Importance sampling with data-driven proposal distributions is widely used in practice. A common workflow first generates an auxiliary sample of size $N$ from an approximation of the target distribution, constructs a density estimate $\hat q$ such as a kernel density estimator (KDE), and then draws $n$ importance samples from this learned proposal. Despite its practical relevance, the theoretical properties of this hierarchical procedure remain poorly understood, since classical importance sampling theory assumes a fixed proposal. We address this gap by deriving non-asymptotic error bounds for standard, defensive, and self-normalized importance sampling estimators with random proposals. Our results separate the Monte Carlo error, scaling as $n^{-1/2}$, from the proposal approximation error measured through the mean integrated absolute and squared errors (MIAE and MISE) of $\hat q$. To obtain explicit convergence rates in $(N,n)$, we establish MIAE and MISE bounds for KDEs constructed from geometrically ergodic Markov chains in stationary and non-stationary regimes. Combining these results yields quantitative guarantees for importance sampling with KDE-based proposals. Our theory provides practical guidance for selecting defensive mixture weights in a nonparametric importance sampling framework.

This experimental particle-physics thesis reports the latest developments on the GasPM, a novel gaseous photodetector aimed at suppressing beam-induced backgrounds in the electromagnetic calorimeter for a potential upgrade of the Belle~II experiment. The GasPM technology is based on combining a photocathode with a resistive-plate chamber offering high efficiency, excellent time resolution, and cost-effective scalability. A further advantage is that, combined with a radiator, the GasPM offers precise Cherenkov-based charged-particle identification. As part of a project launched in 2017, this work aims at addressing the degradation in time resolution observed in a previous beam test over what was achieved earlier with laser light. I focus specifically on ultraviolet-photon emission during excitation and de-excitation of the gas molecules, which leads to a secondary signal that in turn spoils time resolution (photon feedback). I design and execute an improved beam test that, along with several configuration changes, newly introduces single-vs-multiple electron discrimination and high-frequency signal readout. In addition, I probe through a cosmic-ray test the quantum efficiency of a new LaB$_6$ photocathode resistant to damage from ions drifting backwards, for use in future beam tests. The principal results are the development of an algorithm to efficiently suppress photon feedback; a preliminary calibration of a novel digitiser; the achievement of discrimination between single- and multiple-electron events; and an early qualification of a LaB$_6$ photocathode. These results are being prepared for showing at the 7th International Workshop on New Photon Detectors organized in Bologna in December 2025 and pave the way for an upcoming beam test of an improved GasPM prototype.

We study the production of the conventional quarkonium states in ultrarelativistic heavy-ion collisions using an open quantum system framework based on the Lindblad equation. Starting from the complex-valued in-medium potential, we derive the dissociation temperature and thermal decay width for each state, and compute their survival probabilities for a system undergoing Bjorken expansion. We then extend the framework to include recombination from thermalized charm and bottom quarks in the quark-gluon plasma, deriving a coalescence model for quarkonia from the Lindblad equation under the adiabatic approximation. The methodology provides a unified, first-principles-inspired description of suppression and recombination for both charmonium and bottomonium.

These are the notes from lectures I gave at the Oberwolfach Seminar "Tensor Triangular Geometry and Interactions" which was held in October 2025. The aim of these notes is twofold: We develop notions of support for triangulated categories, and we apply them to classify thick and localising tensor ideals of categories that arise in modular representation theory of finite groups.

We present the characterization of a photon-number-resolving detection chain based on Silicon photomultipliers (SiPM) coupled to a 14 bit, 1 Gs\s digital acquisition system embedding an FPGA-based signal processing pipeline that performs real-time baseline subtraction, digital deconvolution, and charge integration. Three SiPM models manufactured by Hamamatsu are tested and compared in the mesoscopic intensity regime using both classical coherent states and quantum twin-beam states, enabling a systematic investigation of the effects of pixel pitch, pile-up, and photon detection efficiency on the detector performance.

We study causal optimal transport in continuous time, with Markovian cost, between a finite-state Markov source and a diffusion target. By replacing the source with its conditional law given the observation of the target, we characterize the value of this transport problem through a fully nonlinear parabolic master equation on an enlarged state space. We further show that this value coincides with those of two equivalent stochastic control problems on the simplex: a control of the Kushner--Stratonovich filtering equation with a zero-mean condition, and a state-constrained stochastic optimal control problem. Both formulations give rise to implementable numerical schemes that approximate the value from above and below.

