The math behind UniswapV3 is absolutely beautiful.

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Oh I remember when they switched to V2. Is there any details that you can share? Why you feel impressed, how it effected your user experience?

V3 has been out for a couple years now.

I've just been learning the details in the last few months.

There is lots of impressive stuff, spanning the capital efficiency improvements, the P&L behaviour of the liquidation positions...all supported by this tick-price convention and data structure that allows price to clear but also balance with quantities as price change.

Is this some atomic swap service?

It's an Automated Market Maker, which enables a DEX for spot trading.