The paper constructs unbounded symplectic embeddings of rational homology balls into surfaces.
problem Bounding symplectic embeddings of rational homology balls into surfaces.
method Using Mori's theory of flips and mutations of polygons.
result Unbounded sequences of symplectically embedded rational homology balls into surfaces.
Classifies 3D spaces using specific invariants.
problem Classifying 3D simply connected Mori fibre spaces.
method Uses finitely many numerical invariants.
result Finds finitely many invariants to classify diffeomorphism types.
Regression learns Mori-Zwanzig operators for dynamical systems.
problem Learning Mori-Zwanzig operators for complex dynamical systems.
method Statistical regression to extract Markov and memory operators.
result Regression models improve learning of memory-dependent corrections.
It is the aim of this article to determine curvature quantities of an arbitrary Riemannian monotone metric on the space of positive matrices resp. nonsingular density matrices. Special interest is focused on the scalar curvature due to its expected quantum statistical meaning. The scalar curvature is explained in more …
In the present paper we prove that, on a hyperkähler manifold, walls of the kähler cone and extremal rays of the Mori cone are determined by all divisors satisfying certain numerical conditions.
We give an introduction to the theory of varieties of minimal rational tangents, emphasizing its aspect as a fusion of algebraic geometry and differential geometry, more specifically, a fusion of Mori geometry of minimal rational curves and Cartan geometry of cone structures.
Data-driven model reduction captures non-Markovian dynamics using Koopman and Mori-Zwanzig formalisms.
problem Modeling complex, non-Markovian dynamics efficiently and understanding their underlying mechanisms.
method Formulates data-driven model reduction within Koopman and Mori-Zwanzig formalisms, deriving NARMAX models from dynamical systems.
result Shows how data-driven methods can represent non-Markovian dynamics using Koopman and Mori-Zwanzig formalisms.
The paper bounds Chern numbers of threefolds, generalizing previous work.
problem Bounding Chern numbers of threefolds, especially those with negative Kodaira dimension.
method Analyzing smooth Mori fibre spaces and applying topological bounds.
result Chern numbers of threefolds are bounded by underlying topological manifolds.
Develops a new deep learning formulation using Mori-Zwanzig formalism.
problem Improves deep learning by introducing a new concept of memory.
method Uses Mori-Zwanzig formalism to propagate quantities of interest through neural networks.
result Rigorously transforms deep networks into shallow ones using decay property of memory operator.
We survey our recent papers (some being joint ones) about the relation between the geometry of a compact Kähler manifold and the existence of automorphisms of positive entropy on it. We also use the language of log minimal model program (LMMP) in biraitonal geometry, but not its more sophisticated technical part. We gi…
New method uses neural networks to create models with memory effects.
problem Accurately modeling memory effects in reduced models.
method Analogies between recurrent neural networks and Mori-Zwanzig formalism to develop reduced models with memory.
result The proposed method produces reduced models with good performance on short-term and long-term predictions.
Study shows non-polyhedral structure in moduli spaces for n≥8.
problem Identifying non-polyhedral structure in moduli spaces of pointed stable curves.
method Constructing an extremal non-polyhedral ray via maps on meromorphic strata of differentials.
result Moduli spaces are not Mori Dream Spaces for n≥8.
Study flip graphs for surfaces of infinite type, finding uncountably many connected components.
problem Understanding relationships between triangulations of infinite type surfaces via flips.
method Associate triangulations to flip graphs and study sequences of simultaneous flips.
result Flip graphs for infinite type surfaces have uncountably many connected components.
Finite subgraphs in flip graphs ensure unique surface embeddings.
problem Ensuring unique embeddings of surfaces based on flip graphs.
method Analyzing finite subgraphs within flip graphs of surfaces.
result Injective homomorphisms are uniquely extendable and induced by embeddings.
The curvature tensor and the scalar curvature are computed in the space of positive definite real matrices endowed by the Kubo-Mori inner product as a Riemannian metric.
Study of flip graphs and their automorphism groups for infinite-type surfaces.
problem Understanding automorphism groups of flip graphs for infinite-type surfaces.
method Examined the relationship between mapping class groups and flip graphs for infinite-type surfaces.
result Extended mapping class groups are isomorphic to proper subgroups of automorphism groups of flip graphs.
