# Homepage of G.K. Srinivasan

### MA 108 (Undergraduate Ordinary Differential Equations)

Equipped with his calculus Newton was able to explain with the help of the universal law of gravitation:

• The motion of the planets
• The precession of the equinox
• The formation of tides
• Astronomy which hitherto had been an empirical science transformed into a dynamical science. Laws of physics when expressed in mathematical terms result in differential equations. The two body problem of celestial mechanics results in a system of differential equations while the motion of a simple pendulum is a scalar second order ordinary differential equation.

### MA 205/MA 207 (Complex Analysis/Partial Differential Equations)

MA 207 is a continuation of the course MA 108. Here we study in detail the basic differential equations of mathematical physics:
• The Laplace's equation
• The Wave equation
• The heat equation
• In the presence of spherical or cylinderical symmetries, the study of these can be reduced to the study of important second order ordinary differential equations with variable coefficients such as the equations of Legendre and Bessel. We also look at other equations such as the Hermite's equation and the Airy's equation. This study must be preceded by a study of power series and their basic properties. Later we turn to Fourier series and Fourier transforms and integral representations of the solutions of the wave and heat equations. The course closes with a chapter on celestial mechanics on the inversion of the Kepler equation in terms of a Kaypten series specifically a Fourier sine series whose coefficients are in terms of Bessel functions.

### MA 556 (Differential Geometry)

Differential geometry is central to many parts of mathematics and sciences. Its indispensibility in general relativity is ofcourse well-known. Cartography is an ancient science and mathematical cartography received much impetus from Lambert and Flemish cartographers. One of the most important results in differential geometry - Theorema Egregium - of Gauss finds applications in cartography namely, the impossibility of making a perfect map. Within mathematics differential geometry interacts closely with complex analysis, topology and partial differential equations.

• John McCleary, Geometry from a differential view point, Cambridge university press.
• Julio Benitez and N. Thome, Applications of differential geometry to cartography,
• ### MA 412 (Complex Analysis)

"Vollständige Erkenntiss der Natur einer analytischen Funktion muss auch die Einsieht den imaginären Werthen des Arguments in sich Schliessen"
- Gauss in a letter to Bessel (1811)

These words augured the creation of a distinguished branch of analysis 14 years later by A. L. Cauchy, marked by the appearence of his memoir:
Mémoire sur les intégrales définies, prises entre des limites imaginaires (1825) Paris.

• In his 1812 memoir Gauss regards the hypergeometric function F(α, ζ, γ, x) as a function of the complex variable x. For describing its behaviour at the point x = 1 on the circle of convergence he formulates and proves a delicate test for convergence which is the precursor of a whole heirarchy of tests due to Raabe and others culminating in a very general version due to Weierstrass (1856).
• The Cauchy residue formula may be regarded as the two dimensional analogue of Gauss's theorem in electrostatics concerning the total electric flux over a closed surface enclosing finitely many point charges. Even the proof proceeds along similar lines.
• Notes and problem bank will be made available presently

### MA 5102 (Basic Algebraic Topology)

Linking of geometrical objects or Verschlingung as it is called in German, has tantalized mankind since remote antiquity featuring in Celtic art on the one hand (depicting artistic forms of knots and braids) and molecular biology (structure of DNA) on the other. Alexander duality is the mathematical apparatus designed for a precise understanding of Verschlingung.

• Knots and braids can be seen in the accompanying picture of a part of the Cathedral of St. James in Jerusalem

• A formula for Verschlingungszahl (linking number) for a pair of space curves was given by C. F. Gauss in the course of his researches on electromagnetism and celestial mechanics.

• In his Inaugraldissertation (1851) Bernard Riemann introduces the notion of higher order connectivity of surfaces. In the words of Jean - Claude Pont:

"L'année 1851 marque un tournant dans le dévelopment de l'analysis situs. C'est en effect à cette date que notre science cesse d'être un simple jeu de l'espirit pour devenir, entre les mains de Riemann, un auxiliaire précieux dans l'étude des fonctions analytiques, à laquelle le XIXe siècle mathématique a consecré le meilleur de ses forces."

