Abstracts

Dec. 2 (Saturday)

14:00 -- 14:10   Opening

14:10 -- 15:10 Camille Laurent-Gengoux (Lorraine University)

Koszul-Tate resolutions: an explicit construction

Joint work with Aliaksandr Hancharuk and Thomas Strobl.

Koszul-Tate resolutions naturally appear in theoretical physics and algebraic geometry, in deformation theory in particular. They are an efficient way to remove singularities as I will briefly explain. The main issue is that, although they are known to exist, it is very hard to describe them. We explain how to construct one of them explicitly. Examples and applications are given.

15:30 -- 16:30  Hsuan-Yi Liao (National Tsing Hua University)

Vertical isomorphisms of Fedosov dg manifolds

Fedosov resolutions, which are functions on Fedosov dg manifolds, played a crucial role in globalizing Kontsevich’s formality theorem to smooth manifolds. In the setting of Lie (algebroid) pairs, the construction of Fedosov dg manifolds involves a choice of a splitting and a connection. We prove that, given any two choices of a splitting and a connection, there exists a unique vertical isomorphism, determined by an iteration formula, between the two associated Fedosov dg manifolds. As applications, we obtain iteration formulas for formal transformations between geodesic coordinates and for isomorphisms between Kapranov dg manifolds. In this talk, I'll spend majority of time on explaining our theorem in the setting of manifolds. The talk is based on a joint work with Hua-Shin Chang. 

16:50 -- 17:50 László Fehér (Szeged University) 

Bi-Hamiltonian structures of integrable many-body models from Poisson reduction

We review our results on bi-Hamiltonian structures of trigonometric spin Sutherland  models built on collective spin variables. Our basic observation was that the  cotangent bundle  $T^*\mathrm{U}(n)$ and its holomorphic analogue $T^* \mathrm{GL}(n,{\mathbb C})$, as well as $T^*\mathrm{GL}(n,{\mathbb C})_{\mathbb R}$, carry a natural quadratic Poisson bracket, which is compatible with the canonical linear one. The quadratic bracket arises by change of variables and analytic continuation from an associated Heisenberg double. Then, the reductions of $T^*{\mathrm{U}}(n)$  and $T^*{\mathrm{GL}}(n,{\mathbb C})$ by the conjugation actions of the corresponding groups lead to  the real  and holomorphic spin Sutherland models, respectively, equipped with a bi-Hamiltonian structure. The reduction of $T^*{\mathrm{GL}}(n,{\mathbb C})_{\mathbb R}$ by the group $\mathrm{U}(n) \times \mathrm{U}(n)$ gives a generalized   Sutherland model coupled to two ${\mathfrak u}(n)^*$-valued spins. We also show that a bi-Hamiltonian structure on the associative algebra ${\mathfrak{gl}}(n,{\mathbb R})$ that appeared in the context of Toda models can be interpreted as the quotient of compatible Poisson brackets on $T^*{\mathrm{GL}}(n,{\mathbb R})$. Before our work, all these reductions were studied using the canonical Poisson structures of the cotangent bundles, without realizing the bi-Hamiltonian aspect.

18:30 -- Discussion

Dec. 3 (Sunday)

10:00 -- 14:55 Short talk session

10:00 -- 10:25 Kevin Morand (Sogang University)

Graph complexes and deformation quantization of higher structures

In the formulation of his celebrated Formality conjecture, M. Kontsevich introduced a combinatorial (graph) model of the deformation theory for Poisson structures, the cohomology thereof providing information regarding the existence of quantizations as well as their classification via the Grothendieck-Teichmuller group. More generally, deformation quantization problems for Poisson manifolds and higher structures (e.g. Lie bialgebras, Courant algebroids, etc.) can be partitioned according to the cohomology of suitable graph complexes. In this talk, we will introduce a version of Kontsevich's construction suitable to address the deformation quantization problem for Lie bialgebroid and their `quasi' generalisations.

