Advait Phadnis (He/Him)

Current Research:

Sheaf models for equivariant stable homotopy theory

Equivariant algebra deals with algebraic structures (such as rings, modules, algebras, etc.) with an action of a group. The study of equivariant algebra has direct applications in representation theory to understanding representations of groups, both finite and infinite. G-Mackey functors, which are objects central to equivariant algebra, are generalisations of the real (or complex) representation rings.

There exist results due to Greenlees and May (1995) that classify the category of rational G-Mackey functors, for a finite group G, using module categories, which are purely algebraic. However, recent work by Barnes and Sugrue (2022) has established the existence of an equivalence between the category of rational G-Mackey functors and the category of rational Weyl-G-sheaves in the case where G is a profinite group. Due to the topological properties of sheaves, this demonstrates that the topology of the group G gains importance when G becomes infinite. I intend to further the equivariant study of profinite groups and study the monoidal and homotopical properties of (improvements to) these equivalences, which at the moment remain mostly unmentioned in the literature.

Biography:

I graduated with an "M.Sc. in Mathematics" degree from Chennai Mathematical Institute in 2024. My M.Sc. thesis, titled Stable Homotopy Theory, was supervised by Dr David Barnes during which I studied and got very interested in stable homotopy theory and its related areas. I am currently working in equivariant stable homotopy theory as part of my PhD project which involves studying spectra (objects of interest in stable homotopy theory) along with their symmetries.

Research Interests:

• Equivariant Algebra
• Stable Homotopy Theory
• Category Theory

Supervisors:

Principal supervisor: Dr David Barnes

Secondary supervisor: Dr Hannah Mitchell