This paper uses concave conjugacy theory to analyze self-paced learning.
problem Understanding the intrinsic mechanism of self-paced learning.
method Proposes a concave conjugacy theory to analyze self-paced learning.
result Proves the equivalence of SPL regime and a latent concave objective.
Method constructs CFMMs matching desired payoffs.
problem Creating CFMMs with specific payoff functions.
method Uses convex analysis and Fenchel conjugacy.
result Every concave, nonnegative, nondecreasing, 1-homogeneous payoff has a corresponding convex CFMM.
Study uses braid conjugacy class invariants to find ribbon obstructions for knots.
problem Finding effective obstructions for slice knots to be ribbon.
method Apply Rudolph's work and Khovanov-Lee theory to braid conjugacy class invariants.
result Cannot obtain effective ribbon obstructions using Khovanov-Lee theory.
In this article, we study connections between representation theory and efficient solutions to the conjugacy problem on finitely generated groups. The main focus is on the conjugacy problem in conjugacy separable groups, where we measure efficiency in terms of the size of the quotients required to distinguish a distinc…
We use the theory of group actions on profinite trees to prove that the fundamental group of a finite, 1-acylindrical graph of free groups with finitely generated edge groups is conjugacy separable. This has several applications: we prove that positive, C′(1/6) one-relator groups are conjugacy separable; we provide a…
Study of conjugacy classes in infinite-type surfaces' mapping class groups.
problem Characterizing conjugacy classes in infinite-type surfaces' mapping class groups.
method Model-theoretic methods developed by Kechris, Rosendal, and Truss.
result Detailed classification of conjugacy classes in mapping class groups of infinite-type surfaces.
Study on homeomorphism groups of manifolds using set theory.
problem Relationship between set theory and homeomorphism groups of manifolds.
method First-order rigidity, type versus conjugacy, axiom of constructibility, projective determinacy.
result Under V=L, homeomorphism groups of manifolds are first-order rigid and conjugacy class is determined by type.
This paper explores links from Thompson's group conjugacy classes.
problem Understanding the relationship between Thompson's group conjugacy classes and links.
method Using Jones's construction to link elements of F to unoriented links. result Found sequences of elements from distinct conjugacy classes yielding specific links.
We show that the number of twisted conjugacy classes is infinite for any automorphism of non-elementary, Gromov hyperbolic group . An analog of Selberg theory for twisted conjugacy classes is proposed.
This work improves information concentration for exp-concave distributions, making it dimension-independent.
problem Challenges in information concentration for log-concave distributions with dimension dependence.
method Proves exp-concavity leads to dimension-independent information concentration using a novel variance Brascamp-Lieb inequality.
result Information concentration depends only on the exp-concavity parameter, not the dimension.
Estimates growth of reciprocal classes in Hecke groups.
problem Estimating the growth of reciprocal conjugacy classes in Hecke groups.
method Using free product structure and word lengths of reciprocal elements, with tools from basic probability theory.
result Estimates the asymptotic growth of reciprocal conjugacy classes in Hecke groups.
Classifies theories with eight supercharges using pseudo-periodic maps and Riemann surfaces.
problem Classifying theories with eight supercharges using mathematical tools.
method Assumes theories are given by genus g fibrations of Riemann surfaces, uses pseudo-periodic maps of negative type in mapping class group.
result Identifies dual graphs and 3d mirror quivers, unifies various SCFTs in combinatorial framework.
This thesis consists of three self-contained chapters. The first two concern quantum invariants of links and three manifolds and the third contains results on the word problem for link groups. In chapter 1 we relate the tree part of the Aarhus integral to the mu-invariants of string-links in homology balls thus general…
The paper constructs random concave functions on the unit simplex.
problem Understanding probability measures on spaces of concave functions.
method Constructing random concave functions via a scaled minimum of random hyperplanes.
result There is a transition from deterministic to non-trivial limiting distributions as the number of hyperplanes increases.
Proves log-concavity of cluster algebra coefficients for type An.
problem Log-concavity of cluster algebra coefficients.
method Introduced atomic theta basis and proved log-concavity for type An. result Proved log-concavity of coefficients for cluster algebra variables of type An. The paper studies quandles from symmetric group automorphisms and finds a correspondence with conjugacy classes.
problem Investigating quandles from automorphisms of symmetric groups.
method Developed quandle invariants and proved a one-to-one correspondence between quandles and conjugacy classes.
result There is a one-to-one correspondence between generalized Alexander quandles arising from symmetric groups $\Sf_n$ and the conjugacy classes of $\Sf_n$ for 3≤n≤30 with neq6,15. In this paper we will provide a representation of the penalty term of general dynamic concave utilities (hence of dynamic convex risk measures) by applying the theory of g-expectations.
