Seminars on "Gauge Theory, Topology, and Physics"
TSVP Thematic Program at Okinawa Institute of Science and Technology (OIST)
Embedded Files
Photo: Koji Fujiwara
These seminars are part of TSVP Thematic Program “New Frontiers in Gauge Theory, Topology, and Physics” (TP26GT) at Okinawa Institute of Science and Technology.
Speakers
Andrew Lobb (Durham University)
Jiakai Li (Stanford University)
Deeparaj Bhat (Columbia University)
Jin Miyazawa (RIMS, Kyoto University)
Paul Feehan (Rutgers University)
Cristina Ana-Maria Anghel (Université Clermont Auvergne) online
Michael Lathwood (OIST)
Enric Solé-Farré (BIMSA)
Andrew Lobb (Durham University): September 15, 3:30pm-4:30pm, lecture room L5D23.
Title: Peg problems and symplectic geometry.
Abstract: The unsolved Toeplitz Square Peg Problem was posed in 1911. It asks whether all Jordan curves (subsets of the real plane homeomorphic to a circle) contain four points at the vertices of a square.
We shall discuss this problem and related "peg" problems. They sometimes admit a description in terms of symplectic geometry.
No prior knowledge of pegs or of symplectic geometry assumed. (Joint work with Josh Greene).
https://groups.oist.jp/tsvp/event/seminar-peg-problems-and-symplectic-geometry-andrew-lobb
Jiakai Li (Stanford University): September 16, 2:30pm-3:30pm, lecture room L5D23.
Title: Signs in real monopole Floer homology
Abstract: The computation of the 3-dimensional framed real Seiberg–Witten invariant of the branched double cover of a knot K is instrumental in Miyazawa's construction of exotic P^2-knots. This invariant, known as Miyazawa's degree invariant deg(K), is the Euler characteristic χ(HMR) of framed real monopole Floer homology (HMR). A priori deg(K) is defined only up to an overall sign, since the real Dirac operator has no complex orientation. Worse, signs associated to distinct real spin^c structures are not comparable. Resolving the latter issue is essential for developing computational techniques for real Seiberg--WItten invariants.
I will explain how to fix the sign of the 3-dimensional count, in other words, define an absolute mod-two grading, via a mod-two index-theoretic correction term. This yields a total Euler characteristic \tilde{χ} that sums over all real spin^c structures, the analogue of \hat{χ} in real Heegaard Floer homology (HFR). Recent work of Srivastava provides the motivating results: an absolute mod-two grading in \hat{HFR} and the evaluation of \hat{χ} in terms of the Alexander polynomial at \sqrt{-1}. This is joint work with Ciprian Manolescu.
https://groups.oist.jp/tsvp/event/seminar-signs-real-monopole-floer-homology-jiakai-li
Deeparaj Bhat (Columbia University): September 16, 4pm-5pm, lecture room L5D23.
Title: 2-torsion in Instanton Homology
Abstract: This talk will focus on the presence of 2-torsion in a specific flavour of instanton (knot) homology introduced by Kronheimer--Mrowka. The main results are that the Poincaré homology sphere and fibered knots in 3-manifolds exhibit 2-torsion in their instanton homology. The key gauge theoretic input used in both results is a new surgery exact triangle, which relates instanton homology of three manifolds obtained by surgeries on a knot to singular instanton homology of the knot. Time permitting, we also sketch the use of sutured Floer theory in the second main result. Part of the talk is based on joint work with Zhenkun Li and Fan Ye.
https://groups.oist.jp/tsvp/event/seminar-2-torsion-instanton-homology-deeparaj-bhat
Jin Miyazawa (RIMS, Kyoto University): September 24, 3pm-4pm, lecture room L5D23.
Title: The obstruction of orientability of Real monopole Floer homology
Abstract: The Real Seiberg-Witten theory is one of a stong tool in knot theory and surface knot theory. For example, that can detect exotic P^2-knot and Kang-Park-Taniguchi proved that any non trivial cable of figure eight knot is not slice using Real Seiberg-Witten theory.
This theory is also provide interesting phenomena in index theory, since Real Seiberg-Witten theory can be seen as a "non-linear version" of Atiya's Real K-theory.
For example, there is an obstruction to orient moduli space of Real Seiberg-Witten eqations on closed 4-manifolds. Recently, Bonciocat obtained a obstruction to lift the real Heegaard Floer homology from mod 2 coefficient to Z-coefficient.
