NCMW - Recent advances in stable homotopical computations (2025)

Speakers and Syllabus

Names and affiliations of possible speakers.
    (1) Samik Basu, ISI Kolkata; Initials - SaB (6 hours)
    (2) Somnath Basu, IISER Kolkata; Initials - SoB (4.5 hours)
    (3) David Blanc, University of Haifa, Israel; Initials - DB (6 hours)
    (4) Surojit Ghosh, IIT Roorkee; Initials - SG (4.5 hours)
    (5) Rekha Santhanam, IIT Mumbai; Initials - RS (6 hours)

Names and affiliations of potential tutors.
    (1) Akashdwip Biswas ISI Kolkata; Initials - AB
    (2) Abhinandan Das, ISI Kolkata; Initials - AD
    (3) Mukilraj K, IISER Kolkata; Initials - MK
    (4) Ankit Kumar, IIT Roorkee; Initials - AK

Speaker

Detailed Description

 

Topic A: Classical techniques in homotopy Theory. This series will introduce background materials in basic homotopy theory which includes:

  1. Introduction to spectral sequences
  2. Operadic multiplications
  3. Simplicial objects
  4. Homotopy theory I: the idea of model categories
  5. Homotopy theory II: the idea of ∞-categories.

Somnath Basu

Topic B: Stable homotopy theory. The lectures will begin by motivating spectra through the Freudenthal suspension theorem and discussing properties of spectra. Further, the speaker will discuss the connection between spectra and generalised cohomology, smash product of spectra, Spanier-Whitehead duality, Steenrod algebra, and Adams spectral sequences.

Rekha Santhanam

Topic C: Synthetic Spectra. Synthetic spectra can be considered as a categorical home for the Adams spectral sequence. In this series, the speaker will discuss the construction of synthetic spectra and their properties. Fur- thermore, Spherical sheaves on additive sites, Adams-type homology theories and sheaves, and The t-structure on synthetic spectra will be discussed.

Surojit Ghosh

Topic D: Ideas from motivic homotopy theory. In this series, the speaker will begin with the basic language and results of motivic homotopy theory. Then, the special and generic fibres of some E-based synthetic spec- tra will be analyzed. Finally, the construction of the cellular motivic stable homotopy category will be presented alongside its surprising connections to synthetic spectra. These ideas bridge synthetic and motivic homotopy theory in unexpected ways.

David Blanc

Topic E: Filtered models. This topic explores the rich interplay between synthetic spectra and filtered spectra. The geometric intuition behind de- formations in homotopy theories will be elucidated, providing a conceptual framework for understanding these objects. A significant focus will be on how the synthetic perspective enables the construction of novel spectral sequences that generalize beyond the classical Adams spectral sequence. The synthetic lifts of spectra will be described in detail, highlighting their computational power.

Samik Basu

Topic F: Multiplicative Structures on Moore Spectra. The final topic presents a striking application of synthetic spectra: the existence of enhanced multiplicative structures on Moore spectra beyond what classical methods predict. This result demonstrates the versatility of synthetic techniques in resolving long-standing questions in stable homotopy theory.

 

Note: Topic A provides essential background material for this workshop. Each speaker will deliver one lecture on this topic, as indicated in the timetable below.

 

 

 

Time Table

 

 

10-11:30

11:45-1:15

1:15-2:45

2:45-4:15

4:30-5:30

26/12

A1

SG

A2

SoB

L

A3

RS

Tutorial

SG+AK+MK

27/12

B1

SoB

A4

RS

U

A5

DB

Tutorial

SoB+AK+AB

28/12

B2

SoB

C1

RS

N

B3

SaB

Tutorial

RS+AK+AD

29/12

D1

SG

E1

DB

C

F1

SaB

Tutorial

DB+MK+AB

30/12

C2

RS

F2

SaB

H

E2

DB

Tutorial

SaB+MK+AD

31/12

E3

DB

D2

SG

 

F3

SaB

Tutorial

SaB+AB+AD

 

References

  1. Barnes, David; Roitzheim, Constanze (2020) Foundations of stable homotopy theory. Cambridge Stud. Adv. Math., 185 Cambridge University Press, Cambridge, vi+423 pp.
  2. Adams, J. F. (1974) Stable homotopy and generalised homology. Chicago Lectures in Math. University of Chicago Press, Chicago, IL, 1995. x+373 pp.
  3. Pstragowski, P. (2023). Synthetic spectra and the cellular motivic category. Inven- tiones Mathematicae, 232(2), 553–681.
  4. Lawson, T. (2024). Filtered spaces, filtered objects. Preprint. arXiv.
  5. Burklund, R., Hahn, J., & Senger, A. (2023). On the boundaries of highly connected, almost closed manifolds. Acta Mathematica, 231(2), 205–344.
  6. Burklund, R. (2022). Multiplicative structures on Moore spectra. Preprint. arXiv.
  7. Gheorghe, B., Isaksen, D. C., Krause, A., & Ricka, N. (2022). Motivic modular forms. Journal of the European Mathematical Society, 24(10), 3597–3628.
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