Weekly Bulletin

Moduli spaces often admit several natural compactifications. The comparison of different compactifications is referred to as wall-crossing. Some compactifications are simpler than others, making wall-crossing one of the most powerful established techniques, for example, in the study of moduli spaces of sheaves. I will discuss wall-crossing phenomena for moduli spaces of maps (Gromov–Witten theory) and other related spaces. The first glimpses of wall-crossing in Gromov–Witten theory go back to its beginnings, namely to Givental’s proof of mirror symmetry for the quintic in 1990s. Later, independently of Givental’s work, Manin and Bayer were the first to explore the idea of wall-crossing in the setting of maps, extending Hassett’s ideas on weighted marked curves. Only with the work of Ciocan-Fontanine, Kim, and Maulik from 2010s, which in turn builds on the work of many others, did it become possible to exploit different compactifications of moduli spaces of maps with considerable flexibility for GIT quotients, thereby revealing Givental’s proof as a wall-crossing calculation. Zhou then made the next major step by proving the Ciocan-Fontanine–Kim wall-crossing formulas for GIT quotients in full generality. The goal is to explain how the ideas of Ciocan-Fontanine, Kim, and Zhou can be applied in three different settings: configuration spaces of points, maps to GIT quotients, and Hurwitz theory. Based on these examples, I will try to formulate some general principles underlying such wall-crossing phenomena. The final goal is to combine all three settings to provide a wall-crossing between spaces of maps and Hilbert schemes associated to threefolds of the form Surface × Curve.

<p>DNA-based archival storage is vulnerable to synthesis and sequencing errors driven by long homopolymer runs, GC-content imbalance, and secondary-structure formation. At the same time, codeword design must also guard against nonspecific hybridization, captured by the Reversible (R) and Reversible-Complement (RC) constraints. In this talk, we present an algebraic framework that satisfies all of these requirements simultaneously. DNA block sets \(\mathcal{A} \subseteq \Sigma_{\text{DNA}}^t\) are built by an incremental algorithm that admits a candidate block only if no occurrence of it, within any four-block concatenation, creates a disjoint pair of Secondary-Complement/Reverse-Secondary-Complement (SC/RSC) substrings of length \(\ell\); this local, checkable test on 4-fold concatenations suffices to guarantee that arbitrarily long concatenations of \(\mathcal{A}\) remain \((\ell + 1)\)-free-structure. Specializing this construction to \(t = 2\), \(\ell = 3\) yields the four-element block set \(\mathcal{A}_4 = \{AC, CA, TC, CT\}\), whose concatenations are further shown to be 3-free-homopolymer and exactly GC-balanced. A bijective, distance-preserving map \(\psi : \mathbb{Z}_4 \to \mathcal{A}_4\) then lifts linear codes over \(\mathbb{Z}_4\) into DNA codes, with a simple generator-matrix symmetry condition guaranteeing the R and RC constraints. Using this map, four families are constructed: Modified Simplex DNA codes (Types 1 and 2), Modified Hamming DNA codes, and Reed--Muller-type DNA codes, each with non-vanishing rate or relative minimum distance as the length grows, and several members meeting the quaternary Singleton and Gilbert--Varshamov bounds.</p>

We study space-time Galerkin--Petrov formulations for parabolic evolution problems and their relation to classical implicit time-stepping schemes. Although such schemes are stable in the usual time-stepping sense, their interpretation as space-time operator equations may lead to conditional stability, with constants depending on the relation between temporal and spatial mesh sizes. We revisit this phenomenon for the continuous Galerkin method of Aziz and Monk, which yields the Crank--Nicolson scheme in the lowest-order case, and provide a detailed space-time error analysis for solutions of both high and low regularity. In particular, the space-time framework allows us to analyze the deteriorated behaviour of classical time-stepping methods for nonsmooth initial data. By applying integration by parts in time, we derive an adjoint space-time formulation that incorporates the initial condition in a natural variational way. In the lowest-order case, this formulation leads to a Rannacher-type smoothing of the initial data. The theoretical results are complemented by numerical experiments. Joint work with R. Löscher and M. Reichelt.

We discuss a class of graph embeddings into Minkowski space R^{2,2} = C^{1,1}, called t-surfaces, which arise in the study of the planar dimer model. A t-surface consists of a (perfect) t-embedding together with its associated origami map. Perfect t-embeddings were recently introduced as a key tool for proving that the gradient of the dimer height function converges to that of the Gaussian Free Field in a canonically associated metric, under suitable technical assumptions. After introducing the notion of a t-embedding, we describe a construction of perfect t-embeddings for regular hexagons of the hexagonal lattice. For which the corresponding t-surfaces converge to space-like maximal surfaces in Minkowski space R^{2,1}. As a consequence, this construction yields a new proof of the convergence of fluctuations of the dimer height function to the Gaussian Free Field in the conformal structure induced by the limiting maximal surface.

More information: https://eth-its.ethz.ch/activities/its-chalk-cheese-2026-2027.htmlcall_made

Stably exotic fillings of a 3-manifold Y are 4-manifolds bounding Y that are homeomorphic but not stably diffeomorphic rel. Y. We show that every closed, orientable 3-manifold admits such fillings, as do certain families of nonorientable 3-manifolds. In contrast we show that for a 3-manifold containing a 2-sided RP², any two smooth, homeomorphic fillings are stably diffeomorphic. This is joint work with Patrick Orson, Mark Powell, and Arunima Ray.

<p>In this talk, we follow in the footsteps of Leonardo da Vinci. He observed that air bubbles can spiral as they rise through water and described this motion in his notes and sketches. We revisit this phenomenon in an idealised fluid model: the three-dimensional, two-phase incompressible Euler equations with surface tension.</p> <p>We construct bubbles and drops whose centroids trace helical paths, with irrotational flow in each fluid phase. These solutions are steady in a frame translating along and rotating about a fixed axis. The proof is a symmetry-breaking bifurcation argument around the stationary spherical solution. I will explain the main ideas behind the construction and discuss a complementary rigidity result highlighting the role of surface tension.</p> <p>This is joint work with Björn Gebhard and Christian Seis.</p>

In this talk, we consider a time-homogeneous d-dimensional diffusion with no drift coefficient. We give a necessary and sufficient condition for the convexity in the starting position of the distribution of its solution on the path-space when the codomain is endowed with the convex order. To do so, we introduce and check the seemingly stronger notion of coupling-convexity in the starting position.