The seminar Interactions between symplectic geometry, combinatorics and number theory will cover different topics, and is aimed at studying the interactions among them. In particular, we will learn about genera on complex or symplectic manifolds (for instance the Todd and Hirzebruch genus and elliptic genera) and their connections with modular forms, as well as the combinatorics of lattice polytopes, in particular Ehrhart theory and reflexive polytopes.

 

Graduate students, postdocs and professors interested in attending will be encouraged to give explanatory talks that are suitable to an audience with diverse background.

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From time to time we will also invite external speakers to give more advanced/research talks on topics related to those covered in this seminar.

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Tentative list of topics (to be continued and completed...):

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1) Topology part (beginning): 

- Cohomology, vector bundles,

- Chern and Pontrjagin classes, Chern and Pontrjagin numbers.

- Cobordism ring and its behaviour w.r.t. Chern numbers (Chapter 1 of [H] and/or 16 of [MS]), 

- Multiplicative sequences and complex genera ( Chapter 19 of [MS]).

 

2) Number theory part:

- Bernoulli Numbers (see Appendix B [MS], whose content is indeed related to complex genera)

- Elliptic genera (Chapter 2 [H], as well as other sources of course)

 

And after this, back to topology...

 

- Genera related to differential operators (Todd and Hirzebruch genus, as much as it is feasible of Chapter 5 [H])

 

Topology/number theory

 

- Chapter 6 up to 6.4 [H]

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Volunteers are welcome!!!

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Schedule and list of speakers:

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• October 24th, Silvia Sabatini  -- Introduction: Why should we care about this seminar?

• October 31st, Thomas Rot  -- Topological Introduction: cohomology and vector bundle

• November 7th, Milena Pabiniak -- Topological Introduction: Characteristic classes and complex genera.  (A nice introduction to Chern classes is given in Hatcher's notes, Chapter 3 page 73)

• November 14th, no seminar

• November 21st, no seminar

• November 28th, Milena Pabiniak -- Multiplicative sequences. 

• December 5th, Michael Woodbury -- Bernoulli Numbers

• December 12th, Silvia Sabatini -- Delzant polytopes and their cohomology ring(s), defined combinatorially, Part I

• December 19th, Silvia Sabatini -- Delzant polytopes and their cohomology ring(s), defined combinatorially, Part

• January 9th, Markus Schwagenscheidt -- Introduction to modular forms 

• January 16th, Steffen Löbrich -- Lattice Polygons and the Number 12

• January 23rd, Frederik von Heymann -- A combinatorial proof of the generalisation of the 12 formula for Delzant polytopes

• January 30th, Markus Schwagenscheidt -- Introduction to elliptic genera I

• February 6th, Markus Schwagenscheidt -- Introduction to elliptic genera II

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Bibliography:

[Ha] Hatcher, "Vector Bundles and K-theory"

[H] Hirzebruch et al. "Manifolds and modular forms"

[MS] Milnor and Stasheff, "Characteristic classes"

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