Title:   Optimal transport and positive energy conditions in general relativity (Part 3)
Abstract:  

Title:   Rigidity aspects of a cosmological singularity theorem
Abstract:   In 2018 Galloway and I established the following cosmological singularity theorem: If a spacetime satisfying the null energy condition contains a compact Cauchy surface with a positive definite second fundamental form (i.e., it’s expanding in all directions), then the spacetime is past null geodesically incomplete unless the Cauchy surface is a spherical space (i.e., a quotient of the three-sphere). In this talk, some rigidity results of this singularity theorem will be addressed. Namely, we aim to answer the question: if the second fundamental form is only positive semidefinite and the spacetime is past null geodesically complete, then what can be said about the global structure of the Cauchy surface? In this case we show that the Cauchy surface (or a finite cover thereof) is necessarily a surface bundle over a circle with totally geodesic fibers. Our results make use of the positive resolution of the virtual positive first Betti number conjecture, which is a consequence of Agol’s proof of the virtual Haken conjecture. We also identify several instances when passing to a cover is unnecessary. Lastly, we can relax the assumption on the second fundamental form provided the spacetime contains a U(1) isometry group. Several examples of our results will be discussed. This is joint work with Carl Rossdeutscher, Walter Simon, and Roland Steinbauer.

Title:   Regular Lagrangian Flows, 20 years later
Speaker:   Luigi Ambrosio
Abstract:   In this talk I will review the theory of Regular Lagrangian Flows, that allows to establish well posedness results for ODE's first in the case when the ambient space is smooth but the velocity field is not, then when even the ambient space is not smooth (under appropriate synthetic regularity assumptions on the metric measure structure).
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Title:   Harmonic Maps into Euclidean Buildings
Speaker:   Chikako Mese
Abstract:   We study harmonic maps into Euclidean buildings, allowing targets that are not necessarily locally finite. Our main technical result establishes that such maps have singular sets of Hausdorff codimension at least two, thereby extending the regularity theory of Gromov and Schoen to this broader setting. As an application, we obtain a superrigidity theorem for algebraic groups over fields with non-Archimedean valuations. This generalizes the rank-one p-adic superrigidity results of Gromov and Schoen and places the Bader–Furman extension of Margulis’ higher-rank superrigidity theorem in a geometric framework.
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Title:   On the boundary regularity of solutions to the oriented Plateau problem
Speaker:   Reinaldo Resende
Abstract:   We will examine the state of the art in both interior and boundary regularity for solutions to the oriented Plateau problem, specifically in the framework of integral currents. After reviewing recent developments in interior regularity, we will turn to the boundary setting, where we will discuss the most recent and optimal results on boundary regularity, with an emphasis on connected boundaries. This talk is based on joint work with Ian Fleschler.
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Title:   Minimal surfaces in the infinite-dimensional sphere
Speaker:   Antoine Song
Abstract:   The study of minimal surfaces in spheres is a classical topic. Minimal surfaces in spheres, which are also invariant under group actions, are intimately connected to fundamental results in Geometry and Topology. For instance, minimal surfaces in spheres yield one of the simplest proofs of Mostow's rigidity theorem in hyperbolic geometry, and give a variational interpretation of the hyperbolization theorem of Thurston-Perelman. I will explain this viewpoint and how it leads to a conjecture on the existence of n-dimensional minimal surfaces in the infinite dimensional Hilbert sphere invariant by a large group action. Our approach to this conjecture is to study a Plateau problem in infinite dimension. I will then review partial results in this direction, which are based on the theory of metric currents.
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Title:   Decomposition of integral metric currents
Speaker:   Giacomo Del Nin
Abstract:   In the setting of complete metric spaces, we prove that integral currents can be decomposed as a sum of indecomposable components. In the special case of one-dimensional integral currents, we also show that the indecomposable ones are exactly those associated with injective Lipschitz curves or injective Lipschitz loops, therefore extending Federer's characterisation to metric spaces. Time permitting, some applications of our main results will be discussed. The talk is based on a joint work with Paolo Bonicatto (Trento) and Enrico Pasqualetto (Jyväskylä).
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Title:   Trading linearity for ellipticity - a novel approach to global Lorentzian geometry
Speaker:   Nicola Gigli
Abstract:   The concepts of Sobolev functions, elliptic operators and Banach spaces are central in modern geometric analysis. In the setting of Lorentzian geometry, however, unless one restricts the attention to Cauchy hypersurfaces these do not have a clear analogue, due to the signature of the metric tensor. Aim of the talk is to discuss some recent observations in this direction centered around the fact that for $p<1$ the $p$-D’Alambertian is elliptic on the space of time functions.
The talk is mostly based on joint project with Beran, Braun, Calisti, McCann, Ohanyan, Rott, Saemann.
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Title:   The mass angular momentum inequality
Abstract:   In this talk, I will discuss our recent result. Either there is a counterexample to black hole uniqueness, or the following statement holds. Axisymmetric, complete, simply connected, maximal initial data sets for the Einstein equations of nonnegative energy density with ends that are either asymptotically flat or asymptotically cylindrical, admit an ADM mass lower bound given by the square root of total angular momentum. Moreover, equality is achieved only for a constant time slice of an extreme Kerr spacetime. The proof is based on a novel flow of singular harmonic maps with hyperbolic plane target, under which the renormalized harmonic energy is monotonically nonincreasing. Relevant properties of the flow are achieved through a refined asymptotic analysis of solutions to the linearized harmonic map equations. This is joint work with Qing Han, Marcus Khuri, and Jingang Xiong.

