In these lectures we detail the interplay between the low-energy dynamics of quantum field theories with four supercharges and the exact WKB analysis. This exposition may be the first comprehensive account of this connection and includes new arguments and results. The lectures start with the introduction of massive two-dimensional $\mathcal{N}=(2,2)$ theories and their spectra of BPS solitons. We place these theories in a two-dimensional cigar background with supersymmetric boundary conditions labelled by a phase $ζ= e^{i \vartheta}$, while turning on the two-dimensional $Ω$-background with parameter~$ε$. We show that the resulting partition function $\mathcal{Z}_{\mathrm{2d}}^\vartheta(ε)$ can be characterized as the Borel-summed solution, in the direction $\vartheta$, to an associated Schrödinger equation. The partition function $\mathcal{Z}_{\mathrm{2d}}^\vartheta(ε)$ is locally constant in the phase $\vartheta$ and jumps across phases $\vartheta_\textrm{BPS}$ associated with the BPS solitons. Since these jumps are non-perturbative in the parameter~$ε$, we refer to $Z^\vartheta_\mathrm{2d}(ε)$ as the non-perturbative partition function for the original two-dimensional $\mathcal{N}=(2,2)$ theory. We completely determine this partition function $\mathcal{Z}^\vartheta_\mathrm{2d}(ε)$ in two classes of examples, Landau-Ginzburg models and gauged linear sigma models, and show that $\mathcal{Z}^\vartheta_\mathrm{2d}(ε)$ encodes the well-known vortex partition function at a special phase $\vartheta_\textrm{FN}$ associated with the presence of self-solitons. This analysis generalizes to four-dimensional $\mathcal{N}=2$ theories in the $\frac{1}{2} Ω$-background.

We propose applying the adiabatic algorithm to prepare high-energy eigenstates of integrable models on a quantum computer. We first review the standard adiabatic algorithm to prepare ground states in each magnetization sector of the prototypical XXZ Heisenberg chain. Based on the thermodynamic Bethe ansatz, we show that the algorithm circuit depth is polynomial in the number of qubits $N$, outperforming previous methods explicitly relying on integrability. Next, we propose a protocol to prepare arbitrary eigenstates of integrable models that satisfy certain conditions. For a given target eigenstate, we construct a suitable parent Hamiltonian written in terms of a complete set of local conserved quantities. We propose using such Hamiltonian as an input for an adiabatic algorithm. After benchmarking this construction in the case of the non-interacting XY spin chain, where we can rigorously prove its efficiency, we apply it to prepare arbitrary eigenstates of the Richardson-Gaudin models. In this case, we provide numerical evidence that the circuit depth of our algorithm is polynomial in $N$ for all eigenstates, despite the models being interacting.

This paper introduces MonadBFT, a novel Byzantine Fault Tolerant (BFT) consensus protocol that advances both performance and robustness. MonadBFT is implemented as the consensus protocol in the Monad blockchain. As a HotStuff-family protocol, MonadBFT has linear message complexity in the common case and is optimistically responsive, operating as quickly as the network allows. A central feature of MonadBFT is its tail-forking resistance. In pipelined BFT protocols, when a leader goes offline, the previous proposal is abandoned. Malicious leaders can exploit this tail-forking behavior as a form of Maximal Extractable Value (MEV) attack by deliberately discarding their predecessor's block, depriving that proposer of rewards and enabling transaction reordering, censorship or theft. MonadBFT prevents such tail-forking attacks, preserving both fairness and integrity in transaction execution. Another related feature of MonadBFT is its notion of speculative finality, which enables parties to execute ordered transactions after a single round (i.e., a single view), with reverts occurring only in the rare case of provable leader equivocation. This mechanism reduces user-perceived latency. Additionally, we introduce the leader fault isolation property, which ensures that the protocol can quickly recover from a failure. To our knowledge, no prior pipelined, leader-based BFT consensus protocol combines all of these properties in a single design.

We present a formalization of Borel determinacy in the Lean 4 theorem prover. The formalization includes a definition of Gale-Stewart games and a proof of Martin's theorem stating that Borel games are determined. The proof closely follows Martin's "A purely inductive proof of Borel determinacy".

