These are notes based on a series of talks that the author gave at the "Interactions between hyperbolic geometry and quantum groups" conference held at Columbia University in June of 2009.
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These are mostly expository notes based on the course of lectures on arithmetic invariants of hyperbolic manifolds given at the workshop associated with the final "Volume Conference," held at Columbia University, June 2009. Some new results are included.
Replicates deep learning strategy for trading factor residuals, finds strong performance.
problem Exploiting mis-pricing from unexplained cross-sectional variation in factor models.
method Adhering to PIT principles, used CNNs and Transformers on recent data.
result Out-of-sample Sharpe ratios exceeding 10 in certain tests.
In the first of these two lectures, I use a comparison to symplectic Khovanov homology to motivate the idea that the Jones polynomial and Khovanov homology of knots can be defined by counting the solutions of certain elliptic partial differential equations in 4 or 5 dimensions. The second lecture is devoted to a descri…
Our goal is to resolve a problem proposed by Fernholz and Karatzas [On optimal arbitrage (2008) Columbia Univ.]: to characterize the minimum amount of initial capital with which an investor can beat the market portfolio with a certain probability, as a function of the market configuration and time to maturity. We show …
In this lecture notes, we aim at giving an introduction to the Kähler-Ricci flow (KRF) on Fano manifolds. It covers some of the developments of the KRF in its first twenty years (1984-2003), especially an essentially self-contained exposition of Perelman's uniform estimates on the scalar curvature, the diameter, and th…
Functional data analysis is a statistical framework where data are assumed to follow some functional form. This method of analysis is commonly applied to time series data, where time, measured continuously or in discrete intervals, serves as the location for a function's value. Gaussian processes are a generalization o…
We present an efficient algorithm for simultaneously training sparse generalized linear models across many related problems, which may arise from bootstrapping, cross-validation and nonparametric permutation testing. Our approach leverages the redundancies across problems to obtain significant computational improvement…
Is There a Real-Estate Bubble in the US?physics.soc-ph
We analyze the quarterly average sale prices of new houses sold in the USA as a whole, in the northeast, midwest, south, and west of the USA, in each of the 50 states and the District of Columbia of the USA, to determine whether they have grown faster-than-exponential which we take as the diagnostic of a bubble. We fin…
Deep networks are well-known to be fragile to adversarial attacks. We conduct an empirical analysis of deep representations under the state-of-the-art attack method called PGD, and find that the attack causes the internal representation to shift closer to the "false" class. Motivated by this observation, we propose to …
Federated learning predicts financial distress across U.S. states without centralizing data.
problem Predicting financial distress across U.S. states using sensitive data without centralization.
method Cross-silo federated learning, interpretable AI techniques, machine learning model for categorical data.
result Identifies both global and state-specific predictors of financial hardship.
Optimizes electric aircraft deployment for Canadian aviation to reduce emissions.
problem Limited fleet capacity and operational structure hinder electric aircraft transition.
method Multi-period mixed-integer linear programming (MILP) framework.
result Electric aircraft can reduce emissions by over 70% within five years.
Machine learning and deep learning infer surface/groundwater exchange from temperature data.
problem Inferring surface/groundwater exchange from temperature data with high temporal resolution.
method Application of machine learning and deep learning algorithms to infer surface/groundwater exchange flux from subsurface temperature observations.
result DL methods outperform ML methods in interpreting noisy temperature data, especially with a smoothing filter.
Modeling daily river flow distribution with seasonal and long-term trends.
problem Capturing both seasonal and gradual long-term changes in environmental variables.
method Distributional regression using GAMLSS framework to estimate daily distribution of river flows.
result Model successfully captures seasonal variation and long-term trends in river flow data.
Novel approach to universal online learning for bounded losses, closing open problems.
problem Characterizing processes for universal online learning under non-i.i.d. conditions.
method Characterization of processes admitting strong and weak universal learning, introduction of optimistically universal learning rule.
result Introduction of a novel 1NN algorithm that is optimistically universal for bounded losses.
Universal Gaussian parity proven for 2D knots.
problem Proving universal Gaussian parity for 2D knots.
method Analyzing Gaussian parity on free 2D knots.
result Gaussian parity is universal for 2D knots.
GCNNs gain rotation invariance with more training augmentation, making SVD-Universal more effective.
problem Improving robustness of GCNNs to adversarial attacks.
method SVD-Universal technique applied to GCNNs trained with larger rotations.
result SVD-Universal becomes more effective as GCNNs gain rotation invariance.
The study of universal links in 3-manifolds and their properties.
problem Existence and characterization of universal links in 3-manifolds.
method Analyzing branched coverings and distinguishing between universal and complement universal links.
result Closed spherical 3-manifolds are the only ones admitting universal links.
Constructs universal link invariants from intersections in configuration spaces.
problem Globalise topologically all coloured Jones polynomials and ADO polynomials.
method Defines new link invariants from graded intersections in configuration spaces.
result Recover all coloured Jones polynomials and ADO polynomials for links.
We construct universal Lefschetz fibrations, defined in analogy with classical universal bundles. We also introduce the cobordism groups of Lefschetz fibrations, and we see how these groups are quotients of the singular bordism groups via the universal Lefschetz fibrations.
We present a universal knot polynomials for 2- and 3-strand torus knots in adjoint representation, by universalization of appropriate Rosso-Jones formula. According to universality, these polynomials coincide with adjoined colored HOMFLY and Kauffman polynomials at SL and SO/Sp lines on Vogel's plane, and give their ex…
Revives Vogel's diagrammatic technique for universal Lie algebra computations.
problem The universality of Lie algebra quantities remains open, despite many being described.
method Diagrammatic algebra based on Vogel's Λ-algebra.
result Diagrammatic technique enables truly universal computations in Lie theory.
Universal locally univalent functions and universal conformal metrics with constant curvaturemath.CV
We prove Runge-type theorems and universality results for locally univalent holomorphic and meromorphic functions. Refining a result of M. Heins, we also show that there is a universal bounded locally univalent function on the unit disk. These results are used to prove that on any hyperbolic simply connected plane doma…
Simple technique turns any adversarial attack into a universal one using few test examples.
problem Creating universal adversarial attacks with minimal data.
method Universalization technique using few adversarial test examples and spectral properties.
result Simple universalization technique achieves comparable fooling rates to state-of-the-art methods.
Some universal inequalities for Dirichlet eigenvalues of the Laplacian on a Euclidean convex domainmath.SP
Universal inequalities found for Laplacian eigenvalues on convex domains.
problem Finding bounds for Laplacian eigenvalues on convex domains.
method Established two universal inequalities.
result Found new bounds for Laplacian eigenvalues.
Universal inequalities for Laplacian eigenvalues on discrete groups.
problem Proving inequalities for Laplacian eigenvalues on discrete groups.
method Analyzing Laplacian eigenvalues with Dirichlet boundary conditions on subsets of discrete groups.
result Yang-type universal inequalities for Cayley graphs of amenable groups and the d-regular tree.
New maps connect universal circles to ideal sphere for hyperbolic manifolds.
problem Understanding universal circles for Anosov foliations with branching.
method Introduced a new type of Cannon--Thurston map for leftmost universal circles.
result Fundamental group acts on leftmost universal circle with pseudo-Anosov dynamics.
This paper studies universal rates of ERM for binary classification under agnostic learning.
problem The challenge of achieving universal rates of ERM for binary classification under agnostic learning.
method The paper explores the agnostic universal rates of ERM for binary classification, revealing three possible rates: , , or arbitrarily slow.
result The paper provides a complete characterization of which concept classes fall into each of the three categories of agnostic universal rates.
Develops a new approach to establish universality for any-dimensional machine learning models.
problem Understanding universality for models with inputs of varying sizes.
method Identifies any-dimensional functions with a unique function in an infinite-dimensional limit space, using symmetries and relations between inputs of different sizes.
result Establishes universality for several existing architectures and proposes modifications to restore it.
Softmax attention approximates complex functions and subsumes many known universal approximators.
problem Universal approximation of continuous sequence-to-sequence functions.
method Interpolation-based analysis of attention's internal mechanism, showing its ability to approximate ReLU functions.
result Softmax attention is a universal approximator for continuous sequence-to-sequence functions.
MLPs can approximate any function in context, challenging the importance of in-context universality.
problem Understanding why transformers are more effective than classical models.
method Proved MLPs with trainable activation functions are universal in context.
result Transformer success is likely due to factors other than in-context universality.
Solves open problem on universally consistent online learning with unbounded losses.
problem Open problem on universally consistent online learning with unbounded losses.
method Constructs random measurable partitions of the instance space.
result Simple memorization rule is optimistically universal for any unbounded loss.
