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We strengthen a result of Bagaria and Magidor~\cite{MR3152715} about the relationship between large cardinals and torsion classes of abelian groups, and prove that (1) the \emph{Maximum Deconstructibility} principle introduced in \cite{Cox_MaxDecon} requires large cardinals; it sits, implication-wise, between Vop\v{e}nka's Principle and the existence of an $\omega_1$-strongly compact cardinal. (2) While deconstructibility of a class of modules always implies the precovering property by \cite{MR2822215}, the concepts are (consistently) non-equivalent, even for classes of abelian groups closed under extensions, homomorphic images, and colimits.

\emph{Approximation Theory} uses nicely-behaved subcategories to understand entire categories, just as projective modules are used to approximate arbitrary modules in classical homological algebra. We use set-theoretic \emph{elementary submodel arguments} to give new, short proofs of well-known theorems in approximation theory, sometimes with stronger results., Comment: Minor corrections

We provide a new characterization of quasi-analyticity of Denjoy-Carleman classes, related to \emph{Wetzel's Problem}. We also completely resolve which Denjoy-Carleman classes carry \emph{sparse systems}: if the Continuum Hypothesis (CH) holds, \textbf{all} Denjoy-Carleman classes carry sparse systems; but if CH fails, a Denjoy-Carleman class carries a sparse system if and only if it is not quasi-analytic. As corollaries, we extend results of \cite{MR3552748} and \cite{CodyCoxLee} about non-existence of "anonymous predictors" for real functions.

The Spot Prawn trap fishery off the west coast of British Columbia (BC) is managed using a fixed escapement strategy that aims to prevent recruitment overfishing while maximizing expected long-term yield by closing the fishery when the catch rate of spawners, projected to the following spring, drops below 1.7 spawners per trap (the de jure rule). We develop a management strategy evaluation framework for BC's Spot Prawn fishery that examines the expected performance of the management procedure implemented in practice (the de facto rule), which was significantly more conservative than the de jure rule, usually closing the fishery when spawner catch rates were at least twice as high as specified by the de jure rule. Simulations indicate that the de facto spawner index rule using average empirical March 31st targets from 2000 to 2019 maintains most stocks near or above 0.8 BMSY with or without accounting for environmental effects and/or increasing future SST on recruitment. Abundance indices were found to be strongly hyperstable, with fishing efficiency 1.5 to 3.0 times higher under low biomass than high biomass., Comment: 86 pages, 12 figures, 33 appendix figures

Erd\H{o}s \cite{MR168482} proved that the Continuum Hypothesis (CH) is equivalent to the existence of an uncountable family $\mathcal{F}$ of (real or complex) analytic functions, such that $\big\{ f(x) \ : \ f \in \mathcal{F} \big\}$ is countable for every $x$. We strengthen Erd\H{o}s' result by proving that CH is equivalent to the existence of what we call \emph{sparse analytic systems} of functions. We use such systems to construct, assuming CH, an equivalence relation $\sim$ on $\mathbb{R}$ such that any "analytic-anonymous" attempt to predict the map $x \mapsto [x]_\sim$ must fail almost everywhere. This provides a consistently negative answer to a question of Bajpai-Velleman \cite{MR3552748}., Comment: to appear in Forum of Mathematics, Sigma

Hardin and Taylor \cite{MR2384262} proved that any function on the reals -- even a nowhere continuous one -- can be correctly predicted, based solely on its past behavior, at almost every point in time. They showed in \cite{MR3100500} that one could even arrange for the predictors to be robust with respect to simple time shifts, and asked whether they could be robust with respect to other, more complicated time distortions. This question was partially answered by Bajpai and Velleman \cite{MR3552748}, who provided upper and lower frontiers (in the subgroup lattice of $\text{Homeo}^+(\mathbb{R})$) on how robust a predictor can possibly be. We improve both frontiers, some of which reduce ultimately to consequences of H\"older's Theorem (that every Archimedean group is abelian).

