ANNA MACQUARIE ROMANOV
UNIVERSITY OF NEW SOUTH WALES
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I am a lecturer at the University of New South Wales in the pure mathematics group.
Before this, I was a postdoctoral research associate and National Science Foundation postdoctoral fellow at the University of Sydney. I received my PhD in 2018 from the University of Utah under the supervision of Dragan Miličić.
I work in geometric and categorical representation theory. Some objects that I like to think about are Lie algebras, Lie groups, equivariant sheaves, D-modules, Soergel bimodules, and Kazhdan-Lusztig polynomials. Here is my CV.News:
I am currently on parental leave until September 2025. During this time, I may be slow to respond to emails.I run the UNSW Pure Maths Seminar, which meets on Tuesdays at 12pm. For a schedule and abstracts, take a look at the Seminar tab above.My little research group at UNSW is growing. Here's what we're up to.I am helping to organise workshop on diagrammatic categorification in October 2025 at ICERM.In the first half of 2024, Alex Sherman and I ran a joint UNSW/USyd learning seminar on tensor categories and their modules. Notes from all the talks can be found here.From August 2023 - January 2024, Dragan Miličić visited the University of Sydney and UNSW. With Geordie Williamson, he ran a course on D-modules and representation theory. Notes and recorded lectures can be found on the course websiteIn March 2023, I gave a public lecture on categorification at the Frontiers of Science Forum in Sydney. Before the event I chatted with Ian Woolf of Diffusion Radio about all things equality. You can listen to our conversation here.I participated in the University of Sydney's Live from the Lab event in August 2021. I was paired with R&B soul artist Mi-Kaisha, who composed a beautiful piece of original music based off my research. To listen to her creation and hear about the process, check out the Live from the Lab podcast .For much of 2019 and 2020, I typed notes for Geordie Williamson's 18 month lecture series on the Langlands correspondence and Bezrukavnikov's equivalence.Come to the the Informal Friday Seminar!In October 2019, I gave a public lecture as a part of Raising the Bar Sydney. My talk on Hidden Symmetries was recorded and made into a podcast.An undergraduate project that I did with David Allen in 2011 recently appeared in Randall Munroe's new book How To. It's a wonderful book! If you get your hands on a copy, look for our flower-wheeled bicycle.Whittaker modules for semisimple Lie algebras
I am interested in a category of modules over a complex semisimple Lie algebra which includes both Whittaker modules and highest weight modules. My PhD thesis gave a geometric algorithm for computing composition multiplicities of standard modules in this category using Beilinson-Bernstein localization. With Adam Brown, we established further properties of this category, including a classification of contravariant forms on standard modules and highest weight structures on blocks of the category. With Jens Eberhardt, we defined Jantzen filtrations of standard Whittaker modules and showed that they satisfy strong functoriality and semisimplicity properties. This provides a generalisation of the Jantzen conjectures to Whittaker modules. Two proofs of a Jantzen conjecture for Whittaker modules, (with J. Eberhardt). (arXiv)Contravariant pairings between standard Whittaker modules and Verma modules. (with A. Brown). Journal of Algeba, https://doi.org/10.1016/j.jalgebra.2022.06.017, 2022. (arXiv)Contravariant forms on Whittaker modules. (with A. Brown). Proceedings of the American Mathematical Society, 149(1), 37-52, 2020. (arXiv)A Kazhdan-Lusztig algorithm for Whittaker modules. Algebras and Representation Theory, 24(1), 81-133, 2021. (arXiv)
A Kazhdan-Lusztig algorithm for Whittaker modules. PhD thesis, University of Utah (2018).Beilinson-Bernstein localization and category O
