Singularities. Sándor Kovács. May 1, 2007
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1 Singularities Sándor Kovács May 1, 2007
2
3 First Impressions
4 First Impressions Pierre de Fermat ( )
5 First Impressions Fermat: a n + b n = c n for n 3 has no solution with a, b, c non-zero integers.
6 First Impressions Bolyai, János ( )
7 First Impressions Fermat: a n + b n = c n for n 3 has no solution with a, b, c non-zero integers. Bolyai: hyperbolic geometry.
8 First Impressions Évariste Galois ( )
9 First Impressions Fermat: a n + b n = c n for n 3 has no solution with a, b, c non-zero integers. Bolyai: hyperbolic geometry. Galois: solving equations, group theory, field extensions.
10 First Impressions Fermat: a n + b n = c n 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 their contemporaries.
11 First Impressions Fermat: a n + b n = c n 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 their contemporaries. Galois died at a very young age (21).
12 Puzzles
13 Puzzles A snail has fallen in a well that s 50 feet deep. Every day it climbs up 10 feet, but then it takes a nap and slides back 9 feet. How long does it take for the snail to get out of the well?
14 Puzzles A snail has fallen in a well that s 50 feet deep. Every day it climbs up 10 feet, but then it takes a nap and slides back 9 feet. How long does it take for the snail to get out of the well? 41 days.
15 Puzzles A snail has fallen in a well that s 50 feet deep. Every day it climbs up 10 feet, but then it takes a nap and slides back 9 feet. How long does it take for the snail to get out of the well? 41 days. ( ) 2007 =...??...
16 Puzzles A snail has fallen in a well that s 50 feet deep. Every day it climbs up 10 feet, but then it takes a nap and slides back 9 feet. How long does it take for the snail to get out of the well? 41 days. ( ) 2007 =
17 Puzzles A snail has fallen in a well that s 50 feet deep. Every day it climbs up 10 feet, but then it takes a nap and slides back 9 feet. How long does it take for the snail to get out of the well? 41 days. ( ) 2007 = The Japanese kindergarten entry exam...
18 The Bus Puzzle
19 The Bus Puzzle
20 A Few Good Hungarians
21 A Few Good Hungarians Riesz, Frigyes ( )
22 A Few Good Hungarians Fejér, Lipót ( )
23 A Few Good Hungarians Haar, Alfréd ( )
24 A Few Good Hungarians Neumann, János ( )
25 A Few Good Hungarians Péter, Rózsa ( )
26 A Few Good Hungarians Erdős, Pál ( )
27 A Few Good Hungarians Bott, Raoul ( )
28 ...and a few more
29 ...and a few more Lax, Péter ( )
30 ...and a few more Lovász, László ( )
31 ...and a few more Kollár, János ( )
32 Mathematical Impressions
33 Mathematical Impressions axiomatic geometry
34 Mathematical Impressions axiomatic geometry hyperbolic geometry
35 Mathematical Impressions axiomatic geometry hyperbolic geometry projective geometry
36 Mathematical Impressions axiomatic geometry hyperbolic geometry projective geometry finite geometry
37 Mathematical Impressions axiomatic geometry hyperbolic geometry projective geometry finite geometry abstract algebra
38 Mathematical Impressions axiomatic geometry hyperbolic geometry projective geometry finite geometry abstract algebra group theory (finite simple groups)
39 Mathematical Impressions axiomatic geometry hyperbolic geometry projective geometry finite geometry abstract algebra group theory (finite simple groups) commutative algebra
40 Mathematical Impressions axiomatic geometry hyperbolic geometry projective geometry finite geometry abstract algebra group theory (finite simple groups) commutative algebra
41 Mathematical Impressions axiomatic geometry hyperbolic geometry projective geometry finite geometry abstract algebra group theory (finite simple groups) commutative algebra
42 Mathematical Impressions axiomatic geometry hyperbolic geometry projective geometry finite geometry abstract algebra group theory (finite simple groups) commutative algebra algebraic geometry
43 Mathematical Impressions axiomatic geometry hyperbolic geometry projective geometry finite geometry abstract algebra group theory (finite simple groups) commutative algebra algebraic geometry