We prove that, assuming Vop\venka's principle, every small projectivity class in a locally presentable category is accessible., Comment: 11 pages

A diagonal version of the strong reflection principle is introduced, along with fragments of this principle associated to arbitrary forcing classes. The relationships between the resulting principles and related principles, such as the corresponding forcing axioms and the corresponding fragments of the strong reflection principle are analyzed, and consequences are presented. Some of these consequences are ``exact'' versions of diagonal stationary reflection principles of sets of ordinals. We also separate some of these diagonal strong reflection principles from related axioms.

Salce \cite{MR565595} introduced the notion of a \emph{cotorsion pair} of classes of abelian groups, and asked whether every such pair is \emph{complete} (i.e., has enough injectives and projectives). We prove that it is consistent, relative to the consistency of Vop\v{e}nka's Principle (VP), that the answer is affirmative. Combined with a previous result of Eklof-Shelah \cite{MR2031314}, this shows that Salce's Problem is independent of the ZFC axioms (modulo the consistency of VP)., Comment: To appear in Bulletin of the London Mathematical Society

We prove that Vop\v{e}nka's Principle implies that for every class $\mathfrak{X}$ of modules over any ring, the class of \textbf{$\boldsymbol{\mathfrak{X}}$-Gorenstein Projective modules} (\textbf{$\boldsymbol{\mathfrak{X}}$-$\boldsymbol{\mathcal{GP}}$}) is a special precovering class. In particular, it is not possible to prove (unless Vop\v{e}nka's Principle is inconsistent) that there is a ring over which the \textbf{Ding Projectives} ($\boldsymbol{\mathcal{DP}}$) or the \textbf{Gorenstein Projectives} ($\boldsymbol{\mathcal{GP}}$) do not form a precovering class (\v{S}aroch previously obtained this result for the class $\mathcal{GP}$, using different methods). The key innovation is a new "top-down" characterization of \emph{deconstructibility}, which is a well-known sufficient condition for a class to be precovering. We also prove that Vop\v{e}nka's Principle implies, in some sense, the maximum possible amount of deconstructibility in module categories., Comment: accepted version

We study the influence of strong forcing axioms on the complexity of the non-stationary ideal on $\omega_2$ and its restrictions to certain cofinalities. Our main result shows that the strengthening $MM^{++}$ of Martin's Maximum does not decide whether the restriction of the non-stationary ideal on $\omega_2$ to sets of ordinals of countable cofinality is $\Delta_1$-definable by formulas with parameters in $H(\omega_3)$. The techniques developed in the proof of this result also allow us to prove analogous results for the full non-stationary ideal on $\omega_2$ and strong forcing axioms that are compatible with CH. Finally, we answer a question of S. Friedman, Wu and Zdomskyyshow by showing that the $\Delta_1$-definability of the non-stationary ideal on $\omega_2$ is compatible with arbitrary large values of the continuum function at $\omega_2$., Comment: Accepted for publication in the "Monatshefte f\"ur Mathematik". 35 pages

The notion of a \textbf{$\boldsymbol{\mathcal{C}}$-filtered} object, where $\mathcal{C}$ is some (typically small) collection of objects in a Grothendieck category, has become ubiquitous since the solution of the Flat Cover Conjecture around the year 2000. We introduce the \textbf{$\boldsymbol{\mathcal{C}}$-Filtration Game of length $\boldsymbol{\omega_1}$} on a module, paying particular attention to the case where $\mathcal{C}$ is the collection of all countably presented, projective modules. We prove that Martin's Maximum implies the determinacy of many $\mathcal{C}$-Filtration Games of length $\omega_1$, which in turn imply the determinacy of certain Ehrenfeucht-Fra\"iss\'{e} games of length $\omega_1$; this allows a significant strengthening of a theorem of Mekler-Shelah-Vaananen \cite{MR1191613}. Also, Martin's Maximum implies that if $R$ is a countable hereditary ring, the class of \textbf{$\boldsymbol{\sigma}$-closed potentially projective modules} -- i.e., those modules that are projective in some $\sigma$-closed forcing extension of the universe -- is closed under $<\aleph_2$-directed limits. We also give an example of a (ZFC-definable) class of abelian groups that, under the ordinary subgroup relation, constitutes an Abstract Elementary Class (AEC) with L\"owenheim-Skolem number $\aleph_1$ in some models in set theory, but fails to be an AEC in other models of set theory., Comment: minor corrections

If $\kappa$ is regular and $2^{<\kappa}\leq\kappa^+$, then the existence of a weakly presaturated ideal on $\kappa^+$ implies $\square^*_\kappa$. This partially answers a question of Foreman and Magidor about the approachability ideal on $\omega_2$. As a corollary, we show that if there is a presaturated ideal $I$ on $\omega_2$ such that $\mathcal{P}(\omega_2)/I$ is semiproper, then CH holds. We also show some barriers to getting the tree property and a saturated ideal simultaneously on a successor cardinal from conventional forcing methods.

