what is differential geometry?
TRANSCRIPT
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
Zhiqin Lu, UC. Irvine What is Geometry 2/14
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
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:
Z 50 + Z 5
1 + Z 52 + Z 5
3 + Z 54 + 10Z0Z1Z2Z3Z4 = 0
in C5.
Zhiqin Lu, UC. Irvine What is Geometry 4/14
Figure: From Wikipedia, the intersection of the quintic Calabi-Yau threefold toour three dimensional space
Zhiqin Lu, UC. Irvine What is Geometry 5/14
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
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
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
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
A simple example
1
2π
∮xdy − ydx
x2 + y2= 1.
Zhiqin Lu, UC. Irvine What is Geometry 7/14
A simple example
1
2π
∮xdy − ydx
x2 + y2= 1.
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
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.
Zhiqin Lu, UC. Irvine What is Geometry 9/14
Linear algebra
Hodge theory!
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 differentfields like PDE, algebraic geometry, algebraic topology, Lie group theory,
functional analysis, and many others together.
Zhiqin Lu, UC. Irvine What is Geometry 11/14
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.
Zhiqin Lu, UC. Irvine What is Geometry 12/14
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.
Zhiqin Lu, UC. Irvine What is Geometry 12/14
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.
Zhiqin Lu, UC. Irvine What is Geometry 12/14
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.
Zhiqin Lu, UC. Irvine What is Geometry 12/14
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.
Zhiqin Lu, UC. Irvine What is Geometry 12/14
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.
Zhiqin Lu, UC. Irvine What is Geometry 12/14
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.
Zhiqin Lu, UC. Irvine What is Geometry 12/14
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.
Zhiqin Lu, UC. Irvine What is Geometry 12/14
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.
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 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.
Zhiqin Lu, UC. Irvine What is Geometry 13/14
Welcome to UCI!
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