A.Romanov
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  • 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.
    My little research group at UNSW is growing. Here's what we're up to.
    I am helping to organise a 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 website
    In 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. 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 proved a generalisation of the Jantzen conjectures for Whittaker modules. The algebraic tools I use to study Whittaker modules 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.
    Two proofs of a Jantzen conjecture for Whittaker modules, (with J. Eberhardt). Selecta Mathematica. (To appear) (arXiv)
    Contravariant pairings between standard Whittaker modules and Verma modules. (with A. Brown). Journal of Algebra, 609, 145-179, 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).
    A lightning introduction to category O (with S. Taylor). Book chapter, Introduction to Soergel Bimodules. RSME Springer Series, Vol. 5, 2020.
    D-modules and Beilinson-Bernstein localization
    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. Viewing representation-theoretic concepts through the lens of D-modules can provide helpful geometric intuition and simplify arguments. Several of my papers pursue this philosophy. With Dragan Miličić, we reproved some classical theorems of Harish-Chandra about discrete series representations using simple D-module techniques. In 2020, I gave a detailed description of the D-modules corresponding to four families of representations, including principal series representations of SL(2,R) and Whittaker modules for sl(2,C). These computations provided a concrete geometric illustration of several classical families of representations. 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.
    On a geometric approach to the discrete series, (with D. Miličić). Proceedings of the Harish-Chandra Centenary Celebrations 2023, HRA Prayagraj. (To appear)
    An example of the Jantzen filtration of a D-module, (with S. Bohun). Journal of the Australian Mathematical Society. 118(3), 265-296, 2025. (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)
    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). (arXiv)
    Algebraic groups and OTI functors
    The modular representation theory of algebraic groups is deep subject drawing on techniques from compact Lie groups, finite groups, and geometric representation theory, among other things. In 2023, I worked with a team of Sydney-based researchers to develop a new approach to the study of tensor products of modular representations. The approach introduces a functor (the so-called "OTI functor") which exhibits several useful properties. We conjecture that this functor provides an algebraic incarnation of hypercohomology under the Finkelberg-Mirkovic conjecture.
    A remarkable functor on G-modules. (with J. Baine, T. Fell, A. Sherman, and G. Williamson). (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)
  • Teaching
    In 2025, I am on leave.
    In Term 1 2024, I taught MATH 5735, Modules and Representation Theory.
    In 2023, I was on leave.
    In Term 2 2022, I taught MATH 1231, Mathematics 1B (linear algebra + probability).
    In Term 1 2022, I taught MATH 5735, Modules and Representation Theory.
    Student Supervision
    My research group meets every Thursday from 9:30-11:00am 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)

    Past Students:
    Tasman Fell. Singular Soergel bimodules. research masters, UNSW 2025 (Primary supervisor)
    Zach Leong. Unitary dual of SL(2,R). honours, UNSW 2024
    Simon Bohun. Beilinson-Bernstein Localization on base affine space. honours, UNSW 2023
    Victor Zhang. Diagrammatic categories in representation theory. honours, UNSW 2023
    Amy Bradford. Whittaker modules for affine Lie algebras. REU, University of Utah 2021
    Dylan Soller. Cracking points of finite Gelfand pairs. REU, University of Utah 2019
    Faith Pearson. Cracking points of finite Gelfand pairs. REU, University of Utah 2019

  • Research talks

    Here are slides from some of my research 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 modules
    Representation Theory XVII, Dubrovnik, Croatia, October 2022: A Soergel bimodule approach to the character theory of real groups
    Number 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 localisation
    Representation Theory and Number Theory Seminar, University of Utah, January 2022: Costandard Whittaker modules and contravariant pairings
    Representation and Number Theory (RANT) Seminar, Chinese University of Hong Kong, November 2021: A Soergel bimodule approach to the character theory of real groups
    Representation Theory and Number Theory Seminar, University of Utah, October 2021: A Soergel bimodule approach to the character theory of real groups
    Mathematics of Conformal Field Theory II, Australian National University, July 2021: Twisted D-modules on the affine flag variety and Whittaker modules
    Oberseminar, University of Bonn, June 2021: Highest weight structures on Whittaker categories
    AMS Western Spring Sectional Meeting, May 2021: Contravariant duality for Whittaker modules, Twisted D-modules on the affine flag variety and Whittaker modules
    Representation 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 3
    Algebra Seminar, University of Georgia, April 2021: A categorification of the Lusztig-Vogan module
    What 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 localisation
    Informal Friday Seminar, University of Sydney, March 2021: Affine Whittaker modules and twisted D-modules on the affine flag variety
    Representation Theory Seminar, University of Melbourne, December 2020: Jordan-Holder multiplicities of Verma modules with rational highest weight
    AWM Colloquium, University of North Carolina, November 2020: Bridging two worlds: A tour via examples of Beilinson-Bernstein localization
    Categorification Learning Seminar, October 2020: A categorification of the Lusztig-Vogan module
    Categorification Learning Seminar, September 2020: A tour via examples of Beilinson-Bernstein localization (video)
    Informal Friday Seminar, University of Sydney, August 2020: Vogan duality: part I, part II
    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 sl2
    Macquarie 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 modules
    Informal Friday Seminar, University of Sydney, November 2019: The Lusztig-Vogan module of the Hecke algebra: part I, part II
    Flags, Galleries, and Reflection Groups conference, University of Sydney, August 2019: Whittaker modules and parabolic Kazhdan-Lusztig polynomials
    Representation Theory XVI conference, Dubrovnik, Croatia, June 2019: A Kazhdan-Lusztig algorithm for Whittaker modules
    Informal Friday Seminar, University of Sydney, June 2019: Equivariant cohomology: part I, part II
    American Mathematical Society Western Sectional Meeting, University of Hawaii at Manoa, March 2019: Filtrations of Whittaker modules
    Informal 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 pairs
    Informal Friday Seminar, University of Sydney, October 2018: Unitary representations of real reductive groups: part I, part II

    Outreach 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 Symmetry
    University of New South Wales architecture students, April 2021: 4 Dimensional Polytopes
    Infinite Pathways, Commemorating International Women's Day, Program for year 9 students, March 2021: Life as a research mathematician
    Sydney University Mathematics Society, October 2020: The ADE correspondence
    Raising the Bar Sydney (public lecture), October 2019: Hidden Symmetries (podcast)
    Sydney University Mathematics Society, August 2019: A Glimpse of the Fourth Dimension
    ACCESS Program for gender equity in science and engineering, University of Utah, Summer 2018: Four Dimensional Polytopes
    Undergraduate Colloquium, University of Utah, Fall 2017: Representation Theory and the Hydrogen Atom
    Graduate Colloquium Micro Talk, University of Utah, Spring 2017: Geometric Representation Theory: A Tale of Two Islands
    Graduate Colloquium, University of Utah, Fall 2016: Representation Theory and the Hydrogen Atom
    Utah Teachers' Math Circle, Salt Lake City, Spring 2016: Keeping it Platonic
    Undergraduate Colloquium, University of Utah, Fall 2015: Exploring Higher Dimensional Polytopes
    What is Math? Day, University of Utah, Spring 2015: What is Symmetry?
    Association for Women in Mathematics Workshop, University of Utah, Spring 2015: Symmetry and Beading
    Graduate Colloquium, University of Utah, Spring 2015: The ADE Classification
    Undergraduate Colloquium, University of Utah, Fall 2014: Platonic Solids and the ADE Correspondence
    Utah Summer Math Program for High School Students, Summer 2014: Anna's Grab Bag of Mathematical Curiosities
  • ANNA 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.