Zimmermann's forest formula is the corner stone of perturbative renormalization in QFT. By renormalizing individual Feynman graphs, it generates the UV finite S-matrix. This approach to renormalization makes the graph and all its forests center pieces in the theory of renormalization. On the other hand the positive geometry program delegate the role of Feynman graphs as secondary to the amplitude itself, which are generated by canonical forms associated to positive geometries. State of the art in this program is the convergence of S-matrix theory in local QFTs and string theory as the scattering amplitudes in QFT arise as integrals over certain moduli spaces. These integrals are known as curve integrals. For theories such as $\textrm{Tr}(\Phi^{3})$ theory with massive colored scalars, these integrals are divergent in the UV and have to be regularized. It is then natural to ask if there is a ``forest-like formula'' for these integrals which produce a renormalized amplitude without needing to explicitly invoke the forests associated to divergent subgraphs. In this paper, we initiate such a program by deriving forest-like formula for planar massive $\textrm{Tr}(\Phi^{3})$ amplitudes in $D = 4$ dimensions. Our analysis relies on the insightful manifestation of the forest formula derived by Brown and Kreimer in \cite{Brown:2011pj}, that lead us to a definition of ``tropical counter-term'' for the bare amplitude.

The connection between topology and quantum mechanics is one of the cornerstones of modern physics. Several examples of current interest like the Aharonov-Bohm effect in quantum mechanics, monopoles and instantons in quantum field theory, the quantum Hall effect in condensed matter physics, anyons in topological quantum computation, and the AdS-CFT correspondence in string theory illustrate this connection. Since classical mechanics is a limiting case of quantum mechanics, it behooves us to ask how topology impacts classical mechanics. Topological considerations do play an important role in the classical context too, for example in fluid vortices and atmospheric dynamics. With a desire to understand this connection more deeply, we study classical mechanics on finite spaces. Towards this end, we use the formalism developed by Bering and identify the corrections to the Klein-Gordon equation due to the presence of the boundary. We solve the modified equation in various dimensions under suitable assumptions of symmetry.

We develop further previous work on de Sitter extremal surfaces and time entanglement structures in quantum mechanics. In the first part, we first discuss explicit quotient geometries. Then we construct smooth bulk geometries with replica boundary conditions at the future boundary and evaluate boundary Renyi entropies in $dS/CFT$. The bulk calculation pertains to the semiclassical de Sitter Wavefunction and thus evaluates pseudo-Renyi entropies. In 3-dimensions, the geometry in quotient variables is Schwarzschild de Sitter. The 4-dim $dS$ geometry involves hyperbolic foliations and is a complex geometry satisfying a regularity criterion that amounts to requiring a smooth Euclidean continuation. Overall this puts on a firmer footing previous Lewkowycz-Maldacena replica arguments based on analytic continuation for the extremal surface areas via appropriate cosmic branes. In the second part (independent of de Sitter), we study various aspects of time entanglement in quantum mechanics, in particular the reduced time evolution operator, weak values of operators localized to subregions, a transition matrix operator with two copies of the time evolution operator, autocorrelation functions for operators localized to subregions, and finally future-past entangled states and factorization.

Multimodal learning, a rapidly evolving field in artificial intelligence, seeks to construct more versatile and robust systems by integrating and analyzing diverse types of data, including text, images, audio, and video. Inspired by the human ability to assimilate information through many senses, this method enables applications such as text-to-video conversion, visual question answering, and image captioning. Recent developments in datasets that support multimodal language models (MLLMs) are highlighted in this overview. Large-scale multimodal datasets are essential because they allow for thorough testing and training of these models. With an emphasis on their contributions to the discipline, the study examines a variety of datasets, including those for training, domain-specific tasks, and real-world applications. It also emphasizes how crucial benchmark datasets are for assessing models' performance in a range of scenarios, scalability, and applicability. Since multimodal learning is always changing, overcoming these obstacles will help AI research and applications reach new heights.