Management performance of five alternative stock assessment methods was evaluated by using them to set harvest levels targeting multi-species maximum yield in a multi-species flatfish fishery, including single-species and hierarchical multi-species models, and methods that pooled data across species and spatial strata, with catch outcomes of each method under three data scenarios compared to catch under an omniscient manager simulation. Operating models included technical interactions between species intended to produce choke effects often observed in output controlled multi-species fisheries. Hierarchical multi-species models outperformed all other methods under data-poor and data-moderate scenarios, and outperformed single-species models under the data-rich scenario. Hierarchical models were least sensitive to prior precision, sometimes improving in performance when prior precision was reduced. Choke effects were found to both positive and negative effects, sometimes leading to underfishing of non-choke species, but at other times preventing overfishing of non-choke species. We highlight the importance of including technical interactions in multi-species assessment models and management objectives, how choke species can indicate mismatches between management objectives and system dynamics, and recommend hierarchical multi-species models for multi-species fishery management systems., Comment: 39 pages, 9 figures, 6 tables, 3 appendices

Fuchino-Maschio-Sakai~\cite{FuchinoEtAl_DRP_LST} proved that the L\"owenheim-Skolem-Tarski (LST) property of Stationary Logic is equivalent to the Diagonal Reflection Principle on internally club sets ($\text{DRP}_{\text{IC}}$) introduced in \cite{DRP}. We prove that the restriction of the LST property to (downward) reflection of $\Pi^1_1$ formulas, which we call the $\Pi^1_1 \! \! \downarrow$-LST property, is equivalent to the \emph{internal} version of DRP from \cite{Cox_RP_IS}. Combined with results from \cite{Cox_RP_IS}, this shows that the $\Pi^1_1 \! \! \downarrow$-LST Property for Stationary Logic is strictly weaker than the full LST Property for Stationary Logic, though if CH holds they are equivalent., Comment: submitted to RIMS Kokyuroku

We study methods to obtain the consistency of forcing axioms, and particularly higher forcing axioms. We first force over a model with a supercompact cardinal $\theta>\kappa$ to get the consistency of the forcing axiom for $\kappa$-strongly proper forcing notions which are also $\kappa$-lattice, and then eliminate the need for large cardinals. The proof goes through a natural reflection property for $\kappa$-strongly proper forcings. We also produce a model of this forcing axiom with $2^\kappa$ arbitrarily large, and prove the inconsistency of certain natural strengthenings of the axiom., Comment: 14 pages

We survey some old and new results on strong variants of Chang's Conjecture and related topics., Comment: Minor revisions; added open problem section at the end

We prove that Martin's Maximum does not imply the Diagonal Reflection Principle for stationary subsets of $[ \omega_2 ]^\omega$.

Shelah considered a certain version of Strong Chang's Conjecture, which we denote $\text{SCC}^{\text{cof}}$, and proved that it is equivalent to several statements, including the assertion that Namba forcing is semiproper. We introduce an apparently weaker version, denoted $\text{SCC}^{\text{split}}$, and prove an analogous characterization of it. In particular, $\text{SCC}^{\text{split}}$ is equivalent to the assertion that the the Friedman-Krueger poset is semiproper. This strengthens and sharpens the results of Cox, and sheds some light on problems from Usuba and Torres-Perez and Wu.