We propose to adapt the confined pseudo-atomic orbitals underpinning the precalculated Slater-Koster (SK) interaction tables in Density Functional Tight Binding (DFTB) to local atomic environments. We demonstrate significant improvement in electronic structure and energetics in the application to a partially oxidized Ni surface and Li insertion into graphite, where we assign optimal SK parameters to metal atoms in different oxidation states. Further analysis reveals the smoothness of the SK integrals across the varying oxidation states. Exploiting this, we introduce a site-resolved machine-learning scheme for fully adaptive DFTB. Using atomic descriptors and simple regression architectures already established in the context of machine-learning interatomic potentials, our scheme achieves 95% band-structure accuracy across all Ni-O binary compositions in the Materials Project.

In this article we study the Golod property of standard graded algebras. We show that determinantal ideals, binomial edge ideals, and permanental ideals are Golod if and only if they have a linear resolution. Next, we give a characterization of when cover ideals define Golod rings, exploiting some considerations on multidegrees of Koszul cycles and Massey products. Finally, we show that squarefree strongly Golod ideals (and, more generally, lcm-strongly Golod ideals) are Golod, and not just weakly Golod.

We investigate the existence of stationary and traveling wave solutions to the 2D Euler equations near the Couette flow in the infinite channel $\mathbb{R} \times [-1,1]$. For Sobolev spaces $W^{s,p}$ or Hölder spaces $C^s$, we identify the index $s= 1+ \frac1p $ as the vorticity regularity threshold separating flexibility from rigidity. Specifically, for any $s<1+ \frac1p$ we prove the existence of $C^\infty$ smooth, compactly supported steady states and traveling waves arbitrarily close to the Couette flow in all $W^{s,p}$ and $C^{1-}$. Conversely, we establish the non-existence of such relative equilibria in $ W^{s,p}$ with $s>1+ \frac1p$ or $C^{1+}$. A notable feature of the variational construction is that these flexible solutions belong to every Gevrey class strictly below the analytic threshold.

Antireflection coatings are ubiquitous in optical systems, where they maximize transmission and suppress undesirable reflections by impedance-matching uniform interfaces. Extending this principle to metasurfaces, however, is fundamentally more challenging because wavefront control requires a library of geometrically distinct meta-atoms, each locally imposing a prescribed phase that is tethered to its transmittance. Here, we show that vertical integration resolves this constraint by allowing bilayer meta-atoms to operate simultaneously as a phase shifter and an impedance-matching stack. Using an effective thin-film model, we derive a design rule that links the effective indices of two independently patterned layers and identifies antireflective bilayer libraries with full $0$-$2π$ transmission-phase coverage. We realize this concept in a free-standing TiO$_2$/TiO$_2$ metalens operating at 1310 nm, which suppresses reflectance below that of bare glass while preserving diffraction-limited focusing. These results establish bilayer metasurfaces as a framework for co-engineering optical impedance and wavefront response at the meta-atom level.

Type Ia supernovae are bright thermonuclear explosions of one or more white dwarf stars. The exact origin and explosion mechanism for these supernovae is still poorly understood. In the near-Chandrasekhar mass progenitor model, a simmering phase precedes the explosion. During this simmering phase, central carbon burning heats the core and drives convection. A poorly understood aspect of this phase is the convective Urca process, a linking of weak nuclear reactions and convective mixing. Convective Urca has the potential to alter characteristics of the convection zone and thus alter the evolution of the white dwarf. To study the convective Urca process, we use the low Mach number hydrodynamic code MAESTROeX to run 3D simulations of the convection zone. We build off previous work to implement a more comprehensive carbon burning network and include the A=21, A=23, and A=25 Urca pairs in the simulations. We compare simulations with and without the convective Urca process to isolate the direct effects the process has on the convection zone. We find the convective Urca process reduces the efficiency of convective mixing near the the convective boundary, but does not restrict the size of the convection zone. We additionally find the A=23 Urca pair to be the most important Urca pair to the convective Urca process in these simulations. All together, our results better inform our understanding of this complex phenomena as well as demonstrates the range of potential convective structures, particularly at the convective boundary, of a simmering white dwarf.

