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Talk 01 — Mauro Spera (Dipartimento di Matematica e Fisica - Università Cattolica del Sacro Cuore, Brescia, Italy) — [computing…]Local time: click here to view in your time zoneTitle: On the geometry and braiding of optical phases
Abstract
In this talk (based on joint with Gabriele Barbieri and Kristina Frizyuk), an application of holomorphic geometric quantization techniques to quantum optics is presented, together with a discussion of emerging braid group aspects.
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Talk 02 — Andrea Galasso (Università Telematica Pegaso, Italy) — [computing…]Local time: click here to view in your time zoneTitle: Star products on CR manifolds with non-degenerate Levi form
Abstract
This talk concerns quantization problems in the setting of compact CR manifolds with non-degenerate Levi form. We study the algebra of Toeplitz operators naturally associated with the CR structure and show that it induces a star product for an appropriate class of symbols. We also consider situations where the CR manifold carries additional symmetry given by a locally free action of a compact Lie group. In this case, we investigate the algebra of invariant Toeplitz operators and explain how symmetry enters the quantization procedure. As an application, this approach recovers well-known results of deformation quantization for a class of compact pseudo-Kähler manifolds. Finally, we briefly mention ongoing joint work on the Levi-flat case, where new analytical difficulties arise. This talk is based on joint work with Chin-Yu Hsiao.
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Talk 03 — Satyen Patel (IISER Pune, India) — [computing…]Local time: click here to view in your time zoneTitle: Fedosov quantization on smooth manifolds and hidden symmetries
Abstract
We discuss the deformation quantization of an arbitrary manifold M, realized locally as a non- commutative deformation of a Poisson structure on M induced from an ambient symplectic manifold X. The construction utilizes the Fedosov star product, defined using sections of a Weyl bundle over X. We check the compatibility of the induced algebra with symmetries of X acting on the submanifold. We also apply this to study the semiclassical limit of a physical quantum system if time permits. This talk is based on ongoing work with Dr. Rukmini Dey.
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Talk 04 — Discussion — [computing…]Local time: click here to view in your time zoneTitle: Discussion
Abstract
This slot is for discussion.
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Talk 05 — Adrian Chitan (Western University, Canada) — [computing…]Local time: click here to view in your time zoneTitle: Quantization and the Jeffrey-Weitsman-Witten Invariants
Abstract
The Jeffrey-Weitsman-Witten (JWW) invariants of 3-manifolds are derived from the geometric quantization of the moduli space of flat $SU(2)$-connections. Standard quantization approaches often bypass the singular nature of this space. In this talk, I present an approach that respects the singular geometry characterized by stratification. Using the real polarization from the Jeffrey-Weitsman system, we define half-densities supported directly on symplectic strata. I will focus on the evaluation of the stratified Blattner-Kostant-Sternberg (BKS) pairing, demonstrating how integrating across these strata recovers the JWW invariants in basic examples aligns closely with conjecture for a plethora of new manifolds formerly excluded.
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Talk 06 — Michael Astwood (University of Manitoba, Canada) — [computing…]Local time: click here to view in your time zoneTitle: Prequantization of 0-Shifted Symplectic Stacks
Abstract
We present work in progress that examines the prequantization of 0-shifted symplectic foliation Lie groupoids, which can be interpreted as presentations for 0-shifted symplectic differentiable stacks. Our constructions are explicit and expressed in the language of basic differential forms on Lie groupoids, multiplicative vector fields, Lie 2-algebras, and equivariant principal bundle groupoids. We demonstrate that on a 0-symplectic Lie groupoid, the set of invariant Hamiltonian vector fields carries the natural structure of a strict Lie 2-algebra inherited from the Lie 2-algebra of multiplicative vector fields. Furthermore, we show that the associated algebra of observables carries the structure of a strict 2-term homotopy Poisson algebra, and use it to define the quantization map as an invertible butterfly of Lie 2-algebras, relating Hamiltonian vector fields to symmetries of a prequantization.
