Research Summary

Homogeneous dynamics in the space of lattices

My primary academic focus centers on the captivating realm of number theory. This field not only boasts an array of intriguing and elegant results but also offers diverse methodological approaches, ranging from algebra and analysis to geometry, topology and more. Among these methodologies, one particularly fascinating tool is the concept of lattices, which represent meticulously arranged points in space (think of \mathbb{Z}^n within \mathbb{R}^n). Leveraging this framework, numerous significant results in number theory can be formulated and proven. Notably, the study of 2-dimensional lattices unveils profound connections to continued fractions, a cornerstone in the exploration of numerical properties.

Here I will summarize the basics needed to understand this field, and the main results from my research.

  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. For example, counting the times a given integer n appears satisfies:
    \frac{|\{ i : a_i=n , 1\leq i \leq k \}| } {k}  \overset{k \to \infty}{\longrightarrow} \log_2(1+\frac{1}{n(n+2)}).
    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 Continued Fractoins

The standard continued fraction expansion constitutes one of the basic tools to study numbers, and in particular how close a real number to “being rational”. However, as strong a tool as this expansion can be, in many cases, it is really hard to find any usable patterns in it. For example, the continued fraction for \pi is 

\pi =[3;7,15,1,292,1,1,1,2,1,3,...] := 3+\cfrac{1}{7+\cfrac{1}{15+\cfrac{1}{1+\cfrac{1}{292+\ddots}}}}.

Looking for similar expansions with simpler patterns lead to the definition of polynomial continued fractions, where both the numerators and denominators are defined using polynomials. For example, \pi can be written as 

\pi +3 = 6 + \cfrac{1^2}{6 + \cfrac{2^2}{6 + \cfrac{3^2}{6 + \cfrac{4^2}{6+\ddots}}}}.

This interesting “continued fraction” like expansion with extra patterns have an interesting dynamical theory behind it, which I studied as part of my research in the Ramanujan Machine group at the Technion, and more details can be found here.

Polynomial Identities

When I began my mathematical research journey, I first delved into the realm of polynomial identities of algebras. In the “simple” commutative algebra, by definition any assignment of the polynomial p(x,y)=xy-yx returns zero, so we call such a polynomial an identity of the algebra. While this fails for general algebra, other polynomial identities might still hold (e.g., all finite-dimensional algebras possess such identities). The study of such polynomial identities attempts to understand the inherent properties of algebras, where further symmetries using group grading reveals even more interesting structures.

  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.