New England Algebraic Topology and Mathematical Physics Seminar

 A miniconference in topology and quantum field theory

Boston University— November 11-12, 2023

Center for Computing and Data Sciences, Room 548

Overview:
Speakers:
  • Xujia Chen (Harvard)
  • Gurbir Dhillon (Yale)
  • Yan Zhou (Northeastern)
  • Xuwen Zhu (Northeastern)

    Graduate student speakers:

  • Hank Chen (Waterloo)
  • Anh Hoang (University of Minnesota)
  • Sidharth Soundararajan (Boston University)
  • Ju Tan (Boston University)
Registration: Registration is now closed.
Titles and abstracts:
  • Xujia Chen (Harvard) Kontsevich’s invariants as topological invariants of configuration space bundles

    • Kontsevich's invariants (also called “configuration space integrals”) are invariants of certain framed smooth manifolds/fiber bundles. The result of Watanabe(’18) showed that Kontsevich’s invariants can distinguish smooth fiber bundles that are isomorphic as topological fiber bundles. I will first give an introduction to Kontsevich's invariants, and then state my work which provides a perspective on how to understand their ability of detecting exotic smooth structures: real blow up operations essentially depends on the smooth structure, and thus given a space/bundle X, the topological invariants of some spaces/bundles obtained by doing some real blow-ups on X can be different for different smooth structures on X.

  • Gurbir Dhillon (Yale) The Fundamental Local Equivalence and the W-algebra

    • A basic (and pleasant) assertion in the geometric Langlands program is the Satake isomorphism, which identifies a factorization category of certain constructible sheaves on the affine Grassmannian of a reductive group G with the representations of its Langlands dual group G^L. Gaitsgory and Lurie proposed a one parameter deformation of this equivalence, which replaces G^L with the corresponding quantum group, and which plays a similarly basic role in the quantum geometric Langlands correspondence. After giving a gentle introduction to the above circle of ideas for non-specialists, we will describe some joint work with Gaitsgory verifying this conjecture, and emphasize the roles played by various factorization algebras, notably the W-algebra.

  • Yan Zhou (Northeastern) Irregular connections, Stokes geometry, and WKB analysis

    • We study the Riemann-Hilbert map of a class of meromorphic linear ODE systems on the complex projective line with irregular singularities. This class of ODE’s show up in various contexts in geometry and representation theory. The Stokes matrices of these ODE’s encode the generalized monodromy data. First, we study the WKB leading terms of the Stokes matrices and give a definite answer for the degenerate Riemann-Hilbert map. Then, if time permits, we will establish the connection to the work of Gaiotto-Moore-Neitzke and explain how the picture of spectral networks and DT theory simplify near the degenerate Riemann-Hilbert map. The talk is based on ongoing joint work with Anton Alekseev, Andrew Neitzke, and Xiaomeng Xu.

  • Xuwen Zhu (Northeastern) Analysis of ALH* gravitational instantons.

    • Gravitational instantons are non-compact Calabi-Yau metrics with L^2 bounded curvature and are categorized into six types. We will discuss one such type called ALH* metrics which has a non-compact end modelled by the Calabi ansatz with inhomogeneous collapsing near infinity. Such metrics were used recently as the scaling bubble limits for codimension-3 collapsing of K3 surfaces, where the study of its Laplacian played a central role. In this talk I will talk about the Fredholm mapping property and L^2 cohomology of such metrics. This is ongoing work joint with Rafe Mazzeo.

  • Hank Chen (Waterloo) Categorical Quantum Groups and Braided Monoidal 2-Categories

    • It is well-known that the category of representations of quantum group Hopf algebras are braided, and hence captures invariants of knots. This talk is based on the work arXiv:2304.07398, where we developed a quantization of the L_\infty structures underlying higher-dimensional QFTs (cf. Costello-Gwilliam), including homotopy refinements and a rigid braiding given by a “2-R-matrx”. We prove that the 2-representations of such categorical quantum groups/2-Hopf algebras form a cohesive braided monoidal (tensor) 2-category, thereby completing the 4d TFT-braided 2-category side of the triangle introduced by Leon in the previous installment of NEAT MAPS. If time allows, I will discuss some applications of this framework (see JHEP 2023 141).

  • Anh Hoang (University of Minnesota) Configuration spaces and applications in arithmetic statistics

    • In the last dozen years, topological methods have been shown to produce a new pathway to study arithmetic statistics over function fields, most notably in Ellenberg-Venkatesh-Westerland's work on the Cohen-Lenstra conjecture. More recently, Ellenberg, Tran and Westerland proved the upper bound in Malle's conjecture on the enumeration of function fields by studying the homology of configuration spaces with certain exponential coefficients. In this talk, we will give an overview of their framework and extend their techniques to study the homology of various configuration spaces. As an application, we study character sums of the resultant of monic squarefree polynomials over finite fields, answering and generalizing a question of Ellenberg and Shusterman.

  • Sidharth Soundararajan (Boston University) Noncollapsing Degenerations of Gravitational Instantons

    • Gravitational Instantons were introduced in physics by Hawking as an analogue to Yang-Mills instantons. Mathematically, they are noncompact complete Hyperkähler 4-manifolds whose curvature tensor is square-integrable. In recent works, they have been completely classified and the Torelli theorems for all types of instantons have been proved. This talk aims to discuss ongoing work on the non-collapsing limit of gravitational instantons, which aims to provide a partial compactification of their moduli spaces.

  • Ju Tan (Boston University) Mirror symmetry and ADHM correspondence

    • Quivers possesses a rich representation theory. On the one hand, it exhibits a deep connection between instantons and coherent sheaves as illuminated by the ADHM construction and the works of many others. On the other hand, in some cases quivers also relate to the formal deformation space of a Lagrangian submanifold. In this talk, we will discuss these relations more explicitly from the perspective of mirror symmetry. The key ingredient is to see that the ADHM construction can be realized as a mirror symmetry phenomenon. This talk is based on the joint work with Siu-Cheong Lau, and Junzheng Nan as well as a working project in collaboration with Jiawei Hu and Siu-Cheong Lau.

    Organizers:
    Financial support has been provided through NSF DMS Award No. 2329855. Our thanks to the staff in the Boston University math department for additional practical and logistical support.
    Websites from past NEAT MAPS conferences: Spring 2023