Weekly Bulletin

The Poisson additivity theorem of Rozenblyum–Safronov gives an equivalence between the category of n-disk algebras valued in 1-shifted Poisson algebras and the category of (1-n)-shifted Poisson algebras. Seeing this as a topological statement on a coordinate patch, I will discuss a generalisation of this theorem to a non-topological statement on arbitrary (but non-stratified) spacetimes, building on an approach due to Tamarkin. The key novel notion is that of a factorisation Lie algebra. This is joint work in preparation with Giovanni Canepa and Nils Carqueville.

Let Y be a symplectic divisor of X, \omega. In the Kahler setting, Givental's (closed-string) Quantum Lefschetz formula relates certain Gromov-Witten invariants (encoded by the G function) of X and Y. Given an Lagrangian L in (Y, \omega|Y), we can lift it to a Lagrangian L' in neighbourhood NY \subset X. We will introduce the notion of the potential of a Lagrangian, which encodes information of Maslov index 2 J-holomorphic disks with boundary on it. From the work of Biran-Khanevski, we can extract a formula for when (X,L',Y,L) forms a monotone tuple (we will define this notion), and the minimal Chern number of Y is 2. We generalise the formula in this setting when we allow the minimal Chern number to be 1. We can use this to show the existence of infinitely many Lagrangian tori in CP^n, Quadrics, Cubics, and other symplectic manifolds, among other results. Following the work of Tonkonog on gravitational descendants, we recover an explicit Quantum Lefschetz formula appearing in the work of Coates-Corti-Galkin-Kasprczyk. Interestingly, their formula applies in a different context--specifically when X is toric -- which neither contains nor is contained in the monotone tuple setting. Motivated by this, we introduce an alternative set of hypotheses, typically satisfied when X degenerates to a toric manifold, under which a broader open-string Quantum Lefschetz formula applies. The differences between these sets of hypotheses will be discussed. This is joint work with Luis Diogo, Dmitry Tonkonog and Weiwei Wu.

In this talk, I will discuss some scale-invariant curvature energies depending on the first and second fundamental forms, with an emphasis on the regularity theory. We focus on the Dirichlet energy of the mean curvature for four-dimensional hypersurfaces and present a regularity theorem for weak critical points. The proof uses Noether-type conservation laws to rewrite the Euler–Lagrange equation as a lower-order elliptic system, where tools from integrability by compensation and interpolation theory apply; I will also highlight the additional difficulties compared with the two-dimensional Willmore setting. This is based on a joint work with Yann Bernard, Dorian Martino, and Tristan Rivière.

<p>This is joint work with Jacopo De Simoi and Kasun Fernando. We consider a class of sufficiently smooth partially hyperbolic fast-slow systems on the 2-torus, obtained by a size ε perturbation of a trivial extension of a family of expanding circle maps. Such fast-slow systems obey an averaging principle: at time-scale 1/ε the slow part is approximated by the solution of an ODE. Assuming that this ODE has exactly one sink and both Lyapunov exponents of the system are positive, we prove the system admits a unique physical (SRB) measure. Moreover, we establish exponential decay of correlations, with explicit, nearly optimal bounds on the decay rate.<br>This result provides a ‘mostly expanding’ counterpart to the work of De Simoi and Liverani, who treated such systems in the ‘mostly contracting’ case (i.e., where there is one negative Lyapunov exponent).</p>

We show that the fiber product of the period map t: M_g^(ct) -> A_g for curves and the product map A_(g_1) x ... x A_(g_k) -> A_g for any g = g_1 + ... + g_k is reduced. This opens up the possibility of calculating all pullbacks t^*[A_(g_1) x ... x A_(g_k)] in the Chow ring of M_g^(ct).

While compact oriented connected manifolds of dimension 1 (the circle and the interval) and 2 (genus <i>g</i> surfaces with <i>r</i> boundary components) are readily classified, the study of 3-manifolds is an active research area with many competing perspectives, including the celebrated geometrization program initiated by Thurston. Among 3-manifolds, the fibered ones—those with a regular map to the circle S¹—are arguably the simplest to study, as their properties can be fully described in terms of their monodromy: the gluing self-map of the fiber (a surface) of a chosen regular map to S¹. For example, irreducibility and atoroidality (the topological properties of not containing interesting spheres or tori) and hyperbolicity (the geometric property of featuring a metric with sectional curvature –1) are readily discerned from the properties of the monodromy. Famously, Thurston's hyperbolization criterion says a fibered 3-manifold is hyperbolic if and only if the monodromy is neither reducible nor periodic.<br> Based on joint work with Lewark–Orbegozo Rodriguez and Orbegozo Rodriguez, we describe how to associate a monodromy to any irreducible surface (a so-called Haken surface) Σ in a 3-manifold that need not be a fiber of a regular map. Our setup is chosen to allow for an analog of Thurston's hyperbolization criterion. We illustrate our approach by providing new results concerning irreducibility, atoroidality and hyperbolicity for a particularly visualizable class of 3-manifolds: the exteriors of knots in the 3-sphere. In terms of technology, we use classical decomposition theory and the language of product discs and annuli as pioneered by Gabai to define a notion of monodromy that takes the form of a partially defined self-map of the arc and curve graph of Σ.