Study shows flipping a small subset of labels can severely damage machine learning models.
problem Adversarial attacks on distributed machine learning models.
method Formalized label flipping attacks, proposed a greedy algorithm, demonstrated with logistic regression models.
result A budget of only 0.1% of labels at each training step can reduce model accuracy by 6%, and some models can perform worse than random guessing when up to 25% of labels are flipped.
We show that, in finite dimensions, the only monotone metrics for which the (+1) and (-1) affine connections are mutually dual are constant multiples of Bogoliubov-Kubo-Mori metric
We introduce flip points to interpret neural networks, providing detailed explanations and confidence measures.
problem Lack of interpretability in neural networks for important applications.
method Investigating flip points, the boundary between two output classes, to provide detailed interpretation and confidence measures.
result Flip points enable detailed interpretation and measure confidence in neural network outputs.
Connected flip graphs for triangulations on hyperbolic surfaces.
problem Connecting triangulations on hyperbolic surfaces via flips.
method Proving connectedness of flip graphs and giving bounds on edge flips.
result Flip graphs of geometric triangulations are connected.
Novel autoencoder method approximates Koopman operator in low dimensions.
problem Challenges in approximating finite Koopman operators using data-driven methods.
method Mori-Zwanzig autoencoder (MZ-AE) for robust Koopman operator approximation.
result Improved predictive capability and robust long-term statistical performance.
Study of skateboard flips as continuous curves in SO(3) group.
problem Characterize skateboard flip tricks as continuous motions.
method Model flips as curves in SO(3), analyze lifts to S3, derive formulas. result There are only four distinct flip tricks up to continuous deformation.
Flip symmetry on knot diagrams affects Khovanov homology.
problem Understanding the flip map on Khovanov homology.
method Analyzing the behavior of the flip map on unlinks and using it to determine the involution.
result The flip map is the identity map over \(\mathbb{F}_2\), confirming a conjecture.
The flip graph and arc complex of a surface are shown to have finite rigidity.
problem Finite rigidity of flip graph and arc complex for surfaces.
method Embedding the flip graph in the arc complex and leveraging finite rigidity of the flip graph.
result Finite rigidity of the flip graph implies finite rigidity of the arc complex.
Efficiently poisons offline RLHF models by flipping preference labels.
problem Vulnerability of offline RLHF models to preference label flipping attacks.
method Developed two attack methods: BAL-A and BMP-A, solving a structured binary sparse approximation problem.
result Demonstrated that flipping one preference label induces a parameter-independent shift in the DPO gradient, enabling structured binary sparse approximation.
New examples show flip distance and polyhedron triangulation numbers differ, with ratio close to 3/2.
problem Understanding the relationship between flip distance and polyhedron triangulation numbers.
method Provided examples to demonstrate the difference between flip distance and polyhedron triangulation numbers.
result Ratio of flip distance to polyhedron triangulation numbers can be arbitrarily close to 3/2.
We prove that every injective simplicial map F(S)→F(S′) between flip graphs is induced by a subsurface inclusion S→S′, except in finitely many cases. This extends a result of Korkmaz--Papadopoulos which asserts that every automorphism of the flip graph of a surface without boundary is ind…
We prove that for a given flat surface with conical singularities, any pair of geometric triangulations can be connected by a chain of flips.
The authors give a complete classification of projective threefolds admitting a holomorphic normal projective connection. Moreover, they prove a general structure theorem on complex projective manifolds admitting a holomorphic normal projective connection, saying in particular, that any such manifold is either the proj…
We introduce a notion of cross-flips: local moves that transform a balanced (i.e., properly (d+1)-colored) triangulation of a combinatorial d-manifold into another balanced triangulation. These moves form a natural analog of bistellar flips (also known as Pachner moves). Specifically, we establish the following the…
This paper is about the geometry of flip-graphs associated to triangulations of surfaces. More precisely, we consider a topological surface with a privileged boundary curve and study the spaces of its triangulations with n vertices on the boundary curve. The surfaces we consider topologically fill this boundary curve s…
Geodesics count exponentially between triangulations of surfaces with enough topology.
problem Counting geodesics in triangulations of surfaces.
method Analyzing the flip-graph of triangulations and their geodesics.
result The number of geodesics grows exponentially for surfaces with enough topology.
