![Page 1: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/1.jpg)
Singularities
Sándor Kovács
May 1, 2007
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![Page 3: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/3.jpg)
First Impressions
Bolyai: hyperbolic geometry.
Galois: solving equations, group theory, field extensions.
Neither Bolyai nor Galois was recognized by theircontemporaries.
Galois died at a very young age (21).
![Page 4: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/4.jpg)
First Impressions
Pierre de Fermat (1601 - 1665)
Bolyai: hyperbolic geometry.
Galois: solving equations, group theory, field extensions.
Neither Bolyai nor Galois was recognized by theircontemporaries.
Galois died at a very young age (21).
![Page 5: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/5.jpg)
First Impressions
Fermat:an + bn = cn
for n ≥ 3 has no solution with a, b, c non-zero integers.
Bolyai: hyperbolic geometry.
Galois: solving equations, group theory, field extensions.
Neither Bolyai nor Galois was recognized by theircontemporaries.
Galois died at a very young age (21).
![Page 6: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/6.jpg)
First Impressions
Bolyai, János (1802 - 1860)
Bolyai: hyperbolic geometry.
Galois: solving equations, group theory, field extensions.
Neither Bolyai nor Galois was recognized by theircontemporaries.
Galois died at a very young age (21).
![Page 7: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/7.jpg)
First Impressions
Fermat:an + bn = cn
for n ≥ 3 has no solution with a, b, c non-zero integers.
Bolyai: hyperbolic geometry.
Galois: solving equations, group theory, field extensions.
Neither Bolyai nor Galois was recognized by theircontemporaries.
Galois died at a very young age (21).
![Page 8: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/8.jpg)
First Impressions
Évariste Galois (1811 - 1832)
Bolyai: hyperbolic geometry.
Galois: solving equations, group theory, field extensions.
Neither Bolyai nor Galois was recognized by theircontemporaries.
Galois died at a very young age (21).
![Page 9: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/9.jpg)
First Impressions
Fermat:an + bn = cn
for n ≥ 3 has no solution with a, b, c non-zero integers.
Bolyai: hyperbolic geometry.
Galois: solving equations, group theory, field extensions.
Neither Bolyai nor Galois was recognized by theircontemporaries.
Galois died at a very young age (21).
![Page 10: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/10.jpg)
First Impressions
Fermat:an + bn = cn
for n ≥ 3 has no solution with a, b, c non-zero integers.
Bolyai: hyperbolic geometry.
Galois: solving equations, group theory, field extensions.
Neither Bolyai nor Galois was recognized by theircontemporaries.
Galois died at a very young age (21).
![Page 11: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/11.jpg)
First Impressions
Fermat:an + bn = cn
for n ≥ 3 has no solution with a, b, c non-zero integers.
Bolyai: hyperbolic geometry.
Galois: solving equations, group theory, field extensions.
Neither Bolyai nor Galois was recognized by theircontemporaries.Galois died at a very young age (21).
![Page 12: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/12.jpg)
Puzzles
A snail has fallen in a well that’s 50 feet deep. Everyday it climbs up 10 feet, but then it takes a nap andslides back 9 feet. How long does it take for the snail toget out of the well? 41 days.
(15 +√
220)2007 = . . . ?�? . . .
The Japanese kindergarten entry exam...
![Page 13: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/13.jpg)
Puzzles
A snail has fallen in a well that’s 50 feet deep. Everyday it climbs up 10 feet, but then it takes a nap andslides back 9 feet. How long does it take for the snail toget out of the well?
41 days.
(15 +√
220)2007 = . . . ?�? . . .
The Japanese kindergarten entry exam...
![Page 14: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/14.jpg)
Puzzles
A snail has fallen in a well that’s 50 feet deep. Everyday it climbs up 10 feet, but then it takes a nap andslides back 9 feet. How long does it take for the snail toget out of the well? 41 days.
(15 +√
220)2007 = . . . ?�? . . .
The Japanese kindergarten entry exam...
![Page 15: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/15.jpg)
Puzzles
A snail has fallen in a well that’s 50 feet deep. Everyday it climbs up 10 feet, but then it takes a nap andslides back 9 feet. How long does it take for the snail toget out of the well? 41 days.
(15 +√
220)2007 = . . . ?�? . . .
The Japanese kindergarten entry exam...
![Page 16: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/16.jpg)
Puzzles
A snail has fallen in a well that’s 50 feet deep. Everyday it climbs up 10 feet, but then it takes a nap andslides back 9 feet. How long does it take for the snail toget out of the well? 41 days.