In this talk, we give a canonical way to obtain Z- coefficient real monopole Floer homology of double branced cover of a link in homology S^3 in the case when the obstruction vanishes. We also proved that the Bonciocat's obstruction is also obstract Z-coefficient in real monopole, however the proof is different. This is joint work with Jiakai Li.
Paul Feehan (Rutgers University): October 13, 2:30pm-3:30pm, lecture room L5D23.
Title: Towards a Bogomolov-Miyaoka-Yau inequality for symplectic 4-manifolds, II
Abstract: The Bogomolov-Miyaoka-Yau inequality for minimal compact complex surfaces of general type was proved in 1977 independently by Miyaoka, using methods of algebraic geometry, and by Yau, as an outgrowth of his proof of the Calabi conjectures. In this talk, we describe progress in our ongoing program to prove the conjecture that symplectic 4-manifolds obey the Bogomolov-Miyaoka-Yau inequality. Our program uses Morse theory on the gauge theoretic moduli space of non-Abelian monopoles, where the Morse function is a Hamiltonian for a natural circle action and natural two-form. We shall describe generalizations of Donaldson’s symplectic subspace criterion (1996) from finite to infinite dimensions and generalizations of Taubes' perturbation analysis from the Seiberg-Witten to non-Abelian monopole equations. These methods have application to the problem of showing that the fundamental two-form is non-degenerate and thus an almost symplectic form on the moduli space of non-Abelian monopoles. This talk is based on joint work with Tom Leness and the monographs https://arxiv.org/abs/2010.15789 (to appear in AMS Mathematical Surveys and Monographs), https://arxiv.org/abs/2206.14710 and https://arxiv.org/abs/2410.13809.
Cristina Ana-Maria Anghel (Université Clermont Auvergne): October 13, 4pm-5pm, lecture room L5D23 (online talk).
Title: Universal Link Invariants via Configuration Spaces
Abstract: Coloured Jones and Alexander polynomials are quantum invariants originating in representation theory and their geometric information is an important open problem in quantum topology. We present a new topological perspective that unifies these invariants through the topology of configuration spaces. First, for a fixed $\mathscr{N}$, we define new link invariants: "$\mathscr{N}$-th Unified Jones invariant" and "$\mathscr{N}$-th Unified Alexander invariant" globalising all coloured Jones and ADO link polynomials of (multi)-colours bounded by $\mathscr{N}$. Asymptotically, Habiro defined his universal knot invariant globalising coloured Jones polynomials by introducing the Habiro ring. For the link case, such globalisation remained open for both sequences of invariants. We answer this problem coming from representation theory using topological tools. On the representation theory side we define extensions of Habiro type rings. On the topological side, we construct a universal Jones link invariant and a universal Alexander link invariant. Putting these together, our universal invariants of geometrical nature take values in the extended Habiro rings that we construct.
Abstract_Cristina_Palmer-Anghel_OIST.pdfMichael Lathwood (OIST): October 19, 4pm-5pm, lecture room L5D23.
Title: Counting Tensor Invariants with Defect TQFT
Abstract: A tensor transforming under a unitary-orthogonal symmetry group carries finitely many invariant observables of each degree. Counting these so-called tensor invariants is an important problem in the theory of tensor models, and it reduces to counting tuples of permutations modulo a conjugation action.
In this talk, I will explain how this counting can be interpreted as the partition function of an untwisted Dijkgraaf-Witten topological quantum field theory with defects. The permutations groups appearing in the tensor invariant counting problem are exactly the defect data of a stratified manifold, and the tensor invariant is then the groupoid cardinality of the associated gauge groupoid. I conclude with physics applications coming from this perspective of tensor invariants.
This talk is based on forthcoming joint work with Remi Cocou Avhou, Noan Botcazou-Brasse, and Reiko Toriumi.
https://groups.oist.jp/tsvp/event/seminar-counting-tensor-invariants-defect-tqft-michael-lathwood
Venue
Venue
Okinawa Institute of Science and Technology (OIST)
Okinawa Institute of Science and Technology (OIST)
1919-1 Tancha, Onna-son, Kunigami-gun, Okinawa, 904-0495, Japan
Organisers:
Hokuto Konno (The University of Tokyo)
Izar Alonso (Rutgers University)
Sergei Gukov (California Institute of Technology)
Yuuji Tanaka (Beijing Institute of Mathematical Sciences and Applications)
Page updated
Google Sites
Report abuse