Chang's Conjecture (CC) asserts that for every $F:[\omega_2]^{<\omega} \to \omega_2$, there exists an $X$ that is closed under $F$ such that $|X|=\omega_1$ and $|X \cap \omega_1| =\omega$. By classic results of Silver and Donder, CC is equiconsistent with an $\omega_1$-Erdos cardinal. Using stronger large cardinal assumptions (between $o(\kappa) = \kappa^+$ and $o(\kappa) = \kappa^{++}$), we prove that it is consistent to also require that $X$ contains a closed unbounded set of ordinals in $\text{sup}(X \cap \omega_2)$. We denote this stronger principle \textbf{Club-CC}, and also show that, unlike CC, Club-CC implies failure of certain weak square principles., Comment: Theorem 12, and the proof of Claim 13, are not correct (thanks to Omer Ben-Neria for pointing this out). The notion in Definition 7 is inconsistent with ZFC

We prove a variety of theorems about stationary set reflection and concepts related to internal approachability. We prove that an implication of Fuchino-Usuba relating stationary reflection to a version of Strong Chang's Conjecture cannot be reversed; strengthen and simplify some results of Krueger about forcing axioms and approachability; and prove that some other related results of Krueger are sharp. We also adapt some ideas of Woodin to simplify and unify many arguments in the literature involving preservation of forcing axioms., Comment: minor corrections

An emerging approach to data-limited fisheries stock assessment uses hierarchical multi-stock assessment models to group stocks together, sharing information from data-rich to data-poor stocks. In this paper, we simulate data-rich and data-poor fishery and survey data scenarios for a complex of dover sole stocks. Simulated data for individual stocks were used to compare estimation performance for single-stock and hierarchical multi-stock versions of a Schaefer production model. The single-stock and best performing multi-stock models were then used in stock assessments for the real dover sole data. Multi-stock models often had lower estimation errors than single-stock models when assessment data had low statistical power. Relative errors for productivity and relative biomass parameters were lower for multi-stock assessment model configurations. In addition, multi-stock models that estimated hierarchical priors for survey catchability performed the best under data-poor scenarios. We conclude that hierarchical multi-stock assessment models are useful for data-limited stocks and could provide a more flexible alternative to data-pooling and catch only methods; however, these models are subject to non-linear side-effects of parameter shrinkage. Therefore, we recommend testing hierarchical multi-stock models in closed-loop simulations before application to real fishery management systems., Comment: 47 pages, 10 figures, 5 tables, submitted to Canadian Journal of Fisheries and Aquatic Sciences

Burke \cite{MR1472122} proved that the generalized nonstationary ideal, denoted NS, is universal in the following sense: every normal ideal, and every tower of normal ideals of inaccessible height, is a canonical Rudin-Keisler projection of the restriction of $\text{NS}$ to some stationary set. We investigate how far Burke's theorem can be pushed, by analyzing the universality properties of NS with respect to the wider class of \emph{$\mathcal{C}$-systems of filters} introduced by Audrito-Steila \cite{AudritoSteila}. First we answer a question of \cite{AudritoSteila}, by proving that $\mathcal{C}$-systems of filters do not capture all kinds of set-generic embeddings. We provide a characterization of supercompactness in terms of short extenders and canonical projections of NS, without any reference to the strength of the extenders; as a corollary, NS can consistently fail to canonically project to arbitrarily strong short extenders. We prove that $\omega$-cofinal towers of normal ultrafilters---e.g.\ the kind used to characterize I2 and I3 embeddings---are well-founded if and only if they are canonical projections of NS. Finally, we provide a characterization of "$\aleph_\omega$ is Jonsson" in terms of canonical projections of NS.

We provide solutions to several problems of Foreman about ideals, several of which are closely related to Mitchell's notion of \emph{strongly proper} forcing. We prove: 1) Presaturation of a normal ideal implies projective antichain catching, enabling us to provide a solution to a problem from Foreman~\cite{MR2768692} about ideal projections which is more comprehensive and simpler than the solution obtained in \cite{MR3343538}. 2) We solve an older question from Foreman~\cite{MR819932} about the relationship between generic hugeness and generic almost hugeness. 3) Finally, we provide solutions to two technical questions from Foreman~\cite{MR3038554} and \cite{MR2768692} related to his \emph{Duality Theorem}.

