# Young Topologists' Meet at CMI

## Chennai Mathematical Institute

This is a 3-day meeting of mathematics research scholars interested in geometry and topology. The goal of this event is to encourage emerging geometers and topologists to share and exchange active research ideas in their fields. In addition to participants' talks there will be three mini-courses. The venue for the meeting is Chennai Mathematical Institute.

The registration closes May 15th.

The purpose of this meeting is to bring together graduate students in topology and give them the opportunity to give talks, expose them to new topics, and get to know other graduate students in their field. The schedule will consist of three mini-courses and talks given by students. Participants are invited to give talks, 30 minutes long, about topics they have been studying. The subject need not be original research, but simply something the speaker enjoys and wishes to share. Talks should, in particular, be accessible to an audience of graduate students of varying levels.

If you are interested in giving a talk then do register as soon as possible. As of now we have around 15 spots available; depending on the number of requests and the subject areas the organizers will finalize speakers by early June. The selected participants can send their title and abstract after the confirmation.

## Mini-courses

Introduction to contact and symplectic topology
Speaker: Dheeraj Kulkarni (IISER, Bhopal)
Abstract: The study of symplectic and contact structures has grown in into a well-established area by itself. We plan to give a brief survey of some important milestones in the development of this field in the last 50 years. However, the main of this mini-course is to explain how symplectic and contact structures arise naturally in many contexts. We will also discuss topological properties of these structures. Finally, we will also present the current state of the art.

Topology of toric manifolds
Speaker: V. Uma (IIT M)
Abstract: We shall introduce quasitoric manifolds and small covers defined by Davis and Januszkiewicz and describe their topological invariants like cohomology ring , fundamental group and the topological $K$-ring.

LS-category and the topology of motion planning
Speaker: Tulsi Srinivasan (Ashoka University)
Abstract: This mini course will introduce the Lusternik-Schnirelmann category and the related notion of topological complexity. We will look at the history of the LS-category, the reason it continued to be of mathematical interest, and its current applications to motion planning.

## Schedule

### List of speakers

#### Kashyap Rajeevsarathy (IISER Bhopal)

Title: Geometric realizations of cyclic actions on surfaces
Abstract: Let $$\mathrm{Mod}(S_g)$$ denote the mapping class group of the closed orientable surface $$S_g$$ of genus $$g\geq 2$$. Given a finite subgroup $$H$$ of the mapping class group let $$\mathrm{Fix}(H)$$ be the set of fixed points points induced by the action of $$H$$ on the corresponding Teichmüller space. In this talk, we give an explicit description of $$\mathrm{Fix}(H)$$ when $$H$$ is cyclic, thereby providing a complete solution to the Modular Nielsen Realization Problem for this case. Among other applications of these realizations, we derive an intriguing correlation between finite order maps and the filling systems of surfaces. Finally, we will briefly discuss some examples of realizations of two-generator finite abelian actions.

#### Apeksha Sanghi (IISER Bhopal)

Title: Model Categories and Homotopy Colimits
Abstract: In this talk, we will discuss about Model Categories and some examples of Model Categories. Then we will talk about homotopy colimits and give some examples of homotopy colimits. Next, we will see the construction of homotopy colimits for simplicial spaces.

#### Lokenath Kundu (IISER Bhopal)

Title: Complex oriented cohomology theories and formal group laws.
Abstract: In this talk we will talk about connection between generalized cohomology theories and $$\Omega$$ spectra and we will look at some examples. Next we define Complex oriented cohomology theories. Finally we will relate the Complex oriented cohomology theory with the formal group laws .

#### M V Ajay Kumar Nair (IISER Bhopal)

Title: Surface homeomorphisms and Dehn-Lickorish theorem
Abstract: In this talk, my aim will be to prove the famous Dehn-Lickorish Theorem which says that every surface homeomorphism is isotopic to the product of Dehn twists. We will start by defining Mapping Class groups and the special class surface homeomorphisms called Dehn twists. If time permits, I would like to give an application of this exquisite theorem in $$3$$-manifold topology and knot theory.