We are interested in the simulation of open quantum systems governed by the Lindblad master equation in an infinite-dimensional Hilbert space. To simulate the solution of this equation, the standard approach involves two sequential approximations: first, we truncate the Hilbert space to derive a differential equation in a finite-dimensional subspace. Then, we use discrete time-step to obtain a numerical solution to the finite-dimensional evolution. In this paper, we establish bounds for these two approximations that can be explicitly computed to guarantee the accuracy of the numerical results. Through numerical examples, we demonstrate the efficiency of our method, empirically highlighting the tightness of the upper bound. While adaptive time-stepping is already a common practice in the time discretization of the Lindblad equation, we extend this approach by showing how to dynamically adjust the truncation of the Hilbert space. This enables fully adaptive simulations of the density matrix. For large-scale simulations, this approach can significantly reduce computational time and relieves users of the challenge of selecting an appropriate truncation. Furthermore, as a special case, our method naturally applies to Hamiltonian (unitary) dynamics.

We present the first systematic approach to static and dynamic taint analysis for Graph APIs focusing on broken access control. The approach comprises the following. We taint nodes of the Graph API if they represent data requiring specific privileges in order to be retrieved or manipulated, and identify API calls which are related to sources and sinks. Then, we statically analyze whether a tainted information flow between API source and sink calls occurs. To this end, we model the API calls using graph transformation rules. We subsequently use Critical Pair Analysis to automatically analyze potential dependencies between rules representing source calls and rules representing sink calls. We distinguish direct from indirect tainted information flow and argue under which conditions the Critical Pair Analysis is able to detect not only direct, but also indirect tainted flow. The static taint analysis (i) identifies flows that need to be further reviewed, since tainted nodes may be created by an API call and used or manipulated by another API call later without having the necessary privileges, and (ii) can be used to systematically design dynamic security tests for broken access control. The dynamic taint analysis checks if potential broken access control risks detected during the static taint analysis really occur. We apply the approach to a part of the GitHub GraphQL API. The application illustrates that our analysis supports the detection of two types of broken access control systematically: the case where users of the API may not be able to access or manipulate information, although they should be able to do so; and the case where users (or attackers) of the API may be able to access/manipulate information that they should not.

The virtual cohomological dimension of~$\operatorname{Out}(F_n)$ is given precisely by the dimension of the spine of Culler--Vogtmann Outer space. However, the dimension of the spine of untwisted Outer space for a general right-angled Artin group~$A_Γ$ does not necessarily match the virtual cohomological dimension~$\textsc{vcd}(U(A_Γ))$ of the untwisted subgroup~$U(A_Γ) \leq \operatorname{Out}(A_Γ)$. Under certain graph-theoretic conditions, we perform an equivariant deformation retraction of this spine to produce a new contractible cube complex upon which~$U(A_Γ)$ acts properly and cocompactly. Furthermore, we give conditions for when the dimension of this complex realises the virtual cohomological dimension of~$U(A_Γ)$. We finish with two applications of our construction; in particular we show that the difference between the dimension of the untwisted spine and~$\textsc{vcd}(U(A_Γ))$ can be arbitrarily large.

We consider the cyclic representations $Ω_{rs}$ of $ U_q(\widehat{\mathfrak{sl}}_2)$ at $q^N=1$ that depend upon two points $r,s$ in the chiral Potts algebraic curve. We show how $Ω_{rs}$ is related to the tensor product $ρ_r\otimes \barρ_s$ of two representations of the upper Borel subalgebra of $U_q(\widehat{\mathfrak{sl}}_2)$. This result is analogous to the factorization property of the Verma module of $U_q(\widehat{\mathfrak{sl}}_2)$ at generic-$q$ in terms of two q-oscillator representation of the Borel subalgebra - a key step in the construction of the Q-operator. We construct short exact sequences of the different representations and use the results to construct Q operators that satisfy TQ relations for $q^N=1$ for both the 6-vertex and $τ_2$ models.

We introduce and investigate a data access model (approximate sample and query) that is satisfiable by the preparation and measurement of block encoded states, as well as in contexts such as classical quantum circuit simulation or Pauli sampling. We illustrate that this abstraction is compositional and has some computational power. We then apply these results to obtain polynomial improvements over the state of the art in the sample and computational complexity of distributed inner product estimation. By doing so, we provide a new interpretation for why Pauli sampling is useful for this task. Our results partially characterize the power of time-limited fault-tolerant quantum circuits aided by classical computation. They are a first step towards extending the classical data Quantum Singular Value Transform dequantization results to a quantum setting.

