# Research Summary

Here I will give a summary of the research I have done so far.

## Homogeneous dynamics in the space of lattices

1. The space of lattices and diagonal orbits – Introduction: basic definitions and notations on lattices.
2. Equidistribution of divergent orbits – It is well known that for almost every $x\in [0,1]$, if we write $x$ in its continued fraction expansion (c.f.e) $x=[0;a_1 ,a_2 ,a_3 ,...]$, then the statistics of the sequence $a_i$ approaches to the Gauss-Kuzmin statistics (e.g. for every $n$ we have that $\frac{|\{ i : a_i=n , 1\leq i \leq k \}| } {k}$ $\to \log_2(1+\frac{1}{n(n+2)})$ as $k\to \infty$). Indeed, this claim is a special case of the pointwise ergodic theorem. However, this is not true for all $x\in [0,1]$ and in particular it fails for rational numbers, which have finite c.f.e. In this work we formulate and prove a counterpart to the claim above which works for finite continued fractions, where the proof relies on the connection of c.f.e and the dynamics of the space of 2-dimensional lattices. More specifically, we use entropy arguments to show that certain families of divergent geodesics equidistribute, and we give partial results for divergent orbits in high dimensional spaces. Finally we show how to interpret these results using the language of the adeles.
3. Shapes of unit lattices and escape of mass – Given a totally real number field $K/\mathbb{Q}$ of dimension $n$ there is a natural embedding $K \hookrightarrow \mathbb{R}^n$ defined by $\alpha \mapsto (\sigma _1(\alpha),...,\sigma_n(\alpha) )$ where the $\sigma_i$ are the distinct n embeddings of $K$ in $\mathbb{R}$. Under this map, full modules in $K$ (e.g. its ring of integers) are mapped to lattices in $\mathbb{R}^n$. Moreover, their orbits under the group $A$ of diagonal matrices produces a periodic orbit (i.e. compact), and any such orbit can be constructed in this way – indeed, this follows from Dirichlet’s unit theorem which states that after “linearizing” the group of units in “full” rings in $K$ we obtain a lattice. Since $A\cong \mathbb{R}^{n-1}$, the shape of this orbit is defined by the stabilizer in $A$ which is in itself a lattice of dimension $n-1$ and corresponds to the unit group of the ring $\mathcal{O}_M=\{a\in K \mid aM\subseteq M\}$. In this joint work with Uri Shapira we investigate these periodic orbits for $n=3$, and we construct many families of lattices (i.e. cubic rings) for which we can compute the shapes of their orbits (i.e. their group of units).

## Machine Learning

1. Learning and compression – A learning algorithm is an algorithm that receives as input a finite (large) sample $((x_1,f(x_1)),...,(x_n,f(x_n)))$ for some function $f$ where the $x_i$ are generated independently by some distribution $\mathcal{D}$, and outputs a hypothesis function $\tilde{f}$ which is “close” to $f$ as possible (with respect to the probability $\mathcal{D}$).
On the other hand, a compression algorithm receives as an input a sample $((x_1,f(x_1)),...,(x_n,f(x_n)))$, and outputs a subsample from which it can recover a function $\tilde{f}$ which agrees with $f$ on the original sample.
For example, given a sample of (at least two distinct) points $(x_i,f(x_i))$ where $f(x)=ax+b$ is linear, it is easy to recover the function $f$, namely the set of linear functions is learnable. On the other hand, we can always compress each such sample to two points and recover the function from them.
In this joint work with Amir Yehudayoff and Shay Moran, we consider the connection between learnability and several variants of compression algorithms, and conclude that for “natural” definitions, learnability and compressionability are equivalent.

## Polynomial Identities

1. Graded Algebras and PI theory: an Introduction – Some preliminaries about graded algebras and polynomial identities.
2. Group Gradings on Algebras – Given a PI algebra $A$, what can be said on all the groups which grade $A$ under some “nice” conditions? For example, grading on (not necessarily commutative) algebra where the homogeneous elements commute up to nonzero scalars.
3. Embeddings of Graded Simple Algebras – If there is a graded embedding $B\hookrightarrow _G A$, then every graded polynomial identity of $A$ is an identity of $B$, or in other words $Id_G (A) \subseteq Id_G(B)$. Is the other direction also true, namely does $Id_G(A)\subseteq Id_G(B)$ imply a graded embedding $B\hookrightarrow _G A$?

## Generic crossed products and their center

Generic Algebras and Generic Crossed Products  – The central simple algebras, which are matrix algebras over division algebras, are the building blocks of finite dimensional noncommutative algebras. Indeed, modulo the Jacobson radical, any such algebras is a product of central simple algebras.

The generic division algebra is a universal object which is used in order to study these algebras. A more specialized construction is the generic $G$-crossed product which is the universal object in the class of $G$-crossed products. In this work we study the center of the generic crossed product and try to determine how close it is to being a purely transcendental extension of the base field. This involves computing the invariants of some Galois action on a field of rational function in several variables, which is closely related to Noether’s problem and the inverse Galois problem.