2008 Course Descriptions

Beginning Lecture Course

Lecturers:  Genevieve Walsh, Tufts University

                   Maggy Tomova, Rice University

Teaching Assistants:   Yvonne Lai, Ellen Goldstein

Titles for the Beginning Lectures Series: Surfaces, orbifolds and knots

Week one is devoted to surfaces and orbifolds.  We will start off with the geometry of the hyperbolic plane and basic properties of surfaces.

Isometries acting on the hyperbolic plane will lead us to hyperbolic surfaces and orbifolds.  Topics include:  different hyperbolic structures on a surface, the minimal area hyperbolic orbifold, orbifold Euler characteristic, and the classification of Euclidean 2-orbifolds.

Week two is on knots and their complements in the three-sphere.  We will study various classes of knots as well as some knot invariants.  We will also discuss the important surfaces that can be found in knot complements.

Preparation:  a class in topology is helpful but not required.

Advanced Lecture Course

Lecturers:    Rachel Roberts, Washington University, St. Louis

                     Jennifer Schultens, University of California, Davis                

Teaching Assistants: Week 1 - Joan Licata, Szego Assistant Professor at Stanford

Week 2 - Alice Stevens, UC Davis

Titles for the Graduate Lectures Series:  

Week 1:  Foliations and Laminations - Rachel Roberts

In the first week we will discuss the theory of laminations and foliations.  More specifically, we will introduce the Thurston-Nielsen theory of automorphisms of surfaces with an emphasis on the role of geodesic laminations.  We will then move up a dimension and discuss codimension one foliations and laminations of 3-manifolds and how these objects allow one to "see" and work with given 3-manifolds.

Week 2:  Surfaces in 3-manifolds - Jennifer Schultens

In the second week we will develop the theory of Heegaard splittings of 3-manifolds.  In particular, we will introduce the curve complex of Harvey and discuss the ways in which recent deeper understanding of the curve complex has shed new light on our understanding of Heegaard splittings and hence 3-manifolds.

Recommended background for week 1:  Basic Algebraic and Differential Topology

Recommended background for week 2:  Courses in Topology, Algebraic Topology, Differential Topology will be helpful but not required.

Reading list for Foliations and Laminations:

Advanced Lecture Course Reading List:

Candel, Alberto; Conlon, Lawrence Foliations. I. Graduate Studies in Mathematics, 23. American Mathematical Society, Providence, RI, 2000.

Candel, Alberto; Conlon, Lawrence Foliations. II. Graduate Studies inMathematics, 60. American Mathematical Society, Providence, RI, 2003.

Casson, Andrew J.; Bleiler, Steven A. Automorphisms of surfaces afterNielsen and Thurston. London Mathematical Society Student Texts, 9. Cambridge University Press, Cambridge, 1988.

Gabai, David; Oertel, Ulrich Essential laminations in $3$-manifolds. Ann. of Math. (2) 130 (1989), no. 1, 41--73.

Gabai, David; Kazez, William H. Order trees and laminations of the plane.  Math. Res. Lett. 4 (1997), no. 4, 603--616.

Gabai, David; Kazez, William H. Homotopy, isotopy and genuine laminations of $3$-manifolds. Geometric topology (Athens, GA, 1993), 123--138, AMS/IP Stud. Adv. Math., 2.1, Amer. Math. Soc., Providence, RI, 1997.

Hatcher, A. E. Measured lamination spaces for surfaces, from the topological viewpoint. Topology Appl. 30 (1988), no. 1, 63--88.

Kronheimer, P.; Mrowka, T.; Ozsvth, P.; Szabo, Z. Monopoles and lens space surgeries. Ann. of Math. (2) 165 (2007), no. 2, 457--546.

Li, Tao Heegaard surfaces and measured laminations. I. The Waldhausen conjecture. Invent. Math. 167 (2007), no. 1, 135--177.

Li, Tao Heegaard surfaces and measured laminations. II. Non-Haken 3-manifolds. J. Amer. Math. Soc. 19 (2006), no. 3, 625--657

Palmeira, Carlos Frederico Borges Open manifolds foliated by planes.  Ann. Math. (2) 107 (1978), no. 1, 109--131.

Reading list for Surfaces and 3-manifolds:

Books for browsing:

1) Hempel, John: ``3-manifolds'', Reprint of the 1976 original. AMS Chelsea Publishing, Providence, RI, 2004.

2) Stillwell, John: ``Classical Topology and combinatorial group theory''. Second edition. Graduate Texts in Mathematics, 72. Springer-Verlag, New York, 1993.

3) Hatcher, Allen: Notes on Basic 3-Manifold Topology Available online for free!  

Specialized papers:

4) Scharlemann, Martin: Heegaard splittings of compact 3-manifolds. Handbook of geometric topology, 921--953, North-Holland, Amsterdam, 2002.

5) Schultens, Jennifer: The classification of Heegaard splittings for (compact orientable surface)$\,\times\,S\sp 1$. Proc. London Math. Soc. (3) 67 (1993), no. 2, 425--448.

6) Hartshorn, Kevin: Heegaard splittings of Haken manifolds have bounded distance. Pacific J. Math. 204 (2002), no. 1, 61--75.

7) Scharlemann, Martin; Tomova, Maggy: Alternate Heegaard genus bounds distance. Geom. Topol. 10 (2006), 593--617.

8) Minsky, Yair N.: Curve complexes, surfaces and 3-manifolds. International Congress of Mathematicians. Vol. II, 1001--1033, Eur. Math. Soc., Zrich, 2006.

Further inspiration:

9) Scharlemann, Martin; Thompson, Abigail: Thin position for $3$-manifolds. Geometric topology (Haifa, 1992), 231--238, Contemp. Math., 164, Amer. Math. Soc., Providence, RI, 1994.

10) Minsky, Yair N.: A geometric approach to the complex of curves on a surface. Topology and Teichmller spaces (Katinkulta, 1995), 149--158, World Sci. Publ., River Edge, NJ, 1996.

11) Masur, Howard A.; Minsky, Yair N.: Geometry of the complex of curves. I. Hyperbolicity. Invent. Math. 138 (1999), no. 1, 103--149.

12) Masur, H. A.; Minsky, Y. N.: Geometry of the complex of curves. II. Hierarchical structure. Geom. Funct. Anal. 10 (2000), no. 4, 902--974.

13) Li, Tao: Saddle tangencies and the distance of Heegaard splittings. Algebr. Geom. Topol. 7 (2007), 1119--1134.

14) Namazi, Hossein: Big Heegaard distance implies finite mapping class group. Topology Appl. 154 (2007), no. 16, 2939--2949.

http://www.math.cornell.edu/~hatcher/3M/3Mdoublepage.pdf