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Gufang Zhao


University of Melbourne

School of Mathematics and Statistics

Peter Hall Building 175

Parkville, VIC 3010


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

Research Statement.

CV.

Teaching Statement.

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.

Yaping's talk on this project, recorded at the Hausdorff Research Institute for Mathematics in Bonn, Oct. 2017.

Papers and Preprints

Peer-Reviewed Journal Articles

1. (with C. Su and C. Zhong) On the K-theory stable bases of the Springer resolution. 43 pages, to appear in Annales sc. de l'ENS. arxiv: 1708.08013.

2. (with Y. Yang) The cohomological Hall algebra of a preprojective algebra. 44 pages, to appear in Proc. Lond. Math. Soc.. arXiv: 1407.7994.

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

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

4. (with M. Levine and Y. Yang) Algebraic elliptic cohomology theory and flops 1, with an appendix by Joel Riou. 29 pages, to appear in Math. Ann.. arXiv:1311.2159.

See here for an unpublished earlier version (with more background introductory materials for my own reference).

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

6. (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.

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

Submitted

8. (with V. Nandakumar) Categorification via blocks of modular representations of sl_n. 21 pages, (2016). arxiv: 1612.06941.

9. (with Y. Yang) On two cohomological Hall algebras. 26 pages, (2016). arXiv: 1604.01477.

A revised version is available here. Significant expositional changes. Revised on Jul. 28, 2017.

10. (with C. Zhong) Elliptic affine Hecke algebras and their representations. 40 pages, (2015). arXiv: 1507.01245.

A revised version is available here. It contains a duality conjecture. Revised on Aug. 2, 2017.

11. (with M. Levine and Y. Yang) Algebraic elliptic cohomology theory and flops 2: $SL$-cobordism. 24 pages, (2016). arxiv: 1610.00396

12. (with Y. Yang) The PBW theorem for the affine Yangians, 12 pages, (2018). arxiv: 1804.04375

Preprints

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

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

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

Conference Proceedings

16. (with Y. Yang) How to sheafify an elliptic quantum group. MATRIX Annals 2017. arXiv: 1803.06627

Work in Progress

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

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

Theses

19. 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.

20. 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

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.