Memorial Mathematics Working Seminar
Winter Schedule: HH 3017 2–3 pm
| March 23th | Serhii Koval - Introduction to
algebraic deformation theory (part 2) In the first part of this talk, we introduced formal one-parameter deformations of associative algebras and defined Hochschild cohomology. In this talk, we will discuss the interpretation of Hochschild cohomology in low degrees. After that, we will define rigid algebras and study some sufficient conditions for rigidity from the perspective of Hochschild cohomology. Time permitting, we will discuss deformation quantization of Poisson manifolds. |
| March 16th | Serhii Koval - Introduction to
algebraic deformation theory (part 1) Algebraic deformation theory studies “infinitesimal perturbations” of various algebraic structures from the point of view of homological algebra. We begin with a review of elementary notions of formal deformations of associative algebras and illustrate them by examples. Following this, we introduce Hochschild cohomology of algebras and discuss their properties. In the next part of the talk, we will discuss the role of this cohomology in studying deformations and, time permitting, the basics of the theory of deformation quantization of Poisson manifolds. |
| March 9th | Sebastiano Argenti - Polynomial
identities of associative algebras (part 3) In the first parts of this talk we discussed some fundamental theorems about the structure of PI-algebras (“qualitative approach”). In the next talk, we are going to discuss Regev’s solution to the tensor product problem, leading to the definition of codimension sequence and to the “quantitative approach” to PI-theory. |
| March 2nd | Sebastiano Argenti - Polynomial
identities of associative algebras (part 2) In the first part of this talk we reviewed the basic definitions regarding polynomial identities and how they are connected to some important classes of algebras. In the next talk we are going to discuss how the theory of polynomial identities provided the natural framework to attack some classical problems. In particular, we are going to discuss the solution of Kurosch problem for PI-algebras and its connection with the Nagata-Higman theorem and the Shirshov theorem. If time permits, we will introduce Regev’s solution to the tensor product problem, leading to the definition of codimension sequence. |
| February 17th | Christopher Lang - Nahm’s
equations and co-Higgs bundles Nahm’s equations are a reduction of the self-dual Yang–Mills equations to one dimension and play a central role in the Nahm transform, which relates magnetic monopoles to Nahm data. These equations also arise as a vector field on the moduli space of co-Higgs bundles on the projective line. In this talk, I will introduce Nahm’s equations and co-Higgs bundles and examine how Nahm’s equations appear in the study of co-Higgs bundles. I will be following Nigel Hitchin’s “Remarks on Nahm’s equations”. Reference: arXiv:1708.08812 |
| February 3rd | Sebastiano Argenti - Polynomial
identities of associative algebras (part 1) The category of algebras with polynomial identities, or PI-algebras, contains both the category of commutative algebras and of finite dimensional algebras. Moreover, several classical problems found a natural solution in this setting. This observation is the key idea that guided the development of the theory of polynomial identities in the last century. In recent years, a fruitful approach has been the description of numerical invariants of the ideal of identities of a PI-algebra, shedding light on the structure of the algebra itself. The goal of this introductory talk is to give an overview of the fundamental result of PI-theory, providing the motivation to the study of polynomial identities. |
| December 2nd | Graham Cox - Geometry of
infinite-dimensional manifolds (part 3) This is an introductory talk on differential geometry in infinite dimensions. I will try to keep prerequisites to a minimum, just assuming familiarity with basic concepts from differential geometry and functional analysis (in particular, Banach spaces and finite-dimensional smooth manifolds). Many of the definitions and constructions are identical to those in finite dimensions, so I will instead focus on some of the subtleties and differences that arise in infinite dimensions. In a future talk I will describe some classical examples of this theory, such as manifolds of mappings and their application to incompressible fluid flow. |
| November 25th | Serhii Koval - Algebra of
symplectic reduction (part 2) In classical mechanics, symplectic (or Hamiltonian) reduction allows simplifying the equations of motion by considering symmetries of the phase space and its furter foliation into symplectic leaves. In this talk, I will give an introduction to the process of symplectic reduction from an algebraic perspective in the framework of affine schemes and actions of reductive groups. I will give an example of computation of Hamiltonian reduction for the commuting scheme. Time permitting, I will discuss Hamiltonian reduction along coadjoint orbit and give the construction of Calogero-Moser space of Kazhdan, Kostant and Sternberg. This talk is based on the lecture notes of P. Etingof on Calogero-Moser systems. Reference: arXiv:0606233. |
| November 18th | Graham Cox - Geometry of
infinite-dimensional manifolds (part 2) This is an introductory talk on differential geometry in infinite dimensions. I will try to keep prerequisites to a minimum, just assuming familiarity with basic concepts from differential geometry and functional analysis (in particular, Banach spaces and finite-dimensional smooth manifolds). Many of the definitions and constructions are identical to those in finite dimensions, so I will instead focus on some of the subtleties and differences that arise in infinite dimensions. In a future talk I will describe some classical examples of this theory, such as manifolds of mappings and their application to incompressible fluid flow. |
| November 12th | Graham Cox - Geometry of
infinite-dimensional manifolds (part 1) This is an introductory talk on differential geometry in infinite dimensions. I will try to keep prerequisites to a minimum, just assuming familiarity with basic concepts from differential geometry and functional analysis (in particular, Banach spaces and finite-dimensional smooth manifolds). Many of the definitions and constructions are identical to those in finite dimensions, so I will instead focus on some of the subtleties and differences that arise in infinite dimensions. In a future talk I will describe some classical examples of this theory, such as manifolds of mappings and their application to incompressible fluid flow. |
| November 4th | Tom Baird - An introduction to
Cohomological Field Theories (part 2) Having introduced moduli spaces of curves and Gromov-Witten theory last week, this week I will present the abstract definition of a cohomological field theory, explore some of its properties, and describe the Givental-Teleman classification of semisimple cohomological field theories. Reference: arXiv:1712.02528. |
| October 28th | Christopher Lang - An introduction
to topological recursion (part 2) Topological recursion is a construction in algebraic geometry that recursively computes invariants of spectral curves. In this talk, I will (finally) introduce topological recursion, examine the connection between topological recursion and Airy ideals, and explore some applications of this construction. Reference: arXiv:2409.06657. |
| October 21th | Serhii Koval - Algebra of
symplectic reduction (part 1) In classical mechanics, symplectic (or Hamiltonian) reduction allows simplifying the equations of motion by considering symmetries of the phase space and its furter foliation into symplectic leaves. In this talk, I will give an introduction to the process of symplectic reduction from an algebraic perspective in the framework of affine schemes and actions of reductive groups. I will give an example of computation of Hamiltonian reduction for the commuting scheme. Time permitting, I will discuss Hamiltonian reduction along coadjoint orbit and give the construction of Calogero-Moser space of Kazhdan, Kostant and Sternberg. This talk is based on the lecture notes of P. Etingof on Calogero-Moser systems. Reference: arXiv:0606233. |
| October 16th | Tom Baird - An introduction to
Cohomological Field Theories (part 1) In this expository lecture, I will give an introduction to cohomological field theories in the sense of Kontsevich and Manin. Along the way, we will learn about moduli spaces of complex curves, Gromov-Witten theory, and 2D topological field theories. I hope to make my presentation accessible to anyone acquainted with manifolds and cohomology (singular or de Rham). Reference: arXiv:1712.02528. |
| October 7th | Christopher Lang - An introduction
to topological recursion (part 1) Topological recursion is a construction in algebraic geometry that recursively computes invariants of spectral curves. It has applications to enumerative geometry, string theory, knot theory, mathematical physics, and random matrix theory, among others. In this talk, I will introduce some of the components of this construction. Reference: arXiv:2409.06657. |
| September 30th | Organizational
Meeting We will set up a preliminary schedule of speakers for the term. |