#### Manpreet Singh (IISER Mohali)

Title: Free Quandles and knot quandles are residually finite
Abstract: We will define the residual finiteness of quandle, and will prove that free quandles and knot quandles of tame knots are residually finite and Hopfian. Further, residual finiteness of automorphism groups of some residually finite quandles will be discussed. This talk is based on joint work with Prof. Valeriy G. Bardakov and Prof. Mahender Singh.

#### Maya Verma (IISER Bhopal)

Title: Complex topological K-theory
Abstract: I will associate a sequence of abelian groups $$K^n(X)$$ to a locally compact Hausdorff space $$X$$ for each natural number $$n$$. There is an infinite long exact sequence relating the $$K$$-theory of a space to its closed subspace. I will state the Bott periodicity theorem which gives us a $$6$$-term cyclic exact sequence. This will be useful in computing $$K$$-theory groups for some spaces.

#### Neeraj Kumar Dhanwani (IISER Bhopal)

Title: The geometry of Fuchsian groups
Abstract: We begin by describing the upper half plane model $$(\mathbb{H})$$ for two-dimensional hyperbolic space. We briefly discuss the classification of orientation-preserving isometries $$(\cong \text{PSL}_2(\mathbb{R}))$$ of $$\mathbb{H}$$. We show that the discrete subgroups of $$\text{PSL}_2(\mathbb{R})$$, also known as \textit{Fuchsian groups}, correspond to properly discontinuous actions on $$\mathbb{H}$$. After exploring some algebraic properties of Fuchsian groups, we analyze the fundamental domains of the corresponding properly discontinuous actions on $$\mathbb{H}$$.
With this background in place, we define the signature of a Fuchsian group, which is basically a tuple of non-negative integers that uniquely determines a Fuchsian group (up to isomorphism). Furthermore, we will establish the Poincaré theorem, which asserts the existence of Fuchsian groups with certain signatures. Finally, we will discuss some consequences of this theorem, along with some examples.

#### Paramjit Singh (CMI)

Title: A sketch of prime decomposition of orientable closed $$3$$-manifolds.
Abstract: We shall present a sketch of the prime decomposition of orientable closed $$3$$-manifolds, somewhat akin to prime decomposition of numbers, where the operation is connected sum. If time permits, we'll make some statements in the non-orientable case.

#### Praphulla (IISER Thiruvananthpuram)

Title: Geometry of Orbifolds
Abstract: Orbifolds can be thought of as two things; first as spaces they are almost like manifolds (i.e., instead of locally being $\mathbb{R}^n$, they are locally $$\mathbb{R}^n/G$$, $$G$$ a finite group acting linearly). Second, as Lie groupoids in which every point has a finite isotropy group. I will give an overview of these two approaches, how they are related and give an overview of what geometry one can do on orbifolds in similar spirit as in the case of Manifolds.

#### R. Syama (PSGCT Coimbatore)

Title: Abstract Cellular Complex and its Applications in Image Analysis
Abstract: Topology deals with study of structures. The basic notions of topology viz open, closed, frontier, connectedness, boundary, connectivity are closely associated to the field of Image Analysis. Rosenfeld introduced the notion of Digital Topology and he represented a digital image as a neighborhood graph and defined the topological notions by introducing the adjacency relation using the notions of connectivity (4- , 8-) between the pixels of the images. Though these representations are very much useful for the image analysis. But it contains connectivity and boundary paradoxes. To overcome these paradoxes, Kovalevsky developed a consistent topology namely of Abstract Cellular Complex (ACC) by introducing the notions of lower dimensional cells in an image. Further he established that every finite topological space with the T0-Separation property is isomorphic to Abstract Cellular Complex. Using the notions of ACC he proposed a boundary tracing algorithm for images which overcomes the paradoxes. Vijaya and Sai Sundara Krishnan implemented the Kovalevsky’s Boundary tracing algorithm for tracing a boundary of a digital images and compared with already existing algorithm. Our aim is to enhance the study of image analysis in ACC through lower dimensional cells.