10:30 -- 10:55 Ryo Hayami (Nagoya University)

Dg (bi)symplectic geometry and higher (double) Poisson vertex algebras

One-to-one correspondence between Courant-Dorfman algebras and Poisson vertex algebras can be seen as an algebraic generalization of the relation between generalized tangent bundles and Alekseev-Strobl currents. In terms of dg symplectic geometry, Alekseev-Strobl currents are 1-dimensional BFV(Batalin-Fradkin-Vilkovisky) currents whose target datum are degree 2 dg symplectic manifolds(Courant algebroids). In this talk, I will introduce a higher version of the one-to-one correspondence, which is an algebraic generalization of n-1 dimensional BFV currents whose target datum are degree n dg symplectic manifolds.

Moreover, a noncommutative analogue of the correspondence, which is a higher generalization of one-to-one correspondence between double Courant-Dorfman algebras and double Poisson verex algebras, will be discussed in terms of dg bisymplectic algebras.

11:00 -- 11:25 Taika Okuda (Tokyo University of Science)

Explicit star product with separation of variables on G_{2,4}(C)

In this talk, we will briefly explain that the explicit star product with separation of variables on a complex Grassmannian \(G_{2,4}(\mathbb{C})\) is given via the construction method proposed by Hara-Sako. It is known that the construction method is the tool to give an explicit star product on a locally symmetric K\"{a}hler manifold. To construct an explicit star product by this method, it is necessary to solve the recurrence relations. We show that the recurrence relations for \(G_{2,4}(\mathbb{C})\) can be solved by using a Fock representation. We also give the star product with separation of variables on \(G_{2,4}(\mathbb{C})\) from the solution of the recurrence relations. This is the joint work with Akifumi Sako (Tokyo University of Science).

11:30 -- 11:55 Ryo Matsuda (Kyoto University)

On extensions of local quasi-isometric maps

In the Teichmüller space of an infinite type Riemann surface R, there exists a Riemann surface homeomorphic to R on its Bers boundary by using a degenerate quasiconformal map called the David map. Since the quasiconformal map has a quasi-isometric map in the three-dimensional hyperbolic space as an extension, the question arises as to what kind of extension the David map has in the three-dimensional hyperbolic space. In my talk, I will discuss when a locally quasi-isometric map has a continuous map on the infinity boundary as an extension.  

13:30 -- 13:55 Shuhei Yonehara (Osaka University)

Reduction of coKähler manifolds

The notion of cosymplectic manifolds is an odd-dimensional analogue of symplectic manifolds. Albert introduced the concept of Hamiltonian actions on cosymplectic manifolds and proved a reduction theorem. In this talk, we refine this theorem in the case of coKähler manifolds, that is, normal almost contact metric manifolds which have a compatible cosymplectic structure.

14:00 -- 14:25 Naoyuki Kanomata (Tokyo University of Science)

Calogero model from real symmetric Φ^4-matrix model

We investigate a real symmetric Φ^4-matrix model with a kinetic term expressed as Tr(E\Phi^2), where E is a positive diagonal matrix with non-degenerate eigenvalues. Our study has revealed that the partition function of this matrix model corresponds to a zero-energy solution of a Schrodinger-type equation with Calogero-Moser Hamiltonian. Additionally, we find that the partition function satisfies a system of partial differential equations generated by Virasoro algebra. This talk is based on collaborative work with Professor Harald Grosse, Professor Akifumi Sako, and Professor Raimar Wulkenhaar.

14:50 -- 15:50  Atsushi Fujioka (Kansai University)

Equivariant projections between spaces of equicentroaffine curves

In this talk, we consider the spaces of equicentroaffine curves, on which the diffeomorphism group of the line acts. In particular, we will show the transformation rule of the curvatures for equicentroaffine curves by this action, and obtain equivariant projections from the space of curves in higher dimensional space into the space of plane or space curves.

16:10 -- 17:10 Satoru Odake (Shinshu University)

Orthogonal Polynomials and Exactly Solvable Quantum Mechanical Systems

The relationship between orthogonal polynomials and exactly solvable quantum mechanical systems is reviewed. (Basic) Hypergeometric orthogonal polynomials in the Askey scheme and their generalizations can be successfully investigated by using quantum mechanical formulations. We explain quantum mechanical formulations, closure relations, shape invariance, Darboux transformations, and exceptional/multi-indexed orthogonal polynomials.