Alexander Rhys Duncan
Department of Mathematics
University of South Carolina
Columbia, SC 29208
Education
 2007  2011
 Doctor of Philosophy, Mathematics
University of British Columbia
Advisor: Zinovy Reichstein  2005  2007
 Master of Science, Mathematics
University of British Columbia
Advisor: Zinovy Reichstein  2001  2005
 Honours Bachelor of Science, Mathematics and Physics
University of Toronto
Positions
 2020 
 Associate Professor
Department of Mathematics
University of South Carolina, Columbia  2015  2020
 Assistant Professor
Department of Mathematics
University of South Carolina, Columbia  2012  2015
 RTG Postdoctoral Assistant Professor
Department of Mathematics
University of Michigan, Ann Arbor
Mentor: Gopal Prasad  2011  2012
 RTG Assistant Adjunct Professor
Department of Mathematics
University of California, Los Angeles
Mentor: Alexander Merkurjev
Publications
 2024
 M. Ballard, A. Duncan, A. Lamarche, and P. McFaddin. Separable algebras and coflasque resolutions. Preprint. To appear in Adv. Math.
 L. Borisov and A. Duncan. Asymptotics of immaculate line bundles on smooth toric DeligneMumford stacks. Preprint. To appear in Michigan Math. J.
 2023
 J. Blanc, I. Cheltsov, A. Duncan, and Y. Prokhorov. Finite quasisimple groups acting on rationally connected threefolds. Math. Proc. Camb. Philos. Soc. 174 (2023), no. 3, 531568.
 2022
 M. Ballard, A. Duncan, and P. McFaddin. Derived categories of centrallysymmetric smooth toric Fano varieties. Math. Nachr. 295 (2022), no. 2, 218241.
 J. Blanc, I. Cheltsov, A. Duncan, and Y. Prokhorov. Birational selfmaps of threefolds of (un)bounded genus or gonality. Amer. J. Math. 144 (2022), no. 2, 575597.
 2020
 M. Ballard, A. Duncan, A. Lamarche, and P. McFaddin. Consequences of the existence of exceptional collections in arithmetic and rationality. Preprint.
 2019
 M. Ballard, A. Duncan, and P. McFaddin. On derived categories of arithmetic toric varieties. Ann. KTheory 4 (2019), no. 2, 211242.
 I. Dolgachev and A. Duncan. Automorphisms of cubic surfaces in positive characteristic. Izv. Ross. Akad. Nauk Ser. Mat. 83 (2019), no. 3, 1592.
 M. Ballard, A. Duncan, and P. McFaddin. The toric Frobenius morphism and a conjecture of Orlov. Eur. J. Math. 5 (2019), no. 3, 640645.
 2018
 I. Dolgachev and A. Duncan. Regular pairs of quadratic forms on odddimensional spaces in characteristic 2. Algebra Number Theory 12 (2018), no. 1, 99130.
 2016
 I. Dolgachev and A. Duncan. Fixed points of a finite subgroup of the plane Cremona group. Algebr. Geom. 3 (2016), no. 4, 441460.
 A. Duncan. Equivariant unirationality of del Pezzo surfaces of degree 3 and 4. Eur. J. Math. 2 (2016), no. 4, 897916.
 A. Duncan. Twisted forms of toric varieties. Transform. Groups 21 (2016), no. 3, 763802.
 2015
 A. Duncan and Z. Reichstein. Versality of algebraic group actions and rational points on twisted varieties. J. Algebraic Geom. 24 (2015), no. 3, 499530.
 2014
 A. Duncan and Z. Reichstein. Pseudoreflection groups and essential dimension. J. Lond. Math. Soc. (2) 90 (2014), no. 3, 879902.
 2013