We prove a variation of Easton's lemma for strongly proper forcings, and use it to prove that, unlike the stronger principle $\textsf{IGMP}$, $\textsf{GMP}$ together with $2^\omega \le \omega_2$ is consistent with the existence of an $\omega_1$-distributive nowhere c.c.c. forcing poset of size $\omega_1$. We introduce the idea of a weakly guessing model, and prove that many of the strong consequences of the principle $\textsf{GMP}$ follow from the existence of stationarily many weakly guessing models. Using Namba forcing, we construct a model in which there are stationarily many indestructibly weakly guessing models which have a bounded countable subset not covered by any countable set in the model.

We prove that the upper bounds for the consistency strength of certain instances of mutual stationarity considered by Liu-Shelah~\cite{MR1469093} are close to optimal. We also consider some related and, as it turns out, stronger properties.

Let $\mathbb{Q}$ denote the poset which adds a Cohen real then shoots a club through the complement of $\big( [\omega_2]^\omega \big)^V$ with countable conditions. We prove that the version of Strong Chang's Conjecture from \cite{MR2965421} implies semiproperness of $\mathbb{Q}$, and that semiproperness of $\mathbb{Q}$---in fact semiproperness of any poset which is sufficiently \emph{nonreasonable} in the sense of Foreman-Magidor~\cite{MR1359154}---implies the version of Strong Chang's Conjecture from \cite{MR2723878} and \cite{MR1261218}. In particular, semiproperness of $\mathbb{Q}$ has large cardinal strength, which answers a question of Friedman-Krueger~\cite{MR2276627}. One corollary of our work is that the version of Strong Chang's Conjecture from \cite{MR2965421} does not imply the existence of a precipitous ideal on $\omega_1$., Comment: Added a new section (Section 3) about relationship between MM, dagger principle, and the principle $\text{SCC}^{\text{cof}}_{\text{gap}}$

We introduce a stronger version of an $\omega_1$-guessing model, which we call an indestructibly $\omega_1$-guessing model. The principle IGMP states that there are stationarily many indestructibly $\omega_1$-guessing models. This principle, which follows from PFA, captures many of the consequences of PFA, including the Suslin hypothesis and the singular cardinal hypothesis. We prove that IGMP is consistent with the continuum being arbitrarily large.

Given an uncountable regular cardinal $\kappa$, a partial order is $\kappa$-stationarily layered if the collection of regular suborders of $\mathbb{P}$ of cardinality less than $\kappa$ is stationary in $\mathcal{P}_\kappa(\mathbb{P})$. We show that weak compactness can be characterized by this property of partial orders by proving that an uncountable regular cardinal $\kappa$ is weakly compact if and only if every partial order satisfying the $\kappa$-chain condition is $\kappa$-stationarily layered. We prove a similar result for strongly inaccessible cardinals. Moreover, we show that the statement that all $\kappa$-Knaster partial orders are $\kappa$-stationarily layered implies that $\kappa$ is a Mahlo cardinal and every stationary subset of $\kappa$ reflects. This shows that this statement characterizes weak compactness in canonical inner models. In contrast, we show that it is also consistent that this statement holds at a non-weakly compact cardinal.

We develop the theory of layered posets, and use the notion of layering to prove a new iteration theorem (Theorem 6): if $\kappa$ is weakly compact then any universal Kunen iteration of $\kappa$-cc posets (each possibly of size $\kappa$) is $\kappa$-cc, as long as direct limits are used sufficiently often. This iteration theorem simplifies and generalizes the various chain condition arguments for universal Kunen iterations in the literature on saturated ideals, especially in situations where finite support iterations are not possible. We also provide two applications: (1) For any $n \ge 1$, a wide variety of $<\omega_{n-1}$-closed, $\omega_{n+1}$-cc posets of size $\omega_{n+1}$ can consistently be absorbed (as regular suborders) by quotients of saturated ideals on $\omega_n$ (see Theorem 7 and Corollary 8); and (2) For any $n \in \omega$, the Tree Property at $\omega_{n+3}$ is consistent with the Chang's Conjecture $(\omega_{n+3}, \omega_{n+1}) \twoheadrightarrow (\omega_{n+1}, \omega_n)$ (Theorem 9)., Comment: corrected proof of Lemma 4

We prove that a wide class of strongly proper forcing posets have quotients with strong properties. Specifically, we prove that quotients of forcing posets which have simple universal strongly generic conditions on a stationary set of models by certain nice regular suborders satisfy the $\omega_1$-approximation property. We prove that the existence of stationarily many $\omega_1$-guessing models in $P_{\omega_2}(H(\theta))$, for sufficiently large cardinals $\theta$, is consistent with the continuum being arbitrarily large, solving a problem of Viale and Weiss.

