Stephan Tillmann

research projects

If you are interested in doing a research project either as an Honours thesis, a higher degree thesis or over summer, please come and talk to me. Possible topics include:

Low-dimensional topology (knot theory, 3-manifolds, geometric structures)

Hyperbolic geometry (hyperbolic manifolds, invariants)

Group theory (representations of finitely generated groups, group actions on trees)

Computational algebraic geometry

Tropical geometry (foundation, algorithms and applications)

current supervision of postgraduate students

James Morgan, PhD (started 2021), joint supervision with Jonathan Spreer

Grace Garden, PhD (started 2019)

previous supervision of postgraduate students

Yossi Bokor, PhD (2019–2022), joint supervision with Kate Turner, Vanessa Robins, Jonathan Spreer

Carol Badre, MPhil (2019–2020), joint supervision with Anne Thomas

Dominic Tate, PhD (2016–2020)

Dickson Annor, MPhil (2017–2018)

Alex Casella, PhD (2015–2018)

Montek Gill, MSc (2015)

supervision of MMathSci students

Sepehr Saryazdi: The length spectrum of convex projective surfaces (2023-2024)

supervision of honours students

The University of Sydney: • Youheng Yao: Detecting surfaces using representations of 3-manifold groups (2022); • Fariza Rashid (associate supervisor; with Lamiae Azizi): Classification of Stem Cells (2020-2021); • Max Davy: Circle valued discrete Morse theory and veering triangulations (2020); • Jonathan Kim: CAT(0) cube complexes (2019); • Grace Garden: The Mapping Class Group and its Action on the Character Variety of the Once-Punctured Torus (2015); • Dominic Tate: Parametrising the Teichmüller Space of the Punctured Torus (2015); • Montek Gill: The Hyperbolic Gluing Equations Over Commutative Rings and a Residual Property of 3-Manifold Groups (2014); • Sampson Wong: Algorithms for the Epstein-Penner decomposition of a cusped surface (2014); • Alex Casella: Toric varieties associated to ideal triangulations (2013-2014)

The University of Queensland: • Samuel Mellick: Alexander invariants and the Thurston norm (2011); • Mitchell Watt: Morse Theory: Smooth and Discrete (2011)

The University of Melbourne: • Tharatorn Supasiti: Serre's tree for SL(2,F) (2008); • James Wan: Projective Geometry, Projective Structures, and the Hex Metric (2008)

supervision of internships

• Yan Garito (Ecole Normale Superieure de Rennes), 4-month masters internship: Strictly convex projective structures on surfaces (2020)

• Julien Sedro (Ecole Normale Superieure de Cachan), 2-month masters internship: The Epstein-Penner construction (2013)

supervision of first year undergraduate research projects

• Adrian Zhao, Alan Li, Ellie Bland, Martin Ong, Steven Lin with Jasmine Wang as mentor: Science Dalyell Showcase (SCDL1991): Understanding the Shape of Spaces and Data Sets (2022)

• Vin Bush, Ryder Chen, Cameron Eggins, Martin Gossow, Jacob Willson, Jennifer Zhao with Hazel Browne as mentor: Science Dalyell Showcase (SCDL1991): Renewal in Fractals and Random Walks (2018)

• Colin Huang, Jongmin Lim, Zac Sanchez, Will Stewart, Max Tobin, Alec Zhang with Leo Jiang as mentor: TSP Showcase Project (SCTP1907): Births and Deaths of Singularities (2017)

• Kieran Ballurkar, Chunxi Jiao, Cecilia Li, Rachel Wong with Joshua Ciappara as mentor: TSP Showcase Project (SCTP1907) Unravelling the Dodecahedral Spaces (2014)

