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 Triple integrals

Compute ZZZ xdxdydz, W II IV N II 0 where W is the region >< bounded by the planes N II + >< N x = 0, y = 0, and z = 2, and ~ the surface z = x 2 + y 2 and

lying in the quadrant '<: x ≥ 0, y ≥ 0. II ~ '<:

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

 ∂2u ∂2u ∂2u  g ¯ + g ¯ + g ¯ + 1,1 ∂z1∂z¯1 1,2 ∂z1∂z¯2 1,3 ∂z1∂z¯3 ∂2u ∂2u ∂2u det g ¯ + g ¯ + g ¯ +   2,1 ∂z2∂z¯1 2,2 ∂z2∂z¯2 2,3 ∂z2∂z¯3  ∂2u ∂2u ∂2u g ¯ + g ¯ + g ¯ + 3,1 ∂z3∂z¯1 3,2 ∂z3∂z¯2 3,3 ∂z3∂z¯3   g1,1¯ g1,2¯ g1,3¯ F = e det g2,1¯ g2,2¯ g2,3¯ , 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