 A. Duncan. Finite groups of essential dimension 2. Comment. Math. Helv. 88 (2013), no. 3, 555585.
 2010
 A. Duncan. Essential dimensions of \(A_7\) and \(S_7\). Math. Res. Lett. 17 (2010), no. 2, 263266.
 2009
 A. Duncan and Z. Reichstein. SAGBI bases for rings of invariant Laurent polynomials. Proc. Amer. Math. Soc. 137 (2009), no. 3, 835844.
 A. Duncan, M. LeBlanc, and D. Wehlau. A SAGBI basis for \(\mathbb{F}[\mathbb{V}_2 \oplus \mathbb{V}_2 \oplus \mathbb{V}_3]^{C_p}\). Canad. Math. Bull. 52 (2009), no. 1, 7283.
Teaching
University of South Carolina, Columbia
 2023 Fall
 Math 241, Vector Calculus, 22 students.
 Math 241, Vector Calculus, 24 students.
 2023 Spring
 Math 742, Representation Theory, 11 students.
 Math 241, Vector Calculus, 16 students.
 2022 Fall
 Math 701, Algebra I, 5 students.
 Math 580, Elementary Number Theory, 18 students.
 2022 Spring
 Math 142, Calculus II, 46 students.
 Math 548, Geometry, Algebra, and Algorithms, 8 students.
 2021 Fall
 Math 142, Calculus II, 46 students.
 Math 142, Calculus II, 38 students.
 2021 Spring
 Math 300H, Transitions to Advanced Mathematics, 17 students.
 Math 746, Commutative Algebra, 15 students.
 2020 Fall
 Math 142, Calculus II, 60 students.
 Math 748, Computational Algebra(graduate), 9 students.
 2020 Spring
 Math 547H, Algebraic Structures II, 10 students.
 Math 748, Galois Cohomology(graduate), 9 students.
 2019 Fall
 Math 141, Calculus I, 60 students.
 Math 546H, Algebraic Structures I, 16 students.
 2019 Spring
 Math 747, Algebraic Geometry(graduate), 9 students.
 Math 300, Transitions to Advanced Mathematics, 19 students.
 2018 Fall
 Math 546, Algebraic Structures I, 21 students.
 Math 142H, Calculus II, 21 students.
 2018 Spring
 Math 548, Geometry, Algebra, and Algorithms, 8 students.
 2017 Fall
 Math 142, Calculus II, 54 students.
 Math 142, Calculus II, 48 students.
 2017 Spring
 Math 702, Algebra II(graduate), 9 students.
 2016 Fall
 Math 701, Algebra I(graduate), 8 students.
 Math 142H, Calculus II, 18 students.
 2016 Spring
 Math 742, Representation and Invariant Theory(graduate), 6 students.
 Math 300, Transition to Advanced Mathematics, 24 students.
 2015 Fall
 Math 141, Calculus I, 53 students.
University of Michigan, Ann Arbor
 2014 Fall
 Math 217, Linear Algebra, 27 students.
 Math 217, Linear Algebra, 28 students.
 2013 Fall
 Math 217, Linear Algebra, 28 students.
 Math 217, Linear Algebra, 30 students.
 2013 Winter
 Math 217, Linear Algebra, 30 students.
 2012 Fall
 Math 115, Calculus I, 29 students.
University of California, Los Angeles
 2012 Spring
 Math 3B, Calculus for Life Sciences Students, 204 students.
 2012 Winter
 Math 3B, Calculus for Life Sciences Students, 209 students.
 2011 Fall
 Math 115A, Linear Algebra, 36 students.
University of British Columbia
 2009 Fall
 Math 102, Differential Calculus with Applications to Life Sciences, 82 students.
 2008 Winter
 Math 105, Integral Calculus for Social Science and Commerce, 80 students.
 2006 Fall
 Math 184, Differential Calculus for Social Science and Commerce, 60 students.
Invited Talks
 2024