• Courtney Darville, Jack Gregan, Gavin Lee, Nick Miltchinov, Ankur Paul, George Polonski with Sampson Wong as mentor: TSP Showcase Project (SCTP1907) Space: Appearance, Imagination and the Fourth Dimension (2013)

supervision of vacation research projects

The University of Sydney: • Youheng Yao (Denison Research Project): Ptolemy varieties (2021/22); • Sepehr Saryazdi (Denison Research Project): Convex Projective Surface Visualisation Tool (software) (2021/22); • Youheng Yao (AMSI Vacation Research Project co-supervised with Grace Garden): The topology of character varieties of once-punctured torus bundles (2020/21); • Max Mikkelsen (Denison Research Project): Algebraic horocircle packings of cusped real projective surfaces (2020/21); • Cameron Eggins (AMSI Vacation Research Project), Supavit Dumrongprechachan and Mark Rothery (Denison Research Project): Convex projective surfaces (2019/20); • Michael Zhao (AMSI Vacation Research Project) and Tony Wang (Denison Research Project): Trisections of hyperbolic 4-manifolds (2018/19); • Jongmin Lim (Denison Research Project): Ptolemy coordinates for ideal triangulations (2018/19); • Richie Lim, Jonathan Mui, Michael Zhao, Summer Vacation Research Project: Solv manifolds (2017/18); • Asem Wardak (AMSI Vacation Research Project): Negatively curved structures in dimensions three and four via discrete geometry (2016/17); • Mahan Ghobadi (Summer Vacation Research Project): Hyperbolic and Projective Arithmetic Fuchsian Groups (2016/17); • Ennes Mehmedbasic and Edwin Spark (Summer Vacation Research Project co-supervised with Alex Casella): The toric geometry of the hyperbolic gluing equations (2015/16); • Abdul Zreika (AMSI Vacation Research Project), together with Courtney Darville, Lachlan Mcpheat and Asem Wardak (Summer Vacation Research Project co-supervised with Robert Haraway): The space of closed essential normal surfaces in knot complements (2015/16); • Montek Gill (AMSI Vacation Research Project): Cross ratios and Thurston's gluing equations over rings (2012/13)

The University of Queensland: • Tom Chappell (Advanced Study Program in Science): Low-dimensional topology (2010); • Samuel Mellick (Summer Research Project): Gromov norm and scl (2010)

The University of Melbourne: • Stephen Muirhead (Summer Vacation Project co-supervised with Craig Hodgson): Projective structures on surfaces (2008)

Columbia University in the City of New York: • Co-supervised three students in a VIGRE Undergraduate Summer Research Experience (with Walter Neumann and Abhijit Champanerkar): A-polynomials of hyperbolic 3-manifolds (2000)

invited lecture series

• Character varieties, A-polynomials and Knots, Australian National University, Australia (June 2021)

• Character varieties, Okinawa Institute of Science and Technology Graduate University, Japan (March 2019)

• Triangulations and combinatorial differential geometry, MSC Tsinghua University, China (July 2014)

• Real projective manifolds, MSC Tsinghua University, China (July 2012)

• Geometry & Groups, AMSI Summer School, University of New South Wales (January 2012)

• Varieties associated to 3-manifolds, Melbourne Summer School, The University of Melbourne (January 2001)

reading courses

Reading courses are tailored to the interests and background knowledge of the participants. Assessment typically consists of exercises or presentations (40%) and a short report (60%).

The University of Sydney: • Martin Ong, Fuchsian groups, SCDL3991 Dalyell Individual Research Project (Semester 2, 2024); • Declan Zammit, The earthquake theorem in hyperbolic geometry, SCDL3991 Dalyell Individual Research Project (Semester 2, 2023); • Jack Coleman, Geometry and Surfaces, SCDL3991 Dalyell Individual Research Project (Semester 2, 2023); • Sepehr Saryazdi, Decorated projective structures on surfaces, SCDL3991 Dalyell Individual Research Project (Semester 1, 2022); • Youheng Yao, Plane algebraic curves, SCDL3991 Dalyell Individual Research Project (Semester 1, 2021); • Martin Gossow, Arithmetic Surface Groups, SCDL3991 Dalyell Individual Research Project (Semester 2, 2019); • Michael Zhao, Fuchsian Groups, SCTP3753 TSP Project (Semester 2, 2018); • Hazel Browne, Lie Groups and Lie Algebras, SCTP3753 TSP Project (Semester 1, 2018); • Shane Arora and Hazel Browne, Naive Lie Theory, SCTP2649 TSP Project (Semester 1, 2017); • Jack Gregan, Knot theory, SCTP1832 TSP Project (Semester 2, 2013); • Joshua Ciappara, Discrete Groups of Isometries, SCTP2649 TSP Project (Semester 2, 2013); • Montek Gill, Geometry after Klein, SCTP2831 TSP Project (Semester 2, 2012)

