Topological Discrepancy: Theory & Applications

• Topological discrepancy is defined as metrics and invariants quantifying structural imbalances in spaces, data sets, and physical systems.
• It refines singularity classification through methods like Mather discrepancy and arc space analysis, yielding novel invariants such as log-canonical thresholds.
• It informs applications ranging from combinatorial graph tilings and digital topology to persistent homology in quantum and dynamic systems.

Topological discrepancy refers to metrics, invariants, or classification results that detect, measure, or exploit structural differences in topological or geometric properties—such as connectivity, singularity class, or spatial configuration—between mathematical objects, data sets, or physical systems. These metrics typically extend classical notions of discrepancy from combinatorics, analysis, or geometry to the topological setting, examining how “balanced” or “unbalanced” configurations can be, or capturing the “distance” between their topological structures. Topological discrepancy finds application in areas ranging from birational geometry and digital image comparison to dynamic systems, data analysis, and quantum information theory.

1. Topological Discrepancy via Mather Discrepancy and Arc Spaces

Mather discrepancy provides a refined invariant for classifying singularities in algebraic geometry. Unlike traditional discrepancy, which only makes sense for $\mathbb{Q}$-Gorenstein varieties (where a canonical divisor $K_X$ exists), Mather discrepancy is defined for arbitrary varieties and is constructed from the Nash blow-up and the transformation of the top exterior power of the cotangent sheaf. The Mather discrepancy divisor $K^M_{Y/X}$ is defined via the image of the map $f^*\Omega_X^n \to \Omega_Y^n$ for any resolution $f\colon Y \to X$, and for any prime divisor $E$ over $X$, $r_E := \operatorname{ord}_E K^M_{Y/X}$ gives the associated value.

This extension enables the definition of new invariants: the Mather log-canonical threshold (lct$^M$) and Mather minimal log-discrepancy (mld$^M$), both of which can be computed for arbitrary varieties (even non-normal, non-$\mathbb{Q}$-Gorenstein). The geometry of arc spaces ($X_\infty$, spaces of formal arcs) is instrumental in expressing these invariants, with valuations corresponding to divisorial sets and codimension computations in arc space giving log-discrepancy formulas (e.g., $\operatorname{codim}(C_x(v)) = q(k_E+1)$ for a divisorial valuation $v = q \operatorname{val}_E$).

These invariants have direct implications for the classification and paper of singularities, yielding results like the criterion that $\operatorname{mld}(x; X, \mathcal{O}_X) = n$ if and only if $(X,x)$ is nonsingular (dimension $n$), and providing inversion of adjunction statements for broader classes of singularities. The use of Mather discrepancy and arc space approach thus expands the reach of discrepancy-based techniques to topological phenomena undetectable by classical (canonical divisor-based) methods (Ishii, 2011).

2. Topological Discrepancy in Combinatorics and Graph Theory

Discrepancy theory in combinatorics traditionally assigns weights or colorings to parts of a structure (vertices, edges) and then seeks substructures exhibiting high imbalance (discrepancy) in these weights. The discrepancy version of the Hajnal-Szemerédi theorem (Balogh et al., 2020) incorporates this concept for the setting of perfect $K_r$-tilings in graphs.

For a graph $G$ and every labeling $f: E(G) \to \{-1, 1\}$, the discrepancy of a subgraph $H$ (e.g., a $K_r$-tiling $T$) is $\left| \sum_{e \in E(T)} f(e) \right|$. The main theorem guarantees that given a sufficiently high minimum degree (specifically, $\delta(G) \geq (1 - 1/(r+1) +o(1))n$), for every labeling, a perfect $K_r$-tiling exists with discrepancy at least $y n$ for some $y > 0$.

Key proof methods include a variant of the regularity lemma, template switching, and absorption, emphasizing that with minimal degree conditions, one can't “neutralize” or “balance away” the discrepancy—the structure of the graph itself forces substantial topological discrepancy (imbalance) in global spanning substructures. This result generalizes the concept of Ramsey-type bias to spanning structures and connects combinatorial discrepancy to topological robustness in networked systems (Balogh et al., 2020).