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Talk 07 — Mahishanka Withanachchi (University of Calgary, Canada) — CANCELLEDTitle: Ricci Curvature Aware Clustering on Manifold Valued DataThis talk has been cancelled.
Abstract
Standard clustering methods such as KMeans assume that data lie in a flat Euclidean space where similarity is captured by straight line distances. In many applications however data are better modeled as samples from a curved manifold embedded in high dimensions. In this talk I introduce a Ricci curvature aware clustering framework that incorporates intrinsic geometry into the clustering objective. A nearest neighbor graph is constructed to approximate the manifold and Forman Ricci curvature is computed on this graph to detect negatively curved regions that often correspond to cluster boundaries. Geodesic distances are then modified by curvature to penalize paths passing through such regions and clustering is performed using a curvature weighted k medoids objective. I will discuss the mathematical formulation of the method and present comparisons with KMeans and UMAP with DBSCAN on standard manifold benchmarks. The results show that incorporating curvature significantly improves clustering performance when intrinsic geometry plays a decisive role.
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Talk 08 — Ruxandra Moraru (University of Waterloo, Canada) — [computing…]Local time: click here to view in your time zoneTitle: New examples of compact holomorphic symplectic manifolds
Abstract
A holomorphic symplectic manifold is a complex manifold $X$ together with a closed, non-degenerate holomorphic 2-form $\Omega$. The top power of $\Omega$ gives a trivialisation of the canonical bundle so that $X$ has trivial first Chern class. In the context of K\"ahler geometry, such manifolds play a very important role due to the Bogomolov covering theorem, which states that any compact K\"ahler manifold with vanishing first Chern class has a covering that splits as the product of Calabi–Yau manifolds, complex tori and irreducible holomorphic symplectic manifolds. Among these, the last two are, in fact, compact holomorphic symplectic manifolds. Furthermore, irreducible holomorphic symplectic manifolds correspond to compact hyperkahler manifolds in the K\"ahler setting. In general, finding compact holomorphic symplectic manifolds is very difficult. In this talk, I will present new examples of compact holomorphic symplectic manifolds. These manifolds correspond to moduli spaces of sheaves on Kodaira surfaces and are non-K\"ahler.
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Talk 09 — Casey Blacker (Augusta University, USA) — [computing…]Local time: click here to view in your time zoneTitle: A Cartan theorem for simplicial Lie groups
Abstract
A classical result of Cartan establishes an isomorphism between the left-invariant de Rham algebra on a Lie group G and the Chevalley–Eilenberg algebra of Lie(G). In this talk, we discuss a generalization of this result to simplicial Lie groups. Specifically, we describe a relation between the totalization of the left-invariant normalized Bott–Shulman–Stasheff double complex of a simplicial Lie group G and the Chevalley–Eilenberg algebra of the simplicial Lie algebra Lie(G). This is joint work with Ethan Clelland and Michael Merkle.
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Talk 10 — Discussion — [computing…]Local time: click here to view in your time zoneTitle: Discussion
Abstract
This slot is for discussion.
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Talk 11 — Harish Seshadri (Indian Institute of Science, Bangalore, India) — [computing…]Local time: click here to view in your time zoneTitle: On Yau's uniformisation conjecture - Part 1
Abstract
The classical uniformization theorem says that every simply connected Riemann surface is isomorphic to either the Riemann sphere, the complex plane or the unit disc. In the compact case, these are characterized geometrically as Riemann surfaces admitting conformal metrics with constant positive, zero and negative curvature respectively. In the non-compact case too, there is a similar geometric characterization of simply connected Riemann surfaces. Yau's uniformization conjecture is a vast generalization of this classical circle of ideas to higher dimensions. The conjecture states that any complete, non-compact Kahler manifold with positive bisectional curvature is biholomorphic to the standard n-dimensional complex Euclidean space. In this series of three talks, we will report on some recent progress on the conjecture on Kahler surfaces with positive sectional curvature.