The main geometric tool in my work is the Beilinson-Bernstein localization theorem, which provides a link between representation theory of Lie algebras and D-modules on flag varieties. In 2020, I wrote an article for the proceedings of the Representation Theory XVI conference in Dubrovnik, Croatia giving a detailed description of the D-modules corresponding to four classes of representations, including principal series representations of SL(2,R) and Whittaker modules for sl(2,C). In 2024, I wrote a follow-up article with my honours student, Simon Bohun , which lifts these computations to base affine space, and computes the Jantzen filtration on the corresponding D-modules. The algebraic tools I use in my work are similar to those used in the study of category O. With Sean Taylor, I wrote a book chapter on category O for the 2020 textbook Introduction to Soergel Bimodules, which was based on lectures given by Geordie Williamson and Ben Elias at a 2017 MSRI summer school on Soergel bimodules. An example of the Jantzen filtration of a D-module, (with S. Bohun). To appear in the Journal of the Australian Mathematical Society. (arXiv)Four examples of Beilinson-Bernstein localization, Lie Groups, Number Theory, and Vertex Algebras, Contemporary Mathematics, vol. 768, Amer. Math. Soc., Providence, RI, 2021, pp. 65-85. (arXiv)A lightning introduction to category O (with S. Taylor). Book chapter, Introduction to Soergel Bimodules. RSME Springer Series, Vol. 5, 2020.Soergel bimodules and real groups
Irreducible characters of real reductive algebraic groups can be computed using certain polynomials (Kazhdan-Lusztig-Vogan polynomials), which appear as entries in a change-of-basis matrix for the corresponding Lusztig-Vogan module of the Hecke algebra. With Scott Larson, we initiated a Soergel bimodule approach to the character theory of real groups by explaining how a piece of the Lusztig-Vogan module (the trivial block) admits a categorification in terms of bimodules over polynomial rings. Our bimodule category arises as the essential image of the hypercohomology functor applied to a category of constructible sheaves and admits a natural action of Soergel bimodules. A categorification of the Lusztig-Vogan module. (with A. Larson). Submitted. (arXiv)Gelfand pairs
In 2016, I attended an AMS Mathematics Research Community (MRC) workshop on Lie groups, discretization, and Gelfand pairs. In this workshop, I started a project with a team of researchers that established a topological model for the space of spherical unitary representations of a nilpotent Gelfand pair in terms of coadjoint orbits in the dual Lie algebra. After the completion of this MRC project, I ran an REU (Research Experience for Undergraduates) out of the University of Utah on finite Gelfand pairs. With Faith Pearson and Dylan Soller, we studied families of group-subgroup pairs constructed from wreath products of symmetric groups, and established exactly when this construction yields a Gelfand pair. Finite Gelfand pairs and cracking points of the symmetric groups. (with F. Pearson, and D. Soller). Rocky Mountain Journal of Mathematics, 50(5), 1807-1812, 2020. (arXiv)An orbit model for the spectra of nilpotent Gelfand pairs. (with H. Friedlander, W. Grodzicki, W. Johnson, G. Ratcliff, B. Strasser, and B. Wessel). Transformation Groups, 25(3), 859-886, 2019. (arXiv)UNSW Pure Mathematics Seminar
In 2024, the seminar will meet on Tuesdays from 12-1pm in Anita Lawrence (RC) 4082 during the weeks of UNSW's terms. After the seminar, all are invited to join for lunch in The Lounge. Term 3, 2024
10 Sept 2024: The Fundamental Theorems of Calculus and Zinbiel Algebras, J.S. Lemay (Macquarie)
Abstract: In algebra, derivations and integrations generalize the differential operator and integral operator from calculus. Derivations are axiomatized by the Leibniz rule, while integrations are axiomatized by the Rota-Baxter rule, which an integral only version of integration by parts. On the other hand, Zinbiel algebras are a special kind of commutative non-unital associative algebras that capture the notion of riffle shuffle permutations. In this talk, I will explain how giving a Zinbiel algebra is equivalent to giving a derivation and integration which together satisfy the two fundamental theorems of calculus. This talk is based on my paper: https://arxiv.org/abs/2401.08223
17 Sept 2024: From H-decompositions to transitive path decompositions, Ajani De Vas Gunasekara (Notre Dame Australia)
Abstract:MathJax Example Let \(\Gamma\) and \(H\) be graphs. An \(H\)-decomposition of a graph \(\Gamma\) is a partition of its edge set into subgraphs isomorphic to $H$. A transitive decomposition is a special kind of \(H\)-decomposition that is highly symmetrical in the sense that the subgraphs (copies of \(H\)) are preserved and transitively permuted by a group of automorphisms of \(\Gamma\). In this talk, I will explore \(H\)-decompositions in general, providing historical context for these decompositions. Additionally, I will delve into transitive \(H\)-decompositions and present our recent results on transitive path decompositions of the Cartesian product \(K_n \Box K_n\) when \(n\) is an odd prime. Part of the talk is joint work with Alice Devillers, UWA. Based on https://arxiv.org/abs/2308.07684 . No prior knowledge is required.
