Algebraic K-theory invariants are notoriously difficult to compute directly, making trace methods an essential and highly successful tool. In recent years, motivated by the generalized Hilbert's third problem, new combinatorial K-theories have been introduced utilizing formalisms like assemblers. While specific trace maps have recently been constructed for higher scissors congruence groups in assembler K-theory, the broader trace method framework remains incomplete. Drawing on joint work with Sanjana Agarwal extending the notion of traces to assemblers, this talk focuses on defining a universal object that serves as the analogue of Hochschild homology in this setting. We will explore the construction of this target object and demonstrate that previously constructed trace maps factor through this "Dennis trace" map.

Venue: Zoom Link
Time: 7 pm IST (17th, 19th & 23rd March 2026)
Notes

The axioms of a quandle are algebraic translations of the three Reidemeister moves in knot theory. Quandles are commonly used to construct algebraic invariants in knot theory. In the first half of the talk, I will discuss basic quandle theory followed by homology theories related to quandles and their generalizations in relation to classical knot theory. The second half of the talk will focus on algebraic invariants in the recently-introduced field of multi-virtual knot theory, a generalization of virtual knot theory. After discussing preliminary ideas related to the subject, I will generalize the usual coloring invariants in classical knot theory to the multi-virtual setting using operator quandles. The talk will showcase results obtained with several co-authors.

Venue: G08 [offline talk]
Time: 3-4 pm IST
Slides

Donaldson’s program on the adiabatic limit of $K3-$fibered $G_2-$manifolds relates associative submanifolds to certain weighted trivalent graphs on the base called gradient cycles. A key ingredient in this relation, near a trivalent vertex, is the existence and uniqueness of an associative pair of pants in the $G_2-$manifold formed as a product of a $K3$ surface and the Euclidean $3-$plane. This is known as the Donaldson–Scaduto conjecture. Although this conjecture remains open, a local version replacing the $K3$ surface with an ALE or ALF hyperkähler $4-$manifold of type $A_2$, has been shown to exist by Esfahani and Li. In this talk, I will discuss our joint work on proving the uniqueness, showing that no other associative pair of pants satisfies this local version of the conjecture.

Venue: LHC G09, IISER Kolkata
Time: 3 PM IST

Rational homotopy theory studies the free parts of homotopy and homology groups. By ignoring torsion, the remaining structure can be represented by algebraic models. This approach works particularly well for simply connected spaces with cohomology of finite type. The Sullivan model, a special type of differentially graded algebra, is one of the central algebraic models in rational homotopy theory. In this lecture series, I will introduce the construction of the Sullivan model, explain how it encodes topological information, and describe how to reconstruct a space from such an algebra. My current research focuses on understanding when the realization of a non-simply-connected Sullivan algebra preserves cohomology. If time permits, I will also discuss another related topic: a space is called formal if its rational cohomology ring itself can serve as its algebraic model. Although a sphere bundle over a compact formal manifold may not be formal, its formality can be determined arithmetically using the Bianchi-Massey tensor introduced by Crowley and Nordström.

Venue: Zoom Link
Time: 11:30 AM-12:30 PM IST (20th January 2026)
3-4 PM IST (22th January 2026)
11:30 AM-12:30 PM IST (26th January 2026)
Slides