What is Differential Geometry? Zhiqin Lu UCI, Recruitment Day April 3, 2009 Zhiqin Lu, UC. Irvine What is Geometry 1/14 II IV N II 0 >< N II + >< N ~ '<: II ~ '<: Figure: From Vector Calculus, Marsden & Tromba Triple integrals Compute ZZZ xdxdydz; W where W is the region bounded by the planes x = 0; y = 0, and z = 2, and the surface z = x 2 + y 2 and lying in the quadrant x ≥ 0; y ≥ 0. Zhiqin Lu, UC. Irvine What is Geometry 2/14 II IV N II 0 >< N II + >< N ~ '<: II ~ '<: Triple integrals Compute ZZZ xdxdydz; W where W is the region bounded by the planes x = 0; y = 0, and z = 2, and the surface z = x 2 + y 2 and lying in the quadrant x ≥ 0; y ≥ 0. Figure: From Vector Calculus, Marsden & Tromba Zhiqin Lu, UC. Irvine What is Geometry 2/14 How to compute integrations over an n-dimensional object? Zhiqin Lu, UC. Irvine What is Geometry 3/14 An example Zeros of a quintic polynomial: 5 5 5 5 5 Z0 + Z1 + Z2 + Z3 + Z4 + 10Z0Z1Z2Z3Z4 = 0 in C5. Zhiqin Lu, UC. Irvine What is Geometry 4/14 Figure: From Wikipedia, the intersection of the quintic Calabi-Yau threefold to our three dimensional space Zhiqin Lu, UC. Irvine What is Geometry 5/14 Use Linear Algebra Use Abstract Algebra Use the results in all other math/physics fields. How to study high dimensional geometric object? Use Partial Differential Equations; Zhiqin Lu, UC. Irvine What is Geometry 6/14 Use Abstract Algebra Use the results in all other math/physics fields. How to study high dimensional geometric object? Use Partial Differential Equations; Use Linear Algebra Zhiqin Lu, UC. Irvine What is Geometry 6/14 Use the results in all other math/physics fields. How to study high dimensional geometric object? Use Partial Differential Equations; Use Linear Algebra Use Abstract Algebra Zhiqin Lu, UC. Irvine What is Geometry 6/14 How to study high dimensional geometric object? Use Partial Differential Equations; Use Linear Algebra Use Abstract Algebra Use the results in all other math/physics fields. Zhiqin Lu, UC. Irvine What is Geometry 6/14 1 I xdy − ydx = 1: 2π x 2 + y 2 A simple example Zhiqin Lu, UC. Irvine What is Geometry 7/14 A simple example 1 I xdy − ydx = 1: 2π x 2 + y 2 Zhiqin Lu, UC. Irvine What is Geometry 7/14 A non-trivial example How many holes in the quintic Calabi-Yau manifold? Zhiqin Lu, UC. Irvine What is Geometry 8/14 In 1977, S. T. Yau was able to solve the following PDE 0 @2u @2u @2u 1 g ¯ + g ¯ + g ¯ + 1;1 @z1@z¯1 1;2 @z1@z¯2 1;3 @z1@z¯3 @2u @2u @2u det Bg ¯ + g ¯ + g ¯ + C @ 2;1 @z2@z¯1 2;2 @z2@z¯2 2;3 @z2@z¯3 A @2u @2u @2u g ¯ + g ¯ + g ¯ + 3;1 @z3@z¯1 3;2 @z3@z¯2 3;3 @z3@z¯3 0 1 g1;1¯ g1;2¯ g1;3¯ F = e det @g2;1¯ g2;2¯ g2;3¯A ; g3;1¯ g3;2¯ g3;3¯ where gi¯j and F are given functions. After that, we are able to tell the topological properties of the manifold. Zhiqin Lu, UC. Irvine What is Geometry 9/14 Hodge theory! Linear algebra Zhiqin Lu, UC. Irvine What is Geometry 10/14 Linear algebra Hodge theory! Zhiqin Lu, UC. Irvine What is Geometry 10/14 Conclusion: Differential Geometry is not a separate math field, it brought different fields like PDE, algebraic geometry, algebraic topology, Lie group theory, functional analysis, and many others together. Zhiqin Lu, UC. Irvine What is Geometry 11/14 Peter Li, Geometric PDE, Analysis on manifolds; Chuu-Lian Terng, Differential Geometry, Integrable systems Ronald J. Stern,Smooth 4-manifolds, symplectic contact topology and geometry, knot theory Zhiqin Lu, complex geometry, Mirror symmetry. UCI's Geometry / Topology Group Zhiqin Lu, UC. Irvine What is Geometry 12/14 Geometric PDE, Analysis on manifolds; Chuu-Lian Terng, Differential