Dynamic networks are a complex subject. Not only do they inherit the complexity of static networks (as a particular case); they are also sensitive to definitional subtleties that are a frequent source of confusion and incomparability of results in the literature. In this paper, we take a step back and examine three such aspects in more details, exploring their impact in a systematic way; namely, whether the temporal paths are required to be \emph{strict} (i.e., the times along a path must increasing, not just be non-decreasing), whether the time labeling is \emph{proper} (two adjacent edges cannot be present at the same time) and whether the time labeling is \emph{simple} (an edge can have only one presence time). In particular, we investigate how different combinations of these features impact the expressivity of the graph in terms of reachability. Our results imply a hierarchy of expressivity for the resulting settings, shedding light on the loss of generality that one is making when considering either combination. Some settings are more general than expected; in particular, proper temporal graphs turn out to be as expressive as general temporal graphs where non-strict paths are allowed. Also, we show that the simplest setting, that of \emph{happy} temporal graphs (i.e., both proper and simple) remains expressive enough to emulate the reachability of general temporal graphs in a certain (restricted but useful) sense. Furthermore, this setting is advocated as a target of choice for proving negative results. We illustrates this by strengthening two known results to happy graphs (namely, the inexistence of sparse spanners, and the hardness of computing temporal components). Overall, we hope that this article can be seen as a guide for choosing between different settings of temporal graphs, while being aware of the way these choices affect generality.

Given a sequence of $n$ numbers and $k$ parallel First-in-First-Out (FIFO) queues, how close can one bring the sequence to sorted order? It is known that $k$ queues suffice to sort the sequence if the Longest Decreasing Subsequence (LDS) of the input sequence is at most $k$. But, what if the number of queues is too small for sorting completely? - We give a simple algorithm, based on Patience Sort, that reduces the LDS by $k - 1$. We also show, that the algorithm is optimal, i.e., for any $L > 0$ there exists a sequence of LDS $L$ such that the LDS cannot be reduced below $L - k + 1$ with $k$ queues. - Merging two sorted queues is at the core of Merge Sort. In contrast, two sequences of LDS two cannot always be merged into a sequence of LDS two. We characterize when it is possible and give an algorithm to decide whether it is possible. Merging into a sequence of LDS three is always possible. - A down-step in a sequence is an item immediately followed by a smaller item. We give an optimal algorithm for reducing the number of down-steps. The algorithm is online. Our research was inspired by an application in car manufacturing.

We introduce deterministic suffix-reading automata (DSA), a new automaton model over finite words. Transitions in a DSA are labeled with words. From a state, a DSA triggers an outgoing transition on seeing a word ending with the transition's label. Therefore, rather than moving along an input word letter by letter, a DSA can jump along blocks of letters, with each block ending in a suitable suffix. This feature allows DSAs to recognize regular languages more concisely, compared to DFAs. In this work, we focus on questions around finding a "minimal" DSA for a regular language. The number of states is not a faithful measure of the size of a DSA, since the transition-labels contain strings of arbitrary length. Hence, we consider total-size (number of states + number of edges + total length of transition-labels) as the size measure of DSAs. We start by formally defining the model and providing a DSA-to-DFA conversion that allows to compare the expressiveness and succinctness of DSA with related automata models. Our main technical contribution is a method to derive DSAs from a given DFA: a DFA-to-DSA conversion. We make a surprising observation that the smallest DSA derived from the canonical DFA of a regular language L need not be a minimal DSA for L. This observation leads to a fundamental bottleneck in deriving a minimal DSA for a regular language. In fact, we prove that given a DFA and a number k, the problem of deciding if there exists an equivalent DSA of total-size at most k is NP-complete.

We study fair division of indivisible mixed manna (items whose values may be positive, negative, or zero) among agents with additive valuations. Here, we establish that fairness -- in terms of a relaxation of envy-freeness -- and Pareto efficiency can always be achieved together. Specifically, our fairness guarantees are in terms of envy-freeness up to $k$ reallocations (EFR-$k$): An allocation $A$ of the indivisible items is said to be EFR-$k$ if there exists a subset $R$ of at most $k$ items such that, for each agent $i$, we can reassign items from within $R$ (in $A$) and obtain an allocation, $A^i$, which is envy-free for $i$. We establish that, when allocating mixed manna among $n$ agents with additive valuations, an EFR-$(n-1)$ and Pareto optimal (PO) allocation $A$ always exists. Further, the individual envy-free allocations $A^i$, induced by reassignments, are also PO. In addition, we prove that such fair and efficient allocations are efficiently computable when the number of agents, $n$, is fixed. We also obtain positive results focusing on EFR by itself (and without the PO desideratum). Specifically, we show that an EFR-$(n-1)$ allocation of mixed manna can be computed in polynomial time. In addition, we prove that when all the items are goods, an EFR-${\lfloor n/2 \rfloor}$ allocation exists and can be computed efficiently. Here, the $(n-1)$ bound is tight for chores and $\lfloor n/2 \rfloor$ is tight for goods. Our results advance the understanding of fair and efficient allocation of indivisible mixed manna and rely on a novel application of the Knaster-Kuratowski-Mazurkiewicz (KKM) Theorem in discrete fair division. We utilize weighted welfare maximization, with perturbed valuations, to achieve Pareto efficiency, and overall, our techniques are notably different from existing market-based approaches.