(15 +√
220)2007 = . . . 9�9 . . .
The Japanese kindergarten entry exam...
![Page 17: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/17.jpg)
Puzzles
A snail has fallen in a well that’s 50 feet deep. Everyday it climbs up 10 feet, but then it takes a nap andslides back 9 feet. How long does it take for the snail toget out of the well? 41 days.
(15 +√
220)2007 = . . . 9�9 . . .
The Japanese kindergarten entry exam...
![Page 18: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/18.jpg)
The Bus Puzzle
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The Bus Puzzle
![Page 20: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/20.jpg)
A Few Good Hungarians
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A Few Good Hungarians
Riesz, Frigyes (1880 - 1956)
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A Few Good Hungarians
Fejér, Lipót (1880 - 1959)
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A Few Good Hungarians
Haar, Alfréd (1885 - 1933)
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A Few Good Hungarians
Neumann, János (1903 - 1957)
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A Few Good Hungarians
Péter, Rózsa (1905 - 1977)
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A Few Good Hungarians
Erdős, Pál (1913 - 1996)
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A Few Good Hungarians
Bott, Raoul (1923 - 2005)
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...and a few more
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...and a few more
Lax, Péter (1926 - )
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...and a few more
Lovász, László (1948 - )
![Page 31: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/31.jpg)
...and a few more
Kollár, János (1956 - )
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Mathematical Impressions
axiomatic geometryhyperbolic geometryprojective geometryfinite geometryabstract algebra
group theory(finite simple groups)
commutative algebra
algebraic
geometry
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Mathematical Impressions
axiomatic geometry
hyperbolic geometryprojective geometryfinite geometryabstract algebra
group theory(finite simple groups)
commutative algebra
algebraic
geometry
![Page 34: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/34.jpg)
Mathematical Impressions
axiomatic geometryhyperbolic geometry
projective geometryfinite geometryabstract algebra
group theory(finite simple groups)
commutative algebra
algebraic
geometry
![Page 35: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/35.jpg)
Mathematical Impressions
axiomatic geometryhyperbolic geometryprojective geometry
finite geometryabstract algebra
group theory(finite simple groups)
commutative algebra
algebraic
geometry
![Page 36: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/36.jpg)
Mathematical Impressions
axiomatic geometryhyperbolic geometryprojective geometryfinite geometry
abstract algebragroup theory
(finite simple groups)commutative algebra
algebraic
geometry
![Page 37: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/37.jpg)
Mathematical Impressions
axiomatic geometryhyperbolic geometryprojective geometryfinite geometryabstract algebra
group theory(finite simple groups)
commutative algebra
algebraic
geometry
![Page 38: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/38.jpg)
Mathematical Impressions
axiomatic geometryhyperbolic geometryprojective geometryfinite geometryabstract algebra
group theory(finite simple groups)
commutative algebra
algebraic
geometry
![Page 39: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/39.jpg)
Mathematical Impressions
axiomatic geometryhyperbolic geometryprojective geometryfinite geometryabstract algebra
group theory(finite simple groups)
commutative algebra
algebraic
geometry
![Page 40: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/40.jpg)
Mathematical Impressions
axiomatic geometryhyperbolic geometryprojective geometryfinite geometryabstract algebra
group theory(finite simple groups)
commutative algebra
algebraic
geometry
![Page 41: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/41.jpg)
Mathematical Impressions
axiomatic geometryhyperbolic geometryprojective geometryfinite geometryabstract algebra
group theory(finite simple groups)
commutative algebra
algebraicgeometry
![Page 42: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/42.jpg)
Mathematical Impressions
axiomatic geometryhyperbolic geometryprojective geometryfinite geometryabstract algebra
group theory(finite simple groups)
commutative algebra
algebraic
geometry
![Page 43: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/43.jpg)
Mathematical Impressions
axiomatic geometryhyperbolic geometryprojective geometryfinite geometryabstract algebra
group theory(finite simple groups)
commutative algebra
algebraic
geometry
![Page 44: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/44.jpg)
First Papers
First paper → took 5 years to get published.
Second paper → Erdős# = 2
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First Papers
First paper
→ took 5 years to get published.
Second paper → Erdős# = 2
![Page 46: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/46.jpg)
First Papers
First paper → took 5 years to get published.
Second paper → Erdős# = 2
![Page 47: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/47.jpg)
First Papers
First paper → took 5 years to get published.
