https://topogeoiiitd.github.io/Online research talks and lecture series in topology, geometry, geometric analysis, and related areas at IIIT Delhi.https://topogeoiiitd.github.io/series/ruben/#RL-3https://topogeoiiitd.github.io/series/ruben/#RL-3Thu, 24 Sep 2026 08:30:00 GMTRuben Louis. We will present a recent result extending the Courant correspondence to Lie–Rinehart algebras over (C^infty)-rings, based on joint work with Eugene Lerman. https://topogeoiiitd.github.io/series/ruben/#RL-2https://topogeoiiitd.github.io/series/ruben/#RL-2Thu, 10 Sep 2026 08:30:00 GMTRuben Louis. We will introduce the notion of (C^infty)-rings and study Poisson structures in this setting. We will then discuss their relationship with singular foliations and, more generally, with Lie-Rinehart algebras. https://topogeoiiitd.github.io/series/ruben/#RL-1https://topogeoiiitd.github.io/series/ruben/#RL-1Thu, 03 Sep 2026 08:30:00 GMTRuben Louis. We will introduce the notions of regular and singular foliations, discuss their main features, and present some motivating examples. https://topogeoiiitd.github.io/series/non-commutative-witt-vectors-characteristic-map-and-pairing/#RB-2https://topogeoiiitd.github.io/series/non-commutative-witt-vectors-characteristic-map-and-pairing/#RB-2Thu, 27 Aug 2026 03:30:00 GMTRamyak Bilas. If one takes the view of Witt vectors as the (t)-adic completion of (End_0(R)), then one can essentially circumvent the use of the trace property and the Witt vectors with coefficients used in DKNP, and recover every property of the Witt vectors as a consequence of the K-theoretic properties of (End_0(R)). This second talk will focus on an alternate definition of Witt vectors and the characteristic map, and additionally give an external pairing of Witt vectors and (Nil_0(R)). https://topogeoiiitd.github.io/series/non-commutative-witt-vectors-characteristic-map-and-pairing/#RB-1https://topogeoiiitd.github.io/series/non-commutative-witt-vectors-characteristic-map-and-pairing/#RB-1Tue, 25 Aug 2026 03:30:00 GMTRamyak Bilas. For a commutative ring (A), the ring of Witt vectors (W(A)) has been useful in number theory, algebraic topology, and algebraic K-theory. Classical results of Almkvist, Stienstra, Grayson, and Weibel demonstrate that the characteristic map bridges the ring (End_0(R)) to its (t)-adic completion (W(A)), and the nil groups (Nil_0(A)) are modules over (W(A)). Recently Dotto, Krause, Nikolaus, and Patchkoria (DKNP) gave an algebraic description of Witt vectors for a non-commutative… https://topogeoiiitd.github.io/series/existence-of-arbitrary-genus-maxfaces-in-the-lorentz-minkowski-space/#AD-2https://topogeoiiitd.github.io/series/existence-of-arbitrary-genus-maxfaces-in-the-lorentz-minkowski-space/#AD-2Tue, 21 Apr 2026 10:30:00 GMTAnu Dhochak. In this talk, we will discuss the existence of a 1-parameter family of arbitrary genus maxfaces with an arbitrary number of spacelike ends and infinitely many swallowtails. Our primary tool of construction will be node-opening technique. All maxfces of these families are embedded in a wider sense. We will also see families of Lorentzian Costa and Costa-Hoffman-Meeks surfaces, as well as periodic surfaces of infinite genus, as examples. https://topogeoiiitd.github.io/series/existence-of-arbitrary-genus-maxfaces-in-the-lorentz-minkowski-space/#AD-1https://topogeoiiitd.github.io/series/existence-of-arbitrary-genus-maxfaces-in-the-lorentz-minkowski-space/#AD-1Tue, 14 Apr 2026 10:30:00 GMTAnu Dhochak. In this talk, we will discuss minimal and maximal surfaces in three-dimensional Euclidean and Lorentz-Minkowski space respectively. We will explore how they are related and how to construct them. In particular, we will cover Björling problem and Weierstrass-Ennerper representations for them. I will also define the term maxface on which my next two lectures will be. https://topogeoiiitd.github.io/series/diffeology/#DM-3https://topogeoiiitd.github.io/series/diffeology/#DM-3Tue, 24 Mar 2026 10:30:00 GMTDavid Miyamoto. Having completed the introductions, we will summarize some recent progress in using diffeology to understand geometry. Among these will include: a generalization of the Serre-Swan theorem using diffeological vector bundles; an integration of Lie algebroids to diffeological groupoids, even when no integrating Lie groupoid exists; and a diffeological approach to the question of geometric