## Homepage

## Gufang Zhao

University of Melbourne

School of Mathematics and Statistics

Parkville, VIC 3010

Office: Peter Hall Building 224

E-mail (don't copy-paste): gufangz you-know-what-it-is unimelb.edu.au

Tel: +61383441028

Research Interest:

My research lies at the interface between algebraic geometry and representation theory. More specifically, I've been working on projects concerning derived category of coherent sheaves, oriented cohomology theories of algebraic varieties, and their applications in representation theory. I am also fond of varieties of local systems and instantons, quantum integrable systems, and related aspects in mathematical physics.

Erdos Number = 4

CV.

Here is a video of me talking about my joint work with Yaping Yang, recorded in the Mathematical Physics Seminar in the Perimeter Institute, Waterloo, Mar. 2016.

And again about the new progress up to Feb. 2018.

And yet again, new progress up to Feb. 2019.

Yaping's talk on this project, although a different perspective of it, recorded at the Hausdorff Research Institute for Mathematics in Bonn, Oct. 2017.

## Papers and Preprints

Peer-Reviewed Journal Articles

1. (with C. Zhong) Elliptic affine Hecke algebras and their representations. 58 pages, accepted for publication in Adv. Math.. arXiv: 1507.01245.

2. (with C. Su and C. Zhong) Wall-crossings and a categorification of K-theory stable bases of the Springer resolution, 34 pages, accepted for publication in Compo. Math.. arXiv:1904.03769

3. (with M. Levine and Y. Yang) Algebraic elliptic cohomology theory and flops 2: $SL$-cobordism. 52 pages, accepted for publication in Adv. Math.. arxiv: 1610.00396

4. (with V. Nandakumar) Categorification via blocks of modular representations of sl_n. to appear in Canadian Journal of Mathematics. arxiv: 1612.06941.

5. (with M. Rapčák, Y. Soibelman, and Y. Yang) Cohomological Hall algebras, vertex algebras and instantons, to appear in Commun. Math. Phys..(DOI: 10.1007/s00220-019-03575-5) arxiv: 1810.10402.

6. (with Y. Yang) The PBW theorem for the affine Yangians, 12 pages, to appear in Transform. Groups. (DOI: 10.1007/s00031-020-09572-6) arxiv: 1804.04375

7. (with I. Mirković and Y. Yang) Loop Grassmannians of quivers and affine quantum groups. to appear in Interactions between Representation Theory and Algebraic Geometry. 25 pages, (2018). arxiv: 1810.10095

8. (with Y. Yang) On two cohomological Hall algebras. 22 pages, to appear in Proc. Roy. Soc. Edinburgh Sect. A. (DOI:10.1017/prm.2018.162) arXiv: 1604.01477.

9. (with C. Su and C. Zhong) On the K-theory stable bases of the Springer resolution. 43 pages, to appear in Ann. Sci. Éc. Norm. Supér.. arxiv: 1708.08013.

10. (with Y. Yang) The cohomological Hall algebra of a preprojective algebra. Proc. Lond. Math. Soc. (3) 116 (2018), no. 5, 1029–1074. arXiv: 1407.7994.

An earlier ArXiv version of this paper was entitled "Formal cohomological Hall algebra of a quiver".

11. (with Y. Yang) Cohomological Hall algebras and affine quantum groups. Selecta Math., 24(2) (2018), 1093--1119. arXiv: 1604.01865.

12. (with M. Levine and Y. Yang) Algebraic elliptic cohomology theory and flops 1, with an appendix by Joel Riou. 29 pages, Math. Ann. 375 (2019), no. 3-4, 1823–1855. arXiv:1311.2159.

See here for an unpublished earlier version (with more background introductory materials for my own reference).

13. (with C. Zhong) Geometric representations of the formal affine Hecke algebras. Adv. Math. 317 (2017), 50--90. arXiv: 1406.1283.

14. (with A. Suciu and Y. Yang) Homological finiteness of abelian covers. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) Vol. XIV (2015), 101--153. ArXiv:1204.4873.

15. (with A. Suciu and Y. Yang) Intersections of translated algebraic subtori. J. Pure Appl. Algebra 217 (2013), no. 3, 481--494.