This paper implements deep reinforcement learning (DRL) with a safety filter for spacecraft reorientation control with a single pointing keep-out zone. A new state space representation is designed which includes a compact representation of the attitude constraint zone. A reward function is formulated to achieve the control objective while enforcing the attitude constraint. The soft actor-critic (SAC) algorithm is adopted to handle continuous state and action space. A curriculum learning approach is implemented for agent training. To guarantee the compliance of the attitude constraint, a control barrier function (CBF)-based safety filter is implemented for agent deployment. Simulation results demonstrate the effectiveness of the proposed state space presentation and the designed reward function. Monte Carlo simulations underscore that reward shaping alone cannot guarantee the safety during reorientation maneuver. In contrast, with the CBF-based safety filter, the constraint can be guaranteed during maneuvers.

The analytic structure of the partition function in finite-volume systems is investigated at complex chemical potentials in a minimal mean-field effective model of QCD with finite-size effects incorporated. We discuss the temperature dependence of the Lee-Yang zeros and their relation to the edge singularity for various system sizes. Different methods for locating the critical point based on finite-size scaling of Lee-Yang zeros and susceptibility ratios are compared. We demonstrate that these methods can successfully identify the critical point, whereas a careful treatment of corrections from irrelevant operators is crucial for its accurate determination.

This paper addresses the problem of tracking in-plane waves from image sequences using periodic surface patterns. Wave-induced deformation is modeled as a spatial phase modulation of a periodic carrier. We propose ADOPT (Analytical Demodulation of Periodic Texture), a method based on an oriented two-dimensional analytic signal to estimate displacement phase and orientation. The approach relies on a physical model describing longitudinal and transverse in-plane waves. Orientation-selective filtering isolates relevant spectral components, and phase extraction provides a stable reconstruction of the displacement field. A theoretical analysis using the Cramer--Rao bound evaluates performance limits of ADOPT. Simulations show that the proposed method outperforms state-of-the-art Digital Image Correlation (DIC) at high signal-to-noise ratios, especially for small displacements where DIC becomes limited. Moreover, ADOPT is more computationally efficient. Experiments on silicone membranes with periodic patterns confirm accurate estimation of wave fields and dispersion curves under impulsive excitation. Overall, the proposed framework provides a robust and efficient solution for wave-induced displacement estimation.

Phylogenetic networks generalize phylogenetic trees by allowing reticulate evolutionary events such as horizontal gene transfer and hybridization. Among the many subclasses of phylogenetic networks, orchard networks have attracted increasing attention due to their structural and algorithmic properties. In this paper, we study the arc-deletion distance to orchard networks, defined as the minimum number of reticulate arcs whose deletion transforms a phylogenetic network into an orchard network. We prove that computing this distance is NP-hard via a polynomial-time reduction from the Degree-3 Vertex Cover problem. Our result establishes the computational intractability of this proximity measure and contributes to the complexity theory of phylogenetic network transformations.

To verify the robustness of a program or protocol, it is common in the computer science community to rely on the theoretical framework of game theory. In particular, if one seeks to enforce a desired property, or specification, despite an unpredictable environment, a useful abstraction is to model the situation as a two-player zero-sum game. The goal is then to find a strategy for the system that guarantees the specification against any strategy of the environment. However, to model more complex situations, such as multiple systems with different objectives or an environment composed of various agents, the richer framework of multiplayer games must be considered. In this setting, a natural question is to identify equilibria, i.e., strategy profiles that are robust in the sense that no player has an incentive to deviate. The most well-known equilibrium concept is the Nash equilibrium, but several alternatives exist. We study five such notions and, for each of them, we provide complexity results for the constrained existence problem, which consists of deciding whether a given game contains an equilibrium that ensures each player a payoff within a specified interval.

TeV string-mass-scale strings have been excluded experimentally at colliders, as their effects have not been observed at the Large Hadron Collider (CERN). On the other hand, higher-scale string theory, with mass scales typically close to the Planck scale, is often regarded as experimentally inaccessible due to the enormous energies required for direct tests, and far beyond the reach of present or foreseeable particle accelerators. Nevertheless, the early Universe may provide an indirect observational window for high-string scale through imprints left on the Cosmic Microwave Background (CMB). In this work, building on previous studies, we reexamine temperature and polarization angular correlations as probes of the geometry and topology of the pre-inflationary Universe. We focus in particular on two-point correlation functions at large angular scales, where signatures of nontrivial spatial topology may survive as relics of primordial physics. We investigate the observational consequences of toroidal compactification and analyze their impact on the primordial power spectrum of the CMB provided by the Planck satellite. Within the current experimental and theoretical uncertainties, we identify a possible indication closely related to spatial-parity breaking, consistent with the presence of six spatial extra dimensions in the early Universe, compactified at the GUT epoch before the start of inflation. Finally, we extend our framework to B-mode polarization, highlighting its potential as a sensitive probe in forthcoming ground-based and space-borne experiments with unprecedented precision.

