K3 Seminar Spring 2019: Difference between revisions
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'''When:''' Thursday 5-7 pm | '''When:''' Thursday 5-7 pm | ||
'''Where:''' Van Vleck | '''Where:''' Van Vleck B135 | ||
''' | ''' | ||
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| bgcolor="#E0E0E0"| March 7 | | bgcolor="#E0E0E0"| March 7 | ||
| bgcolor="#C6D46E"| Mao Li | | bgcolor="#C6D46E"| Mao Li | ||
| bgcolor="#BCE2FE"|[[ | | bgcolor="#BCE2FE"|[[#March 7| Basics of K3 Surfaces and the Grothendieck-Riemann-Roch theorem]] | ||
|- | |- | ||
| bgcolor="#E0E0E0"| March 14 | | bgcolor="#E0E0E0"| March 14 | ||
| bgcolor="#C6D46E"| Shengyuan Huang | | bgcolor="#C6D46E"| Shengyuan Huang | ||
| bgcolor="#BCE2FE"|[[ | | bgcolor="#BCE2FE"|[[#March 14| Elliptic K3 Surfaces]] | ||
|- | |- | ||
| bgcolor="#E0E0E0"| March 28 | | bgcolor="#E0E0E0"| March 28 | ||
| bgcolor="#C6D46E"| Zheng Lu | | bgcolor="#C6D46E"| Zheng Lu | ||
| bgcolor="#BCE2FE"|[[ | | bgcolor="#BCE2FE"|[[#March 28| Moduli of Stable Sheaves on a K3 Surface]] | ||
|- | |- | ||
| bgcolor="#E0E0E0"| April 4 | | bgcolor="#E0E0E0"| April 4 | ||
| bgcolor="#C6D46E"| Canberk Irimagzi | | bgcolor="#C6D46E"| Canberk Irimagzi | ||
| bgcolor="#BCE2FE"|[[ | | bgcolor="#BCE2FE"|[[#April 4| Fourier-Mukai Transforms]] | ||
|- | |- | ||
| bgcolor="#E0E0E0"| April 11 | | bgcolor="#E0E0E0"| April 11 | ||
| bgcolor="#C6D46E"| Moisés Herradón Cueto | |||
| bgcolor="#BCE2FE"|[[#April 11| Cohomology of Complex K3 Surfaces and the Global Torelli Theorem]] | |||
|- | |||
| bgcolor="#E0E0E0"| April 23 | |||
| bgcolor="#C6D46E"| David Wagner | | bgcolor="#C6D46E"| David Wagner | ||
| bgcolor="#BCE2FE"|[[ | | bgcolor="#BCE2FE"|[[#April 23| Derived Categories of K3 Surfaces]] | ||
|} | |} | ||
</center> | </center> | ||
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| bgcolor="#BCD2EE" | Title: Fourier-Mukai Transforms | | bgcolor="#BCD2EE" | Title: Fourier-Mukai Transforms | ||
|- | |- | ||
| bgcolor="#BCD2EE" | Abstract: I will describe Chow theoretic correspondences as a motivation to derived correspondences. We will then define integral functors on derived categories. The dual abelian variety will be given as a moduli space in terms of its functor of points, leading us to a definition of the universal Poincaré bundle on $A \times \hat{A}$. We will look at the integral transform from $D(A)$ to $D(\hat{A})$ induced by the Poincaré bundle. Cohomology of the Poincaré bundle will be stated and used for the computation of the $K$-theoretic Fourier Mukai transform on elliptic curves. With the help of the base change theorem, we will | | bgcolor="#BCD2EE" | Abstract: I will describe Chow theoretic correspondences as a motivation to derived correspondences. We will then define integral functors on derived categories. The dual abelian variety will be given as a moduli space in terms of its functor of points, leading us to a definition of the universal Poincaré bundle on $A \times \hat{A}$. We will look at the integral transform from $D(A)$ to $D(\hat{A})$ induced by the Poincaré bundle. Cohomology of the Poincaré bundle will be stated and used for the computation of the $K$-theoretic Fourier-Mukai transform on elliptic curves. With the help of the base change theorem, we will describe the Fourier-Mukai duals of homogeneous line bundles on $A$. For an elliptic curve $E$, we will establish the equivalence between | ||
1. the abelian category of semistable bundles of slope 0 on E, and | 1. the abelian category of semistable bundles of slope 0 on $E$, and | ||
2. the abelian category of coherent torsion sheaves on E. | 2. the abelian category of coherent torsion sheaves on $E$. | ||