Analysis and processing of data is a vital part of our modern society and requires vast amounts of computational resources. To reduce the computational burden, compressing and approximating data has become a central topic. We consider the approximation of labeled data samples, mathematically described as site-to-value maps between finite metric spaces. Within this setting, we identify the discrete modulus of continuity as an effective data-intrinsic quantity to measure regularity of site-to-value maps without imposing further structural assumptions. We investigate the consistency of the discrete modulus of continuity in the infinite data limit and propose an algorithm for its efficient computation. Building on these results, we present a sample based approximation theory for labeled data. For data subject to statistical uncertainty we consider multilevel approximation spaces and a variant of the multilevel Monte Carlo method to compute statistical quantities of interest. We provide extensive numerical studies to illustrate the feasibility of the approach and to validate our theoretical results.

Last and first passage percolation in two dimensions are classical discrete models of random directed planar Euclidean metrics in the KPZ universality class. Their scaling limit is described by the directed landscape of Dauvergne-Ortmann-Virág. Random planar maps are classical discrete models of random undirected planar fractal metrics in the LQG universality class. Their scaling limit is described by the (undirected) LQG metric of Ding-Dubédat-Dunlap-Falconet and Gwynne-Miller. We present recent progress on the study of longest and shortest directed paths in the following natural model of directed random planar maps: the uniform infinite bipolar-oriented triangulation (UIBOT), which is the local limit of uniform bipolar-oriented triangulations around a typical edge and is expected to be in the LQG universality class with γ=√4/3. We first explain the analogies between this model and last and first passage percolation. Then, we construct the Busemann function, which measures directed distance to infinity along a natural interface of the UIBOT. We show that, in the case of longest (resp. shortest) directed paths, this Busemann function converges in the scaling limit to a 2/3-stable Lévy process (resp. a 4/3-stable Lévy process). These results imply that in a typical subset of the UIBOT with n edges, longest directed path lengths are of order n^^{3/4} and shortest directed path lengths are of order n^^{3/8}. We conclude the talk by explaining why these results fit into a program to construct the (longest and shortest) directed LQG metrics, two distinct two-parameter families of random fractal directed metrics which generalize the LQG metric and which could conceivably converge to the directed landscape upon taking an appropriate limit. Based on joint work with E. Gwynne.

This talk will be a double-header. First we will discuss the examples from the above arXiv note (arXiv:2506.18066), singular fibrations over the sphere or disk which (perhaps surprisingly) have no sections. Then we will turn to the speaker's recent work with descriptive set theorist Aristotelis Panagiotopoulos, showing that, in a precise sense, exotic R⁴'s are unclassifiable.

<p><span style="caret-color: #000000; color: #000000; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; display: inline !important; float: none;">In this talk, we explore locality properties of a Schrödinger-type PDE describing the dynamics of a many-body bosonic system on a finite volume lattice. Such locality properties translate into two types of propagation bounds</span><br style="caret-color: #000000; color: #000000; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;"><br style="caret-color: #000000; color: #000000; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;"><span style="caret-color: #000000; color: #000000; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; display: inline !important; float: none;">1. Information propagation bounds (IPB) or Lieb-Robinson bounds, which control the dynamics of local operators.</span><br style="caret-color: #000000; color: #000000; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;"><span style="caret-color: #000000; color: #000000; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; display: inline !important; float: none;">2. Particle propagation bounds (PPB), which control the dynamics of particles through the lattice.</span><br style="caret-color: #000000; color: #000000; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;"><br style="caret-color: #000000; color: #000000; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;"><span style="caret-color: #000000; color: #000000; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; display: inline !important; float: none;">Although the PDE under consideration is linear, we have to face two mathematical challenges. First, the dimension of the underlying Hilbert space is very high and we want to derive bounds that thermodynamically stable, i.e. are uniform in such dimension. Second, bosonic interactions are unbounded in operator norm. Thanks to a multi-scale adaptation of the ASTLO ( adiabatic space-time localisation observables) method, which allows to remove the dependence of the error term on far away particles, we establish the first thermodynamically stable bosonic PPB for a class of long-range interactions. We were also able to control higher moments of the number operator. This opens the door to proving the first thermodynamically stable IPB for such bosonic systems. Establishing these bounds for bosonic systems is a long-standing open problem due to the lack of existing methods to control the unbounded nature of the bosonic operators when deriving  IPB.</span></p>