Second paper
→ Erdős# = 2
![Page 48: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/48.jpg)
First Papers
First paper → took 5 years to get published.
Second paper → Erdős# = 2
![Page 49: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/49.jpg)
Erdős #Definition
Erdős’s Erdős# = 0,Anyone, who published a research paper with Erdős hasErdős# = 1,
Anyone, who published a research paper with someonewho has Erdős# = 1, has Erdős# = 2, etc.Someone’s Erdős# is n if they published a researchpaper with someone who has Erdős# = n − 1, buthave not published paper with anyone who hasErdős# < n − 1.
ExampleMy Erdős# = 2.
![Page 50: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/50.jpg)
Erdős #Definition
Erdős’s Erdős# = 0,
Anyone, who published a research paper with Erdős hasErdős# = 1,
Anyone, who published a research paper with someonewho has Erdős# = 1, has Erdős# = 2, etc.Someone’s Erdős# is n if they published a researchpaper with someone who has Erdős# = n − 1, buthave not published paper with anyone who hasErdős# < n − 1.
ExampleMy Erdős# = 2.
![Page 51: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/51.jpg)
Erdős #Definition
Erdős’s Erdős# = 0,Anyone, who published a research paper with Erdős hasErdős# = 1,
Anyone, who published a research paper with someonewho has Erdős# = 1, has Erdős# = 2, etc.Someone’s Erdős# is n if they published a researchpaper with someone who has Erdős# = n − 1, buthave not published paper with anyone who hasErdős# < n − 1.
ExampleMy Erdős# = 2.
![Page 52: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/52.jpg)
Erdős #Definition
Erdős’s Erdős# = 0,Anyone, who published a research paper with Erdős hasErdős# = 1,
Anyone, who published a research paper with someonewho has Erdős# = 1, has Erdős# = 2, etc.
Someone’s Erdős# is n if they published a researchpaper with someone who has Erdős# = n − 1, buthave not published paper with anyone who hasErdős# < n − 1.
ExampleMy Erdős# = 2.
![Page 53: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/53.jpg)
Erdős #Definition
Erdős’s Erdős# = 0,Anyone, who published a research paper with Erdős hasErdős# = 1,
Anyone, who published a research paper with someonewho has Erdős# = 1, has Erdős# = 2, etc.Someone’s Erdős# is n if they published a researchpaper with someone who has Erdős# = n − 1, buthave not published paper with anyone who hasErdős# < n − 1.
ExampleMy Erdős# = 2.
![Page 54: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/54.jpg)
Erdős #Definition
Erdős’s Erdős# = 0,Anyone, who published a research paper with Erdős hasErdős# = 1,
Anyone, who published a research paper with someonewho has Erdős# = 1, has Erdős# = 2, etc.Someone’s Erdős# is n if they published a researchpaper with someone who has Erdős# = n − 1, buthave not published paper with anyone who hasErdős# < n − 1.
ExampleMy Erdős# = 2.
![Page 55: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/55.jpg)
Thesis
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Thesis
“Hey, man, what’s yourthesis about?”
![Page 57: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/57.jpg)
Thesis
“Well, it has something to do withhow the universe is changing
through time, and it states thateither the universe does not changeat all, or there must be times when
black holes exist.”
(This is, of course, a very loose andnon-rigorous interpretation.)
![Page 58: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/58.jpg)
Thesis
“Well, it has something to do withhow the universe is changing
through time, and it states thateither the universe does not changeat all, or there must be times when
black holes exist.”(This is, of course, a very loose and
non-rigorous interpretation.)
![Page 59: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/59.jpg)
Thesis
“Well, it has something to do withhow the universe is changing
through time, and it states thateither the universe does not changeat all, or there must be times when
singularities exist.”
(This is, of course, a very loose andnon-rigorous interpretation.)
![Page 60: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/60.jpg)
Thesis
My thesis through Lun Yi’s eyes
![Page 61: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/61.jpg)
Advertisement
![Page 62: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/62.jpg)
Conics
![Page 63: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/63.jpg)
Conics
“This may not be yourthesis, but this Iunderstand.”
![Page 64: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/64.jpg)
Conics
“What are conics, andwhy are they called
“conics”?”
