Study groups with contracting elements using SCC actions.
problem Understanding the asymptotic geometry of groups with contracting elements.
method Exploiting an extension lemma to prove properties of SCC actions.
result Groups with SCC actions contain large free sub-semigroups, have purely exponential growth, and have barrier-free sets with a growth-tight property.
The paper generalizes the Black-Scholes model and inequality.
problem Developing a new model for call price surfaces.
method Introducing a noncommutative semigroup structure and using convex order compatibility.
result Each one-parameter semigroup corresponds to a unique log-concave probability density.
Homotopy equivalent to spheres, free factor complex of a free group.
problem Understanding the homotopy type of free factor complexes.
method Proving homotopy equivalence and connectivity of relevant complexes.
result Homotopy equivalence of free factor complexes to spheres and other complexes.
The paper creates knot invariants using free groups.
problem Invariants of free knots (virtual knots).
method Constructing invariants valued in free groups.
result Series of invariants for free knots.
Proves incoherence of free-by-free and surface-by-free groups, solving two problems.
problem Proving incoherence of free-by-free and surface-by-free groups.
method Semidirect product construction and virtual algebraic fibering analysis.
result Proves incoherence of free-by-free and surface-by-free groups, answering a question posed by J. Hillman.
The notion of free link is a generalized notion of virtual link. In the present paper we define the group of free braids, prove the Alexander theorem that all free links can be obtained as closures of free braids and prove a Markov theorem, which gives necessary and sufficient conditions for two free braids to have the…
Corrects a 1998 proof about free factors of free groups.
problem Verifying the proof of free factors in free groups.
method Analyzes the geometric realization of free factors.
result Geometric realization is homotopy equivalent to a wedge of spheres.
The pants graph of a free group is constructed and studied.
problem Understanding the structure of free groups through graph theory.
method Developed a pants graph and studied its properties.
result The pants graph of a free group is connected and unbounded.
Foundations for free boundary Brakke flows established.
problem Analyzing free boundary flows through singularities
method Introducing unit-regular and cyclic free boundary flows, proving avoidance principle, and introducing inner and outer flows.
result General tools for analyzing free boundary flows through singularities.
The isomorphism problem for [free abelian]-by-free groups is unsolvable.
Free surface-links are shown to be ribbon links.
problem Characterizing surface-links as ribbon links.
method Four proofs are provided, including a stabilization approach.
result Every free surface-link is a ribbon surface-link.
Defines a new free product for coarse spaces.
problem No specific problem stated; focuses on a new mathematical concept.
method Defines and analyzes free products for coarse spaces.
result Free products preserve coarse properties and have a dimension bound.
A Seifert surface F for a knot K is free if the complement of F is a handlebody (i.e., has free fundamental group). The free genus of K is the minimum genus among all free Seifert surfaces for K. In this paper we show that there exist families of hyperbolic knots with arbitrarily large volume, which each have free genu…
We investigate cobordisms of free knots. Free knots and links are also called homotopy classes of Gauss words and phrases. We define a new strong invariant of free knots which allows to detect free knots not cobordant to the trivial one.
The paper classifies fixed subgroups of endomorphisms in free-abelian times surface groups.
problem Characterizing fixed subgroups of endomorphisms in specific group structures.
method Study of endomorphisms, classification of fixed subgroups, and equivalent conditions for end-fixed subgroups.
result Complete classification of fixed subgroups in free-abelian times surface groups.
We develop the notion of Brakke flow with free-boundary in a barrier surface. Unlike the classical free-boundary mean curvature flow, the free-boundary Brakke flow must "pop" upon tangential contact with the barrier. We prove a compactness theorem for free-boundary Brakke flows, define a Gaussian monotonicity formula v…
Spectral gap theorem for free products of groups.
problem Understanding the spectral gap of elements in free products of groups.
method Analyzing the structure of free products and properties of elements in their commutator subgroup.
result The spectral gap of a non-conjugate element in a free product of groups is at least 1/2.
We prove that the free splitting complex of a finite rank free group, also known as Hatcher's sphere complex, is hyperbolic.
Paper generalizes Minkowski inequality for umbilical hypersurfaces with free boundary.
problem Optimal weighted Minkowski inequality for umbilical hypersurfaces with free boundary.
method Generalization of Xia's result to free boundary case in space forms.
result Free boundary version of optimal weighted Minkowski inequality in space forms.
A simple property of Weyl tensor in shear-free, vorticity-free, acceleration-free velocity fields.
problem Proving a property of the Weyl tensor in specific velocity fields.
method Analyzing the Weyl tensor's divergence and contraction properties in shear-free, vorticity-free, acceleration-free velocity fields.
result The covariant divergence of the Weyl tensor is zero if the contraction of the Weyl tensor with the velocity is zero, and vice versa.
The study shows that certain groups can be uniquely identified by their finite abelian summands.
problem Identifying groups based on their finite abelian summands.
method Analyzing hyperbolic groups as graphs of free groups with cyclic edge groups.
result Free products of free and surface groups are profinitely rigid.
We define risk-free portfolios using three gauge invariant differential operators that require such portfolios to be insensitive to price changes, to be self-financing, and to produce a zero real return so there are no risk-free profits. This definition identifies the risk-free rate as the return of an infinitely diver…
The paper identifies manifolds with free circle actions.
problem Characterizing manifolds with free circle actions.
method Analyzing (n−1)-connected (2n+1)-manifolds with torsion-free homology. result Determining manifolds admitting free circle actions up to almost diffeomorphism.
Proves almost profinite rigidity for certain free-by-cyclic groups.
problem Profinite rigidity of free-by-cyclic groups.
method Analyzes ranks of fibres, characteristic polynomials, and stretch factors of monodromies.
result Generic free-by-cyclic groups are almost profinitely rigid.
Elementarily free groups are the finitely generated groups with the same elementary theory as free groups. We prove that elementarily free groups are subgroup separable, answering a question of Zlil Sela.
We examine free orientation-reversing group actions on orientable handlebodies, and free actions on nonorientable handlebodies. A classification theorem is obtained, giving the equivalence classes and weak equivalence classes of free actions in terms of algebraic invariants that involve Nielsen equivalence. This is app…
The paper defines conditions for a free-by-free group to be hyperbolic.
problem Conditions for a free-by-free group to be relatively hyperbolic.
method Necessary and sufficient conditions involving exponentially growing automorphisms and invariant subgroup systems.
result A subgroup system can be used to construct peripheral subgroups making the extension hyperbolic.
Author provides an alternate proof of the free ribbon lemma.
problem Proving that every free sphere-link in the 4-sphere is a ribbon sphere-link.
method An alternate proof of the free ribbon lemma.
result Provides an alternate proof of the free ribbon lemma.
Conditions for hyperbolic and relatively hyperbolic extensions of free groups using automorphisms with fixed points.
problem Conditions for hyperbolic and relatively hyperbolic extensions of free groups.
method Using dynamics of outer automorphisms on the complex of free factors and investigating the geometry of the extension group.
result Conditions for hyperbolic and relatively hyperbolic extensions of free groups using automorphisms with fixed points.
We show how to derive hyperbolicity of the free factor complex of FN from the Handel-Mosher proof of hyperbolicity of the free splitting complex of FN, thus obtaining an alternative proof of a theorem of Bestvina-Feighn. We also show that under the natural map τ from the free splitting complex to free factor co…
ScheduleFree+ improves large language model training without schedules or learning rates.
problem Scaling up Schedule-Free Learning to large language models.
method Learning-rate-free and schedule-free method for training large language models.
result ScheduleFree+ outperforms SOTA schedules by 31% at 1000 tokens per parameter.
Proves a connectivity conjecture for free groups, showing homotopy type of spheres.
problem Establishing a connectivity conjecture for free groups.
method Provided homotopy-equivalent models of the common basis complex using free factors and sphere systems.
result The common basis complex of a free group of rank n has the homotopy type of a wedge of spheres of dimension 2n-3.
Study proves the critical catenoid is the free boundary minimal annulus.
problem Characterizing free boundary minimal surfaces.
method Holomorphic techniques from Fraser and Schoen.
result Proves the critical catenoid is the free boundary minimal annulus.
New prime amphicheiral knots found with a specific period.
problem Finding amphicheiral knots with a specific free period.
method Constructed new prime amphicheiral knots with free period 2.
result Solved an open question about amphicheiral knots and free periods.
The paper solves sliceness for odd free knots.
problem Determining if odd free knots have a spanning disc.
method Using the pairing of knot diagram chords.
result An explicit criterion for sliceness of odd free knots.
Study on profinite rigidity of direct products of free and surface groups.
problem Profinite rigidity of direct products in residually free groups.
method Rank gradients of pro-p groups, virtual homology of limit groups. result Direct products of free and surface groups are profinitely rigid.
Survey on recent advances in free boundary minimal surfaces in unit ball.
problem Constructing and understanding free boundary minimal surfaces in the unit ball.
method Review of various techniques and examples, uniqueness results, Morse index estimates, eigenvalue estimates, and compactness results.
result Discussion of critical catenoid uniqueness conjecture and sharp area bounds.
Sharp geometric inequalities for free boundary hypersurfaces in balls.
problem Understanding geometric properties of free boundary hypersurfaces in balls.
method Proving a family of sharp geometric inequalities.
result Family of sharp geometric inequalities for free boundary hypersurfaces in balls.
Study shows boundary curves of free boundary minimal surfaces are circles.
problem Characterizing boundary curves of free boundary minimal surfaces.
method Holomorphic techniques applied to the unit ball in Euclidean 3-space.
result Intersection curves of free boundary minimal surfaces with the unit sphere are all circles.
In this paper we first state the classification of the prolongations of complex free fundamental graded Lie algebras. Next we introduce the notion of free pseudo-product fundamental graded Lie algebras and study the prolongations of complex free pseudo-product fundamental graded Lie algebras. Furthermore we investigate…
The free factor complex of rank 4+ fails a combinatorial isoperimetric inequality.
problem Failure of combinatorial isoperimetric inequality in the free factor complex.
method Construction of a coarsely Lipschitz function from the upward link of a free factor to integers.
result A loop in the free factor complex requires linearly growing number of 2-simplices to fill.
Classifies compact multiplicity free quasi-Hamiltonian manifolds.
problem Classifying compact, multiplicity free, quasi-Hamiltonian manifolds.
method Symplectic reductions and Lie group analysis.
result Recover old and find new examples of these structures.
New method achieves optimal performance without needing problem parameters.
problem Parameter-free stochastic optimization in non-convex and convex settings.
method Simple hyperparameter search technique for non-convex setting, and method with stochastic gradients for convex setting.
result Fully parameter-free methods can outperform state-of-the-art algorithms in both non-convex and convex settings.
Estimates the growth of Morse index for free boundary minimal hypersurfaces.
problem Estimating the Morse index of free boundary minimal hypersurfaces.
method Adapting Song's method to the free boundary case.
result The Morse index grows linearly with the sum of Betti numbers.
Study conjugacy growth in free groups and products of finite groups.
problem Counting conjugacy classes of commutators in free groups and products of finite groups.
method Classification of commutators in free groups and products by Wicks, building on Rivin and Sharp's work.
result Asymptotic formula for conjugacy classes of commutators with given word length.
Study shows top homology group isn't dualizing module for automorphism group of free groups.
problem Identifying the dualizing module for automorphism group of free groups.
method Analyzing the top homology group of the free factor complex.
result The top homology group is not the dualizing module for extAut(Fn), especially for n=5. Outer automorphisms of free products are represented by CTs.
problem Representing outer automorphisms of free products by relative train track maps.
method Developed theory of attracting laminations and CTs for free products.
result Outer automorphisms of free products satisfy an index inequality.
Affine connections become simple spaces locally.
problem Understanding affine connections and their properties.
method Combinatorial and geometric argument.
result Affine connections are locally affine spaces.