What is Differential Geometry?
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1 What is Differential Geometry? Zhiqin Lu UCI, Recruitment Day April 3, 2009 Zhiqin Lu, UC. Irvine What is Geometry 1/14
2 Triple integrals Compute W xdxdydz, 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
3 II Triple integrals Compute W xdxdydz, 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. N '<: IV N >< II II >< 0 N II ~ + ~ '<: Figure: From Vector Calculus, Marsden & Tromba Zhiqin Lu, UC. Irvine What is Geometry 2/14
4 How to compute integrations over an n-dimensional object? Zhiqin Lu, UC. Irvine What is Geometry 3/14
5 An example Zeros of a quintic polynomial: in C 5. Z Z Z Z Z Z 0 Z 1 Z 2 Z 3 Z 4 = 0 Zhiqin Lu, UC. Irvine What is Geometry 4/14
6 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
7 How to study high dimensional geometric object? Use Partial Differential Equations; Zhiqin Lu, UC. Irvine What is Geometry 6/14
8 How to study high dimensional geometric object? Use Partial Differential Equations; Use Linear Algebra Zhiqin Lu, UC. Irvine What is Geometry 6/14
9 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
10 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
11 A simple example Zhiqin Lu, UC. Irvine What is Geometry 7/14
12 A simple example 1 xdy ydx 2π x 2 + y 2 = 1. Zhiqin Lu, UC. Irvine What is Geometry 7/14
13 A non-trivial example How many holes in the quintic Calabi-Yau manifold? Zhiqin Lu, UC. Irvine What is Geometry 8/14
14 In 1977, S. T. Yau was able to solve the following PDE g 1, u z 1 z 1 g 1, u z 1 z 2 g 1, u det g 2, u z 2 z 1 g 2, u z 2 z 2 g 2, u g 3, u z 3 z 1 g 3, u z 3 z 2 g 3, u g 1, 1 g 1, 2 g 1, 3 = e F det g 2, 1 g 2, 2 g 2, 3, g 3, 1 g 3, 2 g 3, 3 z 1 z 3 z 2 z 3 z 3 z 3 where g i 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
15 Linear algebra Zhiqin Lu, UC. Irvine What is Geometry 10/14
16 Linear algebra Hodge theory! Zhiqin Lu, UC. Irvine What is Geometry 10/14
17 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
18 UCI s Geometry / Topology Group Zhiqin Lu, UC. Irvine What is Geometry 12/14
19 UCI s Geometry / Topology Group Peter Li, Zhiqin Lu, UC. Irvine What is Geometry 12/14
20 UCI s Geometry / Topology Group Peter Li, Geometric PDE, Analysis on manifolds; Zhiqin Lu, UC. Irvine What is Geometry 12/14
21 UCI s Geometry / Topology Group Peter Li, Geometric PDE, Analysis on manifolds; Chuu-Lian Terng, Zhiqin Lu, UC. Irvine What is Geometry 12/14
22 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
23 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
24 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
25 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
26 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
27 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
28 Welcome to UCI! Zhiqin Lu, UC. Irvine What is Geometry 14/14
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