Finite subgroups of Cremona groups and representation dimension
Séminaire Arithmétique et Géométrie Algébrique
Université ParisSaclay 
Finite subgroups of Cremona groups and representation dimension
KStability, Birational Geometry and Mirror Symmetry
Mathematisches Forschungsinstitut Oberwolfach, Germany  2022

Finite subgroups of Cremona groups and representation dimension
Birational Complexity of Algebraic Varieties
Simons Center for Geometry and Physics 
Separable algebras associated to arithmetic toric varieties
14th Brauer Group Conference
The Academic College of Tel AvivYaffo, Tel Aviv, Israel 
Arithmetic toric varieties, endomorphism algebras and rationality
Algebra Seminar
Michigan State University, East Lansing (Online)  2021

Generation and the toric Frobenius
Derived, Biration, and Categorical Algebraic Geometry
Banff International Research Station (Online) 
Derived categories of arithmetic toric varieties
Algebra and Geometry of Homogeneous Spaces
(Online)  2020

Cohomological invariants and separable algebras
Quadratic forms, linear algebraic groups and beyond
(Online) 
Distinguishing objects with separable algebras
Algebraic Geometry Seminar
Washington University in St. Louis 
Finite subgroups of Cremona groups
Winter school on Cremona Groups, Geometric Topology and Algebraic Geometry
Instituto de Matematicas UNAM, Cuernavaca, Mexico  2019

Indistinguishability and simple algebras
Interactions between Brauer Groups, Derived Categories and Birational Geometry of Projective Varieties
Banff International Research Station 
Finite subgroups of Cremona groups
Geometry/Topology Seminar
University of Chicago 
Finite groups of low essential dimension
Braids, Resolvent Degree and Hilbert’s 13th Problem
Institute for Pure & Applied Mathematics, UCLA  2018

Exceptional collections on arithmetic toric varieties
Affine Algebraic Groups, Motives and Cohomological Invariants
Banff International Research Station 
Automorphisms of del Pezzo surfaces in positive characteristic
Subgroups of Cremona Groups
Mathematisches Forschungsinstitut Oberwolfach, Germany 
Exceptional collections on arithmetic toric varieties
Laboratory of Algebraic Geometry and its Applications
Higher School of Economics, Moscow, Russia 
Automorphism groups of cubic surfaces in arbitrary characteristic
Algebra, algebraic geometry and number theory (95th anniversary of I. R. Shafarevich)
Steklov Institute of Mathematics, Moscow, Russia 
Automorphism groups of cubic surfaces in arbitrary characteristic
Algebraic groups and related topics
Fredericton, Canada 
Exceptional collections on arithmetic toric varieties
Cohomology: a link between numbers and geometry
Fredericton, Canada  2017

Toric varieties over general fields
Coloquio IMAFI
Universidad de Talca, Chile 
Automorphisms of del Pezzo surfaces in small characteristic
Conference on aspects of the algebraic and analytic theory of quadratic forms
University of Georgia, Athens 
Equivariant analogs of the arithmetic of del Pezzo surfaces
Algebra and Number Theory Seminar
Emory University, Atlanta  2016

Automorphisms of cubic surfaces in arbitrary characteristic
Workshop on Algebraic Geometry
Hanga Roa, Chile 
Equivariant analogs of arithmetic questions for rational surfaces
Variety Seminar
Independent University of Moscow, Russia 
Rational surfaces in positive characteristic
Pencils of quadrics
Automorphisms of cubic surfaces
Automorphisms of del Pezzo surfaces of degree 1 and 2
Spring School on Surfaces in Positive Characteristics
Higher School of Economics, Moscow, Russia 
Pairs of quadratic forms in characteristic 2
Algebraic Geometry and Number Theory Seminar
Clemson University, South Carolina  2015

Automorphisms of del Pezzo surfaces in positive characteristic
Cohomological Methods in Quadratic Forms and Algebraic Groups
Montreal, Canada 
Pairs of quadratic forms in characteristic 2
The Use of Linear Algebraic Groups in Geometry and Number Theory
Banff International Research Station 
Equivariant unirationality of surfaces
Laboratory of Algebraic Geometry and its Applications
Higher School of Economics, Moscow, Russia 
Pencils of quadrics in characteristic 2
Steklov Institute of Mathematics, Moscow, Russia 
Automorphisms of pencils of quadrics
Special Session on Quadratic Forms in Arithmetic and Geometry
University of Alabama in Huntsville 
Essential dimension of finite groups
Departmental Colloquium
University of Arizona, Tucson 
Essential dimension of finite groups
Departmental Colloquium
University of South Carolina, Columbia  2014

