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Let $f$ be a newform of prime level $p$ with any central character $\chi\, (\bmod\, p)$, and let $g$ be a fixed cusp form or Eisenstein series for $\hbox{SL}_{2}(\mathbb{Z})$. We prove the subconvexity bound: for any $\varepsilon>0$, \begin{align*} L(1/2, \, f \otimes g) \ll p^{1/2-1/524+\varepsilon}, \end{align*} where the implied constant depends on $g$, $\varepsilon$, and the archimedean parameter of $f$. This improves upon the previously best-known result by Harcos and Michel. Our method ultimately relies on non-trivial bounds for bilinear forms in Kloosterman fractions pioneered by Duke, Friedlander, and Iwaniec, with later innovations by Bettin and Chandee., Comment: 27 pages. Comments welcome!

We develop generalized Petersson/Bruggeman/Kuznetsov (PBK) formulas for specified local components at non-archimedean places. In fact, we introduce two hypotheses on non-archimedean test function pairs $f \leftrightarrow \pi(f)$, called geometric and spectral hypotheses, under which one obtains `nice' PBK formulas by the adelic relative trace function approach. Then, given a supercuspidal representation $\sigma$ of ${\rm PGL}_2(\mathbb{Q}_p)$, we study extensively the case that $\pi(f)$ is a projection onto the line of the newform if $\pi$ is isomorphc to $\sigma$ or its unramified quadratic twist, and $\pi(f) = 0$ otherwise. As a first application, we prove an optimal large sieve inequality for families of automorphic representations that arise in our framework., Comment: v2: Proof that p=2 principal series test function is a newform projector added, and a few very minor typos fixed

Quasi-separatrix layers (QSLs) at the Sun are created from regions where channels of open magnetic flux have footpoints near regions of large-scale closed magnetic flux. These regions are particularly prone to magnetic reconnection at the Sun. In recent simulations of coronal mass ejections (CMEs) with the Magnetohydrodynamic Algorithm outside a Sphere (MAS) model coupled to the Energetic Particle Radiation Environment Module (EPREM) model, common sources of energetic particles were discovered over broad longitudinal distributions in the background solar wind, far from the sites of particle acceleration driven by compressions and shocks in front of CMEs. Further investigation revealed these to be accelerated energetic particles from the QSLs and current sheets. The energy released from magnetic reconnection near the QSL drives reconnection exhausts and field-aligned flows, which in turn accelerate energetic particles. The reconnection process also releases material previously contained within closed magnetic field structures, which are often rich in heavy ions and $^3$He ions, as corroborated by recent PSP observations. Therefore, the seed populations produced by QSLs are expected to be rich in $^3$He and heavy ions. Thus, we present the first global model of energetic particles accelerated from QSLs and above current sheets from the Sun. Our results provide a plausible source for seed populations near the Sun, which likely have $^3$He and heavy ion enhancements. These results aid in the development of predictive solar energetic particle models., Comment: 34 pages and 13 figures

This is an introduction to Hall algebras from the perspective of $2$-Segal spaces or decomposition spaces, as introduced by Dyckerhoff and Kapranov and G\'{a}lvez-Carrillo, Kock and Tonks, respectively. We explain how linearizations of the $2$-Segal space arising as the Waldhausen $\mathcal{S}_{\bullet}$-construction of a proto-exact category recover various previously known Hall algebras. We use the $2$-Segal perspective to study functoriality of the Hall algebra construction and explain how relative variants of $2$-Segal spaces lead naturally to representations of Hall algebras., Comment: 32 pages

We construct three dimensional non-semisimple topological field theories from the unrolled quantum group of the Lie superalgebra $\mathfrak{osp}(1 \vert 2)$. More precisely, the quantum group depends on a root of unity $q=e^{\frac{2 \pi \sqrt{-1}}{r}}$, where $r$ is a positive integer greater than $2$, and the construction applies when $r$ is not congruent to $4$ modulo $8$. The algebraic result which underlies the construction is the existence of a relative modular structure on the non-finite, non-semisimple category of weight modules for the quantum group. We prove a Verlinde formula which allows for the computation of dimensions and Euler characteristics of topological field theory state spaces of unmarked surfaces. When $r$ is congruent to $\pm 1$ or $\pm 2$ modulo $8$, we relate the resulting $3$-manifold invariants with physicists' $\widehat{Z}$-invariants associated to $\mathfrak{osp}(1 \vert 2)$. Finally, we establish a relation between $\widehat{Z}$-invariants associated to $\mathfrak{sl}(2)$ and $\mathfrak{osp}(1 \vert 2)$ which was conjectured in the physics literature., Comment: 38 pages

We obtain a strong bound on the second moment of the $GL_3$ standard $L$-function on the critical line. The method builds on the recent work of Aggarwal, Leung, and Munshi which treated shorter intervals. We deduce some corollaries including an improvement on the error term in the Rankin-Selberg problem, and on certain subconvexity bounds for $GL_3 \times GL_2$ and $GL_3$ $L$-functions. As a byproduct of the method of proof, we also obