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Geometry of numbers was initiated by Hermann Minkowski roughly a hundred years ago. At its heart is the relation between lattices (the group, not the poset) and convex bodies. One of its fundamental results, for example, is Minkowski's theorem: If $L$ is a lattice in $\mathbb{R}^d$ and $C$ a centrally-symmetric convex body, then $\mbox{vol}(C) \geq 2^d \det(L)$ implies that $C$ contains another lattice point than $0$.

While there was a lot of activity in the field until at least 1960, it seems that in recent decades not so many people are working on it anymore. One of the reasons could be that the field is somewhat stuck, or in Gruber's more polite words "It seems that fundamental advance in the future will require new ideas and additional tools from other areas." (see [1]).

I would like to know more about current research trends in the geometry of numbers. What are hot topics right now? In which areas was recently considerable progress achieved? Did maybe even the "fundamental advance", that Gruber mentions, take place?

[1] P. Gruber: Convex and discrete geometry, Springer 2007, p. 353

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By the way, an excellent historical account of the geometry of numbers can be found in the doctoral thesis of Sébastien Gauthier, written under the direction of Catherine Goldstein : La géométrie des nombres comme discipline (1890-1945) (math.univ-lyon1.fr/~gauthier/recherche.html). –  Chandan Singh Dalawat Jan 30 '13 at 2:28
    
This is not new research in the geometry of numbers, but rather an application of classical results to another classical problem, that of determining primes of the form x^2+ny^2: tcnj.edu/~hagedorn/papers/… –  Jeff H Jan 30 '13 at 3:29
    
WADR to Minkowski, the field should have been renamed "geometric number theory" in the 1950s. The part of geometric number theory that should be called "lattice theory" is not called that, because of overloading. Diophantine approximation is another rather questionable name of subdiscipline, but at least is named after a cluster of problems, rather than techniques or objects of study. The answers below suggest that some shifts of perspective are overdue. –  Charles Matthews Jan 30 '13 at 13:44

3 Answers 3

There has indeed been exciting recent work in this area, by Bhargava and Shankar (see this Bourbaki expose by Poonen) and also by Bhargava and Gross. Briefly, the work of Bhargava and Shankar bounds the average rank of the group of rational points of elliptic curves over $\mathbb{Q}$, while the Bhargava and Gross paper does the same for Jacobians of hyperelliptic curves.

Section 4 of the (quite readable) write-up by Poonen explains why I refer to these results as recent advances in the geometry of numbers: both of these results boil down to (subtle!) computations of adelic volumes! It's worth noting that the work of Bhargava and Shankar does not use adelic language, and so is more obviously related to the "classical" geometry of numbers.

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+1, and I'd add that nearly all of Bhargava's work is related to geometry of numbers, fascinating, and beautifully written. "Higher Composition Laws" I-IV is a great place to start. In addition to Shankar, his former students Melanie Matchett Wood and Wei Ho are also doing outstanding related work. Ho gave one of the four "Current Events Bulletin" lectures at the recent AMS/MAA meetings; if this field was ever stuck, it is roaring now. –  Frank Thorne Jan 30 '13 at 0:02

Recently several fundamental works have been done in Geometry of numbers. Beside Bhargava's revolutionary ideas (an of course the contribution of his students), Ergodic theory is a new idea that plays an important role in Modern Geometry of Numbers. It seems to me that several ideas are coming from Margulis and E. Lindenstrauss.

Here are a list of works in this area which I think they are extremely interesting

  1. Minkowski's theorem for random lattices. Margulis (Russian) Problemy Peredachi Informatsii 47 (2011), no. 4, 104--108; translation in Probl. Inf. Transm. 47 (2011), no. 4, 398–402

  2. Logarithm laws for unipotent flows. I, Athreya; Margulis

  3. On the probability of a random lattice avoiding a large convex set, Strömbergsson

  4. A note on sphere packings in high dimension, Venkatesh

  5. On the distribution of angles between the $N$ shortest vectors in a random lattice, Södergren, Anders

  6. Remarks on Euclidean Minima, Uri Shapira, Zhiren Wang

  7. On the Mordell-Gruber spectrum, Uri Shapira, Barak Weiss

  8. A solution to a problem of Cassels and Diophantine properties of cubic numbers, Uri Shapira

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I can't say that what I'll relate is fundamental, but it does fit into the new ideas category. Since I and (my collaborator) Florent Balacheff have given talks on the subject and the paper will be in the ArXiv in a few days I feel free to comment on it. This post is an annoucement of joint work with Florent Balacheff and Kroum Tzanev.

As you comment, the basic result in the geometry of numbers is Minkowski's (first) theorem: If the volume of a $0$-symmetric convex body $K \subset \mathbb{R}^n$ is at least $2^n$, then $K$ contains a non-zero integer point.

But what happens when the body is not $0$-symmetric? It is easy to see that Minkowski's theorem fails completely, but that's because one is not thinking symplectically. By using some Hamiltonian dynamics of the sort Balacheff and I used to study isosystolic inequalities in this paper, we guessed that the "right" result should be the following:

Conjecture. If a convex body in $\mathbb{R}^n$ contains no integer point other than the origin, then the volume of its dual body with respect to the origin is at least (n+1)/n!

In other words, one should have a sort of uncertainty principle: if the origin is localized as the unique integer point inside a convex body, the dual body cannot be too small. In fact, its volume is bounded below by $(n+1)/n!$. Another formulation of the conjecture that seems more elementary goes as follows:

If every hyperplane $m_1x_1 + \cdots m_nx_n = 1$, where the $m_i$ are integers not all equal to zero, intersects a convex body $K \subset \mathbb{R}^n$, then the volume of $K$ is at least $(n+1)/n!$

We proved the conjecture in the case $n = 2$ and the asymptotic version:

Theorem. There exists a (universal) constant $C \leq 1$ such that if a convex body $K \subset \mathbb{R}^n$ contains no integer point other than the origin, then the volume of $K^*$ is at least $C^n(n+1)/n!$.

In fact, this result is equivalent to Bourgain-Milman. Moreover, it easily implies the asymptotic version of a conjecture of Ehrhart:

Theorem. There exists a universal constant $c \geq 1$ such that if $K \subset \mathbb{R}^n$ is a convex body with barycenter at the origin and containing no other integer point, then the volume of $K$ is at most $c^n (n+1)^n/n!$.

However, what is really interesting for us is that at least in the case $n=2$ the result trascends the geometry of numbers and is really a result in Hamiltonian dynamics. I just need a definition:

Definition. A hypersurface in the cotangent bundle of a manifold $M$ is said to be optical if its intersection with every cotangent space is a convex hypersurface enclosing the origin.

To an optical hypersurface in the cotangent of a compact manifold we can associate two numbers: the symplectic volume of the region enclosed by $\Sigma$ and the least action of its periodic characteristics.

Theorem. An optical hypersurface $\Sigma$ in the cotangent space of the two-torus carries a periodic characteristic whose action is less than or equal to the square root of two-thirds the symplectic volume enclosed by $\Sigma$.

The inequality is sharp.

Finsler geometers will be happier if I translate: If the Holmes-Thompson volume of a (non-reversible) Finsler $2$-torus $(T^2,F)$ is $3/2\pi$, then $(T^2,F)$ carries a (non-contractible) periodic geodesic of length at most $1$.

In other words, this is the (non-reversible) Finsler version of Loewner's systolic inequality. The reversible Finsler version (replace $3/2\pi$ by $2/\pi$) is due to Stéphane Sabourau and can be found here.

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