44 First Papers
45 First Papers First paper
46 First Papers First paper took 5 years to get published.
47 First Papers First paper took 5 years to get published. Second paper
48 First Papers First paper took 5 years to get published. Second paper Erdős# = 2
49 Erdős # Definition
50 Erdős # Definition Erdős s Erdős# = 0,
51 Erdős # Definition Erdős s Erdős# = 0, Anyone, who published a research paper with Erdős has Erdős# = 1,
52 Erdős # Definition Erdős s Erdős# = 0, Anyone, who published a research paper with Erdős has Erdős# = 1, Anyone, who published a research paper with someone who has Erdős# = 1, has Erdős# = 2, etc.
53 Erdős # Definition Erdős s Erdős# = 0, Anyone, who published a research paper with Erdős has Erdős# = 1, Anyone, who published a research paper with someone who has Erdős# = 1, has Erdős# = 2, etc. Someone s Erdős# is n if they published a research paper with someone who has Erdős# = n 1, but have not published paper with anyone who has Erdős# < n 1.
54 Erdős # Definition Erdős s Erdős# = 0, Anyone, who published a research paper with Erdős has Erdős# = 1, Anyone, who published a research paper with someone who has Erdős# = 1, has Erdős# = 2, etc. Someone s Erdős# is n if they published a research paper with someone who has Erdős# = n 1, but have not published paper with anyone who has Erdős# < n 1. Example My Erdős# = 2.
55 Thesis
56 Thesis Hey, man, what s your thesis about?
57 Thesis Well, it has something to do with how the universe is changing through time, and it states that either the universe does not change at all, or there must be times when black holes exist.
58 Thesis Well, it has something to do with how the universe is changing through time, and it states that either the universe does not change at all, or there must be times when black holes exist. (This is, of course, a very loose and non-rigorous interpretation.)
59 Thesis Well, it has something to do with how the universe is changing through time, and it states that either the universe does not change at all, or there must be times when singularities exist.
60 Thesis My thesis through Lun Yi s eyes
61 Advertisement
62 Conics
63 Conics This may not be your thesis, but this I understand.
64 Conics What are conics, and why are they called conics?
65 Conics ellipse
66 Conics parabola
67 Conics hyperbola
68 Conics degenerate
69 Conics ellipse
70 Conics parabola
71 Conics hyperbola
72 Conics degenerate
73 Conics DEFORMATIONS
74 Conics deformations
75 Conics deformations
76 Conics INTERSECTIONS
77 Conics intersections
78 Conics intersections
79 Conics intersections
80 Cones SINGULARITIES
81 Singularities: cone
82 Singularities: 2 lines vs. 1 line
83 Singularities: deformation
84 Singularities: 2 lines
85 Singularities: 2 lines vs. 1 line
86 Weird 1 line through the vertex intersects 2 lines through the vertex in only 1 point.
87 Weird 1 line through the vertex intersects 2 lines through the vertex in only 1 point. In how many point does 1 line intersect another (1) line?
88 Weird 1 line through the vertex intersects 2 lines through the vertex in only 1 point. In how many point does 1 line intersect another (1) line? 1/2
89 Non-singular case
90 Non-singular case smoothing
91 Non-singular case 2 lines vs. 1 line
92 Non-singular case deformation
93 Non-singular case 2 lines
94 Non-singular case 2 lines vs. 1 line
95 Research Fermat-Wiles: a n + b n = c n n 3 has no solution with a, b, c non-zero integers.
96 Research Fermat-Wiles: a n + b n = c n n 3 has no solution with a, b, c non-zero integers. Reformulation: ( ) n a + c ( ) n b = 1 n 3 c has no solution with a, b, c non-zero integers.
97 Research Fermat-Wiles: a n + b n = c n n 3 has no solution with a, b, c non-zero integers. Reformulation: x n + y n = 1 n 3 has no solution with x, y non-zero rational numbers.
98 Research Fermat-Wiles: a n + b n = c n n 3 has no solution with a, b, c non-zero integers. Reformulation: x n + y n = 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?