![Page 65: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/65.jpg)
Conics
ellipse
![Page 66: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/66.jpg)
Conics
parabola
![Page 67: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/67.jpg)
Conics
hyperbola
![Page 68: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/68.jpg)
Conics
degenerate
![Page 69: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/69.jpg)
Conics
ellipse
![Page 70: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/70.jpg)
Conics
parabola
![Page 71: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/71.jpg)
Conics
hyperbola
![Page 72: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/72.jpg)
Conics
degenerate
![Page 73: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/73.jpg)
Conics
DEFORMATIONS
![Page 74: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/74.jpg)
Conics
deformations
![Page 75: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/75.jpg)
Conics
deformations
![Page 76: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/76.jpg)
Conics
INTERSECTIONS
![Page 77: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/77.jpg)
Conics
intersections
![Page 78: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/78.jpg)
Conics
intersections
![Page 79: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/79.jpg)
Conics
intersections
![Page 80: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/80.jpg)
Cones
SINGULARITIES
![Page 81: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/81.jpg)
Singularities: cone
![Page 82: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/82.jpg)
Singularities: 2 lines vs. 1 line
![Page 83: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/83.jpg)
Singularities: deformation
![Page 84: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/84.jpg)
Singularities: ∼2 lines
![Page 85: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/85.jpg)
Singularities: 2 lines vs. 1 line
![Page 86: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/86.jpg)
Weird
1 line through the vertex intersects 2 lines through thevertex in only 1 point.
In how many point does 1 line intersect another (1)line? 1/2
![Page 87: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/87.jpg)
Weird
1 line through the vertex intersects 2 lines through thevertex in only 1 point.In how many point does 1 line intersect another (1)line?
1/2
![Page 88: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/88.jpg)
Weird
1 line through the vertex intersects 2 lines through thevertex in only 1 point.In how many point does 1 line intersect another (1)line? 1/2
![Page 89: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/89.jpg)
Non-singularcase
![Page 90: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/90.jpg)
Non-singularcase smoothing
![Page 91: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/91.jpg)
Non-singularcase 2 lines vs. 1 line
![Page 92: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/92.jpg)
Non-singularcase deformation
![Page 93: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/93.jpg)
Non-singularcase ∼2 lines
![Page 94: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/94.jpg)
Non-singularcase 2 lines vs. 1 line
![Page 95: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/95.jpg)
ResearchFermat-Wiles:
an + bn = cn n ≥ 3
has no solution with a, b, c non-zero integers.
Reformulation:(ac
)n
+
(bc
)n
= 1 n ≥ 3
has no solution with a, b, c non-zero integers.Question: If f (x , y) is a polynomial in x , y of degree≥ 3 with integer coefficients, does
f (x , y) = 0
have no solution with x , y non-zero rational numbers?
![Page 96: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/96.jpg)
ResearchFermat-Wiles:
an + bn = cn n ≥ 3
has no solution with a, b, c non-zero integers.Reformulation:(
ac
)n
+
(bc
)n
= 1 n ≥ 3
has no solution with a, b, c non-zero integers.
Question: If f (x , y) is a polynomial in x , y of degree≥ 3 with integer coefficients, does
f (x , y) = 0
have no solution with x , y non-zero rational numbers?
![Page 97: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/97.jpg)
ResearchFermat-Wiles:
an + bn = cn n ≥ 3
has no solution with a, b, c non-zero integers.Reformulation:
xn + yn = 1 n ≥ 3
has no solution with x , y non-zero rational numbers.
Question: If f (x , y) is a polynomial in x , y of degree≥ 3 with integer coefficients, does
f (x , y) = 0
have no solution with x , y non-zero rational numbers?
![Page 98: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/98.jpg)
ResearchFermat-Wiles:
an + bn = cn n ≥ 3
has no solution with a, b, c non-zero integers.Reformulation:
xn + yn = 1 n ≥ 3
has no solution with x , y non-zero rational numbers.Question: If f (x , y) is a polynomial in x , y of degree≥ 3 with integer coefficients, does
f (x , y) = 0
have no solution with x , y non-zero rational numbers?
![Page 99: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/99.jpg)
Geometry
The equation f (x , y) = 0 defines a curve on the plane:
A solution with x , y rational numbers corresponds to a pointon the curve with rational coordinates.
![Page 100: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/100.jpg)
Geometry
The equation f (x , y) = 0 defines a curve on the plane:
A solution with x , y rational numbers corresponds to a pointon the curve with rational coordinates.
![Page 101: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/101.jpg)
Geometry
The equation f (x , y) = 0 defines a curve on the plane:
A solution with x , y rational numbers corresponds to a pointon the curve with rational coordinates.
