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In this talk,we will introduce the Yang-Mills-Higgs-Schrödinger(YMHS) flow, which is the infinite dimensional Hamiltonian flow of the Yang-Mills-Higgs functional defined on a holomorphic fiber bundle. It originates from the Schrödinger-Chern-Simons system and natually extends the Schrödinger map flow introduced by Uhlenbeck-Terng and Ding-Wang to a gauged setting. We will discuss its geometric structures and also local well-posedness. This is a joint work with Bo Chen.

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The L_p Minkowski problem concerns the existence and uniqueness of convex bodies with prescribed L_p-surface-area measures. In this talk, we present our recent work on the uniqueness of S₂-isotropic solutions to the isotropic L_p Minkowski problem for super-critical exponents. Title: Numerics for harmonic 1-forms on real loci of Calabi-Yau manifolds

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Calabi-Yau manifolds are manifolds with a unique Ricci-flat metric. Even though existence is known in many cases, no explicit formulae for these metrics are known. That frequently causes problems when one wants to compute things that depend on the metric, in particular in Physics. One example in maths is the following: does there exist a harmonic 1-form on a real locus of a Calabi-Yau manifold that is nowhere vanishing? No example is known. In the talk I will explain a conjectural example and an interesting non-example. To define the manifolds, some real algebraic geometry is used. We then numerically (approximately) solve the Ricci-flat equation and the harmonic 1-form equation. It turns out that a neural network is good at that and the approximate solution is easily interpretable. This is based on arXiv:2405.19402, which is joint work with Michael Douglas, Yidi Qi, and Rodrigo Barbosa. Time permitting, I will comment on ongoing efforts to turn this into a numerically verified proof that there is a genuine solution near our approximate solution.

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From Monge–Ampère to Special Lagrangians Abstract: Hessian-type equations—such as the complex Monge–Ampère equation, the deformed Hermitian–Yang–Mills/Leung–Yau–Zaslow equation, and the J-equation—play a central role in com- plex and differential geometry. In this talk, I will introduce a new ansatz that transforms these nonlinear PDEs into solvable systems of ordinary differential equations. This approach yields broad families of explicit solutions, including entire special Lagrangian and dHYM solutions of arbitrary subcritical phase.
I will also highlight geometric features of these solutions, such as singular- ity formation and their connections with known complete special Lagrangian submanifolds. The goal is to give an accessible introduction to these construc- tions and to explain how explicit models can shed light on broader problems in geometric analysis. This is joint work with Professor Chung-Jun Tsai and Professor Mu-Tao Wang.

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Classical pluripotential theory was introduced in the 1940's, and found immediate application in complex analysis. In the early 2000's, Har-vey and Lawson showed that both pluripotential theory and certain of these applications make sense in a much broader geometric context. Starting with Calabi, however, it has become clear that pluripotential theory and related results, such as Hodge Theory and the de-debar lemma, are central also to Kahler geometry. We will discuss how parts of this "second life" of pluripo- tential theory also extend to other geometries, hinting towards new research directions regarding calibrated geometry and manifolds with special holonomy. This talk, aimed at non-specialists, will be based on joint work with A. Raffero (Univ. of Torino).

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We establish a priori interior curvature estimates for the special La-grangian curvature equations in both the critical phase and convex cases. The supercritical case, however, is distinct from the special Lagrangian equations. In dimension two, we observe that this curvature equation is equivalent to the equation arising in the optimal transportation problem with a "relative heat cost" function, as discussed in Brenier's paper. When 0 < θ < π/2 (supercriti-cal phase), the equation violates the Ma-Trudinger-Wang condition. However, Loeper's counterexample for general optimal transport problems does not di-rectly apply here, as this concerns a specific optimal transport problem with fixed density functions. Moreover, the interior gradient estimates for this cur-vature equation are simpler than those for the special Lagrangian equations. We have demonstrated that these gradient estimates also hold for subcritical phases. It is worth noting that for the special Lagrangian equation, partic-ularly in subcritical phases, the interior gradient estimate remains an open problem. This is joint work with Xingchen Zhou.