Geometry, Integrable systems Ronald J. Stern,Smooth 4-manifolds, symplectic contact topology and geometry, knot theory Zhiqin Lu, complex geometry, Mirror symmetry. UCI's Geometry / Topology Group Peter Li, Zhiqin Lu, UC. Irvine What is Geometry 12/14 Chuu-Lian Terng, Differential Geometry, Integrable systems Ronald J. Stern,Smooth 4-manifolds, symplectic contact topology and geometry, knot theory Zhiqin Lu, complex geometry, Mirror symmetry. UCI's Geometry / Topology Group Peter Li, Geometric PDE, Analysis on manifolds; Zhiqin Lu, UC. Irvine What is Geometry 12/14 Differential Geometry, Integrable systems Ronald J. Stern,Smooth 4-manifolds, symplectic contact topology and geometry, knot theory Zhiqin Lu, complex geometry, Mirror symmetry. UCI's Geometry / Topology Group Peter Li, Geometric PDE, Analysis on manifolds; Chuu-Lian Terng, Zhiqin Lu, UC. Irvine What is Geometry 12/14 Ronald J. Stern,Smooth 4-manifolds, symplectic contact topology and geometry, knot theory Zhiqin Lu, complex geometry, Mirror symmetry. UCI's Geometry / Topology Group Peter Li, Geometric PDE, Analysis on manifolds; Chuu-Lian Terng, Differential Geometry, Integrable systems Zhiqin Lu, UC. Irvine What is Geometry 12/14 Smooth 4-manifolds, symplectic contact topology and geometry, knot theory Zhiqin Lu, complex geometry, Mirror symmetry. UCI's Geometry / Topology Group Peter Li, Geometric PDE, Analysis on manifolds; Chuu-Lian Terng, Differential Geometry, Integrable systems Ronald J. Stern, Zhiqin Lu, UC. Irvine What is Geometry 12/14 Zhiqin Lu, complex geometry, Mirror symmetry. UCI's Geometry / Topology Group Peter Li, Geometric PDE, Analysis on manifolds; Chuu-Lian Terng, Differential Geometry, Integrable systems Ronald J. Stern,Smooth 4-manifolds, symplectic contact topology and geometry, knot theory Zhiqin Lu, UC. Irvine What is Geometry 12/14 complex geometry, Mirror symmetry. UCI's Geometry / Topology Group Peter Li, Geometric PDE, Analysis on manifolds; Chuu-Lian Terng, Differential Geometry, Integrable systems Ronald J. Stern,Smooth 4-manifolds, symplectic contact topology and geometry, knot theory Zhiqin Lu, Zhiqin Lu, UC. Irvine What is Geometry 12/14 UCI's Geometry / Topology Group Peter Li, Geometric PDE, Analysis on manifolds; Chuu-Lian Terng, Differential Geometry, Integrable systems Ronald J. Stern,Smooth 4-manifolds, symplectic contact topology and geometry, knot theory Zhiqin Lu, complex geometry, Mirror symmetry. Zhiqin Lu, UC. Irvine What is Geometry 12/14 http://math.uci.edu/~ zlu/geom-topo-group/index.php Members of the Geometry & Topology Group at UCI work in many different fields and have expertise in a diverse set of techniques. We have lively and well-attended seminars, and one of our key goals is the cross-pollination of ideas between geometry and topology. Our faculty consists of active researchers in many areas of geometry and low-dimensional topology including geometric PDE, differential geometry, integrable systems, mirror symmetry, smooth 4-manifolds, symplectic and contact topology and geometry, and knot theory and its invariants. Our faculty is highly-regarded. All have NSF Grants and one of its members, Peter Li, was elected to the American Academy of Arts and Sciences. The Geometry/Topology Group at UCI has a long-standing commitment to excellence in graduate and postdoctoral training: we have produced some outstanding graduate students, and we have been fortunate to have recruited and mentored exceptional postdoctoral fellows. Zhiqin Lu, UC. Irvine What is Geometry 13/14 Welcome to UCI! Zhiqin Lu, UC. Irvine What is Geometry 14/14.