24 Sept 2024: Kazhdan-Lusztig theory, Joseph Baine (UNSW)
Abstract: Kazhdan-Lusztig theory seeks to reduce deep questions about the structure of objects in representation-theoretic, geometric, algebraic and diagrammatic categories to concrete questions about bases of Hecke algebras. In this talk, I will give an introduction to some of the categories that arise in this field (Category O, perverse sheaves on flag varieties, Soergel modules, Elias-Williamson categories) and explain the equivalences between them. I’ll conclude by noting how much more difficult these problems become when the categories are k-linear over a field of characteristic p.
1 October 2024: From incidence geometry to quantum mechanics, Po Lam Yung (ANU)
Abstract:MathJax Example Herr and Kwak had recently established sharp estimates for certain exponential sums on the 3 dimensional torus by counting rectangles in the plane. I will try to explain their work and highlight connections to Fourier analysis. One interesting question we will address is the following. Given N points in the plane, how many rectangles can we form so that all vertices are in the given points? Surprisingly, the answer has to do with the topology of \(\mathbb{R}^2\), and is the key to unlocking the aforementioned exponential sum estimates. These exponential estimates in turn provide sharp bounds for solutions to the periodic Schrodinger equation in 2+1 dimensions.
15 October 2024: Totality of Rational points on Modular curves, June Park (University of Sydney)
Abstract: The study of fibrations of curves and abelian varieties over a smooth algebraic curve lies at the heart of the classification theory of algebraic surfaces and rational points on varieties. For the case of elliptic curves, it is natural to want to count elliptic curves over global fields such as the field Q of rational numbers or the field Fq(t) of rational functions over the finite field Fq. To this end, we consider the fact that each E/K corresponds to a K-rational point on the fine moduli stack Mbar_{1,1} of stable elliptic curves, which in turn corresponds to a rational curve on Mbar_{1,1}. In this talk, I will explain the exact counting formula for all elliptic curves over Fq(t) along with an explanation for the geometric origin of lower order main terms, as well as basic context, relevant ideas and methods.
6 November 2024, 11:00am-12:00pm: [Note special day and time] Louis Yudowitz (KHT Royal Institute of Technology)
Abstract:
Student Supervision
My research group meets every other Friday from 10:30am-12:30pm in H13-4082. More information can be found here. Guests are welcome to join.
Current students:
Daniel Dunmore. Module categories over Soergel bimodules. PhD, UNSW (Primary supervisor)Thomas Dumore. Quadratic fusion categories. PhD, UNSW (Secondary supervisor)Victor Zhang. Diagrammatic Lusztig-Vogan categories. PhD, UNSW (Primary supervisor)Tasman Fell. Singular Soergel bimodules. research masters, UNSW (Primary supervisor)Zach Leong. Unitary dual of SL(2,R). honours, UNSW
Past Students:
Faith Pearson. Cracking points of finite Gelfand pairs. REU, University of Utah 2019
Research talks
Here are slides from some of my recent talks.
Categorification in Representation Theory Conference, University of Syndey, February 2023: Higher representations of Soergel bimodules arising from real Lie groups.Algebra Seminar, University of Syndey, October 2022: Filtrations of Whittaker modulesRepresentation Theory XVII, Dubrovnik, Croatia, October 2022: A Soergel bimodule approach to the character theory of real groupsNumber Theory Seminar, UNSW Canberra, August 2022: What is the Sato-Tate conjecture?Colloquium, University of Adelaide, May 2022: A tour via examples of Beilinson- Bernstein localisationRepresentation Theory and Number Theory Seminar, University of Utah, January 2022: Costandard Whittaker modules and contravariant pairingsRepresentation and Number Theory (RANT) Seminar, Chinese University of Hong Kong, November 2021: A Soergel bimodule approach to the character theory of real groupsRepresentation Theory and Number Theory Seminar, University of Utah, October 2021: A Soergel bimodule approach to the character theory of real groupsMathematics of Conformal Field Theory II, Australian National University, July 2021: Twisted D-modules on the affine flag variety and Whittaker modulesOberseminar, University of Bonn, June 2021: Highest weight structures on Whittaker categoriesAMS Western Spring Sectional Meeting, May 2021: Contravariant duality for Whittaker modules, Twisted D-modules