Ravi Fernando Professor Vakil, Math 120 Mathieu-12 challenge November 26, 2010 A Straightforward Solution to the Topsy Turvy Puzzle Let L and R be the elements of S_12 that map the vector (1, 2, …, 12) to (11, 9, 7, 5,

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MATH 245 CLASS 2.5 (DAN EDIDIN): CRASH COURSE IN INTERSECTION THEORY TRANSCRIBED BY RAVI VAKIL Reference: Fulton Chapter 2. Throughout, X is finite type over a field k.

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MATH 245 CLASS 5 (DAN EDIDIN) RAVI VAKIL C ONTENTS 1.

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MATH 245 CLASS 3 (DAN EDIDIN) RAVI VAKIL G acts on X, G ,→ GLn (k). We can then define: G0 (G, X) is the Grothendieck group of coherent sheaves.

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MATH 245 CLASS 2 (DAN EDIDIN) RAVI VAKIL C ONTENTS 1.

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POLYA PROBLEM-SOLVING SEMINAR WEEK 6: PROBABILITY AND EXPECTED VALUES BOB HOUGH AND RAVI VAKIL The Rules. These are way too many problems to consider. Just pick a few problems you like and play around with them.

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MATH 245 CLASS 4 (DAN EDIDIN) RAVI VAKIL C ONTENTS 1. 2.

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POLYA PROBLEM-SOLVING SEMINAR WEEK 3: RECURRENCES BOB HOUGH AND RAVI VAKIL The Rules. These are way too many problems to consider. Just pick a few problems you like and play around with them.

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• Programming idioms
• Algebra
• Recurrence relation
• Number theory
• Fibonacci number
• Recursion
• Factorial
• Mathematics
• Integer sequences
• Theory of computation

A natural smooth compactification of the space of elliptic curves in projective space via blowing up the space of stable maps Ravi Vakil and Aleksey Zinger The moduli space of stable maps Mg,k (X, β) to a complex projec

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• Symplectic topology
• String theory
• Moduli theory
• Complex manifolds
• Gromov–Witten invariant
• Stable map
• Moduli space
• Resolution of singularities
• Projective space
• Geometry
• Abstract algebra
• Algebraic geometry

TWELVE POINTS ON THE PROJECTIVE LINE, BRANCHED COVERS, AND RATIONAL ELLIPTIC FIBRATIONS RAVI VAKIL Abstract. The following divisors in the space Sym12 P1 of twelve points on P1 are actually the same: (A) the possible loc

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• Algebraic surfaces
• Algebraic curves
• Birational geometry
• Vector bundles
• Canonical bundle
• Elliptic surface
• Del Pezzo surface
• Ample line bundle
• Riemann–Roch theorem
• Algebraic geometry
• Geometry
• Abstract algebra