Zinovy Reichstein


Research Classification

Research Interests

Algebraic groups
algebraic geometry

Relevant Thesis-Based Degree Programs



Master's students
Doctoral students
Postdoctoral Fellows

Pure mathematics.

Complete these steps before you reach out to a faculty member!

Check requirements
  • Familiarize yourself with program requirements. You want to learn as much as possible from the information available to you before you reach out to a faculty member. Be sure to visit the graduate degree program listing and program-specific websites.
  • Check whether the program requires you to seek commitment from a supervisor prior to submitting an application. For some programs this is an essential step while others match successful applicants with faculty members within the first year of study. This is either indicated in the program profile under "Admission Information & Requirements" - "Prepare Application" - "Supervision" or on the program website.
Focus your search
  • Identify specific faculty members who are conducting research in your specific area of interest.
  • Establish that your research interests align with the faculty member’s research interests.
    • Read up on the faculty members in the program and the research being conducted in the department.
    • Familiarize yourself with their work, read their recent publications and past theses/dissertations that they supervised. Be certain that their research is indeed what you are hoping to study.
Make a good impression
  • Compose an error-free and grammatically correct email addressed to your specifically targeted faculty member, and remember to use their correct titles.
    • Do not send non-specific, mass emails to everyone in the department hoping for a match.
    • Address the faculty members by name. Your contact should be genuine rather than generic.
  • Include a brief outline of your academic background, why you are interested in working with the faculty member, and what experience you could bring to the department. The supervision enquiry form guides you with targeted questions. Ensure to craft compelling answers to these questions.
  • Highlight your achievements and why you are a top student. Faculty members receive dozens of requests from prospective students and you may have less than 30 seconds to pique someone’s interest.
  • Demonstrate that you are familiar with their research:
    • Convey the specific ways you are a good fit for the program.
    • Convey the specific ways the program/lab/faculty member is a good fit for the research you are interested in/already conducting.
  • Be enthusiastic, but don’t overdo it.
Attend an information session

G+PS regularly provides virtual sessions that focus on admission requirements and procedures and tips how to improve your application.



These videos contain some general advice from faculty across UBC on finding and reaching out to a potential thesis supervisor.

Graduate Student Supervision

Doctoral Student Supervision

Dissertations completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest dissertations.

Essential dimension of representations of algebras (2021)

Let k be a field, A be a finitely generated associative k-algebra, and Rep_A[n] be the functor sending a field K containing k to the set of isomorphism classes of representations of A_K of dimension at most n. In the first part of this thesis, we study the asymptotic behavior of the essential dimension of this functor, i.e., the function r_A(n) := ed_k(Rep_A[n]), as n goes to infinity. In particular, we show that the rate of growth of r_A(n) determines the representation type of A. That is, r_A(n) is bounded from above if A is of finite representation type, grows linearly if A is of tame representation type, and grows quadratically if A is of wild representation type. Moreover, r_A(n) allows us to construct invariants of algebras which are finer than the representation type. In the second part of the thesis, we study the essential dimension of representations of a fixed quiver with given dimension vector. We also consider the question of when the genericity property holds, i.e., when essential dimension and generic essential dimension agree. We classify the quivers satisfying the genericity property for every dimension vector and show that for every wild quiver the genericity property holds for infinitely many of its Schur roots. We also construct a large class of examples, where the genericity property fails. Our results are particularly detailed in the case of Kronecker quivers.

View record

A graph-theoretic approach to a conjecture of Dixon and Pressman (2020)

Given n×n matrices, A_1,...,A_k, define the linear operator L(A_1,...,A_k): Mat_n -> Mat_n by L(A_1,...,A_k)(A_(k+1)) = sum_sigma sgn(sigma) sgn(sigma)A_sigma(1)A_sigma(2)...A_sigma(k+1). The Amitsur-Levitzki theorem asserts that L(A_1,...,A_k) is identically 0 for every k > 2n − 1. Dixon and Pressmanconjectured that if 2
View record

Essential dimension and classifying spaces of algebras (2020)

The overarching theme of this thesis is to assign, and sometimes find, numerical values which reflect complexity of algebraic objects. The main objects of interest are field extensions of finite degree, and more generally, etale algebras of finite degree over a ring.Of particular interest to us is the invariant known as essential dimension. The essential dimension of separable field extensions was introduced by J. Buhler and Z. Reichstein in their landmark paper. A major (still) open problem arising from that work is to determine the essential dimension of a general separable field extension of degree n (or equivalently, the essential dimension of the symmetric group). Loosening the separability assumption we arrive at the case of inseparable field extensions. In the first part of this thesis we study the problem of determining the essential dimension of inseparable field extensions. In the second part of this thesis, we study the essential dimension of the double covers of symmetric groups and alternating groups, respectively. These groups were first studied by I. Schur and their representations are closely related to projective representations of symmetric and alternating groups. In the third part, we study the problem of determining the minimum number of generators of an etale algebra over a ring. The minimum of number of generators of an etale algebra is a natural measure of its complexity.