J.-C. Pont, La topologie algébrique des origines à Poincaré, Presses Universitaires de France, Paris, 1974.
• ### MA 417 and MA 525 (Ordinary Differential Equations/Dynamical Systems)

The picture depicts in the background the system of six ordinary differential equations that govern the motion of a spinning top under the influence of gravity.
Physics gives us three first integrals:

• Conservation of energy
• The conservation of the z-component of angular momentum
• The center of mass moves along a sphere

The divergence of the system is zero and so the system admits an integral invariant (Liouville's theorem). According to Jacobi's theorem if one more independent first integral exists the system is completely integrable. The missing first integral was referred to as the Mathematical Mermaid (Die mathematische Nixe).

See the delightful chapter on the Mathematical Mermaid in
R. Cooke, The mathematics of Sonya Kowalevskya, Springer-Verlag, 1984.

### S.P. sukhatme Excellence in Teaching Award (September 5, 2016):

“No man can reveal to you aught but that which already lies half asleep in the dawning of your knowledge. The teacher who walks in the shadow of the temple, among his followers, gives not of his wisdom but rather of his faith and his lovingness. If he is indeed wise he does not bid you enter the house of his wisdom, but rather leads you to the threshold of your own mind.”
― Khalil Gibran (The Prophet)

### Papers

• Structure of WTC expansions Journal of Physics A, Volume 28 (1995) (with S. Kichenassamy).
• Radius of Convergence and well-posedness of the Painlev\'e expansions of the Korteweg-de Vries equation Nonlinearity, Volume 10 (1997) 71-79 (with N. Joshi).
• Modulation equations of weakly non-linear geomertical optics in media exhibiting mixed nonlinearity, Studies in Applied Math., Volume 110 (2003) 103-122 (with V. D. Sharma).
• A note on the jump conditions in systems of conservation laws Studies in Applied Math., Volume 110 (2003) 391-396 (with V. D. Sharma).
• On weakly nonlinear waves in media exhibiting mixed nonlinearity Journal. Math. Anal. Appl. Volume 285 (2003) 629–641 (with V. D. Sharma).
• Energy dissipated across shocks in weak solutions of conservation laws Studies in Appl. Math. Volume 112 (2004) 281–291 (with V. D. sharma).
• Implosion-time for converging cylindrical and spherical shells Zeit. Angew. Math. Phys. Volume 55 (2004) 974–982 (with V. D. Sharma).
• Wave interaction in a nonequilibrium gas flow Internat. J. Non-Linear Mech Volume 40 (2005) 1031-1040 (with V. D. Sharma).
• The Gamma Function - An Eclectic Tour American Math. Monthly, Volume 114 (2007) 297-315.
• On the horizontal monotonicity of |Γ(s)| Canadian Math. Bulletin Volume 54 (2011) 538-543 (with P. Zvengrowski).
• On a remarkable formula of Ramanujan Archiv der Mathematik (Basel) Volume 99 (2012) 125-135 (with D. Chakrabarti).
• Unified Approach to the integrals of Mellin-Barnes-Hecke type Expositiones Mathematicae, Volume 31 (2013) 151-168.
• Dedekind's proof of Euler's reflection formula via ODEs Mathematics Newsletter, Ramanujan Math. Soc.,Vol 21 (2011) 82-83.
• Exterior Derivative - A direct approach
• Free groups, covering spaces and Artin's theorem
• ### Seminar Talks:

• It was a great honor to be invited to deliver a talk on Dec 22, 2018 on the occasion of Ramanujan's Birthday in the Ramanujan symposium IITMadras.

A confluence of thought:

• Behold the world in a grain of sand
And heaven in a wild flower
Hold infinity in the palm of your hand
And eternity in an hour.
-William Blake
• Ramanujan to Janaki Ammal: "Imagine if you could look so closely we could see each grain, each particle. You see there are patterns in everything..."
- Man Who Knew Infinity

The Man Who Knew Infinity (Movie Trailer)
Lecture Slides

• ### Popular Talks:

• Introductio in Analysin Infinitorum - A heuristic introduction to the theory of convergence: A talk delivered at KVPY camp in IISER Mohali in 2016. Leonard Euler's book with this title ranks among the most influential books on mathematics alongside Newton's Principia and Gauss's Disquisitiones Arithmeticae. The talk is a heuristic introduction to the theory of convergence starting from elementary ideas of continuous compounding to Euler's proof of the infinitude of primes and beyond.

• Lecture Slides

Recommended article: G. L. Alexanderson, About the cover: Euler's Introductio in analysin infinitorum, Bull. American. Math. Soc., Volume 44, Number 4 (2007) 635-639.

### Write up on miscellaneous topics

• The Cauchy-Binet formula