Koszul Duality Seminar - Fall 2013
University of California, Berkeley

Thursdays 2-3pm, 736 Evans Hall

Organisers: Constantin Teleman and George Melvin
CCN: 54549

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Seminar Outline

The aim of this seminar is to become acquainted with various dualities that appear across mathematics - for example, commutative algebra, representation theory, topology - that come under the heading of 'Koszul duality'.

There will be three main focus points for the seminar:

• Koszul algebras; equivalence between derived categories of symmetric and exterior algebras (Bernstein-Gelfand-Gelfand); applications (Eisenbud-Floystad-Schreyer).
• Koszul duality patterns in representation theory, duality in category O and perverse sheaves (Beilinson-Ginzburg-Soergel); duality for toric varieties (Braden, Braden-Lunts).
• Koszul duality and operads (Ginzburg-Kapranov, Loday-Valette); operads and moduli spaces (Getzler); BV operad (Khoroshkin-Markarian-Shadrin).

(Tentative) Schedule of Talks + Speakers

Some talks may be rescheduled to a different time and/or location depending upon requirements of the speaker. Any updates will be posted here - make sure to check the department seminar announcement board. Updated 10/31/13

• 9/26 - Koszul algebras; Koszul duality as derived Morita equivalence; numerical Koszul criterion (Ref:2;4 Section 2, Thm 2.12.6) Speaker: Qiaochu Yuan Abstract. Some notes from the talk. (All errors, inconstistencies and inaccuracies are due to me.)
• 10/3 - SV, EV* are Koszul dual; BGG correspondence; applications to sheaf cohomology (Ref: 1;2;3) Speaker: Harrison Chen Abstract.
• 10/10 - IC, perverse sheaves on G/P; D^b(P(X,W)) = D(X,W) (Ref: 4, Section 3.1-3.4) Speaker: George Melvin Abstract. Some notes from the talk.
• 10/17 - Category O; localisation; parabolic-singular duality (Ref: 4, 3.5-3.7,3.9) Speaker: George Melvin Abstract. Some notes from the talk.
• 10/24 - (Guest Lecture) BGG correspondence, Tate resolutions and computing sheaf cohomology. Speaker: Gunnar Fløystad Abstract. Some notes from the talk. (All errors, inaccuracies and inconsistencies are due to me.)
• 10/31 - Koszul duality for toric varieties (Ref: 5;6) Speaker: Pablo Solis Abstract. Some notes from the talk. (All errors, inconsistencies and inaccuracies are due to me.)
• 11/7 - Rational Homotopy Theory I Speaker: Constantin Teleman Abstract.
• 11/14 - Rational Homotopy Theory II Speaker: Constantin Teleman Abstract.
• 11/21 - Koszul Duality for Operads I (Ref: 7) Speaker: Eric Peterson Abstract.
• 11/26 - Koszul Duality for Operads II Special date/time - 736 Evans, 3.10-4pm ( Ref: 7) Speaker: Constantin Teleman Abstract.
• 12/5 - Applications: topological gravity, BV operads (Ref: 9;11) Speaker: Constantin Teleman Abstract.

References

1. Algebraic Bundles over P^{n} and Problems of Linear Algebra, Bernstein-Gelfand-Gelfand
2. Methods of Homological Algebra, Gelfand-Manin
3. Sheaf Cohomology and Free Resolutions over Exterior Algebras, Eisenbud-Floystad-Schreyer
4. Koszul Duality Patterns in Representation Theory, Beilinson-Ginzburg-Soergel
5. Koszul Duality for Toric Varieties, Braden
6. Equivariant-constructible Koszul duality for dual toric varieties, Braden-Lunts
7. Koszul Duality for Operads, Ginzburg-Kapranov
8. Algebraic Operads, Loday-Valette
9. Operads and Moduli Spaces of Genus 0 Riemann Surfaces, Getzler
10. Algebra+Homotopy = Operad, Valette
11. Hypercommutative Operad as a Homotopy Quotient of BV, Khoroshkin-Markarian-Shadrin