Builds geometric structures for algebraic groups over real closed fields.
problem Characterizing and decomposing algebraic groups over specific valued fields.
method Real algebraic geometry to construct and analyze affine buildings.
result Computed stabilizers and obtained group decompositions.
We prove isoperimetric inequalities for quotients of n-dimensional Affine buildings. We use these inequalities to prove topological overlapping for the 2-dimensional skeletons of these buildings.
In this paper, we give a general group-theoretic construction of affine $\RR$-buildings, and more generally, of affine Λ-buildings, associated to semisimple Lie groups over nonarchimedean real closed fields. The construction of Kleiner-Leeb using the asymptotic cone of a Riemannian symmetric space appears as a specia…
Affine maps reveal higher rank structures in certain spaces.
problem Characterizing spaces with higher rank structures.
method Using Hadamard spaces with geometric group actions and affine maps.
result Affine maps not dilations indicate higher rank structures.
We describe some buildings related to complex Kac-Moody groups. First we describe the spherical building of SLn(C) (i.e. the projective geometry PG(Cn)) and its Veronese representation. Next we recall the construction of the affine building associated to a discrete valuation on the rational function field C(z). Then …
Positive configurations of points in the affine building were introduced in \cite{Le} as the basic object needed to define higher laminations. We start by giving a self-contained, elementary definition of positive configurations of points in the affine building and their basic properties. Then we study the geometry of …
Builds torus fibrations over singular manifolds with specific features.
problem Creating torus fibrations over manifolds with singularities.
method Constructs a topological space X as a torus fibration over an integral affine manifold B with singularities. result The fibration has a discriminant in codimension 2.
The relationship between minimal algebraic Kac-Moody groups and twin buildings is well known as is the relationship between formal completions in one direction and affine buildings. Nevertheless, as the completion of a Kac-Moody group in one direction destroys the opposite BN-pair, there exists no longer a twin buildin…
Geometrically interprets and computes intersection pairings for higher laminations.
problem Understanding and computing intersection pairings for higher laminations.
method Realization of higher laminations as points in the affine building, geometric interpretation of pairings, use of combinatorial results.
result Intersection pairings can be computed as the length of minimal weighted networks in the building.
We give a Thurston-like definition for laminations on higher Teichmuller spaces associated to a surface S and a semi-simple group G for G−SLm and PGLm. The case G=SL2 or PGL2 corresponds to the classical theory of laminations. Our construction involves positive configurations of points in the affine bui…
New method for multiway clustering of 3rd order tensors.
problem Clustering of 3rd order tensors.
method MCAM method based on affinity matrix and clustering.
result Competitive results on synthetic and real datasets.
Let G be a simple algebraic group. Labelled trivalent graphs called webs can be used to product invariants in tensor products of minuscule representations. For each web, we construct a configuration space of points in the affine Grassmannian. Via the geometric Satake correspondence, we relate these configuration spaces…
Affine Artin groups have a finite classifying space.
problem Proving the K(π,1) conjecture for affine Artin groups. method Dual Garside structures, Euclidean isometries, and shellability of noncrossing partitions.
result Affine Artin groups have a finite classifying space.
For spherical Tits buildings of the classical types there are well-known explicit descriptions as flag complexes. Similarly for affine buildings of the classical types there are explicit constructions in terms of lattices. In this article we generalize the flag complex description to twin cities, a generalization of tw…
This abstract reviews recent methods for predicting protein-ligand binding affinity.
problem Predicting protein-ligand binding affinity for various applications in life sciences.
method Traditional and deep learning models for binding affinity prediction.
result Improved predictive performance of AI-driven models.
The paper classifies vector fields on 5D nilpotent Lie groups.
problem Classifying left-invariant affine and projective vector fields on 5D nilpotent Lie groups.
method Algebraic characterization and case-by-case analysis of vector fields.
result All projective vector fields are affine, extending classical results.
We study the possibility of applying a finite-dimensionality argument in order to address parts of the Baum-Connes conjecture for finitely generated linear groups. This gives an alternative approach to the results of Guentner, Higson, and Weinberger concerning the Baum-Connes conjecture for linear groups. For any finit…
Solves non-Archimedean Calabi-Yau equation on complex log pairs.
problem Non-Archimedean Monge-Ampère equation on Berkovich analytification.
method Solves complex Monge-Ampère equation, then adapts to non-Archimedean setting.
result Non-Archimedean analog of Ricci-flat metric potentials on complex affine varieties.
