Study on embedding simplicial complexes in high dimensions with homological constraints.
problem Embedding simplicial complexes in Rd+1 with homological conditions. method Homological obstruction to embedding and deriving upper bounds on top-dimensional faces.
result Existence of an obstruction allowing upper bounds on top-dimensional faces.
It is known that, for a regular riemannian foliation on a compact manifold, the properties of its basic cohomology (non-vanishing of the top-dimensional group and Poincaré Duality) and the tautness of the foliation are closely related. If we consider singular riemannian foliations, there is little or no relation betwee…
Estimates simplicial volume for complex hyperbolic surfaces.
problem Bounding the simplicial volume of complex hyperbolic surfaces.
method Estimates Gromov norm and uses top dimensional class in Hc4. result Explicit upper bound for simplicial volume.
The study proves Liouville-type theorems and Bochner formulas for harmonic maps into CAT(κ) spaces.
problem Analyzing harmonic maps from Riemannian polyhedra into CAT(κ) spaces.
method Computing a target variation formula to derive Liouville-type theorems and Bochner formulas.
result Proves Liouville-type theorems and Bochner formulas for harmonic maps into CAT(1) spaces.
Graph manifolds' Thurston norms are sums of linear functionals, and every such norm can be realized.
problem Understanding Thurston norms of graph manifolds and their realizability.
method Analyzing the structure of Thurston norms as sums of linear functionals and showing realizability.
result Every Thurston norm of a graph manifold can be expressed as a sum of absolute values of linear functionals with rational coefficients.
Study top dimensional cohomology groups of congruence subgroups of SL_n(Z).
problem Investigate cohomology groups of congruence subgroups in top degree.
method Use Tits building and cohomological methods to study the groups.
result Show natural map is surjective but not always injective.
It is shown that the singular set for the Yang-Mills flow on unstable holomorphic vector bundles over compact Kaehler manifolds is completely determined by the Harder-Narasimhan-Seshadri filtration of the initial holomorphic bundle. We assign a multiplicity to irreducible top dimensional components of the singular set …
New method calculates Thurston norm for 3-manifolds with toroidal boundaries.
problem Computing the Thurston norm for 3-manifolds with toroidal boundaries.
method maw dual graph construction and sutured manifold hierarchies.
result Explicit procedure to compute Thurston norm from hierarchies.
We study the bounded fundamental class in the top dimensional bounded cohomology of negatively curved manifolds with infinite volume. We prove that the bounded fundamental class of M vanishes if M is geometrically finite. Furthermore, when M is a R-rank one locally symmetric space, we show that the bou…
Calculates eta-invariants for Berger spheres using special metrics.
problem Calculating eta-invariants for Berger spheres.
method Using special Riemannian metrics on the unit ball, calculating the generating function of the eta-invariants for Berger metrics on odd-dimensional spheres.
result Calculates generating function of eta-invariants for Berger metrics on odd-dimensional spheres.
Non-Euclidean number rings have non-integral Steinberg modules.
problem Characterizing when Steinberg modules are generated by integral elements.
method Analyzing special linear groups over non-Euclidean imaginary number rings.
result Steinberg modules are not generated by integral elements in non-Euclidean rings.
Algorithm checks if geometrically triangulated manifolds are isometric.
problem Determining if two geometric triangulations of manifolds are isometric.
method Sequence of Pachner moves and barycentric subdivisions with bounds on lengths.
result Bounding the length of transformations between triangulations.
In this paper we start the program of constructing generalized special Lagrangian torus fibrations for Calabi-Yau hypersurfaces in toric variety near the large complex limit, with respect to the restriction of a toric metric on the toric variety to the Calabi-Yau hypersurface. The construction is based on the deformati…
Consider a finite connected graph possibly with multiple edges and loops. In discrete geometric analysis, Kotani and Sunada constructed the crystal associated to the graph as a standard realization of the maximal abelian covering of the graph. As an application of what the author showed in an earlier paper with Seshadr…
In \cite{Boed}, C.-F. Bödigheimer constructed a finite cell-complex $\mf{Par}_{g,n,m}$ and a bijective map $\cH: \mf{Dip}_{g,n,m} \to \mf{Par}_{g,n,m}$ (the Hilbert-uniformization) from the moduli space of dipole functions on Riemann surfaces with n directions and m punctures to $\mf{Par}_{g,n,m}$. In \cite{Boed} a…
Anomaly in free fermion theory revealed in functorial field theory.
problem Chiral anomaly in the free fermion theory.
method Detailed construction of anomaly theory as a functor.
result The anomaly theory assigns elements of complex line to manifolds.
Constructs an explicit cycle in arithmetic group cohomology.
problem Cohomology of SLn(Z) at virtual cohomological dimension. method Geometric rigidity of Voronoi tessellations and abstract framework for polyhedral tessellations.
result Explicit canonical cycle in top-dimensional homology of Voronoi complex.
Negative curvature manifolds have vanishing bounded volume class if and only if Cheeger constant is positive.
problem Negative curvature manifolds and their volume classes.
method Integration of volume forms and isoperimetric constants.
result Vanishing of bounded volume class implies positivity of Cheeger constant and vice versa.
We establish combinatorial versions of various classical systolic inequalities. For a smooth triangulation of a closed smooth manifold, the minimal number of edges in a homotopically non-trivial loop contained in the 1-skeleton gives an integer called the combinatorial systole. The number of top-dimensional simplices…
We prove that the cohomological dimension of the Torelli group for a closed connected orientable surface of genus g at least 2 is equal to 3g-5. This answers a question of Mess, who proved the lower bound and settled the case of g=2. We also find the cohomological dimension of the Johnson kernel (the subgroup of the To…
Research connects geometric structures to knot theory and algebraic combinatorics.
problem Understanding the mixed Hodge structure on cohomology of open positroid varieties.
method Relates mixed Hodge structure to Khovanov-Rozansky homology of associated links.
result Rational q,t-Catalan numbers are derived from mixed Hodge polynomials of open positroid varieties. The dualizing module of GL_n(O) varies, affecting cohomology vanishing and nonvanishing.
problem Understanding the dualizing module of GL_n(O) and its impact on cohomology.
method Analyzing the Steinberg module and a variant for GL_n(O), proving vanishing and nonvanishing theorems.
result The dualizing module of GL_n(O) is not always the Steinberg module, but a variant that accounts for orientation.
Springer varieties are studied because their cohomology carries a natural action of the symmetric group Sn and their top-dimensional cohomology is irreducible. In his work on tangle invariants, Khovanov constructed a family of Springer varieties Xn as subvarieties of the product of spheres (S2)n. We show that…
Proves part of the singular set of energy minimizing harmonic maps is a topological manifold.
problem Characterizing the singular set of energy minimizing harmonic maps.
method Analyzes topological and analytic properties of tangent maps.
result Proves part of the singular set is a topological manifold.
We show that if H is a cocommutative Hopf algebra, then there is a natural action of Aut(F_n) on the nth tensor power of H which induces an Out(F_n) action on a quotient \overline{H^{\otimes n}}. In the case when H=T(V) is the tensor algebra, we show that the invariant Tr^C of the cokernel of the Johnson homomorphism s…
New metrics and coordinates for barcode space using group theory.
problem Describing and measuring the space of barcodes.
method Geometric group theory applied to barcodes.
result Stratification of barcode space into regions with similar statistical properties.
We give a surgery formula for the asymptotic behavior of the sequence given by the logarithm of the higher dimensional Reidemeister torsion. Applying the resulting formula to Seifert fibered spaces, we show that the growth of the sequences has the same order as the indices and we give the explicit values for the limits…
The study restricts circle actions on certain manifolds to dimensions 4, 8, and 16.
problem Prohibiting circle actions on manifolds with specific fixed points and topological properties.
method Analyzing the Pontrjagin numbers and signatures of manifolds.
result Only manifolds of dimensions 4, 8, and 16 can have exactly three fixed points under circle actions.
The abstract defines and studies a Tits building for commutative rings and proves a Solomon-Tits theorem under certain conditions.
problem Defining and studying a Tits building for commutative rings.
method Proving a Solomon-Tits theorem for commutative rings under specific conditions, defining Steinberg modules, and computing ranks and lengths.
result Proves a Solomon-Tits theorem for commutative rings satisfying certain conditions.
Optimizes master faces for 2D and 3D face verification using evolutionary algorithms and neural networks.
problem Impersonation attacks using master faces for face-based identity authentication.
method Evolutionary algorithm in latent space of StyleGAN, neural network to direct search, 2D and 3D face reconstruction.
result Obtains high impersonation rates with fewer master faces for 2D and 3D face verification.
Face recognition systems are vulnerable to composite face reconstruction attacks.
problem Vulnerability of face recognition systems to composite face reconstruction attacks.
method Assumed attacker uses composite face parts to reconstruct faces faster and more efficiently.
result Current face recognition systems are extremely vulnerable to random search attacks.
We prove a new structural result for the spherical Tits building attached to SL_n(K) for many number fields K, and more generally for the fraction fields of many Dedekind domains O: the Steinberg module St_n(K) is generated by integral apartments if and only if the ideal class group cl(O) is trivial. We deduce this int…
Unified model for age-invariant face recognition with photorealistic face synthesis.
problem Reliable face recognition across ages remains challenging due to significant intra-class variations.
method Unified deep architecture for cross-age face synthesis and recognition, continuous face rejuvenation/aging, disentangled age-invariant face representations.
result Superior performance on CAFR and other cross-age datasets, promising generalizability to unconstrained face recognition.
This paper converts speech to match a face image and vice versa.
problem Matching speech to a face image and vice versa.
method Proposes a model with speech converter, face encoder/decoder, and voice encoder.
result Trained model converts speech to match a face image and generates a face image that matches the voice of input speech.
Universal adversarial patches prevent face detection in various frameworks.
problem Preventing face detection in state-of-the-art face detection systems.
method Investigated the phenomenon of patches that suppress face detection and proposed optimization-based approaches for automatic design.
result Universal adversarial patches can prevent face detection without introducing false positives.
Generates high-resolution faces based on attributes.
problem Creating realistic face images with user-specified attributes.
method Conditional CycleGAN, handling unpaired data and attribute control.
result Produces realistic face images with user-controlled attributes.
Faces of quasi-arithmetic Coxeter polytopes are also quasi-arithmetic.
problem Characterizing faces of quasi-arithmetic Coxeter polytopes.
method Proof of quasi-arithmetic property of faces and sufficient condition for arithmetic faces.
result Lower-dimensional faces of quasi-arithmetic Coxeter polytopes are quasi-arithmetic.
In this paper, we study face vectors of simplicial posets that are the face posets of cell decompositions of topological manifolds without boundary. We characterize all possible face vectors of simplicial posets whose geometric realizations are homeomorphic to the product of spheres. As a corollary, we obtain the chara…
Study the geometry of matrix multiplication in deep neural networks.
problem Understanding the structure of matrix multiplication in deep neural networks.
method Using quiver representations and equivariant cohomology, determine codimension and irreducible components.
result Codimension and number of top-dimensional irreducible components of matrix multiplication are invariant under permutations and have specific log-canonical thresholds.
Study on critical faces convergence in a Poisson point process.
problem Convergence of point processes associated with critical faces in a Čech filtration.
method Established convergence in M0-topology for critical faces above vanishing threshold. result Obtained limit theorems for positive and negative critical faces.
Estimates the capacity of face representations, providing upper bounds for automatic face recognition.
problem Estimating how many identities a face representation can resolve.
method Formulated as packing bounds on a low-dimensional manifold embedded in a deep representation space, accounting for manifold structure and noise.
result Demonstrated upper bounds of 2.7×10^4 and 8.4×10^4 for FaceNet and SphereFace at a FAR of 1%, respectively.
A new search system helps recall faces with probabilistic methods.
problem Challenging face recall due to large and unstructured search space.
method Probabilistic evolutionary approach using skew-normal distribution.
result Greater granularity, regularized, and realistic results in face recall.
Bayesian optimization generates personalized face stimuli for cognitive neuroscience.
problem Lack of personalized face stimuli in cognitive neuroscience studies.
method Combines GANs with Bayesian optimization to identify individual response patterns to faces.
result Algorithm efficiently generates optimal faces maximizing individual subject's response.
Study on cohomology of SL_n(Z) for n>=3, proving vanishing of certain cohomology groups.
problem Determine the cohomology of SL_n(Z) for n>=3.
method Construct a partial resolution of the Steinberg module to show vanishing of specific cohomology groups.
result Vanishing of codimension-2 rational cohomology group H^{{n \choose 2} -2} for n >= 3.
New method generates fast adversarial faces with high success rate.
problem Vulnerability of face recognition systems to adversarial attacks.
method Fast landmark manipulation method and semantic structure constrained attack.
result 99.86% success rate on state-of-the-art face recognition models.
We find a presentation of symplectic Steinberg modules and show vanishing cohomology for certain groups.
problem Cohomology vanishing for specific groups and modules.
method Presented a symplectic Steinberg module and used it to prove cohomology vanishing.
result Cohomology of Sp2n(Z) vanishes in a specific degree for n≥2. DepthNets learns 3D face geometry and transformations without supervision.
problem Learning 3D face geometry and transformations from a single image.
method Unsupervised learning of facial keypoints depth, using backpropable loss for 3D transformations.
result DepthNets can predict 3D transformations and re-target faces to new poses or geometries.
New method improves Frank-Wolfe for low-rank matrix completion.
problem Low-rank matrix completion problem.
method Extended Frank-Wolfe method with in-face directions.
result Significant speed-ups in computing very low-rank solutions.