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Nalini's research focuses on ways to describe new special functions that arise as models in large classes of applications.
If each function was a number, they would be like the transcendental number π, which cannot be expressed as a finite combination of simpler numbers, yet they appear everywhere in applications.
These special, universal functions describe a wide range of pheneomena, including:
interaction of heavy atomic particlesquantum gravitydistribution of bus arrival times in Cuernavaca, the New York subway system and flight boarding times
the distribution of large prime numbers;
collisions of atomic particles;
aircraft boarding times;
bus arrival times;
the New York subway system;
electric fields in electrolyte solutions;
magnetic fields and waves in plasma physics;
water waves in the oceans;
stationary solutions of Einstein’s equations of general relativity;
cloud formations in the atmosphere.
Nalini describes her research as developing methods to solve mathematical puzzles that allow her to discover and describethese functions. The puzzles involve non-linear differential and difference equations, studied through a wide variety of lenses: including analysis, algebra, algebraic geometry, topology, and asymptotic methods in limits.
The equations she studies are called integrable systems. When they involve one independent variable, they are the Painlevé equations, while in two or more variables, the equations are soliton equations.
The mathematical tools involve a panorama of different perspectives. Instead of describing solutions as functions of an independent variable like time, they can be tracked by curves that go through initial values. The first perspective is like pointing a telescope to one point in the sky at night and taking pictures while time is changing. The second perspective is like tracking one star as it follows circular arcs of light in the sky at night.
The second perspective is called a space of initial values. It turns out that the geometry of this space gives us a great deal of information about the global nature of all possible solutions of the integrable systems and leads to some deep algebraic descriptions of the solutions, given by reflection groups.
The discrete Painlevé equations turn out to be given by translations (or walks) on lattices created by affine reflection groups. The results are very pretty and beautiful.
Nalini's research papers and books can be found on her publication page or her OrCiD page.
Specific research areas: Integrable systems, the Painlevé equations, discrete Painlevé equations, lattice equations, geometric asymptotics, nonlinear dynamics, nonlinear waves, perturbation theory.
Nalini's research focuses on ways to describe new special functions that arise as models in large classes of applications.
If each function was a number, they would be like the transcendental number π, which cannot be expressed as a finite combination of simpler numbers, yet they appear everywhere in applications.
These special, universal functions describe a wide range of pheneomena, including:
interaction of heavy atomic particlesquantum gravitydistribution of bus arrival times in Cuernavaca, the New York subway system and flight boarding times
the distribution of large prime numbers;
collisions of atomic particles;
aircraft boarding times;
bus arrival times;
the New York subway system;
electric fields in electrolyte solutions;
magnetic fields and waves in plasma physics;
water waves in the oceans;
stationary solutions of Einstein’s equations of general relativity;
cloud formations in the atmosphere.
Nalini describes her research as developing methods to solve mathematical puzzles that allow her to discover and describethese functions. The puzzles involve non-linear differential and difference equations, studied through a wide variety of lenses: including analysis, algebra, algebraic geometry, topology, and asymptotic methods in limits.
The equations she studies are called integrable systems. When they involve one independent variable, they are the Painlevé equations, while in two or more variables, the equations are soliton equations.
The mathematical tools involve a panorama of different perspectives. Instead of describing solutions as functions of an independent variable like time, they can be tracked by curves that go through initial values. The first perspective is like pointing a telescope to one point in the sky at night and taking pictures while time is changing. The second perspective is like tracking one star as it follows circular arcs of light in the sky at night.
The second perspective is called a space of initial values. It turns out that the geometry of this space gives us a great deal of information about the global nature of all possible solutions of the integrable systems and leads to some deep algebraic descriptions of the solutions, given by reflection groups.
The discrete Painlevé equations turn out to be given by translations (or walks) on lattices created by affine reflection groups. The results are very pretty and beautiful.
Nalini's research papers and books can be found on her publication page or her OrCiD page.
Specific research areas: Integrable systems, the Painlevé equations, discrete Painlevé equations, lattice equations, geometric asymptotics, nonlinear dynamics, nonlinear waves, perturbation theory.
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NONLINEARITYno. 8 (2023): 3969-4006
COMMUNICATIONS IN MATHEMATICAL PHYSICS (2023)
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arxiv(2023)
Joshua Holroyd,Nalini Joshi
Journal of Physics A: Mathematical and Theoreticalno. 1 (2023): 014002-014002
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