Publication in Pure Mathematics (Algebraic Geometry, Mathematics of String theory)

57. Tyurin degenerations, Derived Lagrangians and Categorification of DT invariants (with Jacob Kryczka), arXiv:2510.20325

56. Tropical Super Gromov-Witten Invariants (with Shing-Tung Yau and Benjamin Zhou), arXiv:2510.17400

55. Towards Non-commutative Crepant resolutions of Affine Toric Gorenstein varieties (with Aimeric Malter), arXiv:2509.11664

54. The D-Geometric Hilbert Scheme, Part II: Hilbert and Quot DG-Schemes (Jacob Kryczka), 50 pages, arXiv:2411.02387 .

53. The D-Geometric Hilbert Scheme, Part I: Involutability and Stability, 87 pages (With Jacob Kryczka), arXiv:2507.07937.

52. Geography of Landau-Ginzburg models and threefold syzygies (with Yang He), 57 pages, arXiv:2506.15427

51. On the normality of commuting scheme for general linear Lie algebra (with Xiaopeng Xia and Beihui Yuan), 30 pages, arXiv:2505.13013

49. Multi-rigidity of Schubert classes in partial flag varieties (with Yuxiang Liu and Shing-Tung Yau), 39 pages, arXiv:2410.21726

48. Relative Monoidal Bondal-Orlov (with Angel Toledo), 36 pages, arXiv:2410.20942

47. Derived moduli Spaces of Non-linear PDEs II: Variational Tricomplex and BV Formalism, (with Jacob Kryczka and Shing-Tung Yau), 59 pages, arXiv:2406.16825

46. Sheaf stable pairs, Quot schemes and Birational geometry, (with Caucher Birkar and Jia Jia), 44 pages, arXiv:2406.00230

45. Rigid Schubert classes in partial flag varieties, (with Yuxiang Liu and Shing-Tung Yau), 36 pges,  arXiv:2401.11375

44. Derived moduli Spaces of Non-linear PDEs I: Singular Propagations, (with Jacob Kryczka and Shing-Tung Yau), 84 pages, arXiv:2312.05226

43. Super Gromov-Witten invariants via torus localizations, (with Enno Kessler and Shing-Tung Yau), 69 pages, arXiv:2311.09074.

42. Torus actions on moduli spaces of super stable maps of genus zero, (with Enno Kessler and Shing-Tung Yau), 41 pages, arXiv:2306.09730.

41. Super quantum cohomology I: Super stable maps of genus zero with Neveu-Schwarz punctures, (with Enno Kessler and Shing-Tung Yau), 55 pages, arXiv:2010.15634. 

40. Global shifted potentials for moduli stacks of sheaves on Calabi-Yau four-folds, (with Dennis Borisov, Ludmil Katzarkov and Shing-Tung Yau), American Journal of Mathematics, Accepted (2025), arXiv:2007.13194. 

39. Shifted Symplectic Structures on Derived Quot-Stacks II- Derived Quot-Schemes as DG manifolds (with Dennis Borisov and Ludmil Katzarkov), Advances in Mathematics, Vol 462, 10092  (2025), arXiv:2312.02815

38. Twisted Quasimaps and Symplectic Duality for Hypertoric Spaces, (with Michael Mcbreen and Shing-Tung Yau), 46 pages, Published version., Annales de L'Institut Fourier (2025) 

37. Higher rank flag shaves on surfaces and Vafa-Witten invariants, (with Shing-Tung Yau), 55 pages, European Journal of Mathematics, Vol 10, 44, (2024). arXiv:1911.00124. 

36. Strictification and gluing of Lagrangian distributions on derived schemes with shifted symplectic forms, (with Dennis Borisov, Ludmil Katzarkov and Shing-Tung Yau), 33 pages, arXiv:1908.00651, Advances in Mathematics, Vol 438, 109477, (2024). 

35. Elliptic stable envelopes and hypertoric loop spaces, (with Michael Mcbreen and Shing-Tung Yau), 21 pages, arXiv:2010.00670, Selecta Mathematica, 29, 73, (2023). 

34. Non-Holomorphic Cycles and Non-BPS Black Branes, (with Cody Long and Cumrun Vafa and Shing-Tung Yau), 57 pages, arXiv:2104.06420. Communications in Mathematical Physics, (2023).

33. 3-manifolds and Vafa-Witten theory, (with Sergei Gukov and Shing-Tung Yau), 27 pages, arXiv:2207.05775, Adv. Theor. Math. Phys, Volume 27, Issue 2, 2023).

32. Shifted symplectic structures on derived Quot-stacks, (with Dennis Borisov and Ludmil Katzarkov), Advances in Mathematics, Vol 403, 34 pages, Published version

31. Super J-holomorphic Curves: Construction of the Moduli Space, (with Enno Kessler and Shing-Tung Yau), 50 pages, Mathematischel Annalen (2021) arXiv:1911.05607.

