This lecture note by Boris Dubrovin focuses on the WDVV (Witten–Dijkgraaf–E.Verlinde–H.Verlinde) equations, which arise in the study of two-dimensional topological field theories (TFTs). The main goal is to explore the geometric and mathematical properties of these equations, particularly their connection to Frobenius manifolds and their applications in various fields such as mirror symmetry and integrable hierarchies. The WDVV equations are a system of nonlinear partial differential equations (PDEs) that describe the structure of an associative algebra with a unity, where the third derivatives of a quasihomogeneous function \( F \) are the structure constants of this algebra. The function \( F \) is defined over variables \( t = (t^1, \ldots, t^n) \), and the equations are overdetermined, making them challenging to solve. Key points covered in the lecture include: 1. **Frobenius Manifolds**: These are manifolds equipped with a flat metric, a nondegenerate inner product, and a grading operator, which satisfy certain differential equations derived from the WDVV equations. Frobenius manifolds are important in understanding the geometry of moduli spaces of TFTs. 2. **Symmetries of WDVV**: The lecture discusses various symmetries of the WDVV equations, including Legendre-type transformations and inversions, which preserve the structure of the equations. These symmetries help in classifying and solving the equations. 3. **Polynomial Solutions**: The lecture explores polynomial solutions to the WDVV equations, particularly in the case of three variables. These solutions are significant for understanding the structure and behavior of the equations. 4. **Twisted Frobenius Manifolds**: These are extensions of Frobenius manifolds where the multiplication of tangent vector fields is globally well-defined, but the inner product and the function \( F \) are defined only locally. The lecture also touches on the physical and mathematical motivations behind the WDVV equations, their role in mirror symmetry, and their connections to integrable hierarchies and matrix integrals. The author aims to provide a comprehensive understanding of the WDVV equations and their applications in various areas of mathematics and physics.This lecture note by Boris Dubrovin focuses on the WDVV (Witten–Dijkgraaf–E.Verlinde–H.Verlinde) equations, which arise in the study of two-dimensional topological field theories (TFTs). The main goal is to explore the geometric and mathematical properties of these equations, particularly their connection to Frobenius manifolds and their applications in various fields such as mirror symmetry and integrable hierarchies. The WDVV equations are a system of nonlinear partial differential equations (PDEs) that describe the structure of an associative algebra with a unity, where the third derivatives of a quasihomogeneous function \( F \) are the structure constants of this algebra. The function \( F \) is defined over variables \( t = (t^1, \ldots, t^n) \), and the equations are overdetermined, making them challenging to solve. Key points covered in the lecture include: 1. **Frobenius Manifolds**: These are manifolds equipped with a flat metric, a nondegenerate inner product, and a grading operator, which satisfy certain differential equations derived from the WDVV equations. Frobenius manifolds are important in understanding the geometry of moduli spaces of TFTs. 2. **Symmetries of WDVV**: The lecture discusses various symmetries of the WDVV equations, including Legendre-type transformations and inversions, which preserve the structure of the equations. These symmetries help in classifying and solving the equations. 3. **Polynomial Solutions**: The lecture explores polynomial solutions to the WDVV equations, particularly in the case of three variables. These solutions are significant for understanding the structure and behavior of the equations. 4. **Twisted Frobenius Manifolds**: These are extensions of Frobenius manifolds where the multiplication of tangent vector fields is globally well-defined, but the inner product and the function \( F \) are defined only locally. The lecture also touches on the physical and mathematical motivations behind the WDVV equations, their role in mirror symmetry, and their connections to integrable hierarchies and matrix integrals. The author aims to provide a comprehensive understanding of the WDVV equations and their applications in various areas of mathematics and physics.