The University of Queensland: • Nirav Shah, Invariants of knots, SCIE3012 (Semester 2, 2010)

undergraduate working seminars

The University of Sydney: • Surfaces: A Junction of Geometry, Topology and Analysis, SCTP3753 TSP Mathematics Activity A Snr (Semester 1, 2014); • Working Seminar B (Special Studies Program), MATH2917 (Semester 2, 2012)

The University of Queensland: • Topology & Groups, MATH6006 Proseminar (Semester 1, 2011)

open learning environments

• Reading and Writing Mathematics, OLET1625 (July 2018, 2019, 2023 and February 2019)

• Symmetry, OLET1630 (July 2018, 2019, 2020, 2021, 2022, 2023 and February 2019, 2020, 2021, 2022)

lecture courses

The University of Sydney, 2012–present: • Geometry, Weeks 1–6 of MATH3061 (Semester 2, 2024); • Discrete Mathematics for Computation, MATH1064 (Semester 2, 2018); • Basic Pharmaceutical Sciences, PHAR1812 (Semester 1, 2017 and 2016); • Topology and Geometry of Three Dimensional Manifolds, PMH6 (Semester 2, 2016); • Algebra (Advanced), MATH2968 (Semester 2, 2016, 2015, 2014, 2013 and 2012); • Discrete Mathematics, MATH1004 (Semester 2, 2015); • Linear Algebra (Advanced), Weeks 1–6 of MATH1902 (Semester 1, 2015); • Differential Geometry (Advanced), Weeks 7–13 of MATH3968 (Semester 2, 2014); • Linear Mathematics, Weeks 1–6 of MATH2061 (Semester 1, 2014, 2013 and 2012); • Fields and Galois Theory (Advanced), Weeks 7–13 of MATH3962 (Semester 1, 2013)

The University of Queensland, 2009–2011: • Functional Analysis III/IVH, MATH4404 (Semester 2, 2011 and 2009); • Differential Geometry, MATH3405 (Semester 2, 2011); • Functional Analysis, MATH3402 (Semester 1, 2011); • Discrete Mathematics, MATH1061 (Semester 2, 2010); • Measure Theory, MATH4405 (Semester 2, 2010)

The University of Melbourne, 2007–2008: • Linear Algebra, 620-156, The University of Melbourne (Semester 2, 2008); • Complex Analysis, 620-413, The University of Melbourne (Semester 2, 2007)

UQAM, 2004–2006: • Algèbre linéaire I, UQAM (Hiver 2004, 2005 et 2006)

McGill University 2004–2005: • Advanced Calculus, MATH 264, McGill University (Fall 2005); • Vectors, Matrices and Geometry, MATH 133, McGill University (Fall 2004)

undergraduate tutorial teaching / recitation classes

The University of Sydney, 2012–present: • Discrete Mathematics, MATH1004 (Semester 2, 2015); • Algebra (advanced), MATH2968, Tutorial (Semester 2, 2012 and 2013) and Practice class (Semester 2, 2012, 2013, 2014, 2015 and 2016); • Rings, Fields and Galois Theory (Advanced), MATH3962 (Semester 1, 2013); • Two tutorial groups of Linear Mathematics and Vector Calculus, Weeks 2–7 of MATH2061 (Semester 1, 2012)

