degree four, and reviews the classification theory with emphasis on the application of the ideal theory Shaw gives an interesting review about L2 theory and the regularity for on the Hartogs triangle, and function theory for the related Hartogs triangle in the complex projective space CP 2. Hu provides an affirmative answer to a question raised by Harvey and Lawson on the direct connection between Witten’s deformation and finite volume flows. It is one of the most fundamental and interesting problems in geometry to study the relationship between curvature and topology of Riemannian manifolds and Riemannian submanifolds. The famous Synge Theorem showed that a compact orientable even-dimensional Riemannian manifold with positive sectional curvature has no nonconstant stable closed geodesics hence its fundamental group vanishes. Howard and Wei use an extrinsic average variational method in the calculus of vii
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