Title:   Minimizing surfaces in the Heisenberg group
Speaker:   Robert Young
Abstract:   The Heisenberg group is the simplest noncommutative nilpotent group, and a key example of sub-Riemannian geometry. Since a construction of Pauls in the early 2000s, it has been known that perimeter-minimizing surfaces in the Heisenberg group need not be smooth. Indeed, there are perimeter-minimizing surfaces in the Heisenberg group whose singular set has positive measure. In this talk, we will introduce the Heisenberg group, survey some results, conjectures, and questions on the regularity and irregularity of minimizing surfaces and explain how to construct your own examples.
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Coffee Tea in Lobby: 3:15pm
Lecture 3:45pm

Prof Didier Queloz: "Exoplanet and Life in the Universe"

Title:   The Spacetime Penrose Inequality with Suboptimal Constant
Speaker:   Brian Allen
Abstract:   In this talk we will discuss the spacetime Penrose inequality formulated in the context of initial data sets. By taking advantage of the Jang equation, spacetime harmonic functions, and an inequality established by Conghan Dong and Antoine Song, we are able to give a proof of the spacetime Penrose inequality with a suboptimal constant. This is joint work with Edward Bryden, Demetre Kazaras, and Marcus Khuri.
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Title:   Curvature Lower Bounds and Orientability
Speaker:   Elia Brue
Abstract:   In this talk I will discuss orientability of nonsmooth spaces with Ricci curvature bounded below. I will present new characterizations of Honda's notion of orientability for Ricci limit spaces, a stability result, and some applications.
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Title:   Coarse tangent fields
Speaker:   Sylvester Eriksson-Bique
Abstract:   Alberti-Csörnyei-Preiss constructed a tangent line field for measure zero subsets of the plane. This is an assignment of a tangent line to each point of the set in such a way, that every rectifiable curve is tangent to this direction a.e. where it intersects the set. In other words, they ``guess'' correctly the tangent line of every curve at almost every point. In this talk, I will discuss a quantitative, and higher dimensional, version of this field, and explain how even general doubling subsets of Hilbert space possess a coarse tangent field with dimension controlled by the Nagata dimension of the subset. The quantitative version is motivated by the Jones traveling salesman theorem, and may help in constructing interesting embeddings of the space to Euclidean space. This is joint work with Raanan Schul and Guy C. David.
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Title:   Positive Scalar Curvature and Convergence of Manifolds
Speaker:   Paul Sweeney
Abstract:   Does a smooth limit of a sequence of closed Riemannian manifolds with nonnegative scalar curvature necessarily have nonnegative scalar curvature? In this talk, I will address this question for intrinsic flat convergence. I will present results showing that the answer is negative in all dimensions n≥3. In fact, I will describe how curvature can change drastically under intrinsic flat convergence. The main result produces a sequence of manifolds with uniformly positive scalar converging to a smooth manifold with negative Ricci curvature. We will also compare this result with the results of Lee and Topping in dimensions n≥4 and of Kazaras and Xu in dimension n=3. This is based on joint work with Jared Krandel.
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Title:   Capacity, semicontinuity, and general relativistic mass
Speaker:   Jeff Jauregui