Submitted

16. The cohomological Hall algebras of a preprojective algebra with symmetrizer. 16 pages, (2019). arxiv: 1911.02689

17. (with Y. Yang) Quiver varieties and elliptic quantum groups, 43 pages, (2017). arxiv: 1708.01418

This paper is under revision, in collaboration with Dougal Davis. Please write to me for an update if you are interested.

18. (with M. Rapčák, Y. Soibelman, and Y. Yang) Cohomological Hall algebras and perverse coherent sheaves on toric Calabi-Yau 3-folds. 60 pages, (2020). arxiv: 2007.13365

19. (with V. Nandakumar) Categorification via blocks of modular representations II. 24 pages, (2020). arXiv:2005.08248

Preprints

20. (with Yalong Cao and Zijun Zhou) A degeneration formula of Donaldson-Thomas theory on Calabi-Yau 4-folds, 65 pages, arXiv:2402.16103.

21. (with Changlong Zhong) A Langlands duality of elliptic Hecke algebras, arxiv: 2310.13460

22. (with Cristian Lenart and Changlong Zhong) Elliptic classes via the periodic Hecke module and its Langlands dual, arXiv2309.09140

23. (with Yalong Cao) Quasimaps to quivers with potentials, arXiv:2306.01302

24. (with Yaping Yang) Frobenii on Morava E-theoretical quantum groups. 60 pages, (2021). arXiv: 2105.14681

25. Stability conditions for noncommutative symplectic resolutions. 37 pages, (2014). arXiv: 1405.1898.

26. (with J. Weyman) Noncommutative desingularization of orbit closures for some representations of $GL_n$. 44 pages, (2012). arXiv: 1204.0488.

Conference Proceedings

27. (with Y. Yang) How to sheafify an elliptic quantum group. MATRIX Annals 2017, 675-691. arXiv: 1803.06627

Work in Progress

28. (with I. Mirković and Y. Yang) Higher dimensional loop Grassmannians via fusion.

29. (with Z. Lin) Dense orbit ind-constructible abelian categories.

Theses

30. Derived category and cohomology of resolution of singularities: examples from representation theory. Ph.D. Thesis, Northeastern University, (2014).

This is a cumulative thesis. e-version available here. See the slides I used in my defense.

31. Remarks on some classical theorems about characteristic classes. B.Sc. Thesis, Zhejiang University, (2009).

Expository talks with Notes

Arithmetic Fourier transform and proof of the first Kac conjecture. Spring school on Kac conjectures and quiver varieties, Wuppertal, Mar. 2015.

Procesi bundles and symplective McKay equivalence. Joint M.I.T.-Northeastern Student Seminar on Quantum cohomology and Representation theory, Boston, Mar. 2014.

Derived Beilinson-Bernstein localization in positive characteristic and affne braid group action. Joint M.I.T.-Northeastern Student Seminar on Quantum cohomology and Representation theory, Boston, Oct. 2013. Notes for Part 1, Part 2

BLvdB's characteristic free tilting bundles on Grassmannians and noncommutative desingularizations. Quiver and Invariant Theory Seminar, Spring 2012.

Derived Category of Coherent Sheaves and Non-commutative Resolutions. Tapas Seminar, Summer 2011.

## Selected talks I have given

Representation theory underlying PT-invariants of the resolved conifold, Represen- tation Theory Seminar, Melbourne, Australia, Mar. 2020.

Cohomological Hall algebras and their representation theories, Mathematics and String Theory Seminar, the Kavli Institute for Physics and Mathematics of the Universe, Japan. Jan. 2020

Elliptic representation theory, Differential Geometry Seminar Adelaide, Australia, Oct. 2019

Quantum Frobenii on Morava E-theoretical Hall algebras, Topology Seminar, Melbourne, Australia, Oct. 2019

From generalized geometric Satake to AGT correspondence, Representation Theory Seminar, Melbourne, Australia. Apr. 2019

Comultiplications on the cohomological Hall algebras and vertex algebras, Cohomological Hall Algebras in Mathematics and Physics, Perimeter Institute, Waterloo, ON. Feb. 2019

Towards higher loop Grassmannians via fusion, A conference on character varieties and quantizations, Auckland, New Zealand. Dec. 2018

Lecture series on elliptic cohomology after Hopkins-Lurie and applications into representation theory, Elliptic Festival Weeks, Melbourne, Australia. Nov. 2018

Cohomological Hall algebras, vertex algebras and instantons, Representation Theory Seminar, Melbourne, Australia. Oct. 2018

A Borel-Weil-Bott theorem for toroidal algebras, Representation Theory Seminar, Melbourne, Australia. Aug. 2018

From conformal field theories to representations of quantum groups, working seminar, IST Austria, Klosterneuburg, Austria. Jun. 2018

Quiver varieties and elliptic quantum groups, the 3rd Kyoto Top Global University, Mathematics Workshop, Kyoto, Japan. Feb. 2018.

