Regular meetings: Mondays at 1:30–3 pm.

20.02 (Friday!) – Sveta

Overview and organisational.

23.02 – Sveta 

Talk 1. Characteristic-free formal calculus and definitions of vertex (super) algebras.

02.03 – Sveta 

Talk 2. Zhu algebra of a vertex algebra. Example: Heisenberg vertex algebras.

There is a classical definition of Heisenberg vertex algebras over C. It has been observed that Heisenberg and lattice vertex algebras admit integral models, using which Q. Mu defined lattice vertex algebras over a field of characteristic p>0 by base change. In an ongoing collaboration, I found an axiomatic definition of Heisenberg vertex algebras over an arbitrary ring, which gives rise to interesting combinatorial recurrences.

09.03 – no seminar (Canberra day)

Reschedule to a different day same week – Sveta

Talk 3. Vertex (co)algebra structure on (co)homology of an H-group.

This example is motivated by the vertex algebra structure on homology of moduli spaces of objects in an abelian category due to Joyce.

16.03 – 

23.03 – Leo

"Associated varieties" coming from vertex algebras.

Reference: Arakawa, Moreau, "Arc spaces and vertex algebras".

30.03 – 

06.04 – no seminar (teaching break)

13.04 – no seminar (teaching break)

20.04 – Tony or Uri

Basics of modular forms.

27.04 – no seminar (ANZAC Day)

04.05 – 

11.05 – 

18.05 – James T.

Motivation: Monstrous moonshine, mathematical physics.

25.05 –

Regular meetings: Thursdays at 2:15 pm.

18.09 – Sveta

Overview and organisational.

25.09 – Vigleik 

Review of Hochschild (co)homology of algebras.

02.10 – Vigleik 

Review of Hochschild (co)homology of algebras, part 2.

09.10 – Sveta

Introduction to DG categories and their modules.

Reference: Keller, "On DG categories".

16.10 – Sveta

Introduction to DG categories and their modules.

Reference: Keller, "On DG categories".

23.10 – Hoel

Derived categories of DG categories.

Reference: Keller, "Deriving DG categories".

30.10 – Ian

Derived functors. Hochschild (co)homology of DG categories. Alternative definition of HH for schemes and HKR.

06.11 – Asilata

Cyclic homology I.

Reference: Kaledin's lectures 3–4, §§1.1–1.3 or 1.4.

13.11 – Leo

Cyclic homology II.

Reference: Kaledin's lectures 3–4, §§1.5–1.7.

20.11 – Anand 

Hochschild cohomology and deformation theory.

Reference: Kaledin's lectures 5–6, §§1.2–1.3.

28.11 (Friday!) – Anand 

Deformations of Db(X).

Reference: Toda's 2005 preprint.

Additional topics and references

• Noncommutative Hodge theory in Edinburgh: https://hodge.maths.ed.ac.uk/?page_id=229.

• Hochschild cohomology of gentle algebras.

  • Chaparro, Schroll, Solotar, Suarez-Alvarez, "The Hochschild cohomology and the Tamarkin-Tsygan calculus of gentle algebras".

• Homological dimension of f.d. monomial algebras.

  • Igusa, Zacharia. "On the cyclic homology of monomial relation algebras".

  • Igusa, Zacharia. "Syzygy pairs in a Monomial Algebra".

• Mirror Symmetry and HMS.

  • Kontsevich’s two papers where HMS was proposed, one is his ICM talk and the other is "Homological algebra of mirror symmetry".

  • Two papers on how HMS implies the numerical MS by Ganatra, etc.: https://arxiv.org/pdf/1304.7312, https://arxiv.org/pdf/1510.03839.