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Talk 12 — Vamsi Pritham Pingali (Indian Institute of Science, Bangalore, India) — [computing…]Local time: click here to view in your time zoneTitle: On Yau's uniformisation conjecture - Part 2
Abstract
The classical uniformization theorem says that every simply connected Riemann surface is isomorphic to either the Riemann sphere, the complex plane or the unit disc. In the compact case, these are characterized geometrically as Riemann surfaces admitting conformal metrics with constant positive, zero and negative curvature respectively. In the non-compact case too, there is a similar geometric characterization of simply connected Riemann surfaces. Yau's uniformization conjecture is a vast generalization of this classical circle of ideas to higher dimensions. The conjecture states that any complete, non-compact Kahler manifold with positive bisectional curvature is biholomorphic to the standard n-dimensional complex Euclidean space. In this series of three talks, we will report on some recent progress on the conjecture on Kahler surfaces with positive sectional curvature.
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Talk 13 — Ved Datar (Indian Institute of Science, Bangalore, India) — [computing…]Local time: click here to view in your time zoneTitle: On Yau's uniformisation conjecture - Part 3
Abstract
The classical uniformization theorem says that every simply connected Riemann surface is isomorphic to either the Riemann sphere, the complex plane or the unit disc. In the compact case, these are characterized geometrically as Riemann surfaces admitting conformal metrics with constant positive, zero and negative curvature respectively. In the non-compact case too, there is a similar geometric characterization of simply connected Riemann surfaces. Yau's uniformization conjecture is a vast generalization of this classical circle of ideas to higher dimensions. The conjecture states that any complete, non-compact Kahler manifold with positive bisectional curvature is biholomorphic to the standard n-dimensional complex Euclidean space. In this series of three talks, we will report on some recent progress on the conjecture on Kahler surfaces with positive sectional curvature.
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Talk 14 — Ritwik Mukherjee (National Institute of Science Education and Research (NISER), Bhubaneswar, India) — [computing…]Local time: click here to view in your time zoneTitle: A fibre bundle version of Gromov-Witten invariants
Abstract
We consider a very concrete question in enuemrative geometry: how many genus zero degree d curves are there in CP^2, passing through 3d+1-m generic points that have an m-fold point? This question was solved earlier using excess intersection theory for m=3. It was practically unapporachable for higher values of m. We describe a solution to this problem by developing a fibre bundle version of Gromov-Witten invariants and solve this problem for all m.
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Talk 15 — Sam K Mathew (PhD Student, ICTS Bangalore India) — [computing…]Local time: click here to view in your time zoneTitle: Height Function Transformations of Minimal Surfaces
Abstract
In this talk, I present a complete classification of minimal graph surfaces that admit transformations of their height functions into new minimal graph surfaces. While trivial transformations such as translations and reflections are well-understood, the existence of non-trivial cases has remained largely unexplored. I define this as the Non-Trivial Minimal Graph Transformation Problem, which I reduce from a coupled system of nonlinear PDEs to a single problem involving a harmonic function. By employing a complex variable approach and a weakening technique, the analysis is further simplified to an ODE dependent on a real parameter k. I explicitly solve this ODE for all k using elliptic integrals and identities, providing a full classification of admissible transformations and yielding several families of minimal surfaces which, to the best of my knowledge, are new.
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Talk 16 — Vasilisa Shramchenko (Université de Sherbrooke, Canada) — [computing…]Local time: click here to view in your time zoneTitle: Deformations of Hill and Toda algebraic curves
Abstract
Hill and Toda hyperelliptic curves can be defined using properties of some appropriate meromorphic differentials on the associated compact Riemann surfaces. We define particular families of hyperelliptic curves over which these properties do not change. It turns out that such families can be described by differential equations for the parameters of the algebraic curves, and these equations have a rational structure. The obtained families of curves have a close relationship with algebro-geometric solutions to the KdV and Toda equations, respectively.