New algorithms solve word and conjugacy problems in braid group B3.
problem Word and conjugacy problems in braid group B3.
method Classical interpretation of braid group B3 as central extension of modular group, theory of continued fractions.
result Simple and efficient algorithms to solve word and conjugacy problems in braid group B3.
The Links-Gould polynomial of alternating knots is shown to be log-concave and positive.
problem Verifying the positivity and log-concavity of the Links-Gould polynomial for alternating knots.
method Formulated a conjecture and verified it computationally for all 51.3 million knots with up to 19 crossings.
result All but 544 knots satisfy a stronger log-concavity condition.
Paper develops algorithms for solving non-convex non-concave problems with applications in GAN training.
problem Solving non-convex non-concave min-max saddle-point problems.
method Inexact proximal point method with strongly monotone mappings.
result First-order convergence to a nearly stationary solution of the original min-max problem.
The paper explains the concave shape of yield curves from trading perspectives.
problem Lack of explanation for the concavity of yield curves from economics theory.
method Explains the concavity of yield curves from trading perspectives.
result Offers an explanation for the concave shape of yield curves.
New lower bounds for sampling from log-concave distributions in higher dimensions.
problem Proving lower bounds for sampling from log-concave distributions in higher dimensions.
method Multiscale construction inspired by geometric measure theory and reduction to block Krylov algorithms.
result Query lower bounds for sampling from log-concave distributions in higher dimensions are established.
The paper proves complex Monge-Ampère equations with new singularity types and confirms log-concavity of volume.
problem Existence and uniqueness of solutions to complex Monge-Ampère equations with prescribed singularities.
method Proves existence and uniqueness of solutions to complex Monge-Ampère equations with general model type singularities in big cohomology classes.
result Log-concavity of volume of closed positive (1,1)-currents is confirmed.
Developing a non-symmetric strainer theory for spaces with non-negative curvature beyond Alexandrov geometry.
problem Studying spaces with non-Riemannian curvature beyond Alexandrov geometry.
method Introducing a weak quadruple comparison principle and developing a strainer theory.
result Spaces have constant integer dimension, measure contraction property, and unique Banach tangent cones.
The paper develops inequalities for log-concave functions and related surface areas.
problem Understanding log-concave functions and their inequalities.
method Establishing new inequalities through f-divergences and functional affine surface areas.
result New inequalities on functional affine surface area and bounds for Kullback-Leibler divergence.
We characterize in terms of the Goldman Lie algebra which conjugacy classes in the fundamental group of a surface with non empty boundary are represented by simple closed curves. We prove the following: A non power conjugacy class X contains an embedded representative if and only if the Goldman Lie bracket of X with th…
Optimizes portfolios using CPT utility via convex optimization.
problem Maximizing CPT utility in portfolio selection.
method Minorization-maximization (MM) algorithm and convex-concave (CC) procedure.
result Problems can be solved globally and efficiently.
New bound on sliding circuit set size in braid groups.
problem Bounding the size of sliding circuit sets in braid groups.
method Constructing examples of braids with multiple subsurfaces to suggest a geometric property.
result Found a family of braids with a sliding circuit set of at most C⋅LN−2 elements, suggesting a geometric property. The conformal module of conjugacy classes of braids implicitly appeared in a paper of Lin and Gorin in connection with their interest in the 13. Hilbert Problem. This invariant is the supremum of conformal modules (in the sense of Ahlfors) of certain annuli related to the conjugacy class. This note states that the conf…
Extends earthquake and horocycle flows to new measures.
problem Ergodic theory of earthquake flow on measured laminations.
method Generalizes shear coordinates to arbitrary measured laminations.
result Classifies ergodic measures for P action on bundle of quadratic differentials.
We prove that fundamental groups of non-orientable 3-manifolds have a solvable conjugacy problem, and construct an algorithm. Together with our earlier work on the conjugacy problem in groups on orientable geometrizable 3-manifolds, all π1 of (geometrizable) 3-manifolds have a solvable conjugacy problem. In corollar…
New bounds for generative models under weaker assumptions.
problem Establishing convergence guarantees for generative models under weak assumptions.
method Non-asymptotic 2-Wasserstein distance bounds for probability flow ODEs under weak log-concavity and Lipschitz continuity.
result Concrete convergence rates for generative models, including non-log-concave distributions.