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Conifold transitions are a mechanism in which a Calabi-Yau 3-fold is deformed into another by contracting curves and smoothing out the resulting conical singularities. It is fantasized that all Calabi-Yau 3-folds can be linked by a sequence of these transitions, however they do not preserve the Kähler condition. In this talk, I will discuss a string-theoretic generalization of the (Ricci-flat) Kähler condition and a proposed method to obtain these structures known as the Anomaly flow. In particular, I will touch upon results that concern the geometrization of conifold transitions and another that determines whether we can extend the Anomaly flow past a certain interval. This is based in part on joint work with B. Friedman and S. Picard.

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3D mirror symmetry posits that certain algebraic symplectic varieties arise in dual pairs, with "equivariant" and "Kahler" data being swapped under the duality. We explain an arithmetic aspect of 3D mirror symmetry, based on the formulation of isomorphism of certain D-modules in positive characteristic involving the quantization of the symplectic variety on one side and Gromov--Witten theory on the other side. The isomorphism is enriched by the actions of "large centers" in quantizations a la Bezrukavnikov--Kaledin and deformations of Steenrod operations on respective D-modules. Joint with Shaoyun Bai.

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In this talk, we will first explain the generic characteristics of contact Hamiltonian dynamics, and explain its quantitative aspects in relation to thermondynamics and contact topology. We then discuss geometric analysis of a new analytical machinery of contact instantons and its Hamiltonian perturbations, and several applications thereof to the study of the quantitative contact topology and others.

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In this joint work with Xingzhe Li, we prove the strong Morse inequalities for the area functional in the space of embedded tori and spheres in 3-spheres. As a consequence, we prove that in the three dimensional sphere with positive Ricci curvature and bumpy metrics, it contains at least 9 distinct embedded minimal tori with index 5,6,6,7,7,7,8,8,9. The proof relies on a multiplicity one theorem for the Simon-Smith min-max theory proved by the myself and X. Zhou.

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In this talk, we prove Bombieri-de Giorgi-Miranda-type gradient estimate for anisotropic minimal graphs with free boundary. As a result, we prove that any anisotropic minimal graph over half-space with free boundary must be flat, provided that the graph function has at most one-sided linear growth. This yields a similar result for minimal graphs with capillary boundary. This is joint work with Guofang Wang, Wei Wei and Xuwen Zhang.

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WKB analysis is a kind of asymptotic analysis for solutions to certain ODE on a Riemann surface with a parameter.
The Stokes curves living in the Riemann surface play important roles in the WKB analysis.
In this talk, I will discuss some aspect of the Stokes curves in the WKB analysis from the point of view of Lagrangian Floer theory.
This is ongoing joint work with Tatsuki Kuwagaki.

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Very stable Higgs bundles were introduced by Hausel and Hitchin motivated by the study of certain mirror symmetry phenomena in the moduli spaces of Higgs bundles over a smooth projective complex curve. They arise among the fixed points of the natural scaling action and their Białynicki-Birula attracting cells provide examples of closed BAA-branes which serve as simple candidates for the study of their mirror brane. After reviewing these aspects in the context of G-Higgs bundles for a semisimple complex Lie group G, we will explain how to classify very stable Higgs bundles with a generically regular Higgs field using certain Hecke modifications parametrised via affine Grassmannians. In the process we will see that these ideas adapt naturally to moduli spaces of parabolic Higgs bundles.

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In joint work with Patrik Coulibaly, we show that Chekanov's exotictori are not Hamiltonian stationary

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In this talk, I will present a weak compactness theorem for the moduli space of closed Ricci flows with uniformly bounded entropy, each equipped with a natural spacetime distance, under pointed Gromov-Hausdorff convergence. A structure theory for the corresponding Ricci flow limit spaces and some characterizations of the singular sets are given, together with applications to four-dimensional Ricci flows. This is joint work with Hanbing Fang.