View record

Group actions on curves over arbitrary fields (2015)

This thesis consists of three parts. The common theme is finite group actions on algebraic curves defined over an arbitrary field k.In Part I we classify finite group actions on irreducible conic curves defined over k. Equivalently, we classify finite (constant) subgroups of SO(q) up to conjugacy, where q is a nondegenerate quadratic form of rank 3 defined over k. In the case where k is the field of complex numbers, these groups were classified by F. Klein at the end of the 19th century. In recent papers of A. Beauville and X. Faber, this classification is extended to the case where k is arbitrary, but q is split. We further extend their results by classifying finite subgroups of SO(q) for any base field k of characteristic ≠ 2 and any nondegenerate ternary quadratic form q.In Part II we address the Hyperelliptic Lifting Problem (or HLP): Given a faithful G-action on ℙ¹ defined over k and an exact sequence 1 → μ₂ → Gʹ→ G → 1, determine the conditions for the existence of a hyperelliptic curve C/k endowed with a faithful Gʹ-action that lifts the prescribed G-action on the projective line. Alternatively, this problem may be regarded as the Galois embedding problem given by the surjection Gʹ ↠ G and the G-Galois extension k(ℙ¹)/k(ℙ¹)G. In this thesis, we find a complete solution to the HLP in characteristic 0 for every faithful group action on ℙ¹ and every exact sequence as above.In Part III we determine whether, given a finite group G and a base field k of characteristic 0, there exists a strongly incompressible G-curve defined over k. Recall that a G-curve is an algebraic curve endowed with the action of a finite group G. A faithful G-curve C is called strongly incompressible if every dominant G-equivariant rational map of C onto a faithful G-variety is birational. We prove that strongly incompressible G-curves exist if G cannot act faithfully on the projective line over k. On the other hand, if G does embed into PGL₂ over k, we show that the existence of strongly incompressible G-curves depends on finer arithmetic properties of k.

View record

Essential Dimension and Linear Codes (2014)

The essential dimension of an algebraic group G is a measure of the complexity of G-torsors. One of the central open problems in the theory of essential dimension is to compute the essential dimension of PGL_n, whose torsors correspond to central simple algebras up to isomorphism. In this thesis, we study the essential dimension of groups of the form G/μ, where G is a reductive algebraic group satisfying certain properties, and μ is a central subgroup of G. In particular, we consider the case G=GL_(n₁) × ⋯ × GL_(n_r ) where each n_i is a power of a single prime p, which is a generalization of the group PGL_(p^a )=GL_(p^a )/G_m. We will see that torsors for G/μ correspond to tuples of central simple algebras satisfying certain properties. Surprisingly, computing the essential dimension of G/μ becomes easier when r≥3.Using techniques from Galois cohomology, representation theory and the essential dimension of stacks, we give upper and lower bounds for the essential dimension of G/μ. To do this, we first attach a linear ‘code’ to the central subgroup μ, and define a weight function on this code. Our upper and lower bounds are given in terms of a minimal weight generator matrix for the code. In some cases we can determine the exact value of the essential dimension of G/μ.

View record

Finite groups of low essential dimension (2011)

Informally, essential dimension is the minimal number of parameters required to define an algebraic object. This invariant has numerous connections to Galois cohomology, linear algebraic groups and birational geometry. In particular, the essential dimension of finite groups has connections to the Noether problem, inverse Galois theory and the simplification of polynomials via Tschirnhaus transformations.This thesis studies finite groups of low essential dimension using methods from birational geometry. Specifically, the main results are a classification of finite groups of essential dimension 2, and a proof that the alternating and symmetric groups on 7 letters have essential dimension 4.

View record

Essential dimension of algebraic groups (2010)

We study the essential dimension of linear algebraic groups. For a group G, essential dimension is a measure for the complexity of G-torsors or, more generally, the complexity of any algebraic or geometric structure with automorphism group G. This makes essential dimension a powerful invariant with many interesting and surprising connections to problems in algebra and geometry.We show that for various classes of groups, including finite (algebraic) groups and algebraic tori, the essential dimension is related to minimal faithful representations. In many cases this renders the exact value of the essential dimension computable and we explore several of its consequences. An important open problem is the essential dimension of the projective linear group PGLn. This topic is closely related to the structure theory of central simple algebras, which may be viewed as twisted forms of the algebra of n x n matrices.We study central simple algebras with additional structure such as a distinguished Galois subfield. We prove new bounds on the essential dimension of these algebras and, as a corollary, of the group PGLn.