![Page 102: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/102.jpg)
Geometry
The equation f (x , y) = 0 defines a curve on the plane:
A solution with x , y rational numbers corresponds to a pointon the curve with rational coordinates.
![Page 103: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/103.jpg)
Geometry
The equation f (x , y) = 0 defines a curve on the plane:
A solution with x , y rational numbers corresponds to a pointon the curve with rational coordinates.
![Page 104: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/104.jpg)
Geometry
The equation f (x , y) = 0 defines a curve on the plane:
A solution with x , y rational numbers corresponds to a pointon the curve with rational coordinates.
![Page 105: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/105.jpg)
From Arithmetic to Geometry
The original Fermat problem asks to find integersolutions to equations with integer coefficients.
Replace “integer” with “multiple of t”.Consider equations with a free parameter t and askwhether there are solutions among “multiples of t”.Example: Consider the equation
y 2 − x5 +
(
5t
)4
x − 4t
4
= 0.
Are there solutions that can be expressed aspolynomials of t? Let x = 5t and y = 2t2.
![Page 106: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/106.jpg)
From Arithmetic to Geometry
The original Fermat problem asks to find integersolutions to equations with integer coefficients.Replace “integer” with “multiple of t”.
Consider equations with a free parameter t and askwhether there are solutions among “multiples of t”.Example: Consider the equation
y 2 − x5 +
(
5t
)4
x − 4t
4
= 0.
Are there solutions that can be expressed aspolynomials of t? Let x = 5t and y = 2t2.
![Page 107: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/107.jpg)
From Arithmetic to Geometry
The original Fermat problem asks to find integersolutions to equations with integer coefficients.Replace “integer” with “multiple of t”.Consider equations with a free parameter t and askwhether there are solutions among “multiples of t”.
Example: Consider the equation
y 2 − x5 +
(
5t
)4
x − 4t
4
= 0.
Are there solutions that can be expressed aspolynomials of t? Let x = 5t and y = 2t2.
![Page 108: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/108.jpg)
From Arithmetic to Geometry
The original Fermat problem asks to find integersolutions to equations with integer coefficients.Replace “integer” with “multiple of t”.Consider equations with a free parameter t and askwhether there are solutions among “multiples of t”.Example: Consider the equation
y 2 − x5 +
(
5t
)4
x − 4t
4
= 0.
Are there solutions that can be expressed aspolynomials of t? Let x = 5t and y = 2t2.
![Page 109: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/109.jpg)
From Arithmetic to Geometry
The original Fermat problem asks to find integersolutions to equations with integer coefficients.Replace “integer” with “multiple of t”.Consider equations with a free parameter t and askwhether there are solutions among “multiples of t”.Example: Consider the equation
y 2 − x5 +
(
5t
)4
x − 4t
4
= 0.
Are there solutions that can be expressed aspolynomials of t?
Let x = 5t and y = 2t2.
![Page 110: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/110.jpg)
From Arithmetic to Geometry
The original Fermat problem asks to find integersolutions to equations with integer coefficients.Replace “integer” with “multiple of t”.Consider equations with a free parameter t and askwhether there are solutions among “multiples of t”.Example: Consider the equation
y 2 − x5 + (5t)4x − 4t4 = 0.
Are there solutions that can be expressed aspolynomials of t?
Let x = 5t and y = 2t2.
![Page 111: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/111.jpg)
From Arithmetic to Geometry
The original Fermat problem asks to find integersolutions to equations with integer coefficients.Replace “integer” with “multiple of t”.Consider equations with a free parameter t and askwhether there are solutions among “multiples of t”.Example: Consider the equation
y 2 − x5 + (5t)4x − 4t4 = 0.
Are there solutions that can be expressed aspolynomials of t? Let x = 5t and y = 2t2.
![Page 112: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/112.jpg)
Mordell Conjecture
By changing the parametrization there will always be asolution.
There is still something interesting to ask:
Question: Is it true that for any given parametrizationthere are only finitely many solutions (in t)?
This is known as Mordell’s Conjecture and wasconfirmed by Manin in 1963.
![Page 113: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/113.jpg)
Mordell Conjecture
By changing the parametrization there will always be asolution.
There is still something interesting to ask:
Question: Is it true that for any given parametrizationthere are only finitely many solutions (in t)?
This is known as Mordell’s Conjecture and wasconfirmed by Manin in 1963.
