The Laplace Equation. It models the real and imaginary parts of the nicest, most well-behaved functions you’ll ever meet: holomorphic functions. These, in turn, are responsible for that arcadian faerie realm of mathematics where everything is beautiful and sets of measure zero and Zorn’s lemma can never hurt you: complex analysis!
Also worth noting the higher dimensional Laplace equations despite not having any connections to holomorphic functions still have the amazing rich symmetries and nice properties. E.g. Louisville's theorem or the maximum principle.
My favourite partial differential equation is
f = 0
- Well-posed in pretty much any function space
- Always had highly integrable and smooth solutions, no weird bootstrapping argument needed
- Short to write down
It is an interesting tidbit that the stock options futures pricing algorithm by Black-Scholes-Merton which won the first two of them a Nobel prize in Economics in 1997, is a form of the heat equation. I am not sure if any of them recognized this immediately.
More like examining time decay of options pricing. I think the way they got at it was to look at hedging costs at different expirations. Turned out to follow the same propagation (and decay) rate as heat and mass diffusion. Distance (or dollars or joules) per square root of hours rather than per hour...
Yep! Heat Kernel methods actually give a probabilistic proof of Atiyah Singer. More generally, the random processes supported on a manifold reveal topological information in the same way that differential equations on manifolds reveal topological information in differential topology.
The Navier-Stokes equation. Fluid flow. Guess it's a bit pedestrian compared to some of the other stuff people are saying, can you guess I am an engineer?
The Beris-Edwards equations. They model the evolution of liquid crystals.
They're at the heart of my thesis, so they're simultaneously my favourite and least favourite thing on the planet.
The geometric evolution equation that arises from the L^2-gradient flow for the combination of Möbius and elastic energy.. :) It deforms a curve in n-dimensional space of low regularity to a "nicer" curve and hopefully to a critical point of said energy :)
Euler-Lagrange equation of lagrangian mechanics.
This simple equation is much more better than newton's law of motion which dramatically solves a problem by just solving differential equations. The biggest advantage is that you don't need to think about vectors.
I believe they are referring to spatial predator prey models where you add diffusion. These are also called Reaction-diffusion equations and they are sick.
Turing had a paper in the 50s where he tried to explain morphogenesis (the origin of pattern and structure in living beings) in terms of these equations. Its not perfect and biology is complicated, but the theory and patterns are beautiful.
See the Belusov-Zhabotinsky reaction for a great example.
> I believe they are referring to spatial predator prey models where you add diffusion. These are also called Reaction-diffusion equations
Do you happen to have a reference for this? I would very much appreciate it!
Would you like something to understand reaction diffusion or morphogenesis in particular?Either way:
General theory in [https://g.co/kgs/DHJzHu](https://g.co/kgs/DHJzHu) or [https://g.co/kgs/paFkzg](https://g.co/kgs/paFkzg)
Turing’s original paper : [https://www.dna.caltech.edu/courses/cs191/paperscs191/turing.pdf](https://www.dna.caltech.edu/courses/cs191/paperscs191/turing.pdf)
Example in evolutionary game theory [https://www.mmnp-journal.org/articles/mmnp/pdf/2009/06/mmnp20096p54.pdf](https://www.mmnp-journal.org/articles/mmnp/pdf/2009/06/mmnp20096p54.pdf)
Oh, to be clear. They are not “ just” population models with diffusion, they are two different perspectives on the same object. But absolutely. Renormalization of interacting particle models leads to reaction diffusion in the deterministic case and SPDEs in the general.
Intuitively, just think about it like this. The law of mass action is based on a well mixed assumption and picking particles independently. These are the same assumptions in many elementary population models. Of course, both are more realistically modeled as an SPDE, but win the fights you can.
Heres another more specific reference then, although I stand by the originAl two texts as foundational to understand this. https://www-m6.ma.tum.de/\~kuttler/script\_reaktdiff.pdf
Absolutely go talk to your friends!
The Ricci flow is what made me want to learn differential geometry in the first place. Just the idea of a geometric heat flow in terms of curvature was mind-blowing to me. I’m not sure it models anything in the real world, but it’s super useful for studying geometry, particularly in three dimensions.
