New England Algebraic Topology and Mathematical Physics Seminar
A miniconference in topology and quantum field theory
Providence College— November 2-3, 2024
Ruane Center for the Humanities, Room 205
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Overview:
- The miniconference is targeted at researchers and graduate students working on the interface between topology and mathematical physics. Topics include the theory of factorization algebras, factorization homology, and the theory of configuration spaces; functorial TQFT and its applications to knot and manifold invariants; and connections to geometric representation theory.
- All talks are intended to be accessible to graduate students with an interest in topology and mathematical physics. In addition to the invited talks there will be a small number of additional contributed talks from graduate students.
- There is limited funding available to cover the local and travel
expenses of graduate students based in the New England area.
If you would like to be considered for funding, please indicate so in the registration form, link here. Registration must be submitted by October 4 to be considered for funding.
- Transportation: there is both a commuter rail and Amtrak line from Boston to downtown Providence.
Speakers:
- Élie Casbi (Northeastern University),
Representations of affine quantum groups and equivariant homology of affine Grassmannians
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Let G be a semisimple complex Lie group and N a maximal unipotent subgroup of G. In their study of the equivariant homology of the affine Grassmannian Gr_{G\vee}, Baumann-Kamnitzer-Knutson introduced an algebraic morphism \overline{D} on the coordinate ring \mathbb{C}[N] providing a powerful tool to compare distinguished bases of this algebra, such as the Mirkovic-Vilonen basis arising from the geometric Satake correspondence. In this talk we will focus onthe simply-laced case and present an alternative description of \overline{D} proposed in a joint workwith Jian-Rong Li, that relies on Hernandez-Leclerc’s categorification of the cluster structure of \mathbb{C}[N] via finite-dimensional representations of affine quantum groups. We will then present our second work recently posted on the arxiv (also joint with Jian-Rong Li) where we establish a large family of non-trivial rational identities obtained by applying our construction to Frenkel-Reshetikhin’s q-characters. If time allows, we will discuss possible interpretations of these identities in terms of equivariant homology, raising the question of natural geometric models associated to representations of affine quantum groups.
- Ivan Contreras (Amherst College),
Frobenius objects in 2-Span, 2-Segal sets and the symplectic category
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2-dimensional Topological Quantum Field Theories (TQFT) can be classified via (commutative) Frobenius algebras. In this talk, we will describe Frobenius objects in a category where the objects are sets and the morphisms are isomorphism classes of spans, which is a suitable model for the symplectic category. Our key result is that it is possible to construct a simplicial set (with some additional conditions) that encodes the data of the Frobenius structure. The simplicial sets that arise in this way can be characterized by a few relatively nice conditions. As an application (in progress), we recover a Frobenius structure from the reduced, and possibly singular, reduced phase space of the Poisson sigma model. This is based on work with Rajan Mehta and Molly Keller, and Mehta and Walker Stern.
- Sarah Harrison (Northeastern University),
Liouville Theory and Weil-Petersson Geometry
- Two-dimensional conformal field theory is a powerful tool to understand the geometry of surfaces. Liouville conformal field theory in the classical (large central charge) limit encodes the geometry of the moduli space of Riemann surfaces. I describe an efficient algorithm to compute the Weil--Petersson metric to arbitrary accuracy using Zamolodchikov's recursion relation for conformal blocks, focusing on examples of a sphere with four punctures and generalizations to other one-complex-dimensional moduli spaces. Comparison with analytic results for volumes and geodesic lengths finds excellent agreement.
In the case of M_{0,4}, I discuss numerical results for eigenvalues of the Weil-Petersson Laplacian and connections with random matrix theory.
Based on work with K. Coleville, A. Maloney, K. Namjou, and T. Numasawa.
- Andy Neitzke (Yale University)
A new approach to Virasoro conformal blocks
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Given a Riemann surface C one can define the vector space of Virasoro conformal blocks on C. This vector space is important in conformal field theory (it describes in some sense the "universal part" of the correlation functions) and also of some mathematical interest (related e.g. to tau-functions, instanton moduli spaces, representations of mapping class groups and skein algebras.) I will give an elementary definition of the space of Virasoro blocks, and describe a new scheme for constructing them via "abelianization".
- Katherine Novey (University of Notre Dame),
Ordinary TQFTs Cannot Distinguish Simply Connected 6-Manifolds
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Work by David Reutter and Chris Schommer-Pries has shown that TQFTs can distinguish stable diffeomorphism classes of closed, connected, even-dimensional manifolds subject to certain finiteness conditions. In particular, simply connected closed 6-manifolds with finite \pi_2 are diffeomorphic if and only if they cannot be distinguished by TQFTs, and it was conjectured that this result holds for all closed, simply connected 6-manifolds. We will present a pair of non-diffeomorphic closed, simply connected 6-manifolds with infinite \pi_2 that are indistinguishable by TQFTs, disproving the conjecture.
- Sam Panitch (Yale)
3d Quantum Trace Map
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In 2010, Bonahon and Wong constructed the 2d quantum trace map, connecting two different quantizations of the SL_2(C) character variety of an ideally triangulated hyperbolic surface S. The classical limit of this map expresses the trace of the holonomy of a closed immersed curve in S as a Laurent polynomial in the (square roots of the) shear coordinates for the hyperbolic structure defined with respect to the triangulation. In this talk, we will discuss a new construction of a 3d quantum trace map for ideally triangulated 3-manifolds, focusing on the geometric aspects. This is joint work with Sunghyuk Park.
- Ryan Gelnett (University of Albany)
The Topology of Configuration Spaces of Circles in the Plane
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We consider the space of all configurations of finitely many (potentially nested) circles in the plane and compute the fundamental group of each of its connected components. These groups can be viewed as “braided” versions of the automorphism groups of finite rooted trees. This is joint work with Justin Curry and Matthew Zaremsky.
- Clair Xinle Dai (Harvard)
Sectorial Decompositions of Symmetric Products and Homological Mirror Symmetry
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Symmetric products of Riemann surfaces play a crucial role in symplectic geometry and low-dimensional topology. They are essential ingredients for defining Heegaard Floer homology and serve as important examples of Liouville manifolds when the surfaces are open. In this talk, I will discuss ongoing work on the symplectic topology of these spaces through Liouville sectorial methods, along with an example illustrating the application of this decomposition to homological mirror symmetry.
Registration:
Registration is now closed.
Organizers:
Financial support has been provided through NSF DMS Award No. 2329855.
Our thanks to the staff at Providence College for practical and logistical support.
Websites from past NEAT MAPS conferences: spring 2023, fall 2023, spring 2024.