Low-dimensional topology and geometry.

  In particular

• Hyperbolic Geometry

• Teichm ̈uller theory

• Mapping class groups

• Moduli spaces 

• Topological graph theory

• Differential topology

• Combinatorial topology

• Systolic topology and geometry

Submitted.

• Length of Filling Pairs on Punctured Surfaces (With B N Saha)

arXiv:2409.05483 

• Complexity in Bolza surface  (With B N Saha)

arXiv:2311.18483 

Published/Accepted:

• Isoperimetric inequality for disconnected regions (with Arya Vadnere)

Accepted for publication in Geometriae Dedicata (2024)

DOI: 10.1007/s10711-024-00961-9 

journal version 

• Filling with separating curves (with B N Saha)

Journal in topology and analysis, Vol. 17, No. 06, pp. 1841-1867 (2025)

DOI: https://doi.org/10.1142/S1793525324500213

• Filling systems on surfaces  (with Shiv Parsad) 

Journal of Topology and Analysis,  arXiv:1708.06928

DOI. https://doi.org/10.1142/S1793525324500055 (2023)

• Self-intersecting filling curves on surfaces (with Shiv Parsad) 

Journal of Knot Theory and its ramification (JKTR), Vol 31, No. 07, 2250050 (2022) 

https://doi.org/10.1142/S021821652250050X

• A Conjecture on the Lengths of Filling Pairs (with Arya Vadnere)

Geometriae Dedicata, Volume 213, issue 1, 359-373(2021).

DOI: 10.1007/s10711-020-00586-8

arXiv Version 

Journal Version 

• Embedding of metric graphs on hyperbolic surfaces

Bulletin of the Australian Mathematical Society, Volume 99, Issue 3, pp. 508–520, (June, 2019), 

DOI–10.1017/S0004972719000145

arXiv Version

Journal Version 

• Geometric realizations of cyclic actions on surfaces (with Shiv Parsad, Kashyap Rajeevsarathy)

Journal of Topology and Analysis, vol 11, No. 04, pp. 929 - 964 (2018), 

DOI–10.1142/S1793525319500365

arXiv Version

Journal Version 

• Graphs of systoles on hyperbolic surfaces (with Siddhartha Gadgil)

Journal of Topology and Analysis, vol 11, No. 01, pp. 1 - 20 (March, 2019), 

DOI–10.1142/S1793525319500018

arXiv Version

Journal Version 

• Filling of closed surfaces

Journal of Topology and Analysis, Vol. 10, No. 04, pp. 897 - 913 (December, 2018), 

DOI 10.1142/S1793525318500309

arXiv Version

Journal Version 

• Systolic fillings of surfaces

Bulletin of the Australian Mathematical Society, Volume 98, Issue 3, pp. 502–511, (December 2018), 

DOI–10.1017/S0004972718000862

 Journal Version 

Arxiv version: Not available 

Email ID: bidyut@iitk.ac.in

Office Phone: 0512-259-2085 

Office Address: Room no 587, Faculty Building

• PhD, Department of Mathematics, IISc Bangalore, India, (August 2009 - July 2014).

    Thesis Title: Shortest length geodesics on closed hyperbolic surfaces.

    Thesis Advisor - Prof. Siddhartha Gadgil.

• MSc, Department of Mathematics and Statistics, IIT Kanpur, India,  August 2007 - July 2009

• BSc, Jadavpur University, Kolkata, India, 2004 - 2007

• Associate Professor, Department of Mathematics and Statistics, IIT Kanpur

    July, 2024 to Present

• Assistant Professor, Department of Mathematics and Statistics, IIT Kanpur

    November, 2018 to June, 2024

• Postdoctoral Fellow, IMSc, Chennai.

    December, 2016 - November, 2018

• Postdoctoral Fellow, IISER Bhopal.

    August, 2016 – December, 2016.

• Postdoctoral Fellow, RKMVERI, Belur.

    July, 2015 - July, 2016.

• Research Associate, IISc Bangalore.

   August 2014 - June 2015.

Semester: 2025-26(I)

MTH112:

Past: 

Semester: 2024-25(II)

MTH305A: Several variable calculus and differential geometry. 

FCH

B. Lecture notes

1. Lecture 1: Introduction to the course

2. Lecture 2: Euclidean spaces, norm, inner-product and matrix representation of linear transformations

3. Lecture 3: Heine-Borel Theorem and Tube lemma

4. Lecture 4: Continuous functions and characterization theorem, action of continuous functions on compact sets, derivative of several variable functions and uniqueness theorem.

5. Lecture 5: Chain rule, Properties of derivatives, the derivative of a function in terms of its component functions, Derivative of binary operations.