We present a novel parallel implementation for large-scale three-dimensional electromagnetic inversion based on a Gauss-Newton framework combined with a rational near-best approximation of the matrix exponential for transient simulations. The method employs parallel direct solvers for the shifted linear systems arising from the partial fraction representation of the rational approximation and demonstrates efficient parallel execution on a shared-memory architecture using MPI. A key property of the approach is that the time dependence is entirely contained in the residuals of the employed rational functions, such that the computation of forward responses and sensitivities becomes effectively independent of the number of desired observation times. Model regularization is done with smoothness constraints, formulated with Raviart-Thomas elements. The linearized inverse problems are solved using LSQR, using an implicit parallel Jacobian operator. Numerical experiments demonstrate the successful recovery of a synthetic 3-D conductivity structure with approximately 700,000 degrees of freedom. The study further discusses computational bottlenecks related to memory consumption and shared-memory scalability arising from the simultaneous storage of multiple sparse matrix factorizations. Possible improvements based on preconditioned iterative solvers and distributed high-performance computing architectures are outlined. The implementation in the Julia programming language is released as open-source software to support reproducible research and further development by the geophysical inversion community.

Using a recursive solution of the Einstein equations, we consider the perturbative expansion of the metric corresponding to a Kerr black hole. Because the metric is a function of two parameters, Newton's constant G and the Kerr spin parameter a, the perturbation theory naturally becomes a double expansion. In harmonic gauge the recursion relations can be solved to arbitrarily high orders in these two expansion parameters but to re-sum the series into the closed-form harmonic gauge metric requires the introduction of terms that are redundant and correspond to the addition of harmonic functions to the coordinates. Issues related to dimensional regularization of Fourier transforms are explained in detail.

We investigate normalized groundstates for mixed $(p,2)$-Laplacian equations \begin{align*} \begin{cases} -Δ_p u-Δu+λu=f(u) & \text{in } \mathbb{R}^2, \displaystyle \int_{\mathbb{R}^2}|u|^2\,\mathrm{d}x=m, u\in H^1(\mathbb{R}^2)\cap D^{1,p}(\mathbb{R}^2), \end{cases} \end{align*} where $Δ_p$ denotes the $p$-Laplacian with $1<p<2$, $λ\in\mathbb{R}$ represents a Lagrange multiplier and the nonlinerity $f$ exhibits exponential critical growth. Compared to the single-Laplacian case, the lack of regularity here precludes the Pohozaev identity, and the exponential critical growth severely compromises the restoration of compactness. To address these issues, we introduce a refined Moser iteration technique adapted to exponential critical growth, which establishes the Pohozaev identity for weak solutions under the mere assumption of $C_{\mathrm{loc}}^{1,α}$-regularity. By combining constrained minimization on the Pohozaev manifold within a closed $L^2$-ball with a minimax characterization, we prove the existence of normalized groundstates for any prescribed mass $m>0$. Notably, our approach works independently of the sign of the Lagrange multiplier $λ$, thereby surmounting the fundamental barrier in recovering compactness for mixed Laplacian problems.

Gradient flows of unit vector fields arise in a wide range of physical models such as harmonic map heat flows, nematic liquid crystals, and magnetization dynamics. Designing numerical schemes that simultaneously preserve the unit length constraint and dissipate energy is essential for reliable simulations of such systems. Although projection methods can effectively enforce the unit length constraint, ensuring energy dissipation under projection, especially in high-order schemes, remains challenging. Unlike traditional implicit-explicit Runge-Kutta (IMEX-RK) methods, in this work we propose a general methodology for constructing product-type IMEX-RK schemes that offers greater adaptability to various models with the goal of designing structure-preserving numerical schemes. For gradient flows of unit vector fields with Dirichlet energy, we design a linear and second-order numerical scheme that simultaneously preserves energy dissipation and the unit length constraint by using product-type IMEX-RK methods and projection techniques. Numerical experiments verify the accuracy, stability, and structure-preserving properties of the scheme. According to our best knowledge, this is the first second-order linear scheme that can preserve both the unit length and the original Dirichlet energy for harmonic map heat flows.

