1. 3-manifolds with finite fundamental group: We discuss Hopf’s list of all groups which act freely and orthogonally on $S^3$.
2. Groups with periodic cohomology: We discuss work of Smith, Cartan-Eilenberg, Milnor, and Davis which give necessary conditions on for a finite group to act freely on $S^n$. We state a theorem of Madsen-Thomas-Wall which gives necessary and sufficient conditions for a group to act freely on $S^n$ for some $n$.
3. The Swan finiteness obstruction: We discuss work of Swan which characterizes the finite groups which are the fundamental groups of finite dimensional complexes whose universal cover is homotopy equivalent to $S^3$. We discuss work of Milgram, Davis, and Nicholson which studies the question of determining which finite groups are the fundamental groups of finite complexes whose universal cover is homotopy equivalent to $S^3$.

Reference: The book of Davis-Milgram “A Survey of the Spherical Space Form Problem”, available on J. Davis’ webpage.

Venue: Zoom Link (Passcode- Jyph8g)
Time: 8 pm Indian Time (9:30 pm USA time)
Talk 1 Notes
Talk 2 Notes

1. (Talk 1) Virtual knots, parity and invariants for virtual knots: Virtual knot theory is one of the generalizations of knot theory. In virtual knot theory, virtual knots and links can be considered as circles smoothly embedded in thickened surfaces or diagrams on oriented surfaces. It follows that one can study virtual knots by using several properties derived from surfaces. One of the important tools to study virtual knots is the parity. In particular, it is known that the parity is closely related to the underlying surfaces of virtual knots. In this talk, we introduce virtual knots, flat knots, free knots, and basic theorems. We introduce the definition of parity and the simplest application of parity for virtual and free knots.
2. (Talk 2) Classification of knots in $S_{g} \times S^{1}$ with small number of crossings: In knot theory, not only classical knots, which are embedded circles in $S^{3}$ up to isotopy, but also knots in other 3-manifolds are interesting for mathematicians. In particular, virtual knots, which are knots in thickened surface $S_{g} \times [0,1]$ with an orientable surface $S_{g}$ of genus $g$, are studied and they provide interesting properties. In this talk, we will talk about knots in $S_{g} \times S^{1}$ where $S_{g}$ is an oriented surface of genus $g$. We introduce basic notions and properties for them. In particular, for knots in $S_{g} \times S^{1}$ one of important information is “how many times a half of a crossing turns around $S^{1}$”, and we call it the winding parity of a crossing. We extend this notion more generally and introduce a topological model. In the end, we apply it to classify knots in $S_{g}\times S^{1}$ with a small number of crossings.

Venue: Zoom Link (Passcode- 1mqXVj)
Time: 2:30 pm Indian Time (5 pm China time)
Talk 1 Notes
Talk 1 Notes

The study of free loop spaces, particularly their homology, has broad applications in topology and geometry. This series of talks will describe a new approach to the homology of free loop spaces via topological co-Hochschild homology (coTHH), a recently developed invariant of coalgebras in spectra. I will introduce coTHH and discuss how computational tools for coTHH yield new computations of the homology of free loop spaces. In particular, I will introduce a spectral sequence computing the homology of free loop spaces which has an interesting algebraic structure. These talks will include work that is joint with Anna Marie Bohmann and Brooke Shipley.

Venue: Zoom Link (Passcode- T1uU22)
Time: 6 pm Indian Time (8:30 am US Eastern time)

The ring of $\mathbb{Z}$-valued functions on the symmetric group $\Sigma_k$ may be described in a simple manner using certain generators and relations. This is called the Varchenko-Gelfand ring. We will discuss how this ring appears in various avatars as the cohomology ring of certain types of spaces.

Venue: LHC 211
Time: 2 pm

In this talk, I will introduce Borsuk graphs, (circular) chromatic numbers, and anti-Vietoris-Rips complexes. I will discuss the homotopy types of the anti--VR complexes and metric thickenings built on $S^n$ up to certain scales. Using topological contradictions, I will show the absence of graph homomorphisms from the Borsuk graph on $S^n$ to the Borsuk graph on $S^1$ at a certain scale. This gives us a new proof of the lower bound $n+2$ of the chromatic numbers of Borsuk graphs on $S^n$. This is joint work with Henry Adams, Alex Elchesen, and Michael Moy.

Venue: Google Meet
Time: 7:30 pm Indian Time (10 am USA time)