What is Differential Geometry?

What is Differential Geometry?

Zhiqin Lu

UCI, Recruitment Day

April 3, 2009

Zhiqin Lu, UC. Irvine What is Geometry 1/14

Triple integrals

Compute∫∫∫W

xdxdydz ,

where W is the regionbounded by the planesx = 0, y = 0, and z = 2, andthe surface z = x2 + y2 andlying in the quadrantx ≥ 0, y ≥ 0.

N II IV

~ N

><

II

II

>< 0

N

+ ~

'<:

'<: II

Figure: From Vector Calculus,Marsden & Tromba


Triple integrals

Compute∫∫∫W

xdxdydz ,

where W is the regionbounded by the planesx = 0, y = 0, and z = 2, andthe surface z = x2 + y2 andlying in the quadrantx ≥ 0, y ≥ 0.

N II IV

~ N

><

II

II

>< 0

N

+ ~

'<:

'<: II

Figure: From Vector Calculus,Marsden & Tromba


How to compute integrations over an n-dimensional object?


An example

Zeros of a quintic polynomial:

Z 50 + Z 5

1 + Z 52 + Z 5

3 + Z 54 + 10Z0Z1Z2Z3Z4 = 0

in C5.


Figure: From Wikipedia, the intersection of the quintic Calabi-Yau threefold toour three dimensional space


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.




Use Linear Algebra






Use Linear Algebra






Use Linear Algebra




A simple example

1

2π

∮xdy − ydx

x2 + y2= 1.


A simple example

1

2π

∮xdy − ydx

x2 + y2= 1.


A non-trivial example

How many holes in the quintic Calabi-Yau manifold?


In 1977, S. T. Yau was able to solve the following PDE

det

g1,1̄ + ∂2u∂z1∂z̄1

g1,2̄ + ∂2u∂z1∂z̄2

g1,3̄ + ∂2u∂z1∂z̄3

g2,1̄ + ∂2u∂z2∂z̄1

g2,2̄ + ∂2u∂z2∂z̄2

g2,3̄ + ∂2u∂z2∂z̄3

g3,1̄ + ∂2u∂z3∂z̄1

g3,2̄ + ∂2u∂z3∂z̄2

g3,3̄ + ∂2u∂z3∂z̄3

= eF det

g1,1̄ g1,2̄ g1,3̄

g2,1̄ g2,2̄ g2,3̄

g3,1̄ g3,2̄ g3,3̄

,

where gi j̄ and F are given functions. After that, we are able to tell thetopological properties of the manifold.


Linear algebra

Hodge theory!


Linear algebra

Hodge theory!Zhiqin Lu, UC. Irvine What is Geometry 10/14

Conclusion:

Differential Geometry is not a separate math field, it brought differentfields like PDE, algebraic geometry, algebraic topology, Lie group theory,

functional analysis, and many others together.


UCI’s Geometry / TopologyGroup

Peter Li, Geometric PDE, Analysis on manifolds;

Chuu-Lian Terng, Differential Geometry, Integrable systems

Ronald J. Stern,Smooth 4-manifolds, symplectic contact topologyand geometry, knot theory

Zhiqin Lu, complex geometry, Mirror symmetry.



Peter Li,

Geometric PDE, Analysis on manifolds;













Chuu-Lian Terng,

Differential Geometry, Integrable systems













Ronald J. Stern,

Smooth 4-manifolds, symplectic contact topologyand geometry, knot theory













Zhiqin Lu,

complex geometry, Mirror symmetry.








http://math.uci.edu/˜ zlu/geom-topo-group/index.php

Members of the Geometry & Topology Group at UCI work in manydifferent fields and have expertise in a diverse set of techniques. We havelively and well-attended seminars, and one of our key goals is thecross-pollination of ideas between geometry and topology.Our faculty consists of active researchers in many areas of geometry andlow-dimensional topology including geometric PDE, differential geometry,integrable systems, mirror symmetry, smooth 4-manifolds, symplectic andcontact topology and geometry, and knot theory and its invariants.Our faculty is highly-regarded. All have NSF Grants and one of itsmembers, Peter Li, was elected to the American Academy of Arts andSciences. The Geometry/Topology Group at UCI has a long-standingcommitment to excellence in graduate and postdoctoral training: wehave produced some outstanding graduate students, and we have beenfortunate to have recruited and mentored exceptional postdoctoralfellows.


Welcome to UCI!


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What is Differential Geometry?