6. Lecture 6: Partial derivatives, smooth functions, Max/min, Jacobian matrix in terms of partial derivatives. 

7. Lecture 7: A sufficient condition for differentiability, Inverse function theorem: Motivation and Statement. 

8. Lecture 8: Proof of the inverse function theorem

9. Lecture 9: The implicit function theorem

10. Lecture 10: Mean value theorem

11. Lecture 11: Equality of mixed partial derivatives

12. Lecture 12: Taylor Formula

13. Lecture 13: Extrema

14.  Lecture 14

15. Lecture 15

16. Lecture 16: Definition and examples of parameterized curves, tangent and tangent vector, speed, trace of curves, examples, characterization of straight lines.

17. Lecture 17: Positive re-parameterization, negative re-parameterization and reparameterization of curves, chain rule for curves, arc-length function and  unit-speed reparameterization of regular curves.

18. Lecture 18: Change of ordered basis matrix, Orientation of an order basis, Vector product in Euclidean space of dimension three, properties of vector product. for any two linearly independent vectors u and v, the ordered basis consisting of u, v and the cross product of u and v is positively oriented.

19. Lecture 19: Curvature (determines the extent to which a curve is not a straight line) of (i) unit-speed curve and (ii) regular curves; some examples

20. Lecture 20: Curvature of curves, signed curvature of plane curves and its geometric interpretation, fundamental theorem of curvature.

21. Lecture 21: The fundamental theorem of plane curves, Torsion of unit speed space curves

22. Lecture 22: Torsion of unit-speed space curves and any regular curves with no-where vanishing curvature; Torsion of a curve is zero if and only if the curve is planar; Serret-Frenet equations.

23. Lecture 23: Application of Serret-Frenet equations, the fundamental theorem of space curves

24. Lecture 24: Global Properties of curves, simple closed smooth curves on the plane.

25. Lecture 25: Area of the region bounded by simple closed plane curves, isoperimetric inequality

26. Lecture 26: Proof of the isoperimetric inequality in Euclidean geometry, Differential geometry of surfaces-an introduction and the definition of regular surfaces

27. Lecture 27: Regular surfaces--Definitions: Local coordinates, coordinate neighbourhood, charts, atlas, transition functions, examples of regular surfaces.

28. Lecture 28: Graphs of real valued smooth maps on open subset of plane are regular surfaces, subsets those are locally graph of  real valued smooth maps on open subset of plane are regular surfaces, regular points, regular values, critical points and critical values

29. Lecture 29: Pre-image (regular value) theorem, regular surfaces are locally graph of some smooth function. 

30. Lecture 30: Transition functions are diffeomorphisms, The notion of differentiable functions on regular surfaces

31. Lecture 31: Smooth maps between surfaces, diffeomorphisms, local diffeomorphisms, tangent space to regular  surfaces

32. Lecture 32: Tangent plane to regular surfaces, Derivative of differentiable functions between surfaces

33. Lecture 33: Derivative of maps between surfaces and derivative of real-valued maps on surfaces, orientability of regular surfaces.

34. Lecture 34: Orientations of regular surfaces.  

35. Lecture 35: The first fundamental form

36. Lecture 36: Length of curves, angle of intersection of curves and surface area 

37. Lecture 37: Isometry between regular surfaces

38. Lecture 38: Gauss Map and its differential

C. Assignments and solutions 

1. Assignment 1

2. Assignment 2

3. Assignment 3

4. Assignment 4

5. Assignment 5

6. Assignment 6

7. Assignment 7

Bonus: Isometries of the Euclidean plane

8. Assignment 8

D. Quiz papers

E. Midsem and Endsem 

Semester: 2014-25(I)

 MTH633A: An introduction to hyperbolic geometry

FCH

Lectures:

• Lecture 1: Introduction

• Lecture 2: Upper half-plane model, hyperbolic length of curves

• Lecture 3: M\"obius transformations of the upper half-plane: Group structure, action on circles and lines, action on boundary at infinity, isometry of hyperbolic plane; Hyperbolic geodesics. 

• Lecture 4: Cross-ratio, action of M\"obius transformation on cross-ratio, formula for hyperbolic distance function.

• Lecture 5: Euclid's parallel postulate fails in the hyperbolic plane, Angles, conformal transformations: Mobius transformations are conformal, Hyperbolic Pythagorus theorem, Hyperbolic area.

• Lecture 6: Hyperbolic area, Mobi\"us transformations preserve area, Gauss-Bonnet theorem: computation of hyperbolic area of polygons, angle sum of a triangle is less than pi, Tessellation of the hyperbolic plane by regular polygons, some hyperbolic trigonometry of right-angled triangles.

