## Max Weinreich I am a fifth-year Ph.D student at Brown University, working at the intersection of dynamical systems, algebraic geometry, and number theory. My advisor is Joe Silverman. I am applying for junior academic positions starting Fall 2022.

Pronouns: he/him

### Contact Info

Department of Mathematics
Box 1917
Brown University
151 Thayer Street
Providence, RI 02912

Email: max [underscore] weinreich [at] brown [dot] edu

I study arithmetic dynamics, which is the study of iteration of functions in number theory. My particular interests include moduli spaces, integrable systems, finite fields, and projective configurations.

For an idea of what I study, consider any function involving integers or rational numbers, and ask what happens when you apply the function over and over. For instance, does the function f(x) = x^2 - 3, after repeating many times, take any rational numbers right back to themselves? What's the maximum number of steps that could take? This gives a way of introducing a notion of time into the world of numbers. My thesis studies the pentagram map, a simple geometric operation with fascinating dynamics, from this arithmetic perspective. I presented my thesis research in a short talk at the virtual conference AGNES 2021. A recording of my talk is publicly available on the conference page.

In Fall 2021, I am organizing an online seminar on every kind of moduli space. Join us!

### Papers

5. GIT stability of linear maps on projective space with marked points. (32 pages.) 2021. Preprint. arXiv:2111.06351.

We define a moduli space for projective linear maps with marked points. Our moduli spaces are constructed using geometric invariant theory, in the style of Mumford. We interpret the stability and semistability conditions for points in the moduli space via the polyhedral combinatorics of the full root polytope of type A_N, arising in Lie theory.

4. Dynamical moduli spaces and polynomial endomorphisms of configurations. (26 pages.) With Talia Blum, John Doyle, Trevor Hyde, Colby Kelln, and Henry Talbott. 2021. Preprint. arXiv:2108.10777.

A portrait is a combinatorial model for a discrete dynamical system on a finite set. We study the geometry of portrait moduli spaces, whose points correspond to equivalence classes of point configurations on the affine line for which there exist polynomials realizing the dynamics of a given portrait. We present results and pose questions inspired by a large-scale computational survey of intersections of portrait moduli spaces for polynomials in low degree.

3. The algebraic dynamics of the pentagram map. (38 pages.) 2021. Preprint. arXiv: 2104.06211.

The pentagram map, introduced by Schwartz in 1992, is a dynamical system on the moduli space of polygons in the projective plane. Its real and complex dynamics have been explored in detail. We study the pentagram map over an arbitrary algebraically closed field of characteristic not equal to 2. We prove that the pentagram map on twisted polygons is a discrete integrable system: it is birational to a self-map of a family of abelian varieties.

2. Automorphism groups of endomorphisms of P^1(F_p). (34 pages.) Julia Cai and Benjamin Hutz and Leo Mayer and Max Weinreich. Preprint. 2020.

We study automorphisms of self-maps of the projective line over the algebraic closure of a finite field. We show that every finite subgroup of PGL(F_q) can arise as an automorphism group, and prove some bounds on the degrees. We also completely describe the automorphism loci of degree 2 maps for all characteristics.

1. Counting arcs in projective planes via Glynn's algorithm. (17 pages.) Kaplan, N., Kimport, S., Lawrence, R., Peilen, P., and Weinreich, M. J. Geom. (2017) 108: 1013.

Arcs are collections of points in linear general position, a fundamental concept in classical and algebraic geometry. We count arcs in combinatorial projective planes in terms of highly determined configurations of points and lines.

### In preparation

Counting 10-arcs in the projective plane over a finite field. Joint work with Kelly Isham, Nathan Kaplan, Sam Kimport, Rachel Lawrence, and Luke Peilen.

An n-arc in a projective plane over a finite field is a set of n points, no three collinear. For each n up to 9, there are known formulas for the number of n-arcs in the plane, given as a polynomial in the order q of the underlying finite field depending only on the residue class of q mod 30. Such functions are called polynomials on residue classes, or quasipolynomial. We show that the counting function for 10-arcs can be expressed in terms of polynomials in q and point-counting functions of varieties of dimension at most 3. By studying the arithmetic of these varieties, we show that the counting function for 10-arcs is not quasipolynomial.

### Pictures of the pentagram map

Here are some pictures that came up in my research on the pentagram map. In some sense, these are images of 2-dimensional tori living in 4 or 6 dimensions.      