Categorification in Geometry, Topology, and Physics

The emergent mathematical philosophy of categorification is reshaping our view of modern mathematics by uncovering a hidden layer of structure in mathematics, revealing richer and more robust structures capable of describing more complex phenomena. Categorification is a powerful tool for relating various branches of mathematics and exploiting the commonalities between fields. It provides a language emphasizing essential features and allowing precise relationships between vastly different fields. This volume focuses on the role categorification plays in geometry, topology, and physics. These articles illustrate many important trends for the field including geometric representation theory, homotopical methods in link homology, interactions between higher representation theory and gauge theory, and double affine Hecke algebra approaches to link homology. The companion volume (Contemporary Mathematics, Volume 683) is devoted to categorification and higher representation theory. Cover -- Title page -- Contents -- Preface -- Geometry and categorification -- 1. Introduction -- 2. K-theory -- 3. The function-sheaf correspondence -- 4. Symplectic resolutions -- References -- A geometric realization of modified quantum algebras -- 1. Introduction -- 2. Preliminary, I -- 3. Preliminary, II -- 4. Convolution product -- 5. Defining relation -- 6. Algebra \KK_{ } -- 7. Relation with the work [Zh08] -- 8. BLM case -- References -- The cube and the Burnside category -- 1. Introduction -- 2. The cube -- 3. The Burnside category -- 4. Functors from the cube to B -- 5. Properties of such functors -- 6. The Khovanov functor -- 7. Spaces -- 8. Some questions -- Acknowledgments -- References -- Junctions of surface operators and categorification of quantum groups -- 1. Introduction -- 2. Junctions of Wilson lines and quantum groups -- 2.1. Junctions of line operators -- 2.2. Web relations -- 2.3. Skew Howe duality and the quantum group -- 2.4. Why "categorification = surface operators"--3. Junctions of surface operators -- 3.1. Junctions in 4d \CN=4 theory -- 3.2. Line-changing operators in class \CS and network cobordisms -- 3.3. Junctions in 4d \CN=2 theory -- 3.4. Junctions in 4d \CN=1 theory -- 3.5. OPE of surface operators and the Horn problem -- 3.6. OPE and Schubert calculus -- Domain walls in 4d \CN=2 SQCD -- 4. Categorification and the Landau-Ginzburg perspective -- 4.1. Physics perspectives on categorification -- 4.2. LG theory on "time x knot"--4.3. Junctions and LG interfaces -- 4.4. Junctions and matrix factorizations -- 4.5. Junctions and categorification of quantum groups -- 5. What's next? -- Acknowledgments -- Appendix A. Wilson lines and categories Web -- Appendix B. Domain walls, junctions and Grassmannians -- Appendix C. LG Interfaces and the cohomology of Grassmannians 9.1. Uncolored Trefoil-prime -- 9.2. Colored/iterated Examples -- 9.3. Generalized Twisting -- 9.4. Some Examples -- 9.5. Toward The Skein -- Appendix A. Links And Splice Diagrams -- A.1. Links, Cables And Splices -- A.2. Splice Diagrams -- A.3. Operations On Links -- A.4. Equivalent Diagrams -- A.5. Connection With Daha -- References -- Back Cover