Proves universal pairing result for 2-complexes, showing lack of positivity.
problem Detecting equivalence relations in 2-complexes.
method Analogous to Freedman et al. for manifolds, but for 2-complexes.
result Universal pairing does not detect simple homotopy equivalence vs 3-deformations for 2-complexes.
The paper explores universal circles for Anosov foliations and their uniqueness.
problem Exploring the uniqueness of universal circles for Anosov foliations.
method Using the flow space of an Anosov flow to parameterize the circle bundle at infinity of the foliations.
result Several constructions of a universal circle are typically distinct and not conjugate.
The paper tightens bounds on distances between Reeb graphs.
problem Certifying quasi-universality of distances between Reeb graphs.
method Establishes tight bi-Lipschitz bounds for various distances.
result Proves strict universality of the functional contortion distance for contour trees and coincides with interleaving distance for merge trees.
A universal LSTM model outperforms asset-specific models in forecasting stock volatilities.
problem Forecasting stock volatilities across different assets.
method Trained an LSTM network on a pooled dataset of liquid stocks to forecast daily realized volatilities.
result The LSTM model consistently outperforms other asset-specific parametric models in volatility forecasting.
The universal Liouville action equals the renormalized volume of a hyperbolic 3-manifold.
problem Understanding the geometric significance of the universal Liouville action.
method Analyzing the Weil-Petersson universal Teichmüller space and its relation to hyperbolic 3-manifolds.
result The gradient flow of the universal Liouville action converges to the origin, providing a bound on Weil-Petersson distance.
Using deep neural networks that are either invariant or equivariant to permutations in order to learn functions on unordered sets has become prevalent. The most popular, basic models are DeepSets [Zaheer et al. 2017] and PointNet [Qi et al. 2017]. While known to be universal for approximating invariant functions, DeepS…
MAT combines meta-learning and adversarial training to defend against universal patches.
problem Defending against universal patches that fool models in various contexts.
method Meta adversarial training (MAT) integrates meta-learning with adversarial training.
result MAT increases robustness against universal patch attacks on image classification and traffic-light detection.
Study of Gödel Universe as Lie group with specific metric.
problem Characterize geodesics in the Gödel Universe.
method Geometric theory of optimal control applied to Lie groups with left-invariant Lorentz metrics.
result No closed timelike or isotropic geodesics in the Gödel Universe.
The universal sl_2 invariant of string links has a universality property for the colored Jones polynomial of links, and takes values in the h-adic completed tensor powers of the quantized enveloping algebra of sl_2. In this paper, we exhibit explicit relationships between the universal sl_2 invariant and Milnor invaria…
We introduce a class of metrics on gauge theoretic moduli spaces. These metrics are made out of the universal matrix that appears in the universal connection construction of M. S. Narasimhan and S. Ramanan. As an example we construct metrics on the c_{2}=1 SU(2) moduli space of instantons on R^4 for various universal m…
We extend a recently proposed 1-nearest-neighbor based multiclass learning algorithm and prove that our modification is universally strongly Bayes-consistent in all metric spaces admitting any such learner, making it an "optimistically universal" Bayes-consistent learner. This is the first learning algorithm known to e…
Study classifies twist knots with maximal self-linking number in S^3.
problem Classifying transverse-universal knots in S^3.
method Classification of transverse twist knots with maximal self-linking number.
result Obtained an infinite family of non-transverse-universal knots.
Universal approximation for stochastic processes using Brownian motion.
problem Approximating stochastic processes with linear functionals.
method Establishing -type universal approximation theorems for rough path spaces.
result Linear functionals on the signature of time-extended Brownian motion can approximate any -integrable stochastic process.
Deep networks have recently been shown to be vulnerable to universal perturbations: there exist very small image-agnostic perturbations that cause most natural images to be misclassified by such classifiers. In this paper, we propose the first quantitative analysis of the robustness of classifiers to universal perturba…
Universal geometry organizes heterotic moduli spaces.
problem Understanding the structure of heterotic compactifications.
method Fibering compactification data over moduli space and analyzing universal curvatures.
result Deformations are components of universal curvatures, incorporating corrections.
Proves DCNNs with expansive convolution are strongly universally consistent.
problem Theoretical consistency of deep convolutional neural networks (DCNNs).
method Empirical risk minimization on DCNNs with expansive convolution (with zero-padding).
result DCNNs with expansive convolution are strongly universally consistent.