The Advanced Training in Mathematics Schools (ATM Schools) were launched by the National Board for Higher Mathematics (NBHM) in May 2004. The purpose of these schools is to provide training in core subjects in Mathematics to Ph.D. students, young researchers and teachers. The emphasis in these schools is on learning mathematics by doing it. IIT Bombay and TIFR have jointly established the National Centre for Mathematics (NCM) in 2011. The instructional schools and workshops which were earlier planned by an NBHM committee on ATM Schools are now being organized under the supervision of the Apex Committee of the NCM.

Hochschild homology and cohomology of associative algebras is a classical topic which is pursued on its own as well as for its connection with algebraic topology, homotopy theory and K-theory. Cyclic homology is connected to noncommutative geometry and was initiated independently by Connes and Tsygan. In the world of free loop spaces, both of these become connected in a nice manner.

Dates: 4 Dec 2023 to 16 Dec 2023

For more details please click : https://www.atmschools.org/school/2023/AIS/hh

Speakers and Syllabus

 (co)homology, cyclic homology and their refinements as well as explore some of the connections of these to other areas.

Target Audience:
The School is meant for PhD students working in Topology. The topics may also interest students working in algebra and algebraic geometry especially those who are interested in homotopy theoretic techniques in these subjects. The participants must be familiar with basic algebraic topology (basic knowledge of homotopy, fundamental groups, homology theory and cohomology theory) as well as basic algebra (rings, modules, projective modules, tensor products).

Names and affiliations of possible speakers.
(1) Surojit Ghosh, IIT Roorkee; Initials – SG
(2) Somnath Basu, IISER Kolkata; Initials – SoB
(3) Samik Basu, ISI Kolkata; Initials – SaB
(4) Charanya Ravi, ISI Bangalore; Initials – CR
(5) Anita Naolekar, ISI Bangalore; Initials – AN
(6) Rekha Santhanam, IIT Mumbai; Initials – RS

Names and affiliations of potential tutors.
(1) Mukilraj K, IISER Kolkata; Initials – MK
(2) Sandip Samanta, IISER Kolkata; Initials – SS
(3) Aritra Bhowmick, IISER Kolkata; Initials – AB
(4) Abhinandan Das, ISI Kolkata; Initials – AD

2. Syllabus
Lectures and tutorials are either 90 minutes and 60 minutes respectively.
Topic A – Hochschild homology and cohomology, properties, structures
Hochschild homology arising from deformation of algebraic structures and via group homology; bar resolution; basic properties including Kunneth formula, Morita invariance.
Topic B – Equivariant (co)homology, classifying space
Construction of classifying space (for circle bundles), S1-equivariant homology, Borel construction revisited.
Topic C – Cyclic homology, connections to de Rham theory & K-theory
S1-spaces vs cyclic sets, Hochschild-Kostant-Rosenberg Theorem, Goodwillie’s result on Hochschild homology of chains on based loop space being identifiable to the cohomology of free loop space.
Topic D – Topological Hochschild homology
Definition and properties of spectra, infinity algebras and operads; topological refinement of Hochschild homology; specific computations of THH for Zp and Z.
Topic E – Topological cyclic homology
Connes B operator and cyclic homology; definition of cyclotomic spectra; definition and properties of topological cyclic homology; Denis trace map and connections to algebraic K-theory.
We plan to hold a couple of special lectures as well as some participants lectures. Topics (A), (B) and (C) will be covered in the first week, while (D) and (E) will be covered in the second week. The tutorial in the second week may also be used for participants lectures.

Convener(s)