99 Geometry The equation f (x, y) = 0 defines a curve on the plane:
100 Geometry The equation f (x, y) = 0 defines a curve on the plane:
101 Geometry The equation f (x, y) = 0 defines a curve on the plane:
102 Geometry The equation f (x, y) = 0 defines a curve on the plane:
103 Geometry The equation f (x, y) = 0 defines a curve on the plane:
104 Geometry The equation f (x, y) = 0 defines a curve on the plane: A solution with x, y rational numbers corresponds to a point on the curve with rational coordinates.
105 From Arithmetic to Geometry The original Fermat problem asks to find integer solutions to equations with integer coefficients.
106 From Arithmetic to Geometry The original Fermat problem asks to find integer solutions to equations with integer coefficients. Replace integer with multiple of t.
107 From Arithmetic to Geometry The original Fermat problem asks to find integer solutions to equations with integer coefficients. Replace integer with multiple of t. Consider equations with a free parameter t and ask whether there are solutions among multiples of t.
108 From Arithmetic to Geometry The original Fermat problem asks to find integer solutions to equations with integer coefficients. Replace integer with multiple of t. Consider equations with a free parameter t and ask whether there are solutions among multiples of t. Example: Consider the equation y 2 x 5 + 5t x 4t = 0.
109 From Arithmetic to Geometry The original Fermat problem asks to find integer solutions to equations with integer coefficients. Replace integer with multiple of t. Consider equations with a free parameter t and ask whether there are solutions among multiples of t. Example: Consider the equation y 2 x 5 + 5t x 4t = 0. Are there solutions that can be expressed as polynomials of t?
110 From Arithmetic to Geometry The original Fermat problem asks to find integer solutions to equations with integer coefficients. Replace integer with multiple of t. Consider equations with a free parameter t and ask whether there are solutions among multiples of t. Example: Consider the equation y 2 x 5 + (5t) 4 x 4t 4 = 0. Are there solutions that can be expressed as polynomials of t?
111 From Arithmetic to Geometry The original Fermat problem asks to find integer solutions to equations with integer coefficients. Replace integer with multiple of t. Consider equations with a free parameter t and ask whether there are solutions among multiples of t. Example: Consider the equation y 2 x 5 + (5t) 4 x 4t 4 = 0. Are there solutions that can be expressed as polynomials of t? Let x = 5t and y = 2t 2.
112 Mordell Conjecture By changing the parametrization there will always be a solution.
113 Mordell Conjecture By changing the parametrization there will always be a solution. There is still something interesting to ask:
114 Mordell Conjecture By changing the parametrization there will always be a solution. There is still something interesting to ask: Question: Is it true that for any given parametrization there are only finitely many solutions (in t)?
115 Mordell Conjecture By changing the parametrization there will always be a solution. There is still something interesting to ask: Question: Is it true that for any given parametrization there are only finitely many solutions (in t)? This is known as Mordell s Conjecture and was confirmed by Manin in 1963.
116 Shafarevich s Conjecture In 1968 Parshin realized that this is related to another conjecture made by Shafarevich in 1962.
117 Shafarevich s Conjecture In 1968 Parshin realized that this is related to another conjecture made by Shafarevich in The connection is somewhat tricky, but the point is that instead of looking for solutions (in t) the question focuses on studying the total space of the curves:
118 Shafarevich s Conjecture In 1968 Parshin realized that this is related to another conjecture made by Shafarevich in The connection is somewhat tricky, but the point is that instead of looking for solutions (in t) the question focuses on studying the total space of the curves: As t varies, there is a curve C t in the plane that is defined by the equation f (x, y, t) = 0. The union of these curves form a surface, which is fibred over the t-line :
119 Shafarevich s Conjecture
120 Shafarevich s Conjecture
121 Shafarevich s Conjecture Shafarevich s Conjecture says that under certain (well-defined) conditions there are only finitely many families satisfying the conditions.
122 Shafarevich s Conjecture Shafarevich s Conjecture says that under certain (well-defined) conditions there are only finitely many families satisfying the conditions. Parshin s trick shows that Shafarevich s Conjecture implies Mordell s Conjecture.