Abstract:   Capacity, which uses harmonic functions to understand the "size" of a set, has found many applications in the study of asymptotically flat manifolds with nonnegative scalar curvature and their total ("ADM") mass. We will discuss some of these connections and provide intuition for how the capacity functional ought to behave with respect to the background metric. Joint work with R. Perales and J. Portegies will be presented on establishing upper semicontinuity of the capacity for local integral current spaces converging in the pointed Sormani--Wenger intrinsic flat sense. We will conclude with a discussion of total mass in the low regularity setting.
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Title:   Uniformly rectifiable metric spaces
Speaker:   Raanan Schul
Abstract:   In their 1991 and 1993 foundational monographs, David and Semmes characterized uniform rectifiability for subsets of Euclidean space in a multitude of geometric and analytic ways. The fundamental geometric conditions can be naturally stated in any metric space and it has long been a question of how these concepts are related in this general setting. In joint work with D. Bate and M. Hyde, we prove their equivalence. Namely, we show the equivalence of Big Pieces of Lipschitz Images, Bi-lateral Weak Geometric Lemma and Corona Decomposition in any Ahlfors regular metric space. Loosely speaking, this gives a quantitative equivalence between having Lipschitz charts and approximations by nice spaces. After giving some background, we will explain the main theorems and outline some key steps in the proof (which will include a discussion of Reifenberg parameterizations). We will also mention some open questions.
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Title:   From almost smooth spaces to RCD spaces
Speaker:   Shouhei Honda
Abstract:   We provide various characterizations for a given almost smooth space to be an RCD space. Applications include a characterization of Einstein 4-orbifolds. This talk is based on a joint work with Song Sun (Zhejiang University).
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Title:   Reduction Arguments via Asymptotic Correspondence of Initial Data Sets
Speaker:   Anna Sancassani
Abstract:   Motivated by the work of Cha and Khuri, who proposed a strategy to prove the positive mass theorem and other geometric inequalities in the asymptotically AdS-hyperbolic setting, we studied the effects of the correspondence between constant mean curvature (CMC) initial data sets and charges defined on pairs of initial data with corresponding backgrounds. I will introduce a definition of asymptotically corresponding initial data sets and prove a result concerning the associated geometric invariants, as formulated by Michel.
These findings enable us to simplify the strategy proposed by Cha and Khuri and prove a positive mass theorem on asymptotically AdS-hyperbolic initial data, under certain additional conditions. If time permits, I will discuss potential methods to remove these supplementary conditions. This analysis simplifies the above-cited approach and shows how these reduction arguments can be generalized beyond the asymptotically hyperbolic case.
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Title:   Rigidity of mass-preserving 1–Lipschitz maps from integral current spaces into Euclidean space
Speaker:   Raquel Perales
Abstract:   We will prove that given an n–dimensional integral current space and a 1–Lipschitz map, from this space onto the n–dimensional Euclidean ball, that preserves the mass of the current and is injective on the boundary, the map has to be an isometry. We deduce as a consequence the stability of the positive mass theorem for graphical manifolds as originally formulated by Huang–Lee–Sormani. (Joint work with G. Del Nin.)
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Title:   TBA
Speaker:   Colin Defant [Harvard University]