quantization. https://topogeoiiitd.github.io/series/diffeology/#DM-2https://topogeoiiitd.github.io/series/diffeology/#DM-2Tue, 17 Mar 2026 10:30:00 GMTDavid Miyamoto. We will introduce two other approaches to singular and infinite-dimensional spaces, namely Lie groupoids (a step towards differentiable stacks), and convenient infinite-dimensional manifold structures. We will then explore how relevant diffeological spaces appear in both contexts. https://topogeoiiitd.github.io/series/diffeology/#DM-1https://topogeoiiitd.github.io/series/diffeology/#DM-1Tue, 10 Mar 2026 10:30:00 GMTDavid Miyamoto. We will introduce the category of diffeological spaces, and give several examples: the cusp (x^2 = y^3), as a subset of (R^2); the irrational tori (R/(Z + alphaZ)), for irrational (alpha); and the diffeomorphism group of the real line, (Diff(R)). These will illustrate how the category of diffeological spaces is closed under taking, respectively, subsets, quotients, and mapping spaces, and help give a sense of the data contained within a diffeological structure. https://topogeoiiitd.github.io/series/knot-theory/#SK-3https://topogeoiiitd.github.io/series/knot-theory/#SK-3Tue, 24 Feb 2026 10:30:00 GMTSeongjeong Kim. The goal of this lecture is to explain how closed 3-manifolds can be constructed via surgery along framed links. We briefly review the notion of handle decompositions of manifolds and then describe how framed knots and links give rise to 3-manifolds. https://topogeoiiitd.github.io/series/knot-theory/#SK-2https://topogeoiiitd.github.io/series/knot-theory/#SK-2Tue, 17 Feb 2026 10:30:00 GMTSeongjeong Kim. We introduce the Kauffman bracket and the Jones polynomial, which arise naturally in quantum topology and statistical mechanics. As an extension of the Jones polynomial, we also introduce the colored Jones polynomial. https://topogeoiiitd.github.io/series/knot-theory/#SK-1https://topogeoiiitd.github.io/series/knot-theory/#SK-1Tue, 10 Feb 2026 10:30:00 GMTSeongjeong Kim. We introduce basic definitions and fundamental results in knot theory. One of the simplest knot invariants is the coloring invariant, which can detect certain nontrivial knots. As a generalization of the coloring invariant, we define an algebraic structure called a quandle. https://topogeoiiitd.github.io/series/on-the-groups-of-self-homotopy-equivalences/#SS-3https://topogeoiiitd.github.io/series/on-the-groups-of-self-homotopy-equivalences/#SS-3Tue, 03 Feb 2026 13:30:00 GMTSandip Samanta. Here we will briefly see the ingredients of rational homotopy theory, like rationalization of spaces, Sullivan (minimal) models, LS category, etc, and compute some basic examples. Finally, we will relate these to our case of self-homotopy equivalences, like how E(X) and E(X_P) are related, some other results and possible open directions. https://topogeoiiitd.github.io/series/on-the-groups-of-self-homotopy-equivalences/#SS-2https://topogeoiiitd.github.io/series/on-the-groups-of-self-homotopy-equivalences/#SS-2Tue, 27 Jan 2026 12:30:00 GMTSandip Samanta. We survey explicit computations of the group of self-homotopy equivalences E(X) for key spaces in homotopy theory. This includes results for pseudo-projective planes, with emphasis on lens spaces, as well as for Moore spaces M(G, n) and twisted Eilenberg-MacLane spaces. We also discuss the structure of E(X) for real, complex, and quaternionic projective spaces. Additionally, we present computations of E_H(X) for rank one H-spaces, specifically S^1, S^3, S^7, RP^3, RP^7, and for… https://topogeoiiitd.github.io/series/on-the-groups-of-self-homotopy-equivalences/#SS-1https://topogeoiiitd.github.io/series/on-the-groups-of-self-homotopy-equivalences/#SS-1Tue, 20 Jan 2026 10:30:00 GMTSandip Samanta. We provide an overview of the group of self-homotopy equivalences, E(X) = [f] mid f: X simeq X, focusing on key subgroups such as those acting trivially on homotopy or cohomology groups. We summarize the known structures of E(X) for spaces like spheres and low-dimensional complexes. The discussion includes exact sequences relating E(X) for products and wedge sums. Finally, we explore various generalizations obtained by considering free-, fiber-, equivariant-, and H-homotopy… https://topogeoiiitd.github.io/series/coarse-cohomology/#SR-5https://topogeoiiitd.github.io/series/coarse-cohomology/#SR-5Wed, 03 Dec 2025 10:00:00 GMTSergio Romero Alba. It