In quantum thermodynamics, the decomposition of energy exchanges into heat and work remains an open problem beyond weak-coupling and slow-driving regimes. Recent formulations have shown that quantum coherence introduces additional energy contributions whose thermodynamic interpretation is still under debate, raising fundamental questions about the structure of the quantum first law. In this work, we investigate this problem through a time-dependent perturbative framework applied to the first law of quantum thermodynamics. By expanding the thermodynamic quantities up to second order, we derive explicit perturbative corrections for work, heat, and coherence contributions. Our results show that the coherence term can be consistently decomposed into coherent heat and coherent work, demonstrating that quantum coherence does not require the introduction of an independent energetic contribution beyond heat and work. The formalism resolves inconsistencies associated with previous formulations of the quantum first law, including the interpretation of coherence contributions and their connection with entropy fluxes. At second order, the perturbative corrections become directly connected to transition rates governed by Fermi's golden rule, establishing a bridge between microscopic quantum transitions and macroscopic thermodynamic quantities. These results provide a physically transparent framework to investigate coherence-driven thermodynamic processes and offer new perspectives for the analysis of driven quantum systems and nonequilibrium quantum technologies.

This algorithm is designed to perform numerical transforms to convert data from the temporal domain into the spectral domain. This algorithm obtains the spectral magnitude and phase by studying the Coefficient of Determination of a series of artificial sinusoidal functions with the temporal data, and normalizing the variance data into a high-resolution spectral representation of the time-domain data with a finite sampling rate. What is especially beneficial about this algorithm is that it can produce spectral data at any user-defined resolution, and this highly resolved spectral data can be transformed back to the temporal domain.

We propose the rare SM Higgs decay $H\to b\bar b b\bar b$ as a probe of the structure of Higgs interactions with bottom quarks and gauge bosons, and as a baseline for searches for new physics producing four-bottom final states in Higgs decays. We compute the leading contributions to this decay, including the dominant $H\to b\bar b g\to b\bar b b\bar b$ topology, the sizeable $H\to ZZ^\ast\to b\bar b b\bar b$ channel, and the loop-induced $H\to gg\to b\bar b b\bar b$ contribution. We find a branching ratio of order $1.6\times10^{-3}$ and show that destructive interference among the leading amplitudes is phenomenologically relevant. We demonstrate that this decay can be probed in associated Higgs production at both the HL-LHC and the ILC. For $pp\to WH\to Wb\bar b b\bar b$ at $\sqrt{s}=14$ TeV, we use a multivariate analysis based on boosted decision trees to exploit correlations among the four-$b$ kinematic observables. At $3000~{\rm fb}^{-1}$, the statistical significance reaches about $3.5$, while a tighter high-purity working point gives $S/B\simeq5\%$ with significance close to $3σ$. A combined high-luminosity LHC dataset could therefore make this rare decay observable. For $e^+e^-\to ZH\to Zb\bar b b\bar b$ at the ILC with $\sqrt{s}=250$ GeV, we demonstrate that the cleaner collider environment gives a high-purity signal sample. In the nominal setup, the multivariate analysis gives a significance above $5σ$ already at $300~{\rm fb}^{-1}$. At integrated luminosities of order $1-3~{\rm ab}^{-1}$, the branching ratio can be measured with several-percent precision.

We study the large-space and large-time asymptotic behavior of the soliton gas of genus $2n-1$ for the mKdV equation with $n\in \mathbb{N}_+$. As $x \to +\infty$, we show that the large-space asymptotics of the mKdV soliton gas can be expressed with the Riemann-theta function of genus $2n-1$. For large $t$, based on the nonlinear steepest descent method and $g$-function approach, we establish a global large-time asymptotic description of the mKdV soliton gas. The half-plane $\{(x,t):-\infty<x<+\infty, t>0\}$ is divided into $2n+1$ separated regions. In each region, the large-time asymptotics of the mKdV soliton gas is given by using the Riemann-theta functions and uniform error estimation.