• Lecture 7: Some hyperbolic trigonometry of right-angled triangles, existence of hyperbolic polygon for given interior angles 

• Lecture 8: Poincar\'e Disk model of the hyperbolic plane

• Lecture 9: Classification of M\"obius transformations of the upper half-plane

• Lecture 10: Isometry group of the hyperbolic plane, classification, continued fraction

• Lecture 11: Farey tessellation, Continued fraction

• Lecture 12: Fuchsian groups

• Lecture 13: Fundamental domains

• Lecture 14: Dirichlet domains

• Lecture 15: Fuchsian groups-Algebraic properties, characterization theorem

• Lecture 16: Commutative Fuchsian groups, normalizer of Fuchsian groups, Elementary groups, Fundamental domain.

• Lecture 17: Dirichlet domain, Computation of Fundamental domains of groups of (i) integer translations, (ii) rotations and (iii) modular group. Isometric circles and some properties of isometric circles, 

• Lecture 18 

• Lecture 19 

• Lecture 20 

• Lecture 21

• Lecture 22: Computing area of Fuchsian groups of known signature and Poincare Theorem-The existence of a Fuchsian group for a given signature. 

Home-works

• HW1

• HW2 

Assignments

• 1 

• 2 

• 3 

• 4

• 5

Semester: 2023-24(II)

 (MTH114M) Ordinary differential equations

Lecture notes:

1. Lecture 1: Definitions, examples, geometric interpretation, orthogonal and oblique trajectories 

2. Lecture 2: IVP, Separable of variables, reducible to separable, exact equations.

3. Lecture 3: Exact ODEs, integrating factors 

4. Lecture 4: Linear ODEs, Bernoulli's equation, reducible second order ODEs 

5. Lecture 5: Picard's existence and uniqueness theorem, Picard's iteration, Numerical methods: Euler's and improved Euler's methods 

6. Lecture 6: Numerical methods: Euler's method and Improved Euler's method; Second order linear ODEs 

7. Lecture 7: Wronskian, solution space of 2nd order homogeneous linear ODEs, Fundamental system

8. Lecture 8: Reduction of order of 2nd order linear homogeneous equations, homogeneous equations with constant coefficients 

9. Lecture 9: Non-homogeneous linear equations: Method of undetermined coefficients 

Algebraic topology (I) MTH649A 

Semester: 2023-24(I)

FCH 

Lecture notes.

1. Lecture 1: Quotient topology 

2. Lecture 2: Quotient topology and Category 

Assignment 1 

3. Lecture 3: Category, Functor, Homotopy congruence 

4. Lecture 4: Congruence, quotient category, homotopy and more. 

Assignment 2 

5. Lecture 5: Path homotopy fixing endpoints and Fundamental Groupoid  

6. Lecture 6: Algebraic properties of path homotopy and Fundamental groups.

Assignment 3 

7. Lecture 7: Covering spaces, lifting problem, local lifting lemma, path lifting 

8. Lecture 8: Path lifting lemma, homotopy lifting lemma and application  

Quiz 1 

Assignment 4 

9. Lecture 9: Fundamental group of circle, functoriality and based homotopy relation 

10. Lecture 10: Fundamental groups and dependence of base points, Homotopy equivalence of based spaces. Deformation retraction, retraction, fundamental groups of product spaces, No-retraction theorem.

Assignment 5 

11. Lecture 11: Fundamental groups of product spaces, Brouwer fixed point theorem, R^2 and R^3 are topologically not the same 

12. Lecture 12: Weak product of groups, generators, relations and free abelian groups 

Assignment 6

13. Lecture 13: Free abelian groups and free groups 

14. Lecture 14: Free product of groups and Seifert Van-Kampen theorem 

Mid-semester Exam: 

                 Date and time: Sept 21, 2023, 18:00-20:00 hrs

                 Venue and seating plan: L9 ERES

xQuestion paper: Mid-semester 

15. Lecture 15: Seifert Van-Kampen Theorem 

16. Lecture 16: Surfaces:  Orientability, Connected sum and classification theorem.

 Assignment 7 

17. Lecture 17: Surfaces: Connected-sum, Classification theorem, Polygonal representation, Triangulation and Euler characteristic 

18. Lecture 18: Surfaces: Triangulation, Euler characteristic, Fundamental group of Klein bottle, Graphs

 Assignment 8 

19. Lecture 19: Graphs Euler characteristic of surface bounded above by 2, CW complex 

20. Lecture 20: CW complex, Calculating fundamental groups of surfaces, Covering spaces, geometric realization of graphs 

21. Lecture 21: Geometric realization of graphs, Category of graphs and functor to Top, Criterion for covering map, Deck transformation groups 

22. Lecture 22: Deck transformation groups, Regular coverings, Covering space action 

23. Lecture 23: Lifting properties, Classification of covering spaces 

Assignment 9

Quiz 2 

24. Lecture 24: Homology groups--Motivation and examples

25. Lecture 25:  Simplicial homology

26. Lecture 26: Singular homology groups

Assignment 10

Final Exam: November 24, 2023 17:30--20:30 hrs

Venue and seating plan: L8, ERES

• Introduction to Hyperbolic geometry (MTH633A)

Semester. 2022-23 (II)

Syllabus.