is well known that the degree-zero singular homology of a topological space is determined by its path-connected components. Similarly, degree-zero Alexander-Spanier cohomology can be computed from its connected components. We ask whether, for a topological bornological space X, there exists a similar connection between HB^1(X) and the ``components at infinity of X''. This question naturally leads us to give meaning to notions such as rays in X, (path-)connected… https://topogeoiiitd.github.io/series/coarse-cohomology/#SR-4https://topogeoiiitd.github.io/series/coarse-cohomology/#SR-4Wed, 26 Nov 2025 11:30:00 GMTSergio Romero Alba. The category TopBorn of topological bornological spaces is defined, as in [2, §7.1.1]. We propose an invariant HB^* on these spaces, which we call bornological cohomology, and show that it can be determined from the Alexander-Spanier cohomology at infinity. We compute several examples and enrich HB^* with a product, endowing it with the structure of a differential graded algebra. https://topogeoiiitd.github.io/series/coarse-cohomology/#SR-3https://topogeoiiitd.github.io/series/coarse-cohomology/#SR-3Wed, 19 Nov 2025 11:30:00 GMTSergio Romero Alba. We refine the structure of HX^*(X), enhancing it to an algebra by endowing it with a product. Much like the cup product [3, §3.2] allows for a finer distinction between topological spaces, it is natural to seek products in coarse cohomology that sharpen the information it provides. A first strategy is to adapt the cup product to this context, but the resulting product in coarse cohomology turns out to be trivial. However, not all is lost: in such cases, one can define… https://topogeoiiitd.github.io/series/coarse-cohomology/#SR-2https://topogeoiiitd.github.io/series/coarse-cohomology/#SR-2Wed, 12 Nov 2025 11:30:00 GMTSergio Romero Alba. The coarse cohomology HX^*(X) of a space X is introduced as a functor from the coarse category, and its coarse invariance is discussed. We then particularize to metric spaces and study a few examples. https://topogeoiiitd.github.io/series/coarse-cohomology/#SR-1https://topogeoiiitd.github.io/series/coarse-cohomology/#SR-1Wed, 05 Nov 2025 11:30:00 GMTSergio Romero Alba. We review, through examples, the foundational philosophy of Geometric Group Theory, where each group G with a finite generating set S is associated with a metric space (G, dS), translating algebraic properties of the group into geometric features of the space. Although this assignment initially depends on the choice of S, such dependence vanishes when viewing (G, dS) "from infinitely far away". This motivates the introduction of coarse structures. https://topogeoiiitd.github.io/series/spin-geometry-and-scalar-curvature-rigidity/#LS-4https://topogeoiiitd.github.io/series/spin-geometry-and-scalar-curvature-rigidity/#LS-4Wed, 29 Oct 2025 11:30:00 GMTLukas Schönlinner. In this talk, we will discuss another generalization of Llarull's theorem in even dimensions where we allow cone-like singularities in the domain. To do so, we will discuss an abstract functional analytic setup that enables us to show that the Dirac operator on these singular spaces is Fredholm. Moreover, we will apply an index formula of Chou together with a deformation argument to show that the index is non-zero. This is joint work with Simone Cecchini, Bernhard Hanke, and… https://topogeoiiitd.github.io/series/spin-geometry-and-scalar-curvature-rigidity/#LS-3https://topogeoiiitd.github.io/series/spin-geometry-and-scalar-curvature-rigidity/#LS-3Wed, 22 Oct 2025 11:30:00 GMTLukas Schönlinner. The theorem of Llarull is a famous rigidity result in scalar curvature which states the following: Let (M,g) be a closed connected n-dimensional smooth Riemannian spin manifold with ngeq2 even and scal_g geq n(n-1). If fcolon Mto S^n is a smooth 1-Lipschitz map of non-zero degree then f is a Riemannian isometry. We will discuss a generalization of this theorem, which states that it is sufficient to assume that g is a metric of Sobolev regularity W^1,p with p> n and that f is… https://topogeoiiitd.github.io/series/spin-geometry-and-scalar-curvature-rigidity/#LS-2https://topogeoiiitd.github.io/series/spin-geometry-and-scalar-curvature-rigidity/#LS-2Thu, 16 Oct 2025 09:30:00 GMTLukas Schönlinner. In this talk, we will introduce the spin groups and discuss the properties necessary to define spinable manifolds. We