Jacob Lurie

Where did he say this?

The statements are not useless, but they are predictable.

However, they should be predictable. We want the theories to behave as appropriate homtopical generalizations of category theory, commutative algebra, and algebraic geometry. So we could easily guess what the main points should be, even without knowing the details of the constructions.

Developing something like spectral algebraic geometry requires a whole lot of work, but it is a very powerful tool for homotopy theorists.

Who is this cuck?

Is he supposed to be a genius? I dont think a genius would bother wasting time on theory math that wont go anywhere. He should work on physics

says the guy who studies HS physics

Define "real mathematician".

I got a B in Calc 2 last semester, so I'm Eons ahead of any of you "real mathematicians." I pity this board and their LARPing

> Real

Derived Algebraic Geometry does apply to physics

>Someone who adds to humanity's understanding of math
Most math professors wouldn't qualify.

Someone who has published at least one new and non-trivial theorem

>Has anyone ever read his work?
Short answer is no, generally speaking academics in physics and math will not waste their time reading and comprehending a 2k page paper. The consensus being that if your idea takes 2k pages to document its formalism is shit tier or the idea itself is dumpster.

The 1000-2000+ page documents are books, put together from many normal length papers published over many years.

"Higher Algebra" (~1400 pages) being equivalent to rewriting the entire subjects of homological algebra and commutative algebra, from the viewpoint of stable infinity-categories becoming the new derived categories and highly structured ring spectra being the new rings.

"Spectral Algebraic Geometry" (~2300 pages) being equivalent to rewriting algebraic geometry over this new type of commutative algebra. Its like rewriting all of EGA, SGA, and FGA from the homotopy coherent viewpoint.

I know, thats why I said short answer as I assumed op isnt familiar with peer review. Good expansion nonetheless.

>Has anyone ever read his work?
Yep. Been working through this paper
arxiv.org/abs/1705.02240
on the cobordism hypothesis.

>The consensus being that if your idea takes 2k pages to document its formalism is shit tier or the idea itself is dumpster.
Ever heard of EGA?

>homotopy coherent viewpoint.

I want to elaborate on what is meant by this.

A common theme in algebraic topology is instead of dealing with commutative diagrams, we deal with diagrams which commute up to homotopy. i.e. The equality g∘f = f∘g, is replaced with the condition of g∘f being homotopic to f∘g.

A topological category is essentially a category whose Hom sets are topological spaces (and composition is continuous).

So for any diagram in a topological category, we can consider whether the diagram commutes up to homotopy.

Moreover, we can enhance this condition and consider whether a diagram commutes up to "coherent homotopy".

This means not only are the compositions homotopic, but these homotopies are linked by higher homotopies, and those homotopies are linked by even higher homotopies, etc. all in a "coherent" manner. Coherence meaning choices of all higher homotopies form a contractible space.

"Topological categories" together with "homotopy coherent diagrams" are a model for ∞-categories and ∞-functors respectively.

They are the most intuitive model, but one of the hardest to actually work with. Which is why we need the theory outlined in Higher Topos Theory. The theory in that book is less intuitive, but it is relatively easy to work with.

This is generally what is meant when ∞-category theory is referred to as homotopy coherent category theory.

Stable ∞-categories are the ∞-categorical analog of abelian categories.

In the sense that there is a zero object, they have all finite limits&colimits, and all morphisms have fibers (i.e. kernels) and cofibers (i.e. cokernels).

Just like the category of abelian groups is universal amongst abelian categories, the ∞-category of spectra is universal amongst stable ∞-categories.

One way to define a commutative ring is as a commutative monoid object in the category of abelian groups.

So to develop a homotopy coherent theory of algebra, we want to work with a notion of commutative monoid objects in ∞-category of spectra.

This points to why highly structured ring spectra (also called E_∞ - rings) are the right object to build a homotopy coherent theory of algebra out of, as they fit a proper ∞-categorical notion of commutative monoid object in the ∞-category of spectra.

This is essentially what is worked out in Higher Algebra.

Once you have this ∞-category of E_∞ - rings, you can proceed as in usual algebraic geometry and take the opposite category. And thus get the ∞-category of affine spectral schemes. Which is the starting point of Spectral Algebraic Geometry.

>>homotopy coherent viewpoint.
>I want to elaborate on what is meant by this.
>A common theme in algebraic topology is instead of dealing with commutative diagrams, we deal with diagrams which commute up to homotopy. i.e. The equality g∘f = f∘g, is replaced with the condition of g∘f being homotopic to f∘g.
>A topological category is essentially a category whose Hom sets are topological spaces (and composition is continuous).
>So for any diagram in a topological category, we can consider whether the diagram commutes up to homotopy.
>Moreover, we can enhance this condition and consider whether a diagram commutes up to "coherent homotopy".
>This means not only are the compositions homotopic, but these homotopies are linked by higher homotopies, and those homotopies are linked by even higher homotopies, etc. all in a "coherent" manner. Coherence meaning choices of all higher homotopies form a contractible space.
>"Topological categories" together with "homotopy coherent diagrams" are a model for ∞-categories and ∞-functors respectively.
>They are the most intuitive model, but one of the hardest to actually work with. Which is why we need the theory outlined in Higher Topos Theory. The theory in that book is less intuitive, but it is relatively easy to work with.
>This is generally what is meant when ∞-category theory is referred to as homotopy coherent category theory.
>Stable ∞-categories are the ∞-categorical analog of abelian categories.
>In the sense that there is a zero object, they have all finite limits&colimits, and all morphisms have fibers (i.e. kernels) and cofibers (i.e. cokernels).
>Just like the category of abelian groups is universal amongst abelian categories, the ∞-category of spectra is universal amongst stable ∞-categories.

Can you fix the spacing of your posts?

inb4 he gets meme'd on and becomes the next Shinichi "Tamako's biological father" Mochi suki.

But what's the point here?
I'm an undergrad student in mathematics.
Is his work going to help solve something important?

The most significant application so far is constructing topological modular forms.