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.