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

47. Derived moduli Spaces of Non-linear PDEs II: Variational Tricomplex and BV Formalism, arXiv:2406.16825

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

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

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

43. Shifted Symplectic Structures on Derived Quot-Stacks II- Derived Quot-Schemes as DG manifolds, (with Dennis Borisov and Ludmil Katzarkov), arXiv:2312.02815

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

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

40. 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. 

39. Global shifted potentials for moduli stacks of sheaves on Calabi-Yau four-folds, (with Dennis Borisov, Ludmil Katzarkov and Shing-Tung Yau), 32 pages, arXiv:2007.13194. 

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

37. 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). 

36. Twisted Quasimaps and Symplectic Duality for Hypertoric Spaces, (with Michael Mcbreen and Shing-Tung Yau), 46 pages, arXiv:2004.04508., Annales de L'Institut Fourier (To Appear 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:

In gauge theory, the prominent examples of enumerative invariants are Donaldson polynomials and Seiberg-Witten invariants, which help to distinguish different smooth structures on 4 dimensional manifolds. In recent years, other 4-manifold invariants have been introduced by changing the gauge theory (the PDE’s and the “counting” problem) or, by changing the dimension, similar gauge theory invariants were defined on higher-dimensional manifolds. Notable examples include Donaldson-Thomas (DT) invariants for six-dimensional, Calabi-Yau, manifolds. The study of enumerative geometry (counting of algebraic subspaces) of complex surfaces and threefolds has proven to be deeply related to physical structures, e.g. around Gopakumar-Vafa invariants (GV); Gromov-Witten invariants, DT, as well as Pandharipande-Thomas (PT) invariants; and their “motivic lifts". On the other hand, physical dualities in Gauge and String theory, such as Montonen-Olive duality and heterotic/Type II duality have also been a rich source of spectacular predictions about these counting invariants. An example of this is the modularity properties of GW or DT invariants which is proved mathematically in some cases, as suggested by the heterotic/Type II duality. Professor Sheshmani studies geometry of moduli spaces of sheaves and curves on Calabi Yau spaces, some of which arise in the study of mathematics of string theory. In his research he has worked on understanding dualities between geometry of such moduli spaces over complex varieties of dimension 2,3,4 and currently he is working on extension of these projects from derived geometry and geometric representation theory point of view. Besides enumerative algebraic geometry, recently Artan has been studying the derived geometric structure of the moduli space of solutions to Nonlinear PDE's.

Super Algebraic Geometry:

A major challenge in construction of supergeometric analogue of Gromov-Witten invariants is the suitable generalization of intersection theory. In his joint work with Enno Kessler and Shing-Tung Yau, Artan has introduced the notion of super J-holomorphic curves, and provided a functorial construction moduli space of super curves, and super stable maps from super curves to non-super almost Kahler varieties. In order to compute deformation invariants, given as intersection numbers, counting super stable maps, the team has proposed to circumvent the difficulties around construction of super Chow groups, by assuming a virtual torus localization theorem for the odd directions. That is, to construct a super virtual normal bundle to the torus-fixed loci of the moduli space of super stable maps, and compute the super Gromov-Witten invariants, via dividing by the equivariant Euler class of the super virtual normal bundle and intersecting with the virtual class of the torus fixed superstable maps. Artan and collaborators have been able to define the super Gromov-Witten invariants of genus zero which satisfy generalized Kontsevich-Manin axioms. Furthermore, they present a recipe for calculation of super Gromov-Witten invariants of projective space.