Abstract:   TBA
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Title:   Stability for a class of three-tori with small negative scalar curvature
Speaker:   Edward Bryden
Abstract:   We define a flexible class of Riemannian metrics on the three torus. Then, using Stern's inequality relating scalar curvature to harmonic one-forms, we show that any sequence of metrics in this family whose negative part of the scalar curvature tends to zero in $L^{2}$ norm has a subsequence which converges to some flat metric on the three-torus in the sense of Dong-Song.
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Title:   Mass Lower Bounds for asymptotically locally flat 4-manifolds
Speaker:   Marcus Khuri & Jian Wang
Abstract:   The mass is a fundamental global geometric invariant with deep connections to scalar curvature. In this talk, we will present the mass for asymptotically locally flat (ALF) 4-manifolds and establish the corresponding mass inequality. Specifically, we will talk about how the topology at infinity influences the mass within the ALF setting.
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Title:   Gromov's dihedral rigidity and index theory on polyhedra
Speaker:   Jinmin Wang
Abstract:   In this talk, I will present an index-theoretic approach that yields an affirmative answer to Gromov's dihedral rigidity conjecture and outline the key ideas of the proof. As a corollary, we obtain a positive answer to the Stoker problem, showing that the dihedral angles of a convex polyhedron determine its face angles. One of the central ingredients is the resolution to a variant of the angle-shrinking conjecture proposed by Gromov. This talk is based on joint work with Zhizhang Xie and Guoliang Yu.
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Title:   From linear potential theory to the inverse mean curvature flow: monotonicity formulas
Speaker:   Alessandra Pluda
Abstract:   In this talk we consider a family of monotonicity formulas which originate from seemingly different contexts in geometric analysis. On the one side, we have monotonicity formulas in the framework of linear potential theory, tracing back to the work of Colding (“New Monotonicity Formulas for Ricci Curvature and Applications. I”, 2012) and Colding-Minicozzi (“On Uniqueness of Tangent Cones for Einstein Manifolds”, 2014). On the other side, we have the monotonicity of the Hawking mass along the inverse mean curvature flow, proved by Huisken and Ilmanen (“The Inverse Mean Curvature Flow and the Riemannian Penrose Inequality”, 2001).
The connection between these two formulas lies in monotonicity formulas of geometric quantities for the level sets of solutions to the p-Laplacian (formally the IMCF corresponds to the case p=1). These general formulas require some regularity of the level sets of solutions: we show that almost every level set is a curvature varifold for which a Gauss-Bonnet-type theorem is established. To prove the convergence to the inverse mean curvature flow we show that, as p → 1^+, almost every level sets converge in the sense of curvature varifolds and gradients strongly converge in L^q for every finite q. Our monotonicy formulas imply several geometric inequalities, from the Willmore and the p-Minkowski inequalities to the Riemannian Penrose inequality."
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Title:   From linear potential theory to the inverse mean curvature flow: applications to Riemannian Penrose-type inequalities
Speaker:   Luca Benatti
Abstract:   The Riemannian Penrose inequality asserts that the total mass of a time-symmetric spacetime is at least as large as the mass of the black holes it contains. Among the known proofs, two rely on monotonicity formulas arising from distinct theoretical frameworks: the weak inverse mean curvature flow developed by Huisken and Ilmanen (2001), and the nonlinear potential theory approach by Agostiniani, Mantegazza, Mazzieri, and Oronzio (2025).

This talk builds on the new unified perspective on these monotonicity formulas, highlighted in a recent joint work with A. Pluda and M. Pozzetta. This new point of view allows us to remove the additional assumptions required in previous proofs, reducing them to the minimal hypotheses needed for the formulation of the inequality itself. To explore a broader setting, our journey will also lead us through various notions of mass — from the isoperimetric mass introduced by Huisken to the isocapacitary mass defined by Jauregui — and their relationships in light of this unified prespective.
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Title:   Towards a definition of energy and its positivity in spacetimes with an expanding flat de Sitter background
Speaker:   Anachiara Piubello
Abstract:   The positive energy theorems serve as a fundamental pillar in the geometric development of general relativity. Proved by Schoen and Yau, and later by Witten, the original positive energy theorems are stated for asymptotically flat manifolds with either time-symmetric initial data or data whose second fundamental form decays to zero at infinity. This ansatz on the metric and second fundamental form is traditionally motivated by a desire to model an "isolated gravitational system." However, actual astrophysical massive objects are not truly isolated but exist within an expanding cosmological universe. In this talk, we present a definition of energy in the context of such an expanding universe. In this approach, we take the flat expanding de Sitter model as the background spacetime, in contrast to the more conventional Minkowski spacetime, which forces our definition of energy to be quasi-local due to the presence of a cosmological horizon. We prove positivity of the energy provided the cosmological constant associated with the de Sitter background is not too large. This is joint work with Eric Ling and Rodrigo Avalos.
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