The global network of gravitational-wave detectors has completed three observing runs with $\sim 50$ detections of merging compact binaries. A third LIGO detector, with comparable astrophysical reach, is to be built in India (LIGO-Aundha) and expected to be operational during the latter part of this decade. Multiple detectors operating at different parts of the globe will provide several pairs of interferometers with longer baselines and an increased network SNR. This will improve the sky localisation of GW events. Multiple detectors simultaneously in operation will also increase the baseline duty factor, thereby, leading to an improvement in the detection rates and, hence, the completeness of surveys. In this paper, we quantify the improvements due to the expansion of the LIGO Global Network (LGN) in the precision with which source properties will be measured. We also present examples of how this expansion will give a boost to tests of fundamental physics.

As the compact binary catalog continues to grow rapidly, developing and refining tests to probe the nature of compact objects is essential for a comprehensive understanding of both the observed data and the underlying astrophysics of the binary population. We investigate the effectiveness of spin-induced multipole moments (SIQM) and tidal deformability measurements in distinguishing lower mass-gap black hole (BH) binaries from non-BH binaries with different mass and spin configurations. We perform model-agnostic tests on binary BH (BBH) simulations using full Bayesian inference, evaluating the independent and joint measurability of SIQM and tidal parameters across the parameter space. We extend the analysis to simulations of self-interacting spinning boson stars, using synthetic signals that exhibit (a) both SIQM and tidal effects and (b) each effect individually. For case (a), recovery is performed using (i) a BBH model, (ii) a model incorporating both SIQM and tidal effects, and (iii) models including either SIQM or tidal effects. For case (b), we employ (i) a BBH model and (ii) models incorporating either SIQM or tidal effects, consistent with the injection. Simulations employ TaylorF2 waveform model and consider binaries in the low mass gap with varying spin magnitudes. We find that employing an incorrect model to analyze the signal can lead to biases in parameter inference. Notably, when analyzing a simulated binary boson star-like signal with component masses $\rm{(4, 4) \, M_{\odot}}$ using a BBH model, the system is incorrectly identified as having masses $\rm{(8, 2) \, M_{\odot}}$. In contrast, using the correct recovery model that includes both SIQM and tidal deformability effects successfully recovers the true masses, highlighting the significance of waveform model accuracy in performing reliable distinguishability tests for compact objects in the low-mass gap.

The inspiral-merger-ringdown (IMR) consistency test checks the consistency of the final mass and final spin of a binary black hole merger remnant, independently inferred via the inspiral and merger-ringdown parts of the waveform. As binaries are expected to be nearly circularized when entering the frequency band of ground-based detectors, tests of general relativity (GR) currently employ quasicircular waveforms. We quantify the effect of residual orbital eccentricity on the IMR consistency test. We find that eccentricity causes a significant systematic bias in the inferred final mass and spin of the remnant black hole at an orbital eccentricity (defined at $10$ Hz) of $e_0 \gtrsim 0.1$ in the LIGO band (for a total binary mass in the range $65$-$200 \,M_{\odot}$). For binary black holes observed by Cosmic Explorer (CE), the systematic bias becomes significant for $e_0 \gtrsim 0.015$ (for $200$-$600 \,M_{\odot}$ systems). This eccentricity-induced bias on the final mass and spin leads to an apparent inconsistency in the IMR consistency test, manifesting as a false violation of GR. Hence, eccentric corrections to waveform models are important for constructing a robust test of GR, especially for third-generation detectors. We also estimate the eccentric corrections to the relationship between the inspiral parameters and the final mass and final spin; they are shown to be quite small.

Gaussian Processes are widely used for regression tasks. A known limitation in the application of Gaussian Processes to regression tasks is that the computation of the solution requires performing a matrix inversion. The solution also requires the storage of a large matrix in memory. These factors restrict the application of Gaussian Process regression to small and moderate size data sets. We present an algorithm that combines estimates from models developed using subsets of the data obtained in a manner similar to the bootstrap. The sample size is a critical parameter for this algorithm. Guidelines for reasonable choices of algorithm parameters, based on detailed experimental study, are provided. Various techniques have been proposed to scale Gaussian Processes to large scale regression tasks. The most appropriate choice depends on the problem context. The proposed method is most appropriate for problems where an additive model works well and the response depends on a small number of features. The minimax rate of convergence for such problems is attractive and we can build effective models with a small subset of the data. The Stochastic Variational Gaussian Process and the Sparse Gaussian Process are also appropriate choices for such problems. These methods pick a subset of data based on theoretical considerations. The proposed algorithm uses bagging and random sampling. Results from experiments conducted as part of this study indicate that the algorithm presented in this work can be as effective as these methods. Model stacking can be used to combine the model developed with the proposed method with models from other methods for large scale regression such as Gradient Boosted Trees. This can yield performance gains.

We present a topological framework for analysing neural time series that integrates Transfer Entropy (TE) with directed Persistent Homology (PH) to characterize information flow in spiking neural systems. TE quantifies directional influence between neurons, producing weighted, directed graphs that reflect dynamic interactions. These graphs are then analyzed using PH, enabling assessment of topological complexity across multiple structural scales and dimensions. We apply this TE+PH pipeline to synthetic spiking networks trained on logic gate tasks, image-classification networks exposed to structured and perturbed inputs, and mouse cortical recordings annotated with behavioral events. Across all settings, the resulting topological signatures reveal distinctions in task complexity, stimulus structure, and behavioral regime. Higher-dimensional features become more prominent in complex or noisy conditions, reflecting interaction patterns that extend beyond pairwise connectivity. Our findings offer a principled approach to mapping directed information flow onto global organizational patterns in both artificial and biological neural systems. The framework is generalizable and interpretable, making it well suited for neural systems with time-resolved and binary spiking data.

General relativity (GR) has proven to be a highly successful theory of gravity since its inception. The theory has thrivingly passed numerous experimental tests, predominantly in weak gravity, low relative speeds, and linear regimes, but also in the strong-field and very low-speed regimes with binary pulsars. Observable gravitational waves (GWs) originate from regions of spacetime where gravity is extremely strong, making them a unique tool for testing GR, in previously inaccessible regions of large curvature, relativistic speeds, and strong gravity. Since their first detection, GWs have been extensively used to test GR, but no deviations have been found so far. Given GR's tremendous success in explaining current astronomical observations and laboratory experiments, accepting any deviation from it requires a very high level of statistical confidence and consistency of the deviation across GW sources. In this paper, we compile a comprehensive list of potential causes that can lead to a false identification of a GR violation in standard tests of GR on data from current and future ground-based GW detectors. These causes include detector noise, signal overlaps, gaps in the data, detector calibration, source model inaccuracy, missing physics in the source and in the underlying environment model, source misidentification, and mismodeling of the astrophysical population. We also provide a rough estimate of when each of these causes will become important for tests of GR for different detector sensitivities. We argue that each of these causes should be thoroughly investigated, quantified, and ruled out before claiming a GR violation in GW observations.

Lax pairs are a useful tool in finding conserved quantities of some dynamical systems. In this expository article, we give a motivated introduction to the idea of a Lax pair of matrices $(L,A)$, first for mechanical systems such as the linear harmonic oscillator, Toda chain, Eulerian rigid body and the Rajeev-Ranken model. This is then extended to Lax operators for one-dimensional field theories such as the linear wave and KdV equations and reformulated as a zero curvature representation via a $(U,V)$ pair which is illustrated using the nonlinear Schr\"odinger equation. The key idea is that of realizing a (possibly) nonlinear evolution equation as a compatibility condition between a pair of linear equations. The latter could be an eigenvalue problem for the Lax operator $L$ and a linear evolution equation generated by $A$, for the corresponding eigenfunction. Alternatively, they could be the first order linear system stating the covariant constancy of an arbitrary vector with respect to the 1+1 dimensional gauge potential $(V,U)$. The compatibility conditions are then either the Lax equation $\dot L = [L, A]$ or the flatness condition $U_t - V_x + [U, V] = 0$ for the corresponding gauge potential. The conserved quantities then follow from the isospectrality of the Lax and monodromy matrices.

In this paper, we consider the problem of testing properties of joint distributions under the Conditional Sampling framework. In the standard sampling model, the sample complexity of testing properties of joint distributions is exponential in the dimension, resulting in inefficient algorithms for practical use. While recent results achieve efficient algorithms for product distributions with significantly smaller sample complexity, no efficient algorithm is expected when the marginals are not independent. We initialize the study of conditional sampling in the multidimensional setting. We propose a subcube conditional sampling model where the tester can condition on an (adaptively) chosen subcube of the domain. Due to its simplicity, this model is potentially implementable in many practical applications, particularly when the distribution is a joint distribution over $\Sigma^n$ for some set $\Sigma$. We present algorithms for various fundamental properties of distributions in the subcube-conditioning model and prove that the sample complexity is polynomial in the dimension $n$ (and not exponential as in the traditional model). We present an algorithm for testing identity to a known distribution using $\tilde{\mathcal{O}}(n^2)$-subcube-conditional samples, an algorithm for testing identity between two unknown distributions using $\tilde{\mathcal{O}}(n^5)$-subcube-conditional samples and an algorithm for testing identity to a product distribution using $tilde{\mathcal{O}}(n^5)$-subcube-conditional samples. The central concept of our technique involves an elegant chain rule which can be proved using basic techniques of probability theory yet powerful enough to avoid the curse of dimensionality.

Self-dual Yang-Mills and Einstein gravity in Euclidean AdS$_4$ are useful toy models because they can be described by simple scalar Lagrangians exhibiting a new manifestation of the colour/kinematics duality, as recently shown by two of the authors. In this paper, we clarify how the self-dual sectors fit into the full theories. In particular, we explicitly construct the light-cone action for Yang-Mills theory and Einstein gravity in AdS$_4$ in terms of positive and negative helicity fields, where we are able to pinpoint the self-dual sector as expected. We then show that the boundary correlators of these theories take a remarkably simple form in terms of Feynman diagrams in half of flat space, acted on by certain differential operators. We also analyse their soft limits and show that they exhibit Weinberg-like soft factors, where the soft pole which appears in scattering amplitudes is replaced by a derivative with respect to the energy.

We study the \emph{order-finding problem} for Read-once Oblivious Algebraic Branching Programs (ROABPs). Given a polynomial $f$ and a parameter $w$, the goal is to find an order $\sigma$ in which $f$ has an ROABP of \emph{width} $w$. We show that this problem is NP-hard in the worst case, even when the input is a constant degree polynomial that is given in its dense representation. We provide a reduction from CutWidth to prove these results. Owing to the exactness of our reduction, all the known results for the hardness of approximation of Cutwidth also transfer directly to the order-finding problem. Additionally, we also show that any constant-approximation algorithm for the order-finding problem would imply a polynomial time approximation scheme (PTAS) for it. On the algorithmic front, we design algorithms that solve the order-finding problem for generic ROABPs in polynomial time, when the width $w$ is polynomial in the individual degree $d$ of the polynomial $f$. That is, our algorithm is efficient for most/random ROABPs, and requires more time only on a lower-dimensional subspace (or subvariety) of ROABPs. Even when the individual degree is constant, our algorithm runs in time $n^{O(\log w)}$ for most/random ROABPs. This stands in strong contrast to the case of (Boolean) ROBPs, where only heuristic order-finding algorithms are known.

Consider a deep neural network (DNN) that is being used to suggest the direction in which an aircraft must turn to avoid a possible collision with an intruder aircraft. Informally, such a network is well-behaved if it asks the own ship to turn right (left) when an intruder approaches from the left (right). Consider another network that takes four inputs -- the cards dealt to the players in a game of contract bridge -- and decides which team can bid game. Loosely speaking, if you exchange the hands of partners (north and south, or east and west), the decision would not change. However, it will change if, say, you exchange north's hand with east. This permutation invariance property, for certain permutations at input and output layers, is central to the correctness and robustness of these networks. This paper proposes a sound, abstraction-based technique to establish permutation invariance in DNNs with ReLU as the activation function. The technique computes an over-approximation of the reachable states, and an under-approximation of the safe states, and propagates this information across the layers, both forward and backward. The novelty of our approach lies in a useful tie-class analysis, that we introduce for forward propagation, and a scalable 2-polytope under-approximation method that escapes the exponential blow-up in the number of regions during backward propagation. An experimental comparison shows the efficiency of our algorithm over that of verifying permutation invariance as a two-safety property (using FFNN verification over two copies of the network).