The paper develops a theory of C∞-superrings and their superschemes.
problem Developing a theory for C∞-superrings and superschemes. method Proving an equivalence between categories of fair affine C∞-superschemes and fair C∞-superrings. result A key equivalence between fair affine C∞-superschemes and fair C∞-superrings. New framework for probabilistic linear solvers reduces manual effort.
problem Manual implementation of probabilistic iterative methods is laborious.
method Affine Tracing: Automatically constructs PIMs from standard implementations.
result Any realistic affine PIM is calibrated, motivating their adoption.
The geometry of symmetric spaces, polar actions, isoparametric submanifolds and spherical buildings is governed by spherical Weyl groups and simple Lie groups. A natural generalization of semisimple Lie groups are affine Kac-Moody groups as they mirror their structure theory and have good explicitely known representati…
This paper studies closed 3-manifolds which are the attractors of a system of finitely many affine contractions that tile R3. Such attractors are called self-affine tiles. Effective characterization and recognition theorems for these 3-manifolds as well as theoretical generalizations of these results to hig…
New method computes affine normal directions efficiently for sparse polynomials.
problem Computing affine normal directions is computationally expensive in high dimensions.
method Reduces third-order tensor contraction to matrix-free formulation using log-determinant gradient.
result Scalable implementations with near-linear scaling in dimension and sparsity.
Pole ladder improves parallel transport in affine spaces, showing exact results in symmetric spaces.
problem Improving numerical stability and accuracy in parallel transport algorithms.
method Developed a third-order parallel transport scheme using pole ladder in affine connection spaces, showing exact results in symmetric spaces.
result Pole ladder is a third-order scheme in general affine connection spaces and is exact in locally symmetric spaces.
A new ranking algorithm learns data affinity and ranking scores simultaneously.
problem Retrieving similar objects in large databases is challenging.
method Proposes a ranking algorithm that learns data affinity and ranking scores simultaneously, using adaptive neighbors and smoothness constraints.
result The proposed algorithm outperforms existing methods in synthetic and real datasets.
New method builds business taxonomies from corporate reports.
problem Challenging to classify emerging market industries.
method Concept-level hierarchical clustering of annual reports.
result Automatic construction of business taxonomies.
A new method for clustering high-dimensional data into subspaces efficiently and accurately.
problem Inaccurate clustering due to poor intra-subspace similarity in existing methods.
method Iterative Maximum Correlation (IMC) for affinity matrix learning and Piecewise Correlation Estimation (PCE) for densification.
result SDSC framework improves clustering accuracy and efficiency for large-scale data.
New framework discovers non-affine continuous symmetries in neural networks.
problem Lack of efficient methods for detecting non-affine continuous symmetries in neural networks.
method Computational framework for discovering infinitesimal generators of multi-parameter group actions.
result Framework can discover non-affine continuous symmetries in neural networks.
This paper is the first arising from our project announced in math.AG/0211094, "Affine manifolds, log structures, and mirror symmetry." We aim to study mirror symmetry by studying the log structures of Illusie-Fontaine and Kato on degenerations of Calabi-Yau manifolds. The basic idea is that one can associate to certai…
Since the work of Henri Cartan finite dimensional Riemannian symmetric spaces are an important subject of mathematical interest. They are related in a natural way to semisimple Lie groups. In this work we introduce and study their infinite dimensional generalization: Affine Kac-Moody symmetric spaces. Affine Kac-Moody …
The study of logarithmic fields associated with nilmanifolds and their singularities.
problem Understanding logarithmic fields associated with nilmanifolds and their singularities.
method Building a module of an affine Kac Moody vertex algebra and associating logarithmic fields to it.
result Fields associated with specific nilmanifolds have tri-logarithm singularities.
Compact space models automorphisms, proving curvature and linearizability.
problem Modeling automorphisms of affine space.
method Constructing a metric space X with a CW-complex structure and proving curvature properties.
result X is a CAT(0) space for n=3, K of characteristic zero, proving linearizability of automorphisms.
