هندسه و توپولوژی زیرمنفردها و جریانات: کنفرانس هندسه میدوست 2013 (MGC XIX)، 19 اکتبر 2013، دانشگاه ایالت اوکلاهما، استیلواتر، اوکلاهما: کنفرانس هندسه میدوست 2012 (MGC XVIII)، 12-13 مه 2012، دانشگاه اوکلاهما، نورمن، اوکلاهما

The papers in this volume are mainly from the 2013 Midwest Geometry Conference, held October 19, 2013, at Oklahoma State University, Stillwater, OK, and partly from the 2012 Midwest Geometry Conference, held May 12–13, 2012, at the University of Oklahoma, Norman, OK. The papers cover recent results on geometry and topology of submanifolds. On the topology side, topics include Plateau problems, Voevodsky's motivic cohomology, Reidemeister zeta function and systolic inequality, and freedom in 2- and 3-dimensional manifolds. On the geometry side, the authors discuss classifying isoparametric hypersurfaces and review Hartogs triangle, finite volume flows, nonexistence of stable $p$-currents, and a generalized Bernstein type problem. The authors also show that the interaction between topology and geometry is a key to deeply understanding topological invariants and the geometric problems. Cover 1 Title page 4 Contents 6 Preface 8 2013 Midwest Geometry Conference (MGC XIX) Talks 10 Plateau Problems in Metric Spaces and Related Homology and Cohomology Theories 12 1. Introduction 12 2. Compactness and Rectifiability Theorems 18 3. Plateau Problems 20 4. Some One Dimensional Examples 21 5. Homology Theories 22 6. Real Normal Chains and Dual Cochains 24 References 26 Relating Equivariant and Motivic Cohomology via Analytic Currents 30 1. Introduction 30 2. The differential-geometric perspective 36 3. Finite analytic correspondences and ordinary homology 40 4. Equivariant cohomology and the road to the cycle map 43 Appendix A. Regular G-covers 47 References 50 Braids and Symplectic Reidemeister Zeta Functions 52 1. Introduction 52 2. Braids, Knots and SU(2)-representations 53 3. Symplectic Reidemeister Zeta functions 62 References 70 Systoles of Surfaces and 3-Manifolds 72 1. Introduction 72 2. Systolic inequality of surfaces 74 3. Systolic inequality and systolic freedom of 3-manifolds 78 References 89 Ideal Theory and Classification of Isoparametric Hypersurfaces 92 1. Introduction 92 2. A walk through some ideal theory 93 3. How the ideal theory interacts with isoparametric hypersurfaces 104 References 114 The Hartogs Triangle in Complex Analysis 116 Introduction 116 1. Basic properties of the Hartogs triangle in C2 117 2. L2 theory for \db on the Hartogs triangle 118 3. Boundary regularity for \db on the Hartogs triangle 120 4. Holomorphic functions on the Hartogs triangle in CP2 122 References 124 Finite Volume Flows and Witten’s Deformation 128 1. Introduction 128 2. Finite volume flows by Harvey and Lawson 128 3. The Morse Complex 131 4. The Witten deformation 132 5. Applications 134 Acknowledgements 136 References 136 On the Existence and Nonexistence of Stable Submanifolds and Currents in Positively Curved Manifolds and the Topology of Submanifolds in Euclidean Spaces 138 1. Introduction 139 2. Universally mass decreasing sets of vector fields 143 3. Trace formulas for immersed submanifolds of Euclidean Space 148 4. Applications to the topology of hypersurfaces 154 5. Topological vanishing theorems for submanifolds of Euclidean space 157 6. Stable currents in the rank one symmetric spaces 165 7. Mass minimizing currents modulo two in real projective spaces 171 8. The Topology of noncompact stable hypersurfaces in Riemannian manifolds 173 Acknowledgment 176 References 176 Remarks on Stable Minimal Hypersurfaces in Riemannian Manifolds and Generalized Bernstein Problems 180 1. Introduction 180 2. Preliminaries 184 3. Stable Minimal Hypersurfaces 185 4. Generalized Bernstein Problems 188 5. Dualities 191 6. The Unity 193 7. The case p=0: Meeting g “head on” Without Making a Conformal Change 194 8. The Case p∈(2-2/n,∞): Some Interplays Between R and R 195 9. The Rigidity of Stable Minimal Hypersurfaces 196 Acknowledgement 196 References 196 Back Cover 200