20:
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175:(when it is also known as "geometric PDE"), and the application of the theory of partial differential equations to geometry. It incorporates problems involving curves and surfaces, or domains with curved boundaries, but also the study of
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825:
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151:
launched a particularly exciting and productive era of geometric analysis that continues to this day. A celebrated achievement was the solution to the
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216:
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579:
386:
Groups and
Geometric Analysis (Integral Geometry, Invariant Differential Operators and Spherical Functions)
58:
621:
319:
The Ricci Flow in
Riemannian Geometry: A Complete Proof of the Differentiable 1/4-Pinching Sphere Theorem
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703:
206:
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521:
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is sometimes regarded as part of geometric analysis, because differential equations arising from
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159:, completing a program initiated and largely carried out by Richard Hamilton.
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The following is a partial list of major topics within geometric analysis:
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62:
23:
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70:
38:
1109:
241:
1157:
The
Unreasonable Effectiveness of Mathematics in the Natural Sciences
472:
105:
196:
192:
168:
102:
187:
have a strong geometric content. Geometric analysis also includes
69:
to study geometric and topological properties of spaces, such as
1212:
European
Community on Computational Methods in Applied Sciences
776:
594:
1207:
International
Council for Industrial and Applied Mathematics
195:, and the relationship between differential equations and
167:
The scope of geometric analysis includes both the use of
191:, which concerns the study of differential equations on
455:
108:
of
Riemannian manifolds into Euclidean space, work by
826:
Numerical methods for ordinary differential equations
272:
Founder of geometric analysis honored with Abel Prize
30:
are among the objects of study in geometric analysis.
1202:
Société de Mathématiques
Appliquées et Industrielles
1195:
Japan
Society for Industrial and Applied Mathematics
831:
Numerical methods for partial differential equations
545:
1018:
65:. More recently, it refers largely to the use of
1252:
411:(2nd ed.). American Mathematical Society.
1190:Society for Industrial and Applied Mathematics
441:
49:(PDEs), are used to establish new results in
1008:Supersymmetric theory of stochastic dynamics
131:. In the 1980s fundamental contributions by
61:elliptic PDEs dates at least as far back as
448:
434:
341:Riemannian geometry and Geometric Analysis
85:. This approach dates back to the work by
287:
406:
380:
67:nonlinear partial differential equations
18:
313:
47:elliptic partial differential equations
1253:
409:Geometric Analysis on Symmetric Spaces
429:
335:
360:Manifolds and Differential Geometry
357:
13:
457:Industrial and applied mathematics
281:
116:and the Weyl problem, and work by
14:
1282:
687:Stochastic differential equations
299:. International Press of Boston.
297:Lectures on Differential Geometry
1003:Supersymmetric quantum mechanics
118:Aleksandr Danilovich Aleksandrov
885:Stochastic variational calculus
677:Ordinary differential equations
682:Partial differential equations
555:Arbitrary-precision arithmetic
264:
173:partial differential equations
1:
570:Interactive geometry software
390:American Mathematical Society
364:American Mathematical Society
257:
179:in arbitrary dimension. The
41:discipline where tools from
7:
622:Computational number theory
585:Numerical-analysis software
407:Helgason, Sigurdur (2008).
16:Field of higher mathematics
10:
1287:
343:(4th ed.). Springer.
321:(1st ed.). Springer.
1220:
1182:
1166:
1128:
1080:
1028:Algebra of physical space
893:
851:
645:
607:
495:Automated theorem proving
463:
821:Numerical linear algebra
358:Lee, Jeffrey M. (2009).
274:Retrieved 20 March 2019.
270:Jackson, Allyn. (2019).
237:Pseudoholomorphic curves
232:Positive energy theorems
171:methods in the study of
162:
560:Finite element analysis
510:Constraint satisfaction
217:Kähler–Einstein metrics
1271:Differential equations
1266:Calculus of variations
1115:Mathematical economics
789:Multivariable calculus
672:Differential equations
515:Constraint programming
505:Computational geometry
185:variational principles
181:calculus of variations
43:differential equations
31:
1261:Differential geometry
1068:Supersymmetry algebra
1053:Representation theory
1048:Renormalization group
694:Differential geometry
575:Optimization software
547:Mathematical software
55:differential topology
51:differential geometry
22:
1120:Mathematical finance
1105:Social choice theory
1020:Algebraic structures
969:in quantum mechanics
905:Analytical mechanics
871:Stochastic processes
843:Variational calculus
655:Approximation theory
580:Statistical software
252:Yang–Mills equations
227:Minimal submanifolds
177:Riemannian manifolds
99:John Forbes Nash Jr.
