This is the website for the slightly bizarre grad student seminar at UT Austin, on an assortement of topics such as: derived algebraic geometry, representation theory, geometric Langlands, QFT, and who knows what else. It is run by its namesakes: David Ben-Zvi and (in abstentia) Sam Raskin.
The BZ and the R - shifted!
The topic of the 2019 Spring Semester's BZ(R) Seminar will be shifted symplectic geometry.
Meeting:
We meet on Fridays 1pm - 3pm in RLM 11.176
References:
General references for the theory are:
[PTVV] The landmark original paper on shifted symplectic geometry of Pantev, Toen, Vaquie, and Vezzosi.
[CPTVV] The much harder follow-up to PTVV, developing the theory of shifted Poisson geometry.
There is also an abundance of excellent survey articles:
[T1] Toen's wonderful survey of derived algebraic geometry. Section 5 is all about shifted symplectic and Poisson geometry.
[T2] Toen's survey on shifted geometry and quantization
[S1] Safronov's survey on shifted Poisson geometry
Some fun applications:
[S2] Safronov's application of shifted symplectic geometry to representation theory
Notes:
When somebody LaTeXes notes of a talk, we can put them up here for all to see.
[BZ] Rok's (currently 9/10th complete) notes of David's talk
Schedule:
David's Opening Words
David - February 8
Starting off the semester in style, David will share with us an inspiring and motivational tale about shifted symplectic geometry, and its connection to QFT. We might return to several results mentioned in this talk later in the semester for a thorough and detailed treatement.
Crash-course in Derived Algebraic Geometry
Adrian - February 15
Refresher of the basic setup, and objects of study, in derived algebraic geometry - the field to which shifted symplectic geometry (in most incarnations) belongs. This will be the language we shall adopt for the majority of the semester, so even those fluent in it might appreciate a brief review. That said, the central purpose of this talk is to ease the burden of those less familiar with DAG.
Classical Symplectic Geometry, and what it has to do with Physics
Rok - February 22
Since shifted symplectic geometry strives to generalize its non-shifted predecesor. In this talk, we shall undertake a brief tour of that rather classical area of geometry, with emphasis on the various interactions between symplectic geometry and physics. If time permits, we will also describe some physical considerations that motivate the introduction of shifted symplectic geometry.
Outline of Shifted Symplectic Geometry
Alberto - March 1
After two preliminary talks on closely related topics, we will finally enter derived symplectic geometry. This first talk will serve as the introduction of the basic notions, and an overview of the theory, reserving the more technical considerations and details for subsequent talks.
Shifted p-forms, Closed Shifted p-forms, and Shifted Symplectic Structures
Tom - March 8, March 15, March 29
We will start digesting the PTVV paper, going through the technicalities that go into the definition of a shifted symplectic structure. This talk will cover Chapter 1 of PTVV.
The PTVV Main Theorem (AKSZ Construction)
Rok - April 5
Perhaps the cental result of the PTVV paper guarantees the existence of (shifted) symplectic structures on a certain class of mapping stacks. This is an enchancement of the AKSZ Construction in quantum field theory, and is the source of the most exciting examples of symplectic structures on various moduli spaces. In this talk, we shall undertake a careful and detailed proof of this theorem, closely following Chapter 2.1 of PTVV.
The PTVV Main Theorem (AKSZ Construction), pt.2
Rok - April 12
Perhaps the cental result of the PTVV paper guarantees the existence of (shifted) symplectic structures on a certain class of mapping stacks. This is an enchancement of the AKSZ Construction in quantum field theory, and is the source of the most exciting examples of symplectic structures on various moduli spaces. In this talk, we shall undertake a careful and detailed proof of this theorem, closely following Chapter 2.1 of PTVV.
Shifted Lagrangians
Leon - April 19
As Lagrangian submanifolds are a staple of symplectic geometry, so are their shifted analogues for the derived theory. There are some differences though, such as that a shifted Lagrangian structure may be defined on a map which need not itself be an embedding. Therefore intersections of Lagrangians need not be discrete, and in fact there is a rich and interesting theory of Lagrangian intersections. Lastly, both Hamiltonian actions and (shifted) symplectic structures themselves turn out to be special cases of Lagrangians. This talk will mainly follow Chapter 2.2 of PTVV.
RPerf, Examples, and Applications
Tom - April 26
Following Chapters 2.3 and 3 of PTVV, we will discuss the canonical 2-shifted symplectic structure on the moduli stack of perfect complexes. Then we will look at a variety of examples of shifted symplectic spaces, and at the relationship between (-1)-shifted symplectic structures to perfect obstruction theories in the sense of Behrend-Fantechi, one of the key ingredients in the algebraic variant of Gromov-Witten theory.
This talk will conclude our journey through PTVV. What lies ahead? CPTVV? Something else? Stay tuned to find out!
Subsequent talks to be scheduled later in the semester.