Twisted forms of toric varieties
Workshop on Algebraic and Geometric Invariants of Linear Algebraic Groups and Homogeneous Spaces
University of Ottawa, Ottawa, Canada 
Forms of toric varieties
Algebra and Number Theory Seminar
Emory University, Atlanta 
Forms of toric varieties
Special Session on Algebraic Geometry
University of Tennessee, Knoxville 
Equivariant unirationality
Research Institute for Mathematical Sciences
Kyoto University, Japan  2013

Equivariant unirationality
Algebraic and Arithmetic Geometry  "BIRPOL6"
University of Basel, Switzerland 
Essential dimension and versal varieties
Algebra Seminar
Michigan State University, East Lansing 
Toric varieties and SeveriBrauer varieties
Spring School and Workshop on Torsors, Motives and Cohomological Invariants
Fields Institute 
Equivariant rational maps and unirationality
Special Session on Algebraic Geometry
University of Colorado, Boulder 
Gunirationality
Algebraic Geometry Seminar
Queen's University, Kingston  2012

Essential dimension and versal varieties
International Conference on Essential Dimension and Cremona Groups
Chern Institute of Mathematics, Nankai University, Tianjin, China 
Essential dimension
Department of Mathematics Colloquium
University of California, Riverside 
Versal varieties with a twist
Algebraic Geometry Seminar
Rice University  2011

Toric varieties and essential dimension
Algebraic Groups and Related Structures Seminar
University of British Columbia 
Versal varieties and twisting
Southern California Algebraic Geometry Seminar
University of California, Los Angeles 
Introduction to essential dimension I
Introduction to essential dimension II
Finite groups of low essential dimension
Automorphisms of del Pezzo surfaces of degree 1 and 2
Essential Dimension and Birational Geometry

Versal torsors
Ramification in Algebra and Geometry
Emory University 
Finite groups of low essential dimension
Algebraic Geometry Seminar
University of Michigan, Ann Arbor  2009

Finite groups of essential dimension 2
Linear Algebraic Groups and Related Structures
Banff International Research Station 
Finite groups of essential dimension 2
Algebraic Group Actions and Invariant Theory Session
CMS Summer Meeting, Memorial University of Newfoundland
Contributed Talks
 2021

Quasipolynomials
Graduate Colloquium
University of South Carolina, Columbia 
Cremona groups and representation dimension
Algebra, Geometry, and Number Theory Seminar
University of South Carolina, Columbia  2019

The Ring of Periods
Graduate Colloquium
University of South Carolina, Columbia 
Undecidable Problems
Graduate Colloquium
University of South Carolina, Columbia  2017

Cubic surfaces in positive characteristic
Algebraic Geometry, Arithmetic Geometry and Commutative Algebra Seminar
University of South Carolina, Columbia 
Hilbert’s 13th Problem
Graduate Colloquium
University of South Carolina, Columbia  2015

The plane Cremona group
Graduate Colloquium
University of South Carolina, Columbia 
Pairs of quadratic forms in characteristic 2
Algebraic Geometry, Arithmetic Geometry and Commutative Algebra Seminar
University of South Carolina, Columbia 
Normal forms for pencils of quadrics
Postdoc Algebraic Geometry Seminar
University of Michigan, Ann Arbor  2014

Automorphisms of del Pezzo surfaces
Postdoc Algebraic Geometry Seminar
University of Michigan, Ann Arbor  2013

Finite subgroups of the plane Cremona group
Postdoc Algebraic Geometry Seminar
University of Michigan, Ann Arbor 
Undecidable Problems
Undergraduate Math Club
University of Michigan, Ann Arbor  2012