Elliptic quantum groups and affine Grassmannians on an elliptic curve, Mathematical Physics Seminar, Perimeter Institute, Waterloo, ON, Canada. Feb. 2018.

Affine Hecke algebras and their elliptic vs double affine analogues, working seminar, IST Austria, Klosterneuburg, Austria, Dec. 2017.

Elliptic representation theory, working seminar, IST Austria, Klosterneuburg, Austria, Dec. 2017.

From configuration of points on the punctured plane to Higgs bundles on an elliptic curve, Colloquium, SUNY at Buffalo, NY, Sep. 2017.

Quiver varieties and elliptic quantum groups, Representation Theory Seminar, The University of North Carolina at Chapel Hill, NC, Sep. 2017.

Representations of the elliptic affine Hecke algebra and Hitchin systems on an elliptic curve. Representation Theory Seminar, Amherst, MA, Feb. 2017.

Equivarient elliptic cohomology of symplectic resolutions, Algebra Seminar, University of Melbourne, Australia, Dec. 2016.

From points on the punctured plane to Higgs bundles on an elliptic curve, Seminar in Combinatorics, Geometry, and Representation Theory, Smith College, Northampton, MA, Oct. 2016

Cohomological Hall algebras and affine quantum groups, Algebra Seminar, SUNY at Buffalo, NY, Oct. 2016.

Elliptic quantum groups and affine Grassmannians on an elliptic curve, Mathematics Seminar, Zhejiang University, Hangzhou, China, Aug. 2016.

What elliptic cohomology might have to do with other generalized Schubert calculi, Workshop on equivariant generalized Schubert calculus and its applications, Ottowa, Apr. 2016.

Quiver varieties and elliptic quantum groups, Mathematical Physics Seminar, Perimeter Institute, Waterloo, ON, Canada. Mar. 2016.

Quiver varieties and elliptic quantum groups, Algebra and Topology Seminar, State University of New York, Albany, NY. Mar. 2016.

Elliptic affine Hecke algebras and their representations, Algebra Seminar, University of Ottawa, Canada. Nov. 2015.

Preprojective cohomological Hall algebra, Yangian, and Donaldson-Thomas invariants. Algebra Seminar, University of Connecticut, Storrs, CT. Oct. 2015

Preprojective cohomological Hall algebra, Yangian, and Donaldson-Thomas invariants. Valley Geometry Seminar, Amherst, MA. Oct. 2015.

Elliptic affine Hecke algebras and their representations. Park City Institute, Midway, UT, Jul. 2015.

Cohomological Hall algbera of a preprojective algebra. International Conference on Representation Theory and Commutative Algebra, Storrs, CT, Apr. 2015.

Formal group laws and cohomology of quiver varieties. Seminaire Caen Cergy Clermont Paris - Theorie des Representations, Paris, Nov. 2014.

Stability conditions for noncommutative symplectic resolutions. Conference on Geometric Methods in Representation Theory 2, University of Missouri in Columbia, Nov. 2013.

Grassmannian parameterizations of representations (after Huisgen-Zimmermann). Representation Theory Seminar, Northeastern, Oct. 2013.

Noncommutative desingularization of orbit closures for some GLn representations. Graduate Student Representation Theory Conference, Manhattan, KS, Apr. 2013. (Most polished-up slides in all my talks with the same title.)

Noncommutative desingularization of orbit closures for some GLn representations. Conference on Geometric Methods in Representation Theory, University of Missouri in Columbia, Nov. 2012.

A stringy moduli space for noncommutative symplectic resolutions (after Anno-Bezrukavnikov-Mirkovic). Graduate Student Research Seminar, Northeastern, Oct. 2012.

Lefschetz decompositions, wonderful resolutions, and categorical crepant resolution of singularities (after kuznetsov and Abuaf). Quiver and Invariant Theory Seminar, Fall 2012.

Noncommutative desingularization of orbit closures for some GLn-representations. Maurice Auslander International Conference, April 2012.