I'm going to conflate mathematics with statistics as Carlo Beenakker did. Then the earliest application that I know of is that Decision Trees were invented by Breiman et al. to analyze the issue of- ...

IMPORTANT EDIT 12-2015 There is this paper of Tokunaga "Irreducible Plane Curves of Albanese Dimension Two" which based on cited work of Kulikov constructs surfaces in P^2 with isolated singularities ...

For algebraic statistics, I think there are two standard references. The books by Drton, Sturmfels, and Sullivant which can be bought or downloaded as a pdf and 'Algebraic Statistics for Computational ...

Unfortunately it is not true that if $Y$ is smooth, $f$ is proper and the fibers of $f:Y \rightarrow X$ are reduced and connected that $X$ is normal. Unfortunately I am all to familiar with the ...

I was told by my (graduate school) teacher of functional analysis that originally the complex case of the Hahn-Banach theorem was considered a major open problem. It was eventually shown to be such a ...

It is possible for the Jacobian's of non-isomorphic curves to be isomorphic as abelian varieties, but obviously, not as principally polarized abelian varieties. This paper https://arxiv.org/pdf/math/...

Resolution of Singularities. Nowadays, 'simple' proofs are available. The theorem is a huge work horse.

I'm going to assume that by 'high dimensional problems' you specifically mean learning speech recognition and image recognition. As such, I'm afraid that the answer seems to be 'sadly no'. Real ...

Milne's web page contains a number of amusing anecdotes- https://www.jmilne.org/math/apocrypha.html

Related to several answers is the issue of identifiablity and equations for secant varieties. This has actual and real uses. Identifiable means that a point in a secant variety, or at least a ...

The improvement of matrix multiplication from $O(n^3)$ to $O(n^{2.4})$ is based on the Strassen equations for matrix multiplication. Strassen Algorithm 7 multiplications talks about this and gives ...

Speaking of thesis advisers, mine said, "I think something should be called obvious only if it is obvious in the logical sense of if A implies B and if B implies C then A implies C". All else is ...

I believe that on any non-rational and non-hyperelliptic curve , the complete linear system $L = K_X(2p)$ will work. In such a case $2p - p_1 -p_2$ will always be a non-trivial divisor of degree ...

Another, curve related, example would be the secant variety to a smooth curve. If the embedding is 'sufficiently ample', which would mean that it separates 5 points (it's early and no coffee yet, so '...

Since I posted my comment a year ago, I've learned the answer to this question. In 'Geometry of Algbebraic Curves' as a series of exercises one shows the following- i) A general genus 5 curve is the ...

For a very mathematical version of statistics, my favorite is on line lecture notes from two MIT courses. The instructor is named Panchenko and the course is called 'Statistics for Applications'. ...

I have no horse in this race, as I program mostly in R, but a very good friend was very enthusiastic about F sharp as a categorical language. He claimed that it was based on category theory. I would ...

I'm late to the game, but I would like to point out that the answer is systematically no. One class of examples. Suppose g>2 for simplicity. In that case any general line bundle of degree 2g is very ...

The Clifford index comes from Clifford's theorem which bounds the how special a linear system can be in terms of it's degree. I think this is interesting and nice in and of itself. In my dotage ...

This is not an area of expertise for me, so forgive me if I didn't understand the question properly and hence this answer isn't on point. I think the original reference for this might be Lazarsfeld's ...

In statistics in general one considers a class of 'model(s)'. Usually this model has a number of parameters. For example if the model was 'event distribution is gaussian' then the parameters would ...

As has been mentioned, in general 'degree' is not so well/uniquely defined. However, suppose you take a smooth cubic surface in $\bf{ P}^3$ . There are 27 lines and they should all have degree 1. ...

Amplifying on of Speyer's comments, if p is a point on a secant line of C, then the quadrics vanishing on C and p are of codimension one in the space of all quadrics vanishing on C. Such a quadric ...

Am I missing something? Doesn't this hold for arbitrary smooth pairs of varieties using the Kunneth decomposition?

when c= 0 Clifford's them includes the fact that any divisor with Clifford index 0 is a multiple of the hyperelliptic fiber, ie: a sum of fibers of the hyperelliptic map. If c=1 then the curve is ...

I'm not an expert, but I believe that Steven Diaz wrote a number of papers bounding the dimensions of complete subvarieties of $\mathcal{M}_g$. That would certainly bound what you could do with ...

I'm going to assume $L$ is base point free. I think it is clear how to change what I have written in the case where there are base points (numbers goes down by the degree of the base locus). In ...