Identifies minimal training subset to flip a prediction.
problem Flipping predictions in machine learning models.
method Extended influence function for relabeling minimal subset.
result Relabeling fewer than 2% of training points can flip a prediction.
In order to model volatile real-world network behavior, we analyze phase-flipping dynamical scale-free network in which nodes and links fail and recover. We investigate how stochasticity in a parameter governing the recovery process affects phase-flipping dynamics, and find the probability that no more than q% of nodes…
We use flip points to explain and audit deep learning models, revealing decision boundaries and improving model performance.
problem Lack of interpretability in deep learning models hinders their use in important applications.
method Flip points are used to analyze decision boundaries of deep learning models with continuous output scores.
result Flip points reveal the least changes in input that would alter a model's classification, enabling better understanding and improvement of model behavior.
Let Σ be a compact surface. We prove that the set of surface cubications modulo flips, up to isotopy, is in one-to-one correspondence with Z/2Z⊕H1(Σ,Z/2Z).
Study finds flipped classrooms improve student self-concept, enjoyment, but not exam scores.
problem Evaluating the impact of flipped classrooms on higher education outcomes.
method Double/debiased machine learning (DML) approach to analyze student data.
result No significant positive effects on exam scores, passing rates, or knowledge retention.
Study quasisymmetric maps on hyperbolic plane boundaries.
problem Identify quasisymmetric maps corresponding to specific lambda lengths and flip distances.
method Analyze maps on Farey triangulation, relate to shearing coordinates and flip distance.
result Identify quasisymmetric maps corresponding to pinched lambda lengths and flip distances.
We study flip-graphs of triangulations on topological surfaces where distance is measured by counting the number of necessary flip operations between two triangulations. We focus on surfaces of positive genus g with a single boundary curve and n marked points on this curve; we consider triangulations up to homeomor…
Using existing technology, we prove a Masur-Minsky style distance formula for flip- graph distance between two triangulations, expressed as a sum of the distances of the projections of these triangulations into arc graphs of the suitable subsurfaces of S.
We investigate a type of distance between triangulations on finite type surfaces where one moves between triangulations by performing simultaneous flips. We consider triangulations up to homeomorphism and our main results are upper bounds on distance between triangulations that only depend on the topology of the surfac…
Deep Partition Aggregation defends against poisoning attacks with provable certificates.
problem Adversarial poisoning attacks corrupt classifier test-time behavior.
method Deep Partition Aggregation (DPA) is an ensemble method using hash partitions and base models trained on these partitions.
result DPA can certify >= 50% of test images against over 500 poison image insertions on MNIST, and nine insertions on CIFAR-10.
Reductive quotients preserve klt singularities in algebraic geometry.
problem Preserving klt singularities in quotients of klt singularities.
method Proving that the quotient of a klt type singularity by a reductive group is of klt type.
result The quotient of a klt variety by a reductive group results in a klt variety with a suitable boundary.
The paper solves pentagon equations using triangulations and edge transformations.
problem Solving pentagon equations with triangulations and edge transformations.
method General data and transformation rule method applied to triangulations.
result Recovery of initial data after transformations.
New method makes machine learning models robust to label flipping attacks.
problem Machine learning models are vulnerable to label flipping attacks.
method Randomized smoothing over arbitrary functions to build certifiably robust classifiers.
result Linear classifiers are robust to label flipping attacks with deterministic bounds.
We explore several families of flip-graphs, all related to polygons or punctured polygons. In particular, we consider the topological flip-graphs of once-punctured polygons which, in turn, contain all possible geometric flip-graphs of polygons with a marked point as embedded sub-graphs. Our main focus is on the geometr…
Paper uses RL to optimize bit-flipping decoding for binary codes.
problem Improving bit-flipping decoding for binary linear codes.
method Mapped iterative decoding algorithms to MDPs for reinforcement learning.
result Learned BF decoders offer performance-complexity trade-offs and near-optimal performance.
Generalizes Giroux's result to higher dimensions and applies to contact manifolds.
problem Characterizing convex surfaces in contact manifolds.
method Extending Giroux's result to arbitrary dimensions and applying to specific hypersurfaces.
result Closed hypersurfaces are C∞-close to convex hypersurfaces.