The University of Queensland, 2009–2011: • Functional Analysis III/IVH, MATH4404 (Semester 2, 2011); • Measure Theory, MATH4405 (Semester 2, 2010); • Functional Analysis III/IVH, MATH4404 (Semester 2, 2009)

The University of Melbourne, 2008: • Complex Analysis, 620-252 (Semester 2, 2008)

Technische Universität Berlin, 2002: • Linear Algebra and Linear Differential Equations for Engineers

Trinity College at The University of Melbourne, 2000–2003: • Courses include Mathematical Methods, Linear Algebra, Abstract Algebra

The University of Melbourne, 1999–2003: • Courses include Real and Complex Analysis, Vector Analysis, Linear Algebra, Abstract Algebra, Differential Equations

expository lectures

10 May 2018: Three angles on tropical geometry, SUMS Seminar, The University of Sydney
23 April 2015: Scissors congruence and Hilbert's third problem, SUMS Seminar, The University of Sydney
11 September 2014: Three angles on tropical geometry, SUMS Seminar, The University of Sydney
16 August 2010: Circle packing and Ricci flow, GI/LA Seminar, The University of Queensland
26 November 2009: Kneser's Lemma and Decompositions of 3-Manifolds, PhD Journal Club, The University of Queensland
29 October 2009: Introduction to geometric group theory, III (with Murray Elder), PhD Journal Club, The University of Queensland
22 October 2009: Introduction to geometric group theory, II (with Murray Elder), PhD Journal Club, The University of Queensland
15 October 2009: Introduction to geometric group theory, I (with Murray Elder), PhD Journal Club, The University of Queensland
25 June 2007: Two lectures on the A-polynomial, CMS, Zhejiang University, Hangzhou, China
3 November 2005: All about the A-polynomial, Séminaire CIRGET Junior, Montréal, Canada
2 December 2003: Circle packing and Ricci flow, ISM Graduate Students Seminar, Montréal, Canada
27 November 2003: Kneser's lemma and decompositions for 3-manifolds, Séminaire CIRGET Junior, Montréal, Canada

(outreach) primary schools

2013–2019: "Mathematicians in Schools" partnership with Randwick Public School

outreach (lectures and activities for high school and early undergraduate students)

19 June 2023: The shape of space, lecture at MANSW Mathematical Enrichment Day, University of Technology Sydney, Australia
20 June 2022: The shape of space, two lectures at MANSW Mathematical Enrichment Day, The University of Sydney, Australia
24 June 2019: The shape of space, two lectures at MANSW Mathematical Enrichment Day, University of Technology Sydney, Australia
25 June 2018: The shape of space, two lectures at MANSW Mathematical Enrichment Day, Macquarie University, Australia
13 June 2017: The shape of space, two lectures at MANSW "MX1&2 Day," The University of Sydney, Australia
7 June 2016: The shape of space, two lectures at MANSW "MX1&2 Day," The University of Sydney, Australia
14 August 2014: The shape of space, lecture and activity for year 8 students of Scots College, Bellevue Hill, at The University of Sydney, Australia
19 June 2014: The shape of space, lecture at MANSW "Talented Students Day," UNSW, Australia
1 April 2010: Knots, "Advanced Study Program in Science" seminar, The University of Queensland, Australia
19-21 January 2010: The shape of space, a hands-on workshop for "The Science Experience," The University of Queensland, Australia
4 September 2009: Minimal surfaces, a bubbly activity for Year 10 physics classes from Forest Lake College and Alexander State High School as part of the SMP Activities Day, The University of Queensland, Australia
15 February 2008: The shape of space, lecture and activity for K12 Students of Kyneton Secondary College at The University of Melbourne, Australia
7 February 2007: The shape of space, lecture and activity for K12 Students of Kyneton Secondary College at The University of Melbourne, Australia
2 May 2006: Living in hyperbolic space, Marianopolis College, Montréal, Canada
5 April 2006: Living in hyperbolic space, Heritage College, Hull, Canada

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