83:symplectic manifolds
79:Riemannian manifolds
1095:Operations research
964:Perturbation theory
762:Multilinear algebra
733:Functional analysis
590:Numerical libraries
522:Computational logic
222:Mean curvature flow
153:Poincaré conjecture
1232:Mathematics portal
1129:Other applications
853:Probability theory
836:Validated numerics
816:Numerical analysis
709:Geometric analysis
699:Differential forms
532:Information theory
382:Helgason, Sigurdur
35:Geometric analysis
32:
1248:
1247:
1082:Decision sciences
1076:
1075:
1058:Spacetime algebra
750:Harmonic analysis
716:Dynamical systems
660:Clifford analysis
637:Discrete geometry
603:
602:
418:978-0-8218-4530-1
399:978-0-8218-2673-7
373:978-0-8218-4815-9
350:978-3-540-25907-7
328:978-3-642-16285-5
306:978-1-571-46198-8
122:Aleksei Pogorelov
114:Minkowski problem
26:minimal surface.
1278:
1033:Feynman integral
1016:
1015:
976:Potential theory
865:random variables
755:Fourier analysis
738:Operator algebra
665:Clifford algebra
617:Computer algebra
543:
542:
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443:
436:
427:
426:
422:
403:
388:(2nd ed.).
377:
354:
332:
310:
275:
268:
157:Grigori Perelman
149:Richard Hamilton
95:minimal surfaces
28:Minimal surfaces
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1038:Poisson algebra
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743:Operator theory
641:
599:
565:Tensor software
541:
490:Automata theory
459:
454:
419:
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329:
307:
293:Yau, Shing Tung
289:Schoen, Richard
284:
282:Further reading
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189:global analysis
165:
137:Clifford Taubes
133:Karen Uhlenbeck
110:Louis Nirenberg
75:Euclidean space
17:
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726:Control theory
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247:Yamabe problem
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145:Richard Schoen
141:Shing-Tung Yau
15:
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1043:Quantum group
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998:Supersymmetry
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981:String theory
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880:Path integral
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861:Distributions
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633:
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628:
627:Combinatorics
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500:Coding theory
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475:
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465:Computational
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213:
212:Harmonic maps
210:
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186:
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129:hypersurfaces
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111:
107:
104:
100:
96:
92:
91:Jesse Douglas
88:
84:
80:
76:
72:
68:
64:
60:
57:. The use of
56:
52:
48:
45:, especially
44:
40:
36:
29:
25:
21:
1238: /
1234: /
1224:
1100:Optimization
1063:Superalgebra
922:Field theory
895:Mathematical
873: /
721:Chaos theory
708:
704:Gauge theory
632:Graph theory
527:Cryptography
408:
385:
359:
340:
337:Jost, Jürgen
318:
315:Andrews, Ben
296:
266:
207:Gauge theory
201:
166:
71:submanifolds
63:Hodge theory
39:mathematical
34:
33:
24:Saddle tower
1240:topics list
1174:Mathematics
1090:Game theory
991:Topological
957:Topological
952:Statistical
915:Hamiltonian
169:geometrical
1255:Categories
1146:Psychology
1110:Statistics
910:Lagrangian
537:Statistics
473:Algorithms
258:References
242:Ricci flow
106:embeddings
87:Tibor Radó
1151:Sociology
1141:Chemistry
937:Effective
932:Conformal
927:Classical
799:Geometric
772:Geometric
193:manifolds
103:isometric
1226:Category
875:analysis
794:Exterior
767:Exterior
647:Analysis
609:Discrete
483:analysis
384:(2000).
339:(2005).
317:(2010).
295:(2010).
197:topology
1236:outline
1167:Related
1136:Biology
986:Bosonic
947:Quantum
897:physics
863: (
595:Solvers
112:on the
809:Vector
804:Tensor
782:Vector
777:Tensor
478:design
415:
396:
370:
347:
325:
303:
147:, and
126:convex
81:, and
59:linear
942:Gauge
163:Scope
37:is a
413:ISBN
394:ISBN
368:ISBN
345:ISBN
323:ISBN
301:ISBN
120:and
89:and
53:and
155:by
124:on
101:on
93:on
73:of
1257::
392:.
366:.
362:.
291:;
199:.
143:,
139:,
135:,
97:,
77:,
1159:"
1155:"
867:)
449:e
442:t
435:v
421:.
402:.
376:.
353:.
331:.
309:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