Let $Ω$ be a bounded, smooth connected open domain in $\mathbb{R}^n$ with $n\geq 3$. We investigate in this paper compactness properties for the set of sign-changing solutions $v \in H^1_0(Ω)$ of \begin{equation} \tag{*} -Δv+h v =\left|v\right|^{2^*-2}v \hbox{ in } Ω, \quad v = 0 \hbox{ on } \partial Ω \end{equation} where $h\in C^1(\overlineΩ)$ and $2^*:=2n/(n-2)$. Our main result establishes that the set of sign-changing solutions of $(*)$ at the lowest sign-changing energy level is unconditionally compact in $C^2(\overlineΩ)$ when $3 \le n \le 5$, and is compact in $C^2(\overlineΩ)$ when $n \ge 7$ provided $h$ never vanishes in $\overlineΩ$. In dimensions $n \ge 7$ our results apply when $h >0$ in $\overlineΩ$ and thus complement the compactness result of Devillanova-Solimini, Adv. Diff. Eqs. 7 (2002). Our proof is based on a new, global pointwise description of blowing-up sequences of solutions of $(*)$ that holds up to the boundary. We also prove more general compactness results under perturbations of $h$.

Farkas established that a system of linear inequalities has a solution if and only if we cannot obtain a contradiction by taking a linear combination of the inequalities. We state and formally prove several Farkas-like theorems over linearly ordered fields in Lean 4. Furthermore, we extend duality theory to the case when some coefficients are allowed to take "infinite values".

We study a quasi two-zero neutrino texture based on a type-II seesaw model with modular $A_4$ symmetry to evade the cosmological bound on the sum of neutrino masses while keeping some predictability in the neutrino sector. Working on three fixed points for modulus, we discuss predictions of the model and show the allowed points satisfying the cosmological bound on neutrino mass from both CMB and CMB+BAO data.

If we want to transform the quantum state of a system to another using local measurement processes, what is the probability of success? This probability is bounded by quantifying entanglement in both the states. In this paper, we construct a family of local unitary invariants of multipartite states that are monotonic under local operations and classical communication on average. These monotones are constructed from local unitary invariant polynomials of the state and its conjugate, and hence are easy to compute for pure states. Using these measures we bound the success probability of transforming a given state into another state using local quantum operations and classical communication.

We consider variational problem related to entropy maximization in the two-dimensional Euler equations, in order to investigate the long-time dynamics of solutions with bounded vorticity. Using variations on the classical min-max principle and borrowing ideas from optimal transportation and quantitative rearrangement inequalities, we prove results on the structure of entropy maximizers arising in the investigation of the long-time behavior of vortex patches. We further show that the same techniques apply in the study of stability of the canonical Gibbs measure associated to a system of point vortices.

Reliably transmitting quantum information via a noisy quantum channel is a central challenge in quantum information science. While constructing a decoder is crucial to this goal, little was known about quantum circuit implementations of decoders that reach high communication rates. In this paper, we provide two decoders with explicit quantum circuits capable of recovering quantum information when the decoupling condition is satisfied, i.e., when quantum information is in principle recoverable. These are applicable to both entanglement-assisted and non-assisted settings. By developing a technique that relies on a symmetric structure of the decoders, we show that they are applicable to any noise model. As a consequence, for any noisy channel, our decoders can be used to achieve a communication rate arbitrarily close to the quantum capacity by increasing the number of channel uses. To construct the decoders, we employ the fixed-point amplitude amplification (FPAA) based on the quantum singular value transformation (QSVT), extending a previous approach applicable only to erasure noise. Our constructions offer advantages in the computational cost, largely reducing the circuit complexity compared to previous explicit decoders. Through an investigation of the decoding problem, unique advantages of the QSVT-based FPAA are highlighted.

Working in the general context of "modules with an additive dimension," we complete the determination of the minimal dimension of a faithful Alt(n)-module and classify those modules in three of the exceptional cases: 2-dimensional Alt(5)-modules in characteristic 2, 3-dimensional Alt(5)-modules in characteristic 5, and 3-dimensional Alt(6)-modules in characteristic 3. We also highlight the remaining work needed to complete the classification of the faithful Alt(n)-modules of minimal dimension for all n; these open problems seem well suited as projects for advanced undergraduate or master's students.