![Page 114: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/114.jpg)
Mordell Conjecture
By changing the parametrization there will always be asolution.
There is still something interesting to ask:
Question: Is it true that for any given parametrizationthere are only finitely many solutions (in t)?
This is known as Mordell’s Conjecture and wasconfirmed by Manin in 1963.
![Page 115: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/115.jpg)
Mordell Conjecture
By changing the parametrization there will always be asolution.
There is still something interesting to ask:
Question: Is it true that for any given parametrizationthere are only finitely many solutions (in t)?
This is known as Mordell’s Conjecture and wasconfirmed by Manin in 1963.
![Page 116: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/116.jpg)
Shafarevich’s Conjecture
In 1968 Parshin realized that this is related to anotherconjecture made by Shafarevich in 1962.
The connection is somewhat tricky, but the point is thatinstead of looking for solutions (in t) the questionfocuses on studying the “total space” of the curves:
As t varies, there is a curve Ct in the plane that isdefined by the equation f (x , y , t) = 0. The union ofthese curves form a surface, which is “fibred over thet-line”:
![Page 117: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/117.jpg)
Shafarevich’s Conjecture
In 1968 Parshin realized that this is related to anotherconjecture made by Shafarevich in 1962.
The connection is somewhat tricky, but the point is thatinstead of looking for solutions (in t) the questionfocuses on studying the “total space” of the curves:
As t varies, there is a curve Ct in the plane that isdefined by the equation f (x , y , t) = 0. The union ofthese curves form a surface, which is “fibred over thet-line”:
![Page 118: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/118.jpg)
Shafarevich’s Conjecture
In 1968 Parshin realized that this is related to anotherconjecture made by Shafarevich in 1962.
The connection is somewhat tricky, but the point is thatinstead of looking for solutions (in t) the questionfocuses on studying the “total space” of the curves:
As t varies, there is a curve Ct in the plane that isdefined by the equation f (x , y , t) = 0. The union ofthese curves form a surface, which is “fibred over thet-line”:
![Page 119: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/119.jpg)
Shafarevich’s Conjecture
![Page 120: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/120.jpg)
Shafarevich’s Conjecture
![Page 121: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/121.jpg)
Shafarevich’s Conjecture
Shafarevich’s Conjecture says that under certain(well-defined) conditions there are only finitely manyfamilies satisfying the conditions.
Parshin’s trick shows that Shafarevich’s Conjectureimplies Mordell’s Conjecture.
Parshin and Arakelov proved Shafarevich’s Conjecture in1968− 1971.
![Page 122: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/122.jpg)
Shafarevich’s Conjecture
Shafarevich’s Conjecture says that under certain(well-defined) conditions there are only finitely manyfamilies satisfying the conditions.
Parshin’s trick shows that Shafarevich’s Conjectureimplies Mordell’s Conjecture.
Parshin and Arakelov proved Shafarevich’s Conjecture in1968− 1971.
![Page 123: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/123.jpg)
Shafarevich’s Conjecture
Shafarevich’s Conjecture says that under certain(well-defined) conditions there are only finitely manyfamilies satisfying the conditions.
Parshin’s trick shows that Shafarevich’s Conjectureimplies Mordell’s Conjecture.
Parshin and Arakelov proved Shafarevich’s Conjecture in1968− 1971.
![Page 124: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/124.jpg)
Higher dimensional ShafarevichConjecture
The problem of parametrized families make sense inhigher dimensions:
One may studyFamilies of higher dimensional objects (surfaces,threefolds, etc.)Families with more than one parameter.
The problem becomes much more complicated and so itis broken up into three subproblems:
RigidityBoundednessHyperbolicity
![Page 125: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/125.jpg)
Higher dimensional ShafarevichConjecture
The problem of parametrized families make sense inhigher dimensions: One may study
Families of higher dimensional objects (surfaces,threefolds, etc.)Families with more than one parameter.
The problem becomes much more complicated and so itis broken up into three subproblems:
RigidityBoundednessHyperbolicity
![Page 126: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/126.jpg)
Higher dimensional ShafarevichConjecture
The problem of parametrized families make sense inhigher dimensions: One may study
Families of higher dimensional objects (surfaces,threefolds, etc.)
Families with more than one parameter.
The problem becomes much more complicated and so itis broken up into three subproblems:
RigidityBoundednessHyperbolicity
![Page 127: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/127.jpg)
Higher dimensional ShafarevichConjecture
The problem of parametrized families make sense inhigher dimensions: One may study
Families of higher dimensional objects (surfaces,threefolds, etc.)Families with more than one parameter.