View record

Master's Student Supervision

Theses completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest theses.

On the anisotropy theorem of Papadakis and Petrotou (2023)

We study the anisotropy theorem for Stanley-Reisner rings of simplicial homology spheres in characteristic 2 by Papadakis and Petrotou. This theorem implies the Hard Lefschetz theorem as well as McMullen's g-conjecture for such spheres. Our first result is an explicit description of the quadratic form. We use this description to prove a conjecture stated by Papadakis and Petrotou. All anisotropy theorems for homology spheres and pseudo-manifolds in characteristic 2 follow from this conjecture. Using a specialization argument, we prove anisotropy for certain homology spheres over the field Q. These results provide another self-contained proof of the g-conjecture for homology spheres in characteristic 2.

View record

Reduction of structure to parabolic subgroups (2023)

Let G be an affine group over a field of characteristic not two. A G-torsor is called isotropic if it admits reduction of structure to a proper parabolic subgroup of G. This definition generalizes isotropy of affine groups and involutions of central simple algebras. When does G admit anisotropic torsors? Building on work of J. Tits, we answer this question for simple groups. We also give an answer for connected and semisimple G under certain restrictions on its root system.

View record

Birational models of geometric invariant theory quotients (2017)

In this thesis, we study the problem of finding birational models of projective G-varieties with tame stabilizers. This is done with linearizations, so that each birational model may be considered as a (modular) compactification of an orbit space (of properly stable points).The main portion of the thesis is a re-working of a result in Kirwan's paper "Partial Desingularisations of Quotients of Nonsingular Varieties and their Betti Numbers", written in a purely algebro-geometric language. As such, the proofs are novel and require the Luna Slice Theorem as their primary tool.Chapter 1 is devoted to preliminary material on Geometric Invariant Theory and the Luna Slice Theorem.In Chapter 2, we present and prove a version of "Kirwan's procedure". This chapter concludes with an outline of some differences between the current thesis and Kirwan's original paper.In Chapter 3, we combine the results from Chapter 2 and a result from a paper by Reichstein and Youssin to provide another type of birational model with tame stabilizers (again, with respect to an original linearization).

View record

An extension to the Hermite-Joubert problem (2016)

Let E/F be a field extension of degree n. A classical problem is to find a generating element in E whose characteristic polynomial over F is as simple is possible. An 1861 theorem of Ch. Hermite [5] asserts that for every separable field E/F of degree n there exists an element a ∈ E whose characteristic polynomial is of the formf(x) = x⁵ +b₂x³ +b₄x+b₅or equivalently, tr{E/F}(a) = tr{E/F}(a³) = 0. A similar result for extensions of degree 6 was proven by P. Joubert in 1867; see [6].In this thesis we ask if these results can be extended to field extensions of larger degree. Specifically, we give a necessary and sufficient condition for a field F, a prime p and an integer n ≥ 3 to have the following property: Every separable field extension E/F of degree n contains an element a ∈ E such that a generates E over F, and tr{E/F}(a) = tr{E/F}(a^p) = 0.As a corollary we show for infinitely many new values of n that the theorems of Hermite and Joubert do not extend to field extensions of degree n. We conjecture the same for more values of n and provide computational evidence for a large number of these.

View record

Generations And Relations For Ring Of Invariants (2016)

In this thesis we first prove that the algebra of invariants for reductive groups over the base field complex numbers are finitely generated. Then we focus on invariant algebras of finite groups. After showing the natural relation between invariant algebras and reflections in the group we prove the Chevalley-Shephard-Todd theorem. We conclude with classification of complex finite reflection groups and some examples. Throughout the thesis we follow similar arguments as in Springer[8] and Kane[6] where we give full details on the arguments.

View record

Induced maps in Galois cohomology (2011)

Galois cohomology is an important tool in algebra that can be used to classify isomorphism classes of algebraic objects over a field. In this thesis, we show that many objects of interest in algebra can be described in cohomological terms. Some objects that we discuss include quadratic forms, Pfister forms, G-crossed product algebras, and tuples of central simple algebras. We also provide cohomological interpretation to some induced maps that naturally occur in short exact sequences.

View record


If this is your researcher profile you can log in to the Faculty & Staff portal to update your details and provide recruitment preferences.


Get key application advice, hear about the latest research opportunities and keep up with the latest news from UBC's graduate programs.