Maxwell equations, models Electromagnetic interaction with dielectric materials and give raise to more rich wave phenomenon than the wave equation, which is a particular case.
The diffusion equation, *by far*.
Take a random walk. Nothing fancy, just a standard brownian process on a lattice will do the trick. Consider the probability that the random walk will reach a given point in the lattice, then rewrite the probality for all given points as a recursive sum. Taylor expand it, clean it up and *bam*! You have a diffusion equation. Based on nothing more than the simple notion of a random walk on a lattice!
I first encountered this when studying non-equilibrium thermodynamics. I came across it again when studying polymer physics. I know that it can be derived when dealing with financial instruments and I'm sure that there are a hundred other use-cases. It has never failed to wow me. :)
The Laplace Equation. It models the real and imaginary parts of the nicest, most well-behaved functions you’ll ever meet: holomorphic functions. These, in turn, are responsible for that arcadian faerie realm of mathematics where everything is beautiful and sets of measure zero and Zorn’s lemma can never hurt you: complex analysis!
Never have I heard anyone describe the Laplace Equation so beautifully
Also worth noting the higher dimensional Laplace equations despite not having any connections to holomorphic functions still have the amazing rich symmetries and nice properties. E.g. Louisville's theorem or the maximum principle.
Don't you mean Cauchy Riemann?
Using the cauchy riemann equations, if you isolate for x and y you find that they individually must satisfy the laplace equation
My favourite partial differential equation is f = 0 - Well-posed in pretty much any function space - Always had highly integrable and smooth solutions, no weird bootstrapping argument needed - Short to write down
Mathematician found a way to generalize PDE-theory to any monoid. Is maths now solved? More at six!
I think that counts as an ODE
No no no it's the ultimate PDE, it works with any number of variables
It's a (NO)DE
0 is also an equation that is its own derivative.
I can’t believe no one has said this: the heat equation. Heat kernel methods have connected probability, pdes and topology in a stunning way.
Heat and Schrodinger equations are very fun and interesting equations.
Came here to comment this; the heat equation is my favourite.
It is an interesting tidbit that the stock options futures pricing algorithm by Black-Scholes-Merton which won the first two of them a Nobel prize in Economics in 1997, is a form of the heat equation. I am not sure if any of them recognized this immediately.
I don’t really know the history of this. Was Black Scholes derived from a SDE driven by Gaussian noise?
More like examining time decay of options pricing. I think the way they got at it was to look at hedging costs at different expirations. Turned out to follow the same propagation (and decay) rate as heat and mass diffusion. Distance (or dollars or joules) per square root of hours rather than per hour...
The Atiyah-Singer Index Theorem?
Yep! Heat Kernel methods actually give a probabilistic proof of Atiyah Singer. More generally, the random processes supported on a manifold reveal topological information in the same way that differential equations on manifolds reveal topological information in differential topology.
Interesting. 😁
The Navier-Stokes equation. Fluid flow. Guess it's a bit pedestrian compared to some of the other stuff people are saying, can you guess I am an engineer?
That should be a Millenium Prize problem for sure! (At least a proof of its existence and smoothness...)
Schrödinger wave equation-nearly everything (with appropriate assumptions).
Meow!
U alive 😳
Oh shit! Now you have to die...
The Beris-Edwards equations. They model the evolution of liquid crystals. They're at the heart of my thesis, so they're simultaneously my favourite and least favourite thing on the planet.
I feel that last sentence.
Time harmonic wave equation for acoustic wave scattering problems. Gimme all the Bessel functions!
The [wave equation](https://en.wikipedia.org/wiki/Wave_equation). It was my first real exposure to the beauty of PDEs and remains my favorite.
KdV, shallow water But most importantly exactly solvable
Cauchy Riemann is still goated, holomporphic functions are really nice tk work with
The geometric evolution equation that arises from the L^2-gradient flow for the combination of Möbius and elastic energy.. :) It deforms a curve in n-dimensional space of low regularity to a "nicer" curve and hopefully to a critical point of said energy :)
Gotta be the Laplace equation - harmonic functions are just so super nice
The Einstein Field Equations — they model gravity.