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In complex geometry, the Gauss-Manin connection for a family $X$ over $S$ can be described by the sheaf $\Omega^*_{X/S}$ of holomorphic de Rham forms. Motivated by mirror symmetry, we look for an A-model analogue using the framework of relative symplectic cohomology developed by U. Varolgunes. Taking a pre-quantum line bundle $L = O(D)$ over $M$ whose curvature is the symplectic form with fiberwise $S^1$ action, we consider the $S^1$-equivariant Hamiltonian Floer theory on the dual $L^*$ which comes with a Floer Gysin sequence. We construct a chain homotopy between the Floer-theoretic quantum connection, defined as in the work of P. Seidel and D. Pomerleano, and the connecting homomorphism of the Gysin sequence. Afterwards, we can define a relative Floer Gysin sequence on $L^*$. This is a joint work in progress with C.Y. Mak, D. Pomerleano, and U. Varolgunes.

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Existence of free boundary minimal surfaces Mario B. Schulz (based on joint works with Alessandro Carlotto, Giada Franz, Daniel Ketover and David Wiygul)
Free boundary minimal surfaces naturally appear in various contexts, including partitioning problems for convex bodies, capillarity problems for fluids, and extremal metrics for Steklov eigenvalues on manifolds with boundary.
Constructing embedded free boundary minimal surfaces is challenging, especially in ambient manifolds like the Euclidean unit ball, which only allow unstable solutions.
Min-max theory offers a promising avenue for existence results, albeit with the added complexity of requiring control over the topology of the resulting surfaces. This presentation will offer an overview of recent results and applications.

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I will explain how open closed map of Gromov Witten invariant behave with respect to the Lagrangian correspondence and to Kuneth theorem in Lagrangian Floer homology, together its application.

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I will introduce and motivate some of the ideas proposed in the joint work https://arxiv.org/abs/2510.07482 with David Ben-Zvi and Germán Stefanich.

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I will describe a new version of the Hall algebra construction whose output is a (lax) braided monoidal 2-category. This construction can be thought of as a twice-categorified version of the usual Hall algebra story. The resulting object is interesting and rich already in the simplest case of the one-vertex quiver with no loop. Focusing on this example, I will explain why such an object is desirable from the perspective of categorified representation theory and low-dimensional topology. This is joint work in progress with Jonte Goedicke, Yang Hu, and Walker Stern.

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Ricci shrinkers, as critical metrics of Perelman's μ-functional, play a central role in understanding the formation of singularities in the Ricci flow. The rigidity of Ricci shrinkers addresses whether there are nearby Ricci shrinkers in their moduli space. In this talk, I will introduce the deformation theory of compact Ricci shrinkers and discuss the rigidity of non-compact Ricci shrinkers, specifically in the moduli space equipped with the pointed Gromov-Hausdorff topology. As an application, I will also show the uniqueness of the tangent flow for general compact Ricci flows, under the assumption that one of the tangent flows is a generalized cylinder. This talk is based on joint work with Yu Li.

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Given a complex reductive group G and a G-representation N, there is an associated Coulomb branch algebra A_{G,N} defined by Braverman, Finkelberg and Nakajima. In this talk, we discuss a new interpretation of A_{G,N} as the largest subalgebra of the localized equivariant Borel-Moore homology of the affine Grassmannian on which shift operators and their deformations are defined without localization. We give a very general definition of shift operators, and show that if is a smooth semiprojective variety equipped with a G-action, and f is a G-equivariant proper holomorphic map, then the equivariant big quantum cohomology defines a family of closed Lagrangians in the Coulomb branch Spec(A_{G,N}), yielding a transformation of 3d branes in 3d mirror symmetry. This is a joint work with Kwokwai Chan and Ki Fung Chan.