123 Shafarevich s Conjecture Shafarevich s Conjecture says that under certain (well-defined) conditions there are only finitely many families satisfying the conditions. Parshin s trick shows that Shafarevich s Conjecture implies Mordell s Conjecture. Parshin and Arakelov proved Shafarevich s Conjecture in
124 Higher dimensional Shafarevich Conjecture The problem of parametrized families make sense in higher dimensions:
125 Higher dimensional Shafarevich Conjecture The problem of parametrized families make sense in higher dimensions: One may study
126 Higher dimensional Shafarevich Conjecture The problem of parametrized families make sense in higher dimensions: One may study Families of higher dimensional objects (surfaces, threefolds, etc.)
127 Higher dimensional Shafarevich Conjecture The problem of parametrized families make sense in higher dimensions: One may study Families of higher dimensional objects (surfaces, threefolds, etc.) Families with more than one parameter.
128 Higher dimensional Shafarevich Conjecture The problem of parametrized families make sense in higher 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 it is broken up into three subproblems:
129 Higher dimensional Shafarevich Conjecture The problem of parametrized families make sense in higher 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 it is broken up into three subproblems: Rigidity
130 Higher dimensional Shafarevich Conjecture The problem of parametrized families make sense in higher 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 it is broken up into three subproblems: Rigidity Boundedness
131 Higher dimensional Shafarevich Conjecture The problem of parametrized families make sense in higher 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 it is broken up into three subproblems: Rigidity Boundedness Hyperbolicity
132 Higher dimensional results Rigidity
133 Higher dimensional results Rigidity Viehweg-Zuo (2002)
134 Higher dimensional results Rigidity Viehweg-Zuo (2002) Kovács (2002)
135 Higher dimensional results Rigidity Viehweg-Zuo (2002) Kovács (2002) Boundedness
136 Higher dimensional results Rigidity Viehweg-Zuo (2002) Kovács (2002) Boundedness Bedulev-Viehweg (2000)
137 Higher dimensional results Rigidity Viehweg-Zuo (2002) Kovács (2002) Boundedness Bedulev-Viehweg (2000) Viehweg-Zuo (2002), (2003)
138 Higher dimensional results Rigidity Viehweg-Zuo (2002) Kovács (2002) Boundedness Bedulev-Viehweg (2000) Viehweg-Zuo (2002), (2003) Kovács-Lieblich (2006)
139 Higher dimensional results Rigidity Viehweg-Zuo (2002) Kovács (2002) Boundedness Bedulev-Viehweg (2000) Viehweg-Zuo (2002), (2003) Kovács-Lieblich (2006) Hyperbolicity
140 Higher dimensional results Rigidity Viehweg-Zuo (2002) Kovács (2002) Boundedness Bedulev-Viehweg (2000) Viehweg-Zuo (2002), (2003) Kovács-Lieblich (2006) Hyperbolicity Migliorini (1995)
141 Higher dimensional results Rigidity Viehweg-Zuo (2002) Kovács (2002) Boundedness Bedulev-Viehweg (2000) Viehweg-Zuo (2002), (2003) Kovács-Lieblich (2006) Hyperbolicity Migliorini (1995) Kovács (1996), (1997), (2000)
142 Higher dimensional results Rigidity Viehweg-Zuo (2002) Kovács (2002) Boundedness Bedulev-Viehweg (2000) Viehweg-Zuo (2002), (2003) Kovács-Lieblich (2006) Hyperbolicity Migliorini (1995) Kovács (1996), (1997), (2000) Viehweg-Zuo (2002), (2003)
143 Higher dimensional results Rigidity Viehweg-Zuo (2002) Kovács (2002) Boundedness Bedulev-Viehweg (2000) Viehweg-Zuo (2002), (2003) Kovács-Lieblich (2006) Hyperbolicity Migliorini (1995) Kovács (1996), (1997), (2000) Viehweg-Zuo (2002), (2003) Kovács (2003)
144 Higher dimensional results Rigidity Viehweg-Zuo (2002) Kovács (2002) Boundedness Bedulev-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)
145 Higher dimensional results Rigidity Viehweg-Zuo (2002) Kovács (2002) Boundedness Bedulev-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)
146 Higher dimensional results Rigidity Viehweg-Zuo (2002) Kovács (2002) Boundedness Bedulev-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)
147 Acknowledgement This presentation was made using the beamertex L A TEX macropackage of Till Tantau.
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