**Abstract**

The study of the moduli spaces of algebraic curves and coherent sheaves, and their induced invariants over ambient complex kahler varieties of complex dimension 2, 3 and higher, has been a central source of focus for mathematicians in the past 50 years, due to their profound connections to geometry, topology, number theory as well as fruitful contributions to superstring theory. The course aims at introducing these topics and provides discussion of computations of Gromov-Witten and Donaldson-Thomas invariants of complex algebraic varieties.

**Lecturer Intro.**

I am a Professor of pure Mathematics, specialized in Algebraic geometry, Differential Geometry and Mathematics of String Theory. I am a Full Research Fellow (equivalent to Full Professor in the US and Europe) at Yanqi Lake Beijing Institute of Mathematical Sciences and Applications in Beijing, as well as a senior member at Harvard University CMSA, and a visiting professor at Institute for the Mathematical Sciences of the Americas at University of Miami. During the past 5 years I have been a senior personnel at Simons Collaboration Program for Homological Mirror Symmetry at Harvard University Center for Mathematical Sciences and Applications (CMSA), as well as Harvard Physics department (2020-2021), and an Associate Professor of Mathematics at Institut for Mathematik and Center for Quantum Geometry of Moduli Spaces at Aarhus University in Denmark. My work is mainly focused on Gromov Witten theory, Donaldson Thomas theory, Calabi-Yau geometries, and mathematical aspects of String theory. I study 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 my research I have worked on understanding dualities between geometry of such moduli spaces over complex varieties of dimension 2,3,4 and currently I am working on extension of these projects from derived geometry and geometric representation theory point of view. In joint work with Shing-Tung Yau (BIMSA, YMSC, Tsinghua, Harvard Math, Harvard CMSA, and Harvard Physics departments), Cody Long (Harvard Physics), and Cumrun Vafa (Harvard Math and Physics departments) I worked on geometry moduli spaces of sheaves with non-homolomorphic support and their associated non-BPS (non-holomorphic) counting invariants. In 2019 I recieved IRFD "Research Leader" grant (approx $1M) on my project "Embedded surfaces, dualities and quantum number theory". The project has additionally been co-financed by Harvard University CMSA (Approx total. $400K). Detail of IRFD "Research Leader" grant ($1M). --The grant has recently been extended until 2027!