Forms of Algebraic Varieties
Postdoc Algebraic Geometry Seminar
University of Michigan, Ann Arbor 
Geometric approaches to essential dimension
Algebraic Geometry Seminar
University of Michigan, Ann Arbor 
Cox Rings of Toric Varieties
Toric Varieties over Nonalgebraically Closed Fields
Algebra Seminar
University of California, Los Angeles  2011

Finite Groups of Low Essential Dimension
Algebraic Geometry Seminar
University of British Columbia  2010

Solving Equations with Gröbner Bases
Undergraduate Mathematics Colloquium
University of British Columbia 
What is an Undecidable Problem?
Graduate Colloquium
University of British Columbia 
What is Hilbert’s Thirteenth Problem?
Graduate Colloquium
University of British Columbia  2009

Essential Dimensions of \(A_7\) and
\(S_7\)
Algebraic Groups, Galois Cohomology and Related Topics Seminar
University of British Columbia 
Finite Groups of Essential Dimension 2 over \(\mathbb{C}\)
Algebraic Groups, Galois Cohomology and Related Topics Seminar
University of British Columbia  2008

Cubic Surfaces
Graduate Colloquium
University of British Columbia 
Continued Fraction Arithmetic
Graduate Colloquium
University of British Columbia  2006

The Surreal Numbers
Graduate Colloquium
University of British Columbia  2005

Introduction to Gröbner Bases
Graduate Colloquium
University of British Columbia 
Dissection of Rectangles into Squares
Canadian Undergraduate Mathematics Conference
Queen's University, Kingston  2004

The Gamma Function
Canadian Undergraduate Mathematics Conference
Dalhousie University  2003

An Introduction to Invariant Theory
Canadian Undergraduate Mathematics Conference
York University
Awards and Grants
 2019  2024
 Simons Collaboration Grant ($42000 USD)
 2018
 Research in Pairs, Oberwolfach
 2016  2018
 NSA Young Investigator Grant ($40000 USD)
 2011
 NSERC Postdoctoral Fellowship ($80000 CAD) (declined by PI)
 2009  2011
 Four Year Fellowship ($32000 CAD)
 2009
 Pacific Century Graduate Scholarship ($10000 CAD)
 2009
 Graduate Teaching Award, UBC Math. Dept.
 2007  2009
 NSERC Canada Graduate Scholarship D ($70000 CAD)
 2007  2011
 Ph.D. Tuition Fee Award
 2005  2007
 NSERC Postgraduate Scholarship M ($34600 CAD)
 2005
 Graduate Entrance Scholarship ($1000 CAD)
 2005
 Faculty of Science Grant Supplement Award ($4000 CAD)
 2004
 NSERC Undergraduate Student Research Award ($6800 CAD)
 2003
 NSERC Undergraduate Student Research Award ($6400 CAD)
 2003  2004
 James A and Connie P Dickson Scholarship ($500 CAD)
 2002  2003
 Reuben Wells Leonard Scholarship ($500 CAD)
 2002  2003
 Dr John Knowles Colling Memorial Scholarship ($964 CAD)
 2001  2005
 Howard Ferguson Provincial Scholarship ($10000 CAD)
 2001  2002
 University of Toronto Entrance Award ($3000 CAD)
Supervision and Mentorship
Postdoctoral Scholar Supervision
 2016  2017
 Patrick McFaddin (acting mentor)
Doctoral Student Supervision
 2021 
 Shreya Sharma
 2021 
 Pankaj Singh
 2020 
 Bailey Heath
 2020 
 Jonathan Smith
 2017 
 Tracy Huggins
Undergraduate Student Supervision
 2019  2020
 Thomas Elgin
Doctoral Thesis Committee
 2023
 Jack Dalton
 2023
 Grant Fickes
 2022
 Cuyler Warnock
 2021
 Joe Foster
 2020
 Joshua Grice
 2020
 Alicia Lamarche
 2020
 Harsh Mehta
 2020
 Robert Vandermolen
 2019
 Mohammed Alabbood
 2017
 Wilson Harvey
Masters Thesis Committee
 2018
 Erin Hanna
Comprehensive Exam Committee
 2022
 Robert Groth
 2021
 Grant Fickes
 2021
 Bailey Heath
 2021
 Jonathan Smith
 2019
 Jack Dalton
 2019
 Demmas Salim
 2019
 Cuyler Warnock
 2018
 Tracy Huggins
 2017
 Mohammed Alabbood
 2017
 Joshua Grice
 2017
 Robert Vandermolen
 2016
 Harsh Mehta
 2016
 Xiaofei Yi
Service
Department
 2022 
 Undergraduate Program Committee
 2022 
 Peer Review of Teaching Committee
 2022 
 Chair of Committee of Tenured Faculty
 2022 
 Qualifying Exam Committee
 2022
 Undergraduate Advisor
 2020  2022
 Faculty Advisory Council
 2019  2022
 Graduate Advisory Council
 2018  2019
 Math 546 Course Coordinator
 2017  2018
 Faculty Advisor of Graduate Colloquium
 2016 
 Website Committee (Chair)
 2016 
 Organizer of Algebra, Geometry, and Number Theory Seminar
 2016  2017
 Organizer of Graduate Colloquium
 2016  2017
 Graduate Advisory Council
 2015  2020
 Qualifying Exam Committee
 2015  2020
 Undergraduate Advisor
 2015  2018
 Events Committee
University
 2021
 Top Scholar Referee
 2017  2020
 Faculty Senate
Community
 2016  2019
 UofSC High School Math Contest Judge
 2016
 UofSC Practice AP Exam Proctor
Professional
 2021
 Organizer of Algebraic groups and algebraic geometry: in honor of Zinovy Reichtein's 60th birthday. (Online)
Conferences Attended
 2023