The problem becomes much more complicated and so itis broken up into three subproblems:
RigidityBoundednessHyperbolicity
![Page 128: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/128.jpg)
Higher dimensional ShafarevichConjecture
The problem of parametrized families make sense inhigher dimensions: One may study
Families of higher dimensional objects (surfaces,threefolds, etc.)Families with more than one parameter.
The problem becomes much more complicated and so itis broken up into three subproblems:
RigidityBoundednessHyperbolicity
![Page 129: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/129.jpg)
Higher dimensional ShafarevichConjecture
The problem of parametrized families make sense inhigher dimensions: One may study
Families of higher dimensional objects (surfaces,threefolds, etc.)Families with more than one parameter.
The problem becomes much more complicated and so itis broken up into three subproblems:
Rigidity
BoundednessHyperbolicity
![Page 130: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/130.jpg)
Higher dimensional ShafarevichConjecture
The problem of parametrized families make sense inhigher dimensions: One may study
Families of higher dimensional objects (surfaces,threefolds, etc.)Families with more than one parameter.
The problem becomes much more complicated and so itis broken up into three subproblems:
RigidityBoundedness
Hyperbolicity
![Page 131: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/131.jpg)
Higher dimensional ShafarevichConjecture
The problem of parametrized families make sense inhigher dimensions: One may study
Families of higher dimensional objects (surfaces,threefolds, etc.)Families with more than one parameter.
The problem becomes much more complicated and so itis broken up into three subproblems:
RigidityBoundednessHyperbolicity
![Page 132: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/132.jpg)
Higher dimensional results
Rigidity
Viehweg-Zuo (2002)Kovács (2002)
BoundednessBedulev-Viehweg (2000)Viehweg-Zuo (2002), (2003)Kovács-Lieblich (2006)
HyperbolicityMigliorini (1995)Kovács (1996), (1997), (2000)Viehweg-Zuo (2002), (2003)Kovács (2003)Kovács-Kebekus (2005), (2007)
![Page 133: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/133.jpg)
Higher dimensional results
RigidityViehweg-Zuo (2002)
Kovács (2002)Boundedness
Bedulev-Viehweg (2000)Viehweg-Zuo (2002), (2003)Kovács-Lieblich (2006)
HyperbolicityMigliorini (1995)Kovács (1996), (1997), (2000)Viehweg-Zuo (2002), (2003)Kovács (2003)Kovács-Kebekus (2005), (2007)
![Page 134: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/134.jpg)
Higher dimensional results
RigidityViehweg-Zuo (2002)Kovács (2002)
BoundednessBedulev-Viehweg (2000)Viehweg-Zuo (2002), (2003)Kovács-Lieblich (2006)
HyperbolicityMigliorini (1995)Kovács (1996), (1997), (2000)Viehweg-Zuo (2002), (2003)Kovács (2003)Kovács-Kebekus (2005), (2007)
![Page 135: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/135.jpg)
Higher dimensional results
RigidityViehweg-Zuo (2002)Kovács (2002)
Boundedness
Bedulev-Viehweg (2000)Viehweg-Zuo (2002), (2003)Kovács-Lieblich (2006)
HyperbolicityMigliorini (1995)Kovács (1996), (1997), (2000)Viehweg-Zuo (2002), (2003)Kovács (2003)Kovács-Kebekus (2005), (2007)
![Page 136: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/136.jpg)
Higher dimensional results
RigidityViehweg-Zuo (2002)Kovács (2002)
BoundednessBedulev-Viehweg (2000)
Viehweg-Zuo (2002), (2003)Kovács-Lieblich (2006)
HyperbolicityMigliorini (1995)Kovács (1996), (1997), (2000)Viehweg-Zuo (2002), (2003)Kovács (2003)Kovács-Kebekus (2005), (2007)
![Page 137: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/137.jpg)
Higher dimensional results
RigidityViehweg-Zuo (2002)Kovács (2002)
BoundednessBedulev-Viehweg (2000)Viehweg-Zuo (2002), (2003)