on the affine flag variety and Whittaker modulesRepresentation Theory Seminar, University of Melbourne, April-May 2021: The infinite-dimensional geometric story: Kac-Moody groups, affine flag varieties, and D-modules: week 1, week 2, week 3Algebra Seminar, University of Georgia, April 2021: A categorification of the Lusztig-Vogan moduleWhat is...? (WiSe) Seminar, University of Queensland, April 2021: What is a Hecke algebra?Pure Math Seminar, University of New South Wales, April 2021: A tour via examples of Beilinson-Bernstein localisationInformal Friday Seminar, University of Sydney, March 2021: Affine Whittaker modules and twisted D-modules on the affine flag varietyRepresentation Theory Seminar, University of Melbourne, December 2020: Jordan-Holder multiplicities of Verma modules with rational highest weightAWM Colloquium, University of North Carolina, November 2020: Bridging two worlds: A tour via examples of Beilinson-Bernstein localizationCategorification Learning Seminar, October 2020: A categorification of the Lusztig-Vogan moduleCategorification Learning Seminar, September 2020: A tour via examples of Beilinson-Bernstein localization (video)What is ...? Seminar, University of Queensland, August 2020: What is the local Langlands correspondence for GL(2,R)?Geometric Representation Theory conference, Max Planck Institute Bonn/Perimeter Institute Waterloo, June 2020: A categorification of the Lusztig-Vogan module (video)Australian Category Seminar, Macquarie University, May 2020: Representations of representations of sl2Macquarie University Colloquium, March 2020: Bridging two worlds: A tour via examples of Beilinson-Bernstein localization (video)Australian Mathematical Society Annual Meeting, Monash University, December 2019: Contravariant forms on Whittaker modulesInformal Friday Seminar, University of Sydney, November 2019: The Lusztig-Vogan module of the Hecke algebra: part I, part IIFlags, Galleries, and Reflection Groups conference, University of Sydney, August 2019: Whittaker modules and parabolic Kazhdan-Lusztig polynomialsRepresentation Theory XVI conference, Dubrovnik, Croatia, June 2019: A Kazhdan-Lusztig algorithm for Whittaker modulesAmerican Mathematical Society Western Sectional Meeting, University of Hawaii at Manoa, March 2019: Filtrations of Whittaker modulesInformal Friday Seminar, University of Sydney, March 2019: Representation theory of SL(2,R): part I (the geometric classification), part II (the Langlands classification)Australian Mathematical Society Annual Meeting, University of Adelaide, December 2018: An orbit model for nilpotent Gelfand pairsOutreach talks
I like to give math talks to a wide range of audiences. Here are some links to talks I've given to students and the public.
Frontiers of Science Forum, Sydney, Australia, March 2023: What does = mean?Girls Do the Maths, UNSW outreach event, June 2022: Searching for SymmetryUniversity of New South Wales architecture students, April 2021: 4 Dimensional PolytopesInfinite Pathways, Commemorating International Women's Day, Program for year 9 students, March 2021: Life as a research mathematicianSydney University Mathematics Society, October 2020: The ADE correspondenceSydney University Mathematics Society, August 2019: A Glimpse of the Fourth DimensionACCESS Program for gender equity in science and engineering, University of Utah, Summer 2018: Four Dimensional PolytopesUndergraduate Colloquium, University of Utah, Fall 2017: Representation Theory and the Hydrogen AtomGraduate Colloquium Micro Talk, University of Utah, Spring 2017: Geometric Representation Theory: A Tale of Two IslandsGraduate Colloquium, University of Utah, Fall 2016: Representation Theory and the Hydrogen AtomUtah Teachers' Math Circle, Salt Lake City, Spring 2016: Keeping it PlatonicUndergraduate Colloquium, University of Utah, Fall 2015: Exploring Higher Dimensional PolytopesWhat is Math? Day, University of Utah, Spring 2015: What is Symmetry?Association for Women in Mathematics Workshop, University of Utah, Spring 2015: Symmetry and BeadingGraduate Colloquium, University of Utah, Spring 2015: The ADE ClassificationUndergraduate Colloquium, University of Utah, Fall 2014: Platonic Solids and the ADE CorrespondenceUtah Summer Math Program for High School Students, Summer 2014: Anna's Grab Bag of Mathematical CuriositiesANNA MACQUARIE ROMANOV
Office: Anita B Lawrence Centre H13-4078
School of Mathematics and Statistics
University of New South Wales Sydney
NSW 2052 Australia
a.romanov@unsw.edu.au
Background images were taken by Anna Romanov.