Paper develops algorithms for PWA systems with polynomial regret.
problem Learning in piecewise affine systems due to discontinuities.
method Smoothed online learning framework applied to PWA systems.
result First algorithms with polynomial regret in PWA systems.
Efficient superpixel method for real-time segmentation.
problem Real-time superpixel generation for computer vision tasks.
method Two-stage graph-based framework with Deep Affinity Learning and Hierarchical Entropy Rate Segmentation.
result HERS produces superpixels in near real-time.
Paper explores coarse embeddings between symmetric spaces and Euclidean buildings, answering open questions.
problem Understanding coarse embeddings between symmetric spaces and Euclidean buildings.
method Generalization of quasi-isometric embeddings, focusing on coarse embeddings without Euclidean factors.
result Rank is monotonous under coarse embeddings when the domain does not contain a Euclidean factor.
Paper proposes a method to refine embeddings efficiently.
problem Efficiently refining embeddings after their creation.
method Uses a Domain Adversarial Network (DAN) with constraints.
result Significantly outperforms state-of-the-art unsupervised algorithms.
Let F be a real closed field. We define the notion of a maximal framing for a representation of the fundamental group of a surface with values in Sp(2n,F). We show that ultralimits of maximal representations in Sp(2n,R) admit such a framing, and that all maximal framed represen…
New insights into deep learning via max-affine spline operators.
problem Understanding the inner workings of deep neural networks.
method Mapping deep networks to max-affine spline operators (MASOs).
result MASOs reveal signal-dependent templates and orthogonalization improves performance.
For each positive integer n, Khovanov and Rozansky constructed an invariant of links in the form of a doubly-graded cohomology theory whose Euler characteristic is the sl(n) link polynomial. We use Lagrangian Floer cohomology on some suitable affine varieties to build a similar series of link invariants, and we conject…
Single-head attention approximates any function under various norms.
problem Universal approximation of functions using attention mechanisms.
method Interpreting attention as partitioning and summing linear transformations.
result Single-head attention can approximate any continuous function under L∞-norm and Lebesgue integrable functions under Lp-norm. New framework for equivariant neural networks using Lie group decompositions.
problem Limitations of existing equivariant neural network methods for Lie groups.
method Lie group structure and geometry, decomposition into subgroups and submanifolds.
result Equivariant neural networks for affine transformations outperform previous methods.
There exist natural generalizations of the real moduli space of Riemann spheres based on manipulations of Coxeter complexes. These novel spaces inherit a tiling by the graph-associahedra convex polytopes. We obtain explicit configuration space models for the classical infinite families of finite and affine Weyl groups …
Study optimal policies under budget and coverage constraints.
problem Optimal policy learning with budget and coverage constraints.
method Combination of knapsack structure, affine threshold rule, linear programming relaxation, Greedy-Lagrangian (GLC), and rank-and-cut (RC) algorithms.
result GLC closely approximates the optimal solution and achieves near-optimal performance in finite samples; RC is approximately optimal under certain conditions.
Exact LAD line fitting via PALB with linear scaling and speed.
problem Robust line fitting for data with outliers.
method Piecewise Affine Lower-Bounding (PALB) method using supporting lines and subdivision scheme.
result Empirical log-linear scaling and significantly faster than LP and IRLS methods.
Study geodesic and affine Killing completeness in homogeneous affine surfaces.
problem Geodesic and affine Killing completeness in homogeneous affine surfaces.
method Examined using the solution space of the quasi-Einstein equation.
result Characterized geodesic and affine Killing completeness in homogeneous affine surfaces.
Almost Zoll affine surface found on cylinder.
problem Finding surfaces with special geodesic properties.
method Exhibited an affine structure on a cylinder.
result Affine structure on cylinder is almost Zoll.
This thesis explores DAHA representations using stated skein theory.
problem Understanding the representation theory of double affine Hecke algebras.
method Combining stated skein theory with DAHA, focusing on the A1 DAHA. result Constructed a module of Laurent polynomials for the A1 DAHA. We study affine maps between affine manifolds. Even when the fibers are compact and diffeomorphic, two of them can inherit different affine structures from the source space. This leads to a fixed linear holonomy deformation theory of the affine structure of an affine manifold. We found various conditions which make the…