Alexander Rhys Duncan

1523 Greene Street
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 Deligne-Mumford 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, 531-568.
2022
M. Ballard, A. Duncan, and P. McFaddin. Derived categories of centrally-symmetric smooth toric Fano varieties. Math. Nachr. 295 (2022), no. 2, 218-241.
J. Blanc, I. Cheltsov, A. Duncan, and Y. Prokhorov. Birational self-maps of threefolds of (un)-bounded genus or gonality. Amer. J. Math. 144 (2022), no. 2, 575-597.
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. K-Theory 4 (2019), no. 2, 211-242.
I. Dolgachev and A. Duncan. Automorphisms of cubic surfaces in positive characteristic. Izv. Ross. Akad. Nauk Ser. Mat. 83 (2019), no. 3, 15-92.
M. Ballard, A. Duncan, and P. McFaddin. The toric Frobenius morphism and a conjecture of Orlov. Eur. J. Math. 5 (2019), no. 3, 640-645.
2018
I. Dolgachev and A. Duncan. Regular pairs of quadratic forms on odd-dimensional spaces in characteristic 2. Algebra Number Theory 12 (2018), no. 1, 99-130.
2016
I. Dolgachev and A. Duncan. Fixed points of a finite subgroup of the plane Cremona group. Algebr. Geom. 3 (2016), no. 4, 441-460.
A. Duncan. Equivariant unirationality of del Pezzo surfaces of degree 3 and 4. Eur. J. Math. 2 (2016), no. 4, 897-916.
A. Duncan. Twisted forms of toric varieties. Transform. Groups 21 (2016), no. 3, 763-802.
2015
A. Duncan and Z. Reichstein. Versality of algebraic group actions and rational points on twisted varieties. J. Algebraic Geom. 24 (2015), no. 3, 499-530.
2014
A. Duncan and Z. Reichstein. Pseudo-reflection groups and essential dimension. J. Lond. Math. Soc. (2) 90 (2014), no. 3, 879-902.
2013
A. Duncan. Finite groups of essential dimension 2. Comment. Math. Helv. 88 (2013), no. 3, 555-585.
2010
A. Duncan. Essential dimensions of \(A_7\) and \(S_7\). Math. Res. Lett. 17 (2010), no. 2, 263-266.
2009
A. Duncan and Z. Reichstein. SAGBI bases for rings of invariant Laurent polynomials. Proc. Amer. Math. Soc. 137 (2009), no. 3, 835-844.
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, 72-83.

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 Paris-Saclay
Finite subgroups of Cremona groups and representation dimension
K-Stability, 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 Aviv-Yaffo, 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 Severi-Brauer 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
G-unirationality
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 Non-algebraically 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 Mini-Workshop, 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