Arithmetic Geometry & Algebraic Groups
University of Virginia, Charlottesville, Virginia 
Georgia Algebraic Geometry Symposium
University of Georgia, Athens, Georgia 
Workshop on Syzygies and mirror symmetry
AIM, Pasadena, California  2022

Georgia Algebraic Geometry Symposium
Emory University, Atlanta, Georgia  2018

A day of Algebraic Geometry in Savannah
Georgia Southern University, Armstrong Campus, Savannah, Georgia 
Georgia Algebraic Geometry Symposium
Georgia Institute of Technology, Atlanta, Georgia  2017

Emory Conference on Higher Obstructions to Rational Points
Emory University, Atlanta, Georgia 
Beyond Toric Geometry
Casa Matemática Oaxaca, Oaxaca, Mexico 
Algebraic Groups
Mathematisches Forschungsinstitut Oberwolfach, Germany 
Georgia Algebraic Geometry Symposium
Emory University, Atlanta, Georgia  2016

Cremona Conference 2016
Universität Basel, Switzerland 
Georgia Summer Workshop in Algebraic Geometry
University of Georgia, Athens  2015

Georgia Algebraic Geometry Symposium
Emory University, Atlanta, Georgia 
Méthodes cohomologiques dans la théorie des groupes algébriques
Luminy, France  2013

Cremona Groups
FRG MiniWorkshop, University of Utah 
Lie Algebras, Torsors and Cohomological Invariants
Banff International Research Station  2010

Workshop on Torsors, Lie algebras and Galois Cohomology
University of Ottawa  2009

Toric Varieties Workshop
Mathematical Sciences Research Institute  2008

Mini Course 2008: Essential dimension and canonical dimension
Lens, France 
Lie Theory and Geometry: The Mathematical Legacy of Bertram Kostant
University of British Columbia 
ABC Algebra Workshop
Simon Fraser University  2007

PIMS Algebra Summer School
University of Alberta 
ABC Algebra Workshop
University of Alberta  2006

Symmetry and Spaces: On the Occasion of Gerry Schwarz's 60th Birthday
Fields Institute, Toronto 
PIMS/UNAM Algebra Summer School
Banff International Research Station 
ABC Algebra Workshop
University of British Columbia