3. Metrics for Topological Discrepancy in Digital and Data-Driven Settings

Quantifying topological discrepancy also appears in metric design for digital images and geometric data:

• Simplicial Hausdorff Distance (Nnadi et al., 6 Feb 2025): Integrates the combinatorial structure of simplicial complexes with geometric proximity. For simplicial complexes $(X, f)$ and $(Y, g)$ (with measurement maps to $\mathbb{R}^d$), the directed distance considers all $k$-simplices and matches, measuring (in a sup–inf–sup sense) the largest necessary vertex-to-vertex move to geometrically and combinatorially reconcile the two structures. For filtered complexes, this metric captures persistent differences in topological structure across scales.
• Augmented Metrics in Digital Topology (Boxer, 2021): Combines the Hausdorff metric with other pseudometrics such as difference of Euler characteristic, Lusternik–Schnirelmann category, or shortest-path diameter, highlighting that Hausdorff proximity can coexist with significant topological discrepancy, which is detected only by these supplementary metrics. For instance, two digital objects may overlap closely in the Hausdorff sense while being topologically distinct (different number of components, holes, etc.).
• Manifold Topology Divergence (MTop-Divergence) (Barannikov et al., 2021): In topological data analysis, Cross-Barcode and MTop-Divergence quantify spatial discrepancy at the persistent homology level between point clouds, sensitive not only to feature counts (loops, connected components) but to their placement and scale. Formally, for point clouds $P$, $Q$, a barcode is constructed from a Vietoris–Rips complex with modified distances, and the sum of barcode interval lengths gives the divergence: $$\mathrm{MTop\text{-}Div}(P, Q) = \sum_j (d_j - b_j )$$.

Such metrics provide robust, stable, and interpretable means to measure how far data sets are not just in geometry but in intrinsic topology—critical in areas like generative model validation, scientific imaging, or shape comparison.

4. Discrete Structures and Topological Discrepancy in Dynamics

In the context of dynamical systems, topological discrepancy can be both measured and localized using discrete combinatorial constructions (Prishlyak, 1 Feb 2025). Algorithms are provided for:

• Triangulating or cellulating manifolds and computing characteristic vectors: The numbers $c_i$ of $i$-cells encode overall complexity.
• Calculating invariants: Euler characteristic ($\chi = \sum (-1)^i c_i$) and homology ($$H_i = \ker \partial_i / \operatorname{im} \partial_{i+1}$$) distinguish non-homeomorphic spaces.
• Tracking discrete Morse functions and gradient vector fields: Critical points, Morse–Smale complexes, and handle decompositions allow monitoring of topological type under flows or perturbations.
• Poincaré rotation indices: Local winding numbers of vector fields provide local certificates for topological changes.

These tools enable systematic detection, classification, and evaluation of discrepancies in system topology (e.g., through bifurcations, merging or splitting of components, or the birth/death of holes), as the state space or flow evolves. The practical computability of these invariants in discrete settings makes them indispensable in scientific computing and simulation (Prishlyak, 1 Feb 2025).

5. High-Dimensional and Geometric Discrepancy

High-dimensional generalizations of discrepancy engage with topological aspects by considering geometric assignments and imbalances across combinatorially rich structures. For example:

• Discrepancy of Subtrees in Trees (Hollom et al., 5 Dec 2024): Given a tree $T$ and function $f: E(T) \to \mathbb{S}^d$ assigning unit vectors in $\mathbb{R}^d$, the $d$-dimensional discrepancy $D_d(T)$ is the minimum, over $f$, of the maximal norm of the vector sum over any (connected) subtree. Quantitatively, for trees with $\ell$ leaves and as $\ell \to \infty$, the tight bound is

$$D_d(T) = (1 + o(1)) \cdot \frac{\ell}{d B(d/2,1/2)}$$

where $B$ is the Beta function. This result extends classical two-color ($\pm 1$) discrepancy to a continuous setting and confirms sharp conjectures about the minimum achievable imbalance given only the topology of $T$. The framework can also accommodate oriented discrepancy, measuring how rootings and orientations can (or cannot) balance imbalances over all induced rooted subtrees.