Refs. 

1. J. Anderson, Hyperbolic Geometry, 1st ed., Springer Undergraduate Mathematics Series, Springer-Verlag, Berlin, New York, 1999.

2. S. Katok, Fuchsian Groups, Chicago Lecture Notes in Mathematics, Chicago University Press, 1992.

3. A. Beardon, The Geometry of Discrete Groups, Springer-Verlag, Berlin, New York, 1983.

• MTH713A: Differential Topology

    2020 - 2021, Semester II

Course Description: This is an introductory course in Differential Topology. The aim of this course is to introduce basic tools to study the topology and geometry of manifolds. We start with reviewing two key results from several variable calculus, namely the inverse function theorem and implicit function theorem which are essential to study differential manifolds. Throughout this course, we will discuss the theory of manifolds and a way to generalise differential, integral and vector calculus. By the end of the course, we should understand and able to work with manifolds, tangent and co-tangent bundles, transversality, Morse Lemma, Morse function, Whitney embedding theorem, Poincare – Hopf theorem, Sards theorem and its applications and many other things listed below in the course contents.

Course Contents:

 Review of inverse function theorem and implicit function theorem.

 Introduction to differential manifolds, sub-manifolds and manifolds with boundary.
Smooth maps between differential manifolds, tangent space, differential of smooth maps,
(local) diffeomorphism. Immersions, embeddings, regular value, level sets, submersions.

 Tangent and cotangent bundles, vector bundles, vector fields, integral curves.
Transversality, Whitney embedding theorem, Sards theorem, Morse lemma, Morse
functions.

 Oriented intersection Theory: degree, Lefchetz fixed point theory, the Poincare-Hopf
theorem, the Euler characteristic and triangulations (time permitting: Integration on manifolds).

Recommended books:

• 1. Differential Topology, Victor Guillemin and Alan Pollack (AMS Chelsea Book Series)

  2. Topics in Differential Topology, Amiya Mukherjee (Hindustan Book Agency)

  3. Topology from the Differentiable Viewpoint, John W. Milnor (Princeton University Press)

  4. An Introduction to Differentiable Manifolds and Riemannian Geometry, William M.
     Boothby (Elsevier)

  5. An introduction to manifolds, Loring W Tu (Springer).

• MTH305A : Several variable calculus and differential geometry 

  2020 - 2021, Semester I

Syllabus:

1. Differentiation: Definition and examples, Mean value inequality, Tangent planes to level sets of functions; Implicit mapping theorem, Inverse mapping theorem and applications; Taylor's theorem and applications.

2. Curves: Definition and examples, Regular curves, Plane curves, Curvature of plane curves, Isoperimetric inequality for plane curves; Space curves, FrenetSerret formula for space curves; Local existence theorem curves.

3. Surfaces: Definition and examples; Tangent planes, Maps between surfaces; First fundamental and second fundamental forms; Curvature of surface; Hilbert's theorem for compact surfaces; Gauss theorem a Egregium. 

References:

     1. Spivak: Calculus on manifolds, Springer.

     2. M P do Carmo: Differential geometry of curves and surfaces, Prentice Hall.

     3. W. Rudin: Principles of Mathematical Analysis.

     4. Tom M. Apostol: Mathematical Analysis, Narosa Publishing House, India.

     5. A Pressley: Elementary differential geometry, Springer India.

Important Dates:

Course begins: September 1, 2020. 

Ref. for first few lectures Chapter 1 and 2, Spivak: Calculus on manifolds

• MTH 301A -Analysis I

2021-2022, Semester I

Summer Term 2021:

MTH305A: Several variables calculus \& differential geometry 

Notes:

Introduction 

• MTH304A : Topology

    2019 - 2020, Semester II

• MTH713A : Differential Topology

    2019 - 2020, Semester I

• MTH102A Tutor

    2018 - 2019, Semester II

PhD students

current.

1) Bhola Nath Saha

2) Achintya Dey (Joint with Dr Abhijit Pal)

UG Project students.

Current.

......

Past.

1) Nupur Jain 2022-23(I):

Topic: Topological graph theory

Major results-

• Kuratowski's theorem, 

• Cayley Graphs and Frucht’s Theorem

• Existence of regular graphs of a given Girth

 Report

MSc project students

Current. 

Papiya Sur

Topic. Spherical Geometry

 Past.

1) Arghys Sinha (2020-21): Report

Summer research programe:

1) Arya Vadnere (2019-20). 

Topic. Aougab-Huang conjecture

Publication.