will then associate a canonical connection, the so-called spin connection, with every spin manifold M. This connection induces a Dirac bundle structure on M. The associated Dirac operator satisfies the Schrödinger-Lichnerowicz formula, a special case of the Weitzenböck formula. Additionally, we will discuss the Atiyah-Singer index theorem, which links the… https://topogeoiiitd.github.io/series/spin-geometry-and-scalar-curvature-rigidity/#LS-1https://topogeoiiitd.github.io/series/spin-geometry-and-scalar-curvature-rigidity/#LS-1Wed, 08 Oct 2025 11:30:00 GMTLukas Schönlinner. We will define Clifford algebras, Clifford bundles and Dirac bundles. Every Dirac bundle carries an associated Dirac operator and we will prove the Weitzenböck formula in this general setup, which is one of the most important tools in spin geometry. https://topogeoiiitd.github.io/series/invariant-theory-of-hamiltonian-mechanics-and-related-numerical-analysis/#OC-3https://topogeoiiitd.github.io/series/invariant-theory-of-hamiltonian-mechanics-and-related-numerical-analysis/#OC-3Wed, 01 Oct 2025 11:30:00 GMTOscar Cosserat. To finish, we will introduce pre-Lie algebras to solve Hamilton-Jacobi equation at arbitrary order and obtain Poisson integrators discretising Hamiltonian dynamics in that way. Geometric properties will be gathered and explained. This last session contains numerical simulations on concrete examples such as Lotka-Volterra systems and rigid body dynamics. To conclude, we will summarize the results of this geometric approach to numerical analysis and give several perspectives. https://topogeoiiitd.github.io/series/invariant-theory-of-hamiltonian-mechanics-and-related-numerical-analysis/#OC-2https://topogeoiiitd.github.io/series/invariant-theory-of-hamiltonian-mechanics-and-related-numerical-analysis/#OC-2Wed, 24 Sep 2025 11:30:00 GMTOscar Cosserat. We start by defining groupoids and relate them to Poisson manifolds through the concept of symplectic groupoid. Geometrical tools such as Lagrangian bisections will be defined and explained. In the last part of this session, we introduce so-called "bi-realisations". Those are local forms of symplectic groupoids. On bi-realisations, we will explain how a Hamilton-Jacobi allows to approximate Hamiltonian dynamics on Poisson manifolds. https://topogeoiiitd.github.io/series/invariant-theory-of-hamiltonian-mechanics-and-related-numerical-analysis/#OC-1https://topogeoiiitd.github.io/series/invariant-theory-of-hamiltonian-mechanics-and-related-numerical-analysis/#OC-1Wed, 17 Sep 2025 11:30:00 GMTOscar Cosserat. We define Poisson manifolds and review the notions we need for this lecture : Hamiltonian vector fields, first integrals, Casimir functions, symplectic foliation... The end of this first lecture is devoted to explain why traditional numerical tools do not straightforwardly apply to Hamiltonian systems on Poisson manifolds. https://topogeoiiitd.github.io/series/existence-of-higher-extremal-kahler-metrics-on-minimal-ruled-surface/#RSS-3https://topogeoiiitd.github.io/series/existence-of-higher-extremal-kahler-metrics-on-minimal-ruled-surface/#RSS-3Wed, 10 Sep 2025 09:00:00 GMTRajas Sandeep Sompurkar. In this final talk our goal is to state the main results of the above mentioned research work. We will discuss about the various notions of canonical Kähler metrics on a compact Kähler manifold. These are roughly speaking Kähler metrics that satisfy some special curvature properties, and the three well-known and well-studied notions of canonical Kähler metrics are Kähler-Einstein metrics, constant scalar curvature Kähler (cscK) metrics and Calabi’s extremal Kähler… https://topogeoiiitd.github.io/series/existence-of-higher-extremal-kahler-metrics-on-minimal-ruled-surface/#RSS-2https://topogeoiiitd.github.io/series/existence-of-higher-extremal-kahler-metrics-on-minimal-ruled-surface/#RSS-2Wed, 03 Sep 2025 09:00:00 GMTRajas Sandeep Sompurkar. In this talk we will see the definition of a holomorphic vector bundle on a compact Kähler manifold and we will also see some of the properties of it. For the purpose of this series of talks we would be mostly interested in holomorphic line bundles (which are just rank one holomorphic vector bundles), but we will still see some standard examples like the holomorphic tangent and cotangent bundles and most importantly for us the canonical and anticanonical line bundles…