Viale \cite{Viale_GuessingModel} introduced the notion of Generic Laver Diamond at $\kappa$---which we denote $\Diamond_{\text{Lav}}(\kappa)$---asserting the existence of a single function from $\kappa \to H_\kappa$ that behaves much like a supercompact Laver function, except with generic elementary embeddings rather than internal embeddings. Viale proved that the Proper Forcing Axiom (PFA) implies $\Diamond_{\text{Lav}}(\omega_2)$. We strengthen his theorem by weakening the hypothesis to a statement strictly weaker than PFA. We also show that the principle $\Diamond_{\text{Lav}}(\kappa)$ provides a uniform, simple construction of 2-cardinal diamonds, and prove that $\Diamond_{\text{Lav}}(\kappa)$ is quite prevalent in models of set theory; in particular: 1) $L$ satisfies $\Diamond_{\text{Lav}}^+(\kappa)$ whenever $\kappa$ is a successor cardinal, or when the appropriate version of Chang's Conjecture fails. 2) For any successor cardinal $\kappa$, there is a $\kappa$-directed closed class forcing---namely, the forcing from Friedman-Holy \cite{MR2860182}---that forces $\Diamond_{\text{Lav}}(\kappa)$., Comment: To appear in Proceedings of the AMS

Foreman proved the Duality Theorem, which gives an algebraic characterization of certain ideal quotients in generic extensions. As an application he proved that generic supercompactness of $\omega_1$ is preserved by any proper forcing. We generalize portions of Foreman's Duality Theorem to the context of generic extender embeddings and ideal extenders (as introduced by Claverie in his PhD Thesis, Universitat Munster, 2010). As an application we prove that if $\omega_1$ is generically strong, then it remains so after adding any number of Cohen subsets of $\omega_1$; however many other $\omega_1$-closed posets---such as $\text{Col}(\omega_1, \omega_2)$---can destroy the generic strength of $\omega_1$. This generalizes some results of Gitik-Shelah about indestructibility of strong cardinals to the generically strong context. We also prove similar theorems for successor cardinals larger than $\omega_1$.

There are several examples in the literature showing that compactness-like properties of a cardinal $\kappa$ cause poor behavior of some generic ultrapowers which have critical point $\kappa$ (Burke \cite{MR1472122} when $\kappa$ is a supercompact cardinal; Foreman-Magidor \cite{MR1359154} when $\kappa = \omega_2$ in the presence of strong forcing axioms). We prove more instances of this phenomenon. First, the Reflection Principle (RP) implies that if $\vec{\mathcal{I}}$ is a tower of ideals which concentrates on the class $GIC_{\omega_1}$ of $\omega_1$-guessing, internally club sets, then $\vec{\mathcal{I}}$ is not presaturated (a set is $\omega_1$-guessing iff its transitive collapse has the $\omega_1$-approximation property as defined in Hamkins \cite{MR2540935}). This theorem, combined with work from \cite{VW_ISP}, shows that if $PFA^+$ or $MM$ holds and there is an inaccessible cardinal, then there is a tower with critical point $\omega_2$ which is not presaturated; moreover this tower is significantly different from the non-presaturated tower already known (by Foreman-Magidor \cite{MR1359154}) to exist in all models of Martin's Maximum. The conjunction of the Strong Reflection Principle (SRP) and the Tree Property at $\omega_2$ has similar implications for towers of ideals which concentrate on the wider class $GIS_{\omega_1}$ of $\omega_1$-guessing, internally stationary sets. Finally, we show that the word "presaturated" cannot be replaced by "precipitous" in the theorems above: Martin's Maximum (which implies SRP and the Tree Property at $\omega_2$) is consistent with a precipitous tower on $GIC_{\omega_1}$.