30. Donaldson-Thomas invariants, linear systems and punctual Hilbert schemes, (with Amin Gholampour), 10 pages, Mathematical Research Letters (2021), arXiv:1909.02679.

29. Atiyah class and sheaf counting on local Calabi Yau 4 folds, (with Emanuel Diaconescu and Shing-Tung Yau), Advances in Mathematics, Vol 368, 15 July, 2020, 54 pages, arXiv:1810.09382.

28. Nested Hilbert schemes on surfaces: Virtual fundamental class, (with Amin Gholampour and Shing-Tung Yau), 47 pages, Advances in Mathematics, Vol 365, 13, May 2020 arXiv:1701.08899.

27. Localized Donaldson-Thomas theory of surfaces, (with Amin Gholmapour and Shing-Tung Yau), 28 pages, American Journal of Mathematics, Vol 142, 2, April 2020, arXiv:1701.08902.

26. Stacky GKM Graphs and Orbifold Gromov-Witten Theory, (with Melissa Liu), 38 pages, Asian Journal of Mathematics 24 (5) , pp. 48. arXiv:1807.05697.

25. Hilbert Schemes, Donaldson-Thomas theory, Vafa-Witten and Seiberg-Witten theories, 12 pages, Notices of International Congress of Chinese Mathematicians, Volums 7, Issue 2, p. 25-31, arXiv:1911.01796.

24. Donaldson-Thomas Invariants of 2-Dimensional sheaves inside threefolds and modular forms, (with Amin Gholampour), 22 pages, Advances in Mathematics, Vol. 326, No. 21, p. 79-107 arXiv:1309.0050.

23. On topological approach to local theory of surfaces in Calabi-Yau threefolds, (with Sergei Gukov, Melissa Liu and Shing-Tung Yau), 39 pages, Advances in Theoretical and Mathematical Physics, Vol 21, no 7, p. 1679-1728 arXiv:609.04363.

22. Equivariant Gromov-Witten Invariants of Algebraic GKM Manifolds, (with Melissa Liu), 22 pages (2017), SIGMA, 13, 048, pp 1-21, arXiv:1407.1370.

21. Stable pairs on nodal K3 brations, (with Amin Gholampour and Yukinobu Toda), 38 pages, International Mathematical Research Notices, Vol. 2017, No. 00, pp. 1-50, arXiv:1308.4722.

20. Vertical D4-D2-D0 bound states on K3 brations and modularity, (with Vincent Bouchard, Thomas Creutzig, Emanuel Diaconescu, Charles Doran, Callum Quigley), 54 pages, (2016), Communications in Mathematical Physics Volume 350, Issue 3, pp 1069-1121, arXiv: 1601.04030.

19. Weighted Euler characteristic of the moduli space of higher rank Joyce-Song pairs, 48 pages, European Journal of Mathematics (EJM), Vol 2, issue 2 (2016), arXiv:1107.0295.

18. Intersection numbers on the relative Hilbert schemes of points on surfaces, (with Amin Gholampour), 11 pages, Asian Journal of Mathematics, Vol 21, 3, Pp. 531-542 (2016), arXiv:1504.01107.

17. Wall-crossing and invariants of higher rank stable pairs, 31 pages, Illinois Journal of Mathematics, Vol 59, 1, 55-83 (2016), arXiv:1101.2252.

16. Higher rank stable pairs and virtual localization, 40 pages, Communications in Analysis and Geometry, Vol 24, 1 (2016), arXiv:1011.6342.

15. Generalized Donaldson-Thomas Invariants of 2-Dimensional sheaves on local P2, (with Amin Gholampour), 29 pages, Adv. Theor. Math. Phys., Volume 19, Number 3, 673-699 (2015), arXiv:1309.0056.

14. Counting curves on surfaces in Calabi-Yau threefolds, (with Amin Gholampour and Richard P. Thomas), 10 pages, Mathematische Annalen, Volume 360, Issue 1-2, pp 67-78 (2014), arXiv:1309.0051.

13. Introduction to higher rank theory of stable pairs, 11 pages, AMS Proc. Symp. Pure Math. Vol 85, Amer. Math. Soc. (2012), arXiv:1210.4202.

12. Towards studying the higher rank Pandharipande-Thomas theory of stable pairs, 207 pages. Thesis (Ph.D.)-University of Illinois at Urbana Champaign (2011). 209 pp. ISBN, 978-1267-16462-9.