#### Reebhu Bhattacharyya (IIT Bombay)

Title: Universal Bundles and Classifying Spaces
Abstract: We will discuss the concept of classifying space $$BG$$ for a topological group $$G$$, and its relation with universal principal $$G$$-bundles. We will prove a condition for a principal $$G$$-bundle to be universal, namely if $$p:E \to B$$ is a principal $$G$$-bundle such that the total space $E$ is aspherical, then it is a universal bundle.

#### Samir Shukla (IIT Bombay)

Title: Spectral bounds for vanishing of cohomology of simplicial complexes and an application to random neighborhood complexes
Abstract: The Clique complex $$X(G)$$ of a graph $$G$$ is the simplicial complex whose simplices are subsets of vertices of $$G$$ which spans a complete subgraph of $$G$$. In 2005, Aharoni et al. proved the following result which guarantee the vanishing of cohomology of a clique complex, provided the spectral gap of its $$1$$-skeleton is enough large. Let $$G$$ be a graph on $$n$$ vertices and let $$\lambda_2(G)$$ denote the second smallest eigen-value of the Laplacian of $$G$$. If $$\lambda_2(G) > \frac{kn}{k+1}$$, then $$\widetilde{H}^k(X(G); \mathbb{R}) = 0$$.
In this talk, we generalize the above result to a simplicial complex $$X$$ whose $$k$$-skeleton is a clique complex and to a simplicial complex which is a subcomplex of a clique complex having the same $$1$$-skeletons. As an application we show that the cohomology of Neighborhood complexes of a random graphs $$G(n, p)$$ vanishes for certain range of probability $$p$$. This is joint work with D. Yogeshwaran

#### Shuchita Goyal (IIT Bombay)

Title: Lack of Ström-Hurewicz model structure on the category of graphs.
Abstract: In this talk, we will first examine the notions of paths, products, double mapping cylinder object, etc in the category of graphs similar to those in the category of topological spaces. We will show that the mapping cylinder in graphs defined analogously to the one in spaces is not a correct candidate, and with respect to this notion of mapping cylinder, there cannot exist a Ström-Hurewicz model structure on the category of graphs. This is a joint work with Prof. Rekha Santhanam.

#### Soumya Dey (IISER Mohali)

Title Generalized braid groups and their commutators
Abstract: We will discuss about various generalizations of the classical Artin's braid group. We will talk about different geometric and group theoretic properties of these groups which can be concluded through analyzing the structure of the commutator subgroups of these groups. This talk is based on joint work with Dr. Krishnendu Gongopadhyay.

## Participants

• Amit Kumar Paul (IIT Kanpur)
• Aparajita (ISI)
• Aritra Bhowmick (ISI)
• Chandan Singh (Delhi University)
• Debayan Goswami (University of Allahabad)
• Deepti Singh (IIT Kanpur)
• Dinesh Nayak (Central University of Rajasthan)
• Dipali Swain (Central University of Jharkhand)
• Ganesh Satpute (IISER Mohali)
• Gayatri Limaye (IISER Bhopal)
• K. Nivetha (Thiruvallur University)
• Mohammad Aamir Qayyoom (Integral University of Lucknow)
• Prerak Deep (IISER Bhopal)
• Priya Garg (IIT Bombay)
• Priya Sharma (Dayalbag Educational Institute)
• Ramandeep Singh Arora (IISER Mohali)
• Ramya Nair (IISER Pune)
• Neha Nanda (IISER Mohali)
• Rishabh Gupta (IIT Kanpur)
• Sagar Shinde (IISER Bhopal)
• Sambit Senapati (CMI)
• Sayantan Goswami (BHU)
• Shilpa saini (IISER Bhopal)
• Shreya Banerjee (IISER Kolkata)
• Soumyajit Saha (IIT Bhopal)
• Sourayan Banerjee (IISER Bhopal)
• Sunanda Sinha (University of Calcutta)
• Vidit Das (IISER Bhopal)

## Support

• All the participants will be provided accommodation at the CMI hostel from the evening of July 15th till the afternoon of July 19th.
• Breakfast, meals, tea-coffee and snacks will be provided.
• Unfortunately we have very limited funding and hence are unable to provide travel support to all the participants. We request participants to use their contingency. However, feel free to get in touch with the organizers in case of any difficulty.

This conference is supported by CMI.