Kovács-Lieblich (2006)Hyperbolicity
Migliorini (1995)Kovács (1996), (1997), (2000)Viehweg-Zuo (2002), (2003)Kovács (2003)Kovács-Kebekus (2005), (2007)
![Page 138: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/138.jpg)
Higher dimensional results
RigidityViehweg-Zuo (2002)Kovács (2002)
BoundednessBedulev-Viehweg (2000)Viehweg-Zuo (2002), (2003)Kovács-Lieblich (2006)
HyperbolicityMigliorini (1995)Kovács (1996), (1997), (2000)Viehweg-Zuo (2002), (2003)Kovács (2003)Kovács-Kebekus (2005), (2007)
![Page 139: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/139.jpg)
Higher dimensional results
RigidityViehweg-Zuo (2002)Kovács (2002)
BoundednessBedulev-Viehweg (2000)Viehweg-Zuo (2002), (2003)Kovács-Lieblich (2006)
Hyperbolicity
Migliorini (1995)Kovács (1996), (1997), (2000)Viehweg-Zuo (2002), (2003)Kovács (2003)Kovács-Kebekus (2005), (2007)
![Page 140: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/140.jpg)
Higher dimensional results
RigidityViehweg-Zuo (2002)Kovács (2002)
BoundednessBedulev-Viehweg (2000)Viehweg-Zuo (2002), (2003)Kovács-Lieblich (2006)
HyperbolicityMigliorini (1995)
Kovács (1996), (1997), (2000)Viehweg-Zuo (2002), (2003)Kovács (2003)Kovács-Kebekus (2005), (2007)
![Page 141: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/141.jpg)
Higher dimensional results
RigidityViehweg-Zuo (2002)Kovács (2002)
BoundednessBedulev-Viehweg (2000)Viehweg-Zuo (2002), (2003)Kovács-Lieblich (2006)
HyperbolicityMigliorini (1995)Kovács (1996), (1997), (2000)
Viehweg-Zuo (2002), (2003)Kovács (2003)Kovács-Kebekus (2005), (2007)
![Page 142: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/142.jpg)
Higher dimensional results
RigidityViehweg-Zuo (2002)Kovács (2002)
BoundednessBedulev-Viehweg (2000)Viehweg-Zuo (2002), (2003)Kovács-Lieblich (2006)
HyperbolicityMigliorini (1995)Kovács (1996), (1997), (2000)Viehweg-Zuo (2002), (2003)
Kovács (2003)Kovács-Kebekus (2005), (2007)
![Page 143: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/143.jpg)
Higher dimensional results
RigidityViehweg-Zuo (2002)Kovács (2002)
BoundednessBedulev-Viehweg (2000)Viehweg-Zuo (2002), (2003)Kovács-Lieblich (2006)
HyperbolicityMigliorini (1995)Kovács (1996), (1997), (2000)Viehweg-Zuo (2002), (2003)Kovács (2003)
Kovács-Kebekus (2005), (2007)
![Page 144: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/144.jpg)
Higher dimensional results
RigidityViehweg-Zuo (2002)Kovács (2002)
BoundednessBedulev-Viehweg (2000)Viehweg-Zuo (2002), (2003)Kovács-Lieblich (2006)
HyperbolicityMigliorini (1995)Kovács (1996), (1997), (2000)Viehweg-Zuo (2002), (2003)Kovács (2003)Kovács-Kebekus (2005), (2007)
![Page 145: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/145.jpg)
Higher dimensional results
RigidityViehweg-Zuo (2002)Kovács (2002)
BoundednessBedulev-Viehweg (2000)Viehweg-Zuo (2002), (2003)Kovács-Lieblich (2006)
HyperbolicityMigliorini (1995)Kovács (1996), (1997), (2000)Viehweg-Zuo (2002), (2003)Kovács (2003)Kovács-Kebekus (2005), (2007)
![Page 146: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/146.jpg)
Higher dimensional results
RigidityViehweg-Zuo (2002)Kovács (2002)
BoundednessBedulev-Viehweg (2000)Viehweg-Zuo (2002), (2003)Kovács-Lieblich (2006)
HyperbolicityMigliorini (1995)Kovács (1996), (1997), (2000)Viehweg-Zuo (2002), (2003)Kovács (2003)Kovács-Kebekus (2005), (2007)
![Page 147: Singularities - University of Washington › ~kovacs › pdf › mckm.pdf · (finite simple groups) commutative algebra algebraic geometry. Mathematical Impressions axiomatic geometry](https://reader033.vdocuments.site/reader033/viewer/2022060413/5f1143becd494b7a9b25ebb4/html5/thumbnails/147.jpg)
Acknowledgement
This presentation was made using thebeamertex LATEX macropackage of Till Tantau.http://latex-beamer.sourceforge.net