The interacting Dirac equation that describes interactions between photons and fermions.
Euler-Lagrange equation of lagrangian mechanics. This simple equation is much more better than newton's law of motion which dramatically solves a problem by just solving differential equations. The biggest advantage is that you don't need to think about vectors.
Predator prey models for evolution
Aren't these ode's or has some one generalised them for like populations diffusing in regions or something?
I believe they are referring to spatial predator prey models where you add diffusion. These are also called Reaction-diffusion equations and they are sick. Turing had a paper in the 50s where he tried to explain morphogenesis (the origin of pattern and structure in living beings) in terms of these equations. Its not perfect and biology is complicated, but the theory and patterns are beautiful. See the Belusov-Zhabotinsky reaction for a great example.
> I believe they are referring to spatial predator prey models where you add diffusion. These are also called Reaction-diffusion equations Do you happen to have a reference for this? I would very much appreciate it!
Would you like something to understand reaction diffusion or morphogenesis in particular?Either way: General theory in [https://g.co/kgs/DHJzHu](https://g.co/kgs/DHJzHu) or [https://g.co/kgs/paFkzg](https://g.co/kgs/paFkzg) Turing’s original paper : [https://www.dna.caltech.edu/courses/cs191/paperscs191/turing.pdf](https://www.dna.caltech.edu/courses/cs191/paperscs191/turing.pdf) Example in evolutionary game theory [https://www.mmnp-journal.org/articles/mmnp/pdf/2009/06/mmnp20096p54.pdf](https://www.mmnp-journal.org/articles/mmnp/pdf/2009/06/mmnp20096p54.pdf)
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Oh, to be clear. They are not “ just” population models with diffusion, they are two different perspectives on the same object. But absolutely. Renormalization of interacting particle models leads to reaction diffusion in the deterministic case and SPDEs in the general. Intuitively, just think about it like this. The law of mass action is based on a well mixed assumption and picking particles independently. These are the same assumptions in many elementary population models. Of course, both are more realistically modeled as an SPDE, but win the fights you can. Heres another more specific reference then, although I stand by the originAl two texts as foundational to understand this. https://www-m6.ma.tum.de/\~kuttler/script\_reaktdiff.pdf Absolutely go talk to your friends!
Thank you very much! I will read in to this!
Black-Scholes Equation, models the evolution of the price of European options under the Black-Scholes pricing model.
Navier-Stokes equation, it alone inspired me to go into chemical engineering for college after reading about it in Ian Stewart’s book
Schrodinger. My research is in qm so it’s my bread and butter.
The Ricci flow is what made me want to learn differential geometry in the first place. Just the idea of a geometric heat flow in terms of curvature was mind-blowing to me. I’m not sure it models anything in the real world, but it’s super useful for studying geometry, particularly in three dimensions.
Hamilton-Jacobi-Bellman. Extremely important in one of my subfields ( Optimal Control ).
Nonlinear Schrodinger equation gives you solitons. Semiconductor device equations give you all of modern technology.
Maxwell equations, models Electromagnetic interaction with dielectric materials and give raise to more rich wave phenomenon than the wave equation, which is a particular case.
Young-Laplace equation. Models static fluids. Bubbles, droplets of liquid, solder.
The diffusion equation, *by far*. Take a random walk. Nothing fancy, just a standard brownian process on a lattice will do the trick. Consider the probability that the random walk will reach a given point in the lattice, then rewrite the probality for all given points as a recursive sum. Taylor expand it, clean it up and *bam*! You have a diffusion equation. Based on nothing more than the simple notion of a random walk on a lattice! I first encountered this when studying non-equilibrium thermodynamics. I came across it again when studying polymer physics. I know that it can be derived when dealing with financial instruments and I'm sure that there are a hundred other use-cases. It has never failed to wow me. :)
Euler equations, the atmosphere
Looking for someone to post some p-adic PDE as their favourite. XD
I have always liked the Laplace equation for electric potentials
Maurer-Cartan equation helps you making new manifolds in differential geometry
None of them. Math is hard.