Stochastic infection processes are continuous-time Markov chains on graphs that assign each vertex one of multiple states, such as susceptible, infected, or recovered. Depending on the model, vertices change their state based on random transition rates and the states of their neighbors, resulting in a variety of complex dynamics. The body of rigorous literature is rich for processes that consider a single infection. In contrast, the setting with at least two infections, where the same state exists for different types, allows for far more transition combinations, leaving several interesting models entirely unexplored. We address this shortcoming in the literature by defining the IRIR process, in which two SIR processes run on the same graph and each vertex is immune only to its most recent infection. We study the survival time of the IRIR process, that is, the time until no infected vertex remains, with mathematical rigor. Our main result is a tight threshold, known as epidemic threshold, where the survival time rapidly changes from at most quasi-linear in the graph size $n$ to at least super-polynomial in $n$. This result is applicable to perfectly mixed graphs, which are graphs where the density of edges between each non-empty subset of vertices is a given value $p \in (0, 1]$. Our super-polynomial lower bound extends to jumbled graphs, which allow for some more flexibility in the density. In particular, this includes with high probability Erdos-Renyi graphs with an average degree of $k\in ω(\ln^2(n))$. Our proof for the lower bound is based on a potential that transforms the configurations of the IRIR process to a supermartingale with drift in a large region, implying the lower bound. We detail how to systematically derive such a potential, based on a Lyapunov function of the transition equilibrium of the process.

Scanning helium microscopy (SHeM) is attractive for imaging delicate and insulating surfaces because it combines a non-destructive neutral-atom probe with strong surface sensitivity. However, large-working-distance pinhole instruments have so far been limited in spatial resolution. Here we report sub-micron resolution in a large-working-distance pinhole SHeM, with an intrinsic beamwidth of 340nm achieved at working distances of 770 μm to 850 μm. This sixfold improvement over our previous long-working-distance configuration is enabled by constrained optimisation of the atom optics together with a redesigned high-resolution pinhole-plate, a reduced pinhole diameter, an increased source--pinhole distance and a larger detector aperture. Beamwidth measurements agree well with the modified optimisation model and show that geometric, source-size and diffraction terms now contribute on a similar footing, placing the instrument in a near-optimised regime. The resulting combination of sub-micron beam size, useful depth of field and practical sample access is demonstrated on bacterial specimens and eroded diamond. The work establishes large-working-distance pinhole SHeM as a viable sub-micron imaging platform and extends its usefulness for topographic imaging and micro-diffraction applications.

We present a new accurate analysis of the $^3$He$(p,e^+ν_e)$${}^4$He (''hep'') reaction at astrophysical energies. The S-factor is computed using a state-of-the-art method to calculate the four-nucleon scattering and bound-state wave functions (the hyperspherical harmonic expansion), and by using nuclear interactions and accompanying electroweak nuclear currents obtained within the chiral effective field theory framework. Our analysis includes a detailed examination of the theoretical uncertainties coming from two different sources: the truncation of the interaction and current chiral expansions, and the model dependence. Our recommended final theoretical value for the hep S-factor at zero energyis $S(0)=(8.7\pm 0.9)\times 10^{-20}$ keV b. We provide also the energy spectrum of the outgoing hep positrons which may be measured in future experiments. We include also an analysis of the ''sister'' reaction $^3$He$(n,γ)$${}^4$He (''hen'') at low energies, showing that the calculation well reproduce the total cross section from thermal energies to few MeV, validating our results on the hep reaction.

We address the question of whether dualities formulated in continuum field theory can be realised exactly at finite lattice spacing, rather than only emerging in the infrared. In this context, we construct a lattice framework for a genuinely stringy form of T-duality. We extend the exact lattice T-duality of the compact boson to curved backgrounds with non-trivial circle fibrations, where the duality is no longer exhausted by the familiar exchange of momentum and winding, but also involves global topological data. To this end, we define the twisted Villain model, which couples the lattice fibre field to cochains encoding the bundle connection and the fibre-horizontal component of the $B$-field. We realise this structure in lattice models for several fibred backgrounds and recover the characteristic bundle-flux exchange of T-duality. Using a half-gauging procedure, we derive the associated lattice defect action and show that it defines a topological defect. This establishes that the distinctive topological features of T-duality on curved manifolds can be captured exactly in a lattice model, implying that this duality is not tied to a particular continuum representation is present in lattice-regularised models.

We survey two new compactification methods for the KSBA moduli space of general type surfaces so that both of them admit a perfect obstruction theory. Virtual fundamental classes exist on these two moduli spaces, and tautological invariants can be defined on KSBA moduli spaces. This is the starting point to do enumerative geometry on KSBA moduli spaces, and we include some discussions in this direction.