• Average Squared Discrepancy in Cubes (Clément et al., 6 Aug 2025): For point sets in $[0,1]^d$, classical star discrepancy ($L_2^*$) is origin-anchored and may exhibit pathological minimizers (“Pathology II” of Matoušek). The average squared discrepancy is defined by integrating $L_2^*$ over all $2^d$ anchorings (vertices), removing exceptional bias and ensuring that only authentically uniform sets score low. The closed formula,

$$(D_2^{\mathrm{asd}})^2 = \frac{1}{3^d} - \frac{2}{n} \sum_{i=1}^n \prod_{j=1}^d \left( \frac{1 + 2x_{ij} - 2x_{ij}^2}{4} \right ) + \frac{1}{n^2} \sum_{i,i'} \prod_{j=1}^d \left( \frac{1 - |x_{ij} - x_{i'j}|}{2} \right )$$

ensures symmetry and favorable optimization properties, directly addressing topological discrepancy by preventing “degenerate” configurations from achieving artificially low values (Clément et al., 6 Aug 2025).

6. Topological Discrepancy in Multiparameter Persistence and Quantum Information

In multiparameter persistent homology, topological discrepancy quantifies the “gap” between the richness of multiparameter (e.g., bifiltrations) and classical (monoparameter) models. The topological difference is defined as

$$\Delta(\Phi_1, \Phi_2) = d_{\text{match}}(\Phi_1, \Phi_2) - \max\{d_B(f_1, f_2), d_B(g_1, g_2)\}$$

where $d_{\text{match}}$ is the matching distance between multiparameter persistence modules, and $d_B$ is the bottleneck distance between monoparameter diagrams for each coordinate function. The topological correlation further normalizes this difference, providing a measure of interdependence of the filter functions and quantifying when multiparameter persistence yields more structural discrimination than any combination of one-dimensional invariants (Mastroianni et al., 20 Jun 2025).

In quantum information, the geometry of the set of zero-discord quantum states for two qubits yields a nine-dimensional, nonconvex, simply-connected manifold embedded within the fifteen-dimensional set of all states. The “thinness” (co-dimension) of this topological set forces quantum discord to only “vanish” at isolated points in the time evolution of a quantum state, as opposed to entanglement which may vanish over intervals. This topological distinction underpins fundamentally different dynamical behaviors (bouncing, half-life decay) and provides a robust topological explanation for observed physical phenomena in open quantum systems (Nguyen et al., 2013).

7. Applications and Implications

Topological discrepancy frameworks inform a wide range of theoretical and applied areas:

• Singularity theory and birational geometry: New invariants from Mather discrepancy guide the classification and resolution of singularities, even in non-normal, non-$\mathbb{Q}$-Gorenstein spaces.
• Combinatorics and network science: Guarantees on the existence of high-discrepancy substructures reinforce robustness and algorithmic lower bounds.
• Topological data analysis (TDA): New metrics and divergences precisely measure differences in shapes, manifolds, or data sets, with direct applications to generative modeling, clustering, and pattern recognition.
• Dynamic systems and scientific computing: Computable discrete invariants allow detection and localization of topological changes during system evolution.
• Quantum computing and information: Topological constraints explain qualitative differences in information-theoretic quantities (discord, entanglement).

Emergent research directions include generalizing discrepancy measures to hypergraphs, higher-dimensional complexes, or non-Euclidean settings; further integrating TDA with geometry-aware metrics; and exploiting topological discrepancy for robust, interpretable machine learning and modeling in scientific domains.