Simple and indecomposable objects of these categories will be described (with the help of the structure theorem of PIDs) and we will | Simple and indecomposable objects of these categories will be described (with the help of the structure theorem of PIDs) and we will relate this picture to Atiyah’s classification of the indecomposable vector bundles of degree 0. | ||
|} | |} | ||
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | {| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | ||
|- | |- | ||
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' | | bgcolor="#A6B658" align="center" style="font-size:125%" | '''Moisés Herradón Cueto''' | ||
|- | |- | ||
| bgcolor="#BCD2EE" | Title: Cohomology of Complex K3 Surfaces and the Global Torelli Theorem | | bgcolor="#BCD2EE" | Title: Cohomology of Complex K3 Surfaces and the Global Torelli Theorem | ||
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</center> | </center> | ||
== April | == April 23 == | ||
<center> | <center> | ||
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | {| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | ||
|- | |- | ||
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' | | bgcolor="#A6B658" align="center" style="font-size:125%" | '''David Wagner''' | ||
|- | |- | ||
| bgcolor="#BCD2EE" | Title: Derived Categories of K3 Surfaces | | bgcolor="#BCD2EE" | Title: Derived Categories of K3 Surfaces | ||
Latest revision as of 13:29, 22 April 2019
When: Thursday 5-7 pm
Where: Van Vleck B135
Schedule
| Date | Speaker | Title |
| March 7 | Mao Li | Basics of K3 Surfaces and the Grothendieck-Riemann-Roch theorem |
| March 14 | Shengyuan Huang | Elliptic K3 Surfaces |
| March 28 | Zheng Lu | Moduli of Stable Sheaves on a K3 Surface |
| April 4 | Canberk Irimagzi | Fourier-Mukai Transforms |
| April 11 | Moisés Herradón Cueto | Cohomology of Complex K3 Surfaces and the Global Torelli Theorem |
| April 23 | David Wagner | Derived Categories of K3 Surfaces |
March 7
| Mao Li |
| Title: Basics of K3 Surfaces and the Grothendieck-Riemann-Roch theorem |
| Abstract: |
March 14
| Shengyuan Huang |
| Title: Elliptic K3 Surfaces |
| Abstract: |
March 28
| Zheng Lu |
| Title: Moduli of Stable Sheaves on a K3 Surface |
| Abstract: |
April 4
| Canberk Irimagzi |
| Title: Fourier-Mukai Transforms |
| Abstract: I will describe Chow theoretic correspondences as a motivation to derived correspondences. We will then define integral functors on derived categories. The dual abelian variety will be given as a moduli space in terms of its functor of points, leading us to a definition of the universal Poincaré bundle on $A \times \hat{A}$. We will look at the integral transform from $D(A)$ to $D(\hat{A})$ induced by the Poincaré bundle. Cohomology of the Poincaré bundle will be stated and used for the computation of the $K$-theoretic Fourier-Mukai transform on elliptic curves. With the help of the base change theorem, we will describe the Fourier-Mukai duals of homogeneous line bundles on $A$. For an elliptic curve $E$, we will establish the equivalence between
1. the abelian category of semistable bundles of slope 0 on $E$, and 2. the abelian category of coherent torsion sheaves on $E$. Simple and indecomposable objects of these categories will be described (with the help of the structure theorem of PIDs) and we will relate this picture to Atiyah’s classification of the indecomposable vector bundles of degree 0. |
April 11
| Moisés Herradón Cueto |
| Title: Cohomology of Complex K3 Surfaces and the Global Torelli Theorem |
| Abstract: |
April 23
| David Wagner |
| Title: Derived Categories of K3 Surfaces |
| Abstract: |
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