Photon loss rates set an effective upper limit on the size of computations that can be run on current linear optical quantum devices. We present a family of techniques designed to mitigate the effects of photon loss on both output probabilities and expectation values derived from noisy linear optical circuits composed of an input of n photons, an m-mode interferometer, and m single photon detectors. Central to these techniques is the construction recycled probabilities. Recycled probabilities are constructed from output statistics affected by loss, and are designed to amplify the signal of the ideal (lossless) probabilities. Classical postprocessing techniques then take recycled probabilities as input and output a set of loss-mitigated probabilities, or expectation values. Our postprocessing methods result in biased estimators of the lossless probabilities. Nevertheless, we provide both analytical and numerical evidence that these methods can be applied, up to large sample sizes, to produce output probabilities with lower combined bias and statistical errors than the statistical errors of the output probabilities obtained from postselection. Therefore, these methods can outperform postselection - currently the standard method of coping with photon loss when sampling from discrete variable linear optical quantum circuits. In contrast, we provide evidence that the popular zero-noise extrapolation technique cannot improve on the performance of postselection for any photon loss rate.

This paper investigates the impossibility of certain $({n^2+n+k}_{n+1})$ configurations. Firstly, for $k=2$, the result of \cite{gropp1992non} that $\frac{n^2+n}{2}$ is even and $n+1$ is a perfect square or $\frac{n^2+n}{2}$ is odd and $n-1$ is a perfect square is reproved using the incidence matrix $N$ and analysing the form of $N^TN$. Then, for all $k$, configurations where paralellism is a transitive property are considered. It is then analogously established that if $n\equiv0$ or $n\equiv k-1$ mod $k$ for $k$ even, then $\frac{n^2+n}{k}$ is even and $n+1$ is a perfect square or $\frac{n^2+n}{k}$ is odd and $n-(k-1)$ is a perfect square. Finally, the case $k=3$ is investigated in full generality.

We exhibit a new approach to the proofs of the existence of a large family of almost isometric ideals in nonseparable Banach spaces and existence of a large family of almost isometric local retracts in metric spaces. Our approach also implies the existence of a large family of nontrivial projections on every dual of a nonseparable Banach space. We prove three possible formulations of our results are equivalent. Some applications are mentioned which witness the usefulness of our novel approach.

We establish two-term spectral asymptotics for the operator of linear elasticity with mixed boundary conditions on a smooth compact Riemannian manifold of arbitrary dimension. We illustrate our results by explicit examples in dimension two and three, thus verifying our general formulae both analytically and numerically.

We determine the complexity of second-order HyperLTL satisfiability, finite-state satisfiability, and model-checking: All three are equivalent to truth in third-order arithmetic. We also consider two fragments of second-order HyperLTL that have been introduced with the aim to facilitate effective model-checking by restricting the sets one can quantify over. The first one restricts second-order quantification to smallest/largest sets that satisfy a guard while the second one restricts second-order quantification further to least fixed points of (first-order) HyperLTL definable functions. All three problems for the first fragment are still equivalent to truth in third-order arithmetic while satisfiability for the second fragment is $Σ_1^2$-complete, and finite-state satisfiability and model-checking are equivalent to truth in second-order arithmetic. Finally, we also introduce closed-world semantics for second-order HyperLTL, where set quantification ranges only over subsets of the model, while set quantification in standard semantics ranges over arbitrary sets of traces. Here, satisfiability for the least fixed point fragment becomes $Σ_1^1$-complete, but all other results are unaffected.

In this note we examine the proportion of periodic orbits of Anosov flows that lie in an infinite zero density subset of the first homology group. We show that on a logarithmic scale we get convergence to a discrete fractal dimension.

A logic family is a bunch of logics that belong together in some way. First-order logic is one of the examples. Logics organized into a structure occurs in abstract model theory, institution theory and in algebraic logic. Logic families play a role in adopting methods for investigating sentential logics to first-order like logics. We thoroughly discuss the notion of logic families as defined in the recent Universal Algebraic Logic book.

Zero forcing is a dynamic coloring process on graphs. Initially, each vertex of a graph is assigned a color of either blue or white, and then a process begins by which blue vertices force white vertices to become blue. The zero forcing number is the cardinality of the smallest set of initially blue vertices which can force the entire graph to become blue, and the propagation time is the minimum number of steps in such a zero forcing process. In this paper we will determine the zero forcing numbers and propagation times of two infinite classes of graphs called gear graphs and helm graphs.

Given a conjunctive query $Q$ and a database $D$, a direct access to the answers of $Q$ over $D$ is the operation of returning, given an index $k$, the $k$-th answer for some order on its answers. While this problem is $\#\mathcal{P}$-hard in general with respect to combined complexity, many conjunctive queries have an underlying structure that allows for a direct access to their answers for some lexicographical ordering that takes polylogarithmic time in the size of the database after a polynomial time precomputation. Previous work has precisely characterised the tractable classes and given fine-grained lower bounds on the precomputation time needed depending on the structure of the query. In this paper, we generalise these tractability results to the case of signed conjunctive queries, that is, conjunctive queries that may contain negative atoms. Our technique is based on a class of circuits that can represent relational data. We first show that this class supports tractable direct access after a polynomial time preprocessing. We then give bounds on the size of the circuit needed to represent the answer set of signed conjunctive queries depending on their structure. Both results combined together allow us to prove the tractability of direct access for a large class of conjunctive queries. On the one hand, we recover the known tractable classes from the literature in the case of positive conjunctive queries. On the other hand, we generalise and unify known tractability results about negative conjunctive queries -- that is, queries having only negated atoms. In particular, we show that the class of $β$-acyclic negative conjunctive queries and the class of bounded nest set width negative conjunctive queries admit tractable direct access.

We prove the existence of topological solutions to the self-dual Chern-Simons model and the Abelian Higgs system on the lattice graphs Z^n for n>1. This extends the results in Huang, Lin and Yau [HLY20] from finite graphs to lattice graphs.

Dispersion relations let us leverage the analytic structure of scattering amplitudes to derive constraints such as bounds on EFT coefficients. An important input is the large-energy behavior of the amplitude. In this paper, we systematically study how different large-energy behavior affects EFT bounds for the $2 \to 2$ amplitude of complex scalars coupled to photons, gravity, both, or neither. In many cases we find that singly-subtracted dispersion relations (1SDRs) yield exactly the same bounds as doubly subtracted relations (2SDRs). However, we identify another assumption, which we call "$t$-channel dominance," that significantly strengthens the EFT bounds. This assumption, which amounts to the requirement that the $++ \to ++$ amplitude has no $s$-channel exchange, is justified in certain cases and is analogous to the condition that the isospin-2 channel does not contribute to the pion amplitude. Using this assumption in the absence of massless exchanges, we find that the allowed region for the complex scalar EFT is identical to one recently discussed for pion scattering at large-$N$. In the case of gravity and a gauge field, we are able to derive a number of interesting bounds. These include an upper bound for $G$ in terms of the gauge coupling $e^2$ and the leading dispersive EFT coefficient, which is reminiscent of the weak gravity conjecture. In the $e \to 0$ limit, we find that assuming smeared 1SDRs plus $t$-channel dominance restores positivity on the leading EFT coefficient whose positivity was spoiled by the inclusion of gravity. We interpret this to mean that the negativity of that coefficient in the presence of gravity would imply that the global $U(1)$ symmetry must be gauged.

Let $Ω\subseteq \mathbb{R}^{4}$ be a bounded domain with smooth boundary $\partialΩ$. In this paper, we establish the following sharp form of the trace Adams' inequality in $W^{2,2}(Ω)$ with zero mean value and zero Neumann boundary condition: \begin{equation*} S(α)=\underset{\int_Ωudx=0,\frac{\partial u}{\partialν}|_{\partialΩ}=0,\VertΔu\Vert_{2}\leq{1}}{\underset {u\in{W^{2,2}(Ω)\setminus\{0\}}}{\sup}}\int_{\partial Ω} e^{αu^{2}}dσ<\infty \end{equation*} holds if and only if $ α\leq12π^2$. Moreover, we prove a classification theorem for the solutions of a class of nonlinear boundary value problem of bi-harmonic equations on the half space $\mathbb{R}^4_{+}$. With this classification result, we can show that $S({12π^2})$ is attained by using the blow-up analysis and capacitary estimate. As an application, we prove a sharp trace Adams-Onofri type inequality in general four dimensional bounded domains with smooth boundary.

We prove an $r$-variation estimate, $r>4$, in the norm for ergodic averages with respect to three commuting transformations. It is not known whether such estimates hold for all $r\ge 2$ as in the analogous cases for one or two commuting transformations, or whether such estimates hold for any $r<\infty$ for more than three commuting transformations.

We derive a global higher regularity result for weak solutions of the linear relaxed micromorphic model on smooth domains. The governing equations consist of a linear elliptic system of partial differential equations that is coupled with a system of Maxwell-type. The result is obtained by combining a Helmholtz decomposition argument with regularity results for linear elliptic systems and the classical embedding of $H(\mathrm{div};Ω)\cap H_0(\mathrm{curl};Ω)$ into $H^1(Ω)$.