Miscellaneous Publications (Mathematical Computer Science, Mathematical AI, Theoretical Machine Learning, Mathematical Fluid Mechanics)

11. Renormalization Group flow, Optimal Transport and Diffusion-based Generative Model, (with Yi-Zhuang You, Baturalp Buyukates, and Sallman Avestimehr), arXiv:2402.17090, Physica Review E, (2024)

10. Frequency-Domain Diffusion Model with Scale-Dependent Noise Schedule, (with Amir Ziashahabi, Baturalp Buyukates, Yi-Zhuang You and Salman Avestimehr), IEEE International Symposium on Information Theory, (2024). 

9. Categorical Representation Learning and RG flow operators for Algorithmic Classifiers, (with Yizhuang You, and Wenbo Fu and Ahmadreza Azizi), 31 pages, arXiv:2203.07975, Machine Learning: Science and Technology, Volume 4, 015012, (2023).

8. Categorical Representation Learning: Morphism is all you need, (with Yizhuang You), 16 pages, Machine Learning: Science and Technology, Vol 3, Issue 1, (2021) arXiv:2103.14770.

7. Hyrdodynamic advantage of in-line schooling, (with Mehdi Saadat, Florian Berlinger, Radhika Nagpal, George V. Lauder and Hossein Haj-Hariri), Bioinspiration and Biomimetics, pp. 1-21., 20 pages. Publication

6. Structure and Dynamics of Neutrally Buoyant Rigid Sphere Interacting with Thin Vortex Rings, (with Banavara Shashikanth, Scott David Kelly and Wei Mingjun), 18 pages, Journal of Mathematical Fluid Mechanics Vol 12, Issue 3, 335-353, Birkhauser-Verlag. (2008).

5. Hamiltonian structure and dynamics of a neutrally buoyant rigid sphere interacting with thin vortex rings, 18 pages, Proc. ECI conf. inter. Trans. Phen. V, F., Therm., Bio., Mat. Space Sci. (2007). 

4. Hamiltonian structure for a neutrally buoyant rigid body interacting with N vortex rings of arbitrary shape, the case of arbitrary smooth body shape, (with Banavara Shashikanth, Scott David Kelly and Jerrold Marsden), 28 pages. Theoretical and Computational Fluid Dynamics Vol 22 Issue 1, 37-64. (2006).

3. Objectivity of rates of deformation tensors in nonlinear continuum mechanics,(with Reza Naghdabadi), Proc. ASME Conf. (2004).

2. General derivation for conjugate strains of Eshelby-like stress tensors,(with Kambiz Behfar and Reza Naghdabadi), Proc. ASME (2004).

1. A thermo elastic solution for functionally graded beams using stress function, (with Mohsen Asghari), Proc. ICCES Conf. (2004).

Enumerative and Derived Algebraic Geometry, Mirror Symmetry:

My research explores the deep connections between geometry and physics, traversing a landscape from classical counting problems to the quantum foundations of string theory. It begins with Enumerative Algebraic Geometry, the classical art of counting geometric objects. This field rests on the monumental foundation laid by Alexander Grothendieck, whose introduction of scheme theory fundamentally rewrote the language of algebraic geometry. Schemes provide a unified framework where complicated geometric spaces and their underlying algebraic equations are treated as a single entity, turning seemingly intractable geometric problems into more manageable algebraic ones. This powerful perspective is essential for precisely formulating and solving enumerative questions, which serve as concrete testing grounds for new theories and a bridge to physical phenomena.

To solve these classical problems and venture into new territories, one may employ some powerful machineryies developed within modern Algebraic Geometry, such as Deformation theory, Derived Category Theory, Intersection Theory, then Derived Algebraic Geometry and the theory of stacks. Stacks, a conceptual descendant of scheme theory, are the correct language for studying "spaces of spaces," known as moduli spaces. These spaces, which parameterize all possible geometric objects like curves or vector bundles, are often poorly behaved—possessing singularities and symmetries that make them difficult to handle. Stacks gracefully manage this complexity by remembering automorphisms, much like a folder on a computer can remember that it contains multiple copies of the same file. Derived geometry then builds upon this by incorporating homological algebra and homotopy theory directly into the geometric foundations, allowing us to "smooth out" these moduli spaces conceptually and perform calculus on them. This derived framework is essential for rigorously defining the sophisticated invariants that arise in modern theoretical physics.

The profound interplay between these fields culminates in Mirror Symmetry, a phenomenon first discovered in String Theory. This duality posits that for a certain Calabi-Yau space (the geometry of a string theory's extra dimensions), there exists a mirror partner where complex and symplectic geometry are swapped. The full power of this correspondence can only be unlocked using the modern tools descended from Grothendieck's vision. The enumerative predictions on one side are computed using techniques from modern algebraic geometry or derived geometry applied to moduli stacks on the other, providing a powerful computational tool. My work sits at this exciting nexus, using this advanced geometric lexicon to probe the very nature of Mirror Symmetry, thereby using insights from string theory to solve profound mathematical problems and, conversely, using rigorous mathematics to illuminate the underlying structure of our physical universe.