Throughout the WS 2024/2025, we shall explore the vast world of Derived Geometry. Our bigger objective is to understand this newish field of research, particularly, in relation to Differential Geometry and, even more specifically, Poisson Geometry. Thus, our primary aim will be to introduce the different notions involved in the theory and (hopefully) arrive at the statement and proof of the Darboux theorem for shifted symplectic structures. Understandably, this is a very ambitious task for a single semester seminar and, as such, we shall focus more on definitions and examples rather than give a full exploration of the subjects. The following is a tentative list of contents and schedule for the seminar throughout the aforementioned semester, to be better fixed in the following weeks:
05.11. Elementary Category Theory (Jorn):
In this first session, we shall explore the very basics of Category Theory.
References
T. Leinster. ‘Basic Category Theory’. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2014. doi: 10.1017/CBO9781107360068
E. Riehl. ‘Category Theory in Context’. Aurora: Dover Modern Math Originals. Dover Publications, 2016. ISBN: 9780486809038
Extra Sessions (on 05.11, 12.11 & 03.12). Crash-course on Algebraic Geometry (Rodrigo):
So that we are all on the same boat, we shall start by exploring the fundamentals of Algebraic Geometry. Starting on Sheaf theory and going through basic results on Schemes and (quasi-)coherent sheaves, we will also have a short discussion on Kähler differentials and (co)tangent sheaves.
References
Standard references in Algebraic Geometry, e.g. Harthshorne, Shafarevich as well as the Stacks Project.
12.11 & 19.11. Higher Category Theory (Jorn):
The language of Derived Geometry would not be possible without Higher Category Theory. As such, we shall explore this world, starting with the introduction of the notions of ∞-categories, ∞-functors as well as the category arising from these. Then, we explore certain properties of these ∞-categories, based on those known in the usual non-higher Category Theory, and we shall also construct ∞-categories from model categories — in particular, we construct the ∞-category of ∞-categories. With these results, we are then set for the full exploration of Derived Algebraic/Differential Geometry.
References
J. Lurie. ‘Higher Topos Theory’. Annals of Mathematics Studies 170. Princeton University Press, Princeton, 2009. doi: 10.1515/9781400830558
D.-C. Cisinski. ‘Higher Categories and Homotopical Algebra’. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2019. doi: 10.1017/9781108588737
‘relation between quasi-categories and simplicial categories’. nLab. Last revised on the 31st of May, 2023. https://ncatlab.org/nlab/show/relation+between+quasi-categories+and+simplicial+categories
J. Lurie. ‘Kerodon’. Accessible at https://kerodon.net.
26.11, 03.12 & 10.12. Derived Algebraic Geometry (Rodrigo):
Introduced by Lurie, Toën and Vezzosi in a series of papers, Derived Algebraic Geometry has been standardised and used in many different applications in Modern Mathematics. Given its blend of classical Algebraic Geometry, Homotopy Theory and and Higher Category Theory, we obtain a very reach subject which is incredibly useful to study certain problems from deformation theory, intersection theory as well as better understand certain moduli spaces arising in Geometry. Now that we have a better understanding of Algebraic Geometry and after the exploration of the fundamentals of Higher Category theory, we are ready to explore this now important subject, focusing more on the theory of derived schemes, which are sufficiently general for our purposes. These already allow for a great deal of interesting features, such as the (co)tangent complex — that will be essential for the study of Derived Symplectic Geometry — or perfect complexes, derived complexes or even cohomology of (derived) sheaves. Naturally, we will not explore all of these facets but we plan to, at least, give enough to motivate the study of this broad subject.
References
B. Toën. ‘Derived algebraic geometry’. EMS Surv. Math. Sci. 1 (2014), 153–240. doi: 10.4171/EMSS/4.
W. D. Gillam. ‘Simplicial Methods in Algebra and Algebraic Geometry’. Accessible at https://math.mit.edu/~hrm/palestine/gillam-simplicial-methods.pdf.
J. Lurie. ‘Spectral Algebraic Geometry’. Accessible at https://www.math.ias.edu/~lurie/papers/SAG-rootfile.pdf.
D.-C. Cisinski. ‘Higher Categories and Homotopical Algebra’. Cambridge University Press, Cambridge, 2019. doi: 10.1017/9781108588737.
B. Toën and G. Vezzosi. Papers on ‘Homotopical Algebraic Geometry’. arXiv:math/0207028, arXiv:math/0404373.
J. Lurie. Series on ‘Derived Algebraic Geometry’. Accessible at http://www.math.harvard.edu/~lurie/papers/DAG-*number*.pdf (substituting *number* by any roman numeral between I and XIV).
M. Anel. ‘The Geometry of Ambiguity: An Introduction to the Ideas of Derived Geometry’. 2016. Accessible at http://mathieu.anel.free.fr/mat/doc/Anel-DerivedGeometry.pdf.
17.12. Derived Differential Geometry (Rodrigo):
Not as standardised as its algebraic counterpart, Derived Differential Geometry is somewhat still in its infancy. Thus, in the literature, it is possible to find — at least — four different approaches for what a notion of a derived manifold (or some kind of generalisation of it) should be — not all of which necessarily equivalent. Since we do not have the time to properly introduce them all, we shall focus on just two of the more well-known ones: the derived C^∞-schemes, pioneered by the work of Spivak; and d-manifolds and µ-Kuranishi spaces arising from works of Joyce. To properly understand these, we shall also start by introducing C^∞-algebraic Geometry, which is a subject relatively unknown and, essentially, aims to be able to use the powerful results of Algebraic Geometry (and Commutative Algebra) in the context of Differential Geometry, by inserting the differential structure of the smooth functions of the objects of the latter (these could be manifolds/orbifolds with or without boundary or corners) into the (commutative unital) rings used in the former.
References
E. J. Dubuc. ‘C^∞-schemes’. Amer. J. Math. 103 (1981), 683–690.
D. Joyce. ‘Algebraic Geometry over C^∞-rings’. Memoirs of the A.M.S. 260 (2019), no. 1256. doi: 10.1090/memo/1256.
D. I. Spivak. ‘Derived smooth manifolds’. Duke Mathematical Journal 153 (2010), 55–128. arXiv:0810.5174.
D. Joyce. ‘D-manifolds and d-orbifolds: a theory of derived differential geometry’ (survey arXiv:1208.4948, more details in http://people.maths.ox.ac.uk/~joyce/dmanifolds.html).
P. Steffens. ‘Derived C^∞-geometry I: Foundations’, 2023. arXiv:2304.08671.
14.01, 21.01 & 11.02. Derived Symplectic Geometry (Jorn, Rodrigo):
More well-known as shifted symplectic structures, and the crux of the seminar, derived symplectic structures were introduced by Pantev, Toën, Vaquié and Vezzosi and they arise from the intent to extend symplectic structures to the context of Derived Algebraic Geometry, also in an attempt to later quantise these spaces (usually as moduli spaces of some other geometric/physical problem). In these series of three lectures, we will explore this natural extension (also trying to somehow connect to the Derived Differential Geometry side) and state and prove the equivalent formulation of Darboux’s theorem for these derived symplectic structures.
References
D. Calaque. ‘Shifted symplectic geometry by examples’. Lecture Notes for the Poisson 2024 Summer School. Accessible at https://www.imag.umontpellier.fr/~calaque/lecturenotes.html
T. Pantev, B. Toën, M. Vaquié and G. Vezzosi. ‘Shifted symplectic structures’. Publ. Math. Inst. Hautes Études Sci. 117 (2013), 271–328. doi: 10.1007/s10240-013-0054-1.
C. Brav, V. Bussi and D. Joyce. ‘A Darboux theorem for derived schemes with shifted symplectic structure’. J. Amer. Math. Soc. 32 (2019), no.2, 399–443. doi: 10.1090/jams/910.
D. Calaque. ‘Derived Symplectic Geometry’. Encyclopedia of Mathematical Physics, 697–703. Academic Press, 2025. doi: 10.1016/B978-0-323-95703-8.00009-4. arXiv:2308.04210.
ATTENTION: The talk on 11.12 will occur at 10:30h in Seminarraum 3 (314)!
11.02. From Calabi-Yau structures to Shifted Symplectic Structures (Gustavo Jasso):
We finish our seminar with a talk by Prof. Dr. Gustavo Jasso, where we explore the connections between Calabi-Yau structures and the shifted symplectic structures we have been developing.
References
C. Brav, T. Dyckerhoff. ‘Relative Calabi–Yau structures’. Compositio Mathematica 155 (2019), 2, 372–412. doi: 10.1112/S0010437X19007024.
C. Brav, T. Dyckerhoff. ‘Relative Calabi–Yau structures II: shifted Lagrangians in the moduli of objects’. Sel. Math. New Ser. 27, 63 (2021). doi: 10.1007/s00029-021-00642-5.
T. Bozec, D. Calaque, S. Scherotzke. ‘Relative critical loci and quiver moduli’. Annales Scientifiques de l’École Normale Supérieure 57 (2024), 2, 553–614. doi: 10.24033/asens.2579. hal- 02861404.