New information geometry from exponentially concave functions.
problem Optimal transport and information geometry.
method Information geometry tools applied to exponentially concave functions.
result New information geometry on the unit simplex.
Study weak conjugacy in surface homeomorphisms.
problem Understanding weak conjugacy in homeomorphisms of surfaces.
method Exploring the group of homeomorphisms isotopic to the identity.
result New insights into weak conjugacy relations.
Proves tropical Hodge theory for smooth projective varieties, conditional on Laplacian regularity.
problem Proving log-concavity of characteristic polynomials of matroids.
method Combinatorial approach, conditional proof of Kähler package.
result Conditional proof of Kähler package for tropical cohomology.
Bayesian inference becomes tractable with log-concave priors and targets.
problem Transforming samples from a prior to a posterior distribution efficiently.
method Optimal transport theory and convex optimization.
result Log-concave priors and targets allow for efficient Bayesian inference.
Paper analyzes GDA for nonconvex-concave problems, showing efficient convergence.
problem Solving nonconvex-concave minimax problems efficiently.
method Two-time-scale gradient descent ascent algorithm.
result GDA can find stationary points efficiently in nonconvex-concave settings.
Classifies manifolds with dense conjugacy classes in their mapping class groups.
problem Classifying manifolds based on conjugacy classes in their mapping class groups.
method Analyzing connected orientable 2-manifolds and their mapping class groups.
result Mapping class groups of certain manifolds have dense conjugacy classes.
Study conjugacy classes of parabolic diffeomorphisms fixing the origin.
problem Understanding conjugacy classes of parabolic diffeomorphisms fixing the origin.
method Establish results on differentiability classes and order of tangency, focusing on the invariance of residues under low-regular conjugacies.
result Sharp results on invariance of residues under low-regular conjugacies, extending previous work on Schwarzian derivatives.
Develops parabolic pluripotential theory for complex flows.
problem Complex Monge-Ampère equations in degenerate settings.
method Study of semi-concave envelopes and unique solutions.
result Shows semi-concave envelopes as unique solutions.
We show that every non-decreasing function f:N→N bounded from above by an for some a≥1 can be realized (up to a natural equivalence) as the conjugacy growth function of a finitely generated group. We also construct a finitely generated group G and a subgroup H≤G of index 2 such…
The paper extends mixability theory to function-valued forecasts, proving various loss functions are mixable.
problem Efficient aggregation of functional and probabilistic forecasts in online prediction games.
method Adapting mixable and exponentially concave loss functions to function-valued forecasts.
result Various loss functions used for probabilistic forecasting are mixable (exp-concave).
Folded concave penalization methods have been shown to enjoy the strong oracle property for high-dimensional sparse estimation. However, a folded concave penalization problem usually has multiple local solutions and the oracle property is established only for one of the unknown local solutions. A challenging fundamenta…
Study finds optimal investment strategies in incomplete markets.
problem Optimal investment in non-concave, non-smooth utility functions.
method Dynamic programming and measurable selection arguments.
result Existence of optimal portfolio in genericly incomplete markets.
The paper classifies orientation reversing periodic maps on surfaces.
problem Classifying orientation reversing periodic maps on closed surfaces.
method Developed a group of data for each map and criteria to judge conjugacy.
result Given orientation reversing periodic maps on Σ_g with period ≥ 3g are conjugate to specific types of maps.
Criterion for stopping conjugacy class enumeration in triangle groups.
problem Enumerating all conjugacy classes in cocompact triangle groups.
method Encoding by P. Dehornoy and T. Pinsky; stopping criterion based on geometric length.
result Stopping criterion for the generation of conjugacy classes in cocompact triangle groups.
Authors determine the SL(2,C) character variety of a specific knot without computer aid.
problem Determine the SL(2,C) character variety of a specific knot without computational assistance.
method Develop an efficient method for working with conjugacy classes of four elements of SL(2,C).
result Determine the character variety of the knot 8_18 efficiently and software-free.
Paper derives asymptotic formula for conjugacy classes in groups with contracting elements.
problem Counting conjugacy classes in groups with contracting elements.
method Statistical convex-cocompact actions with contracting elements.
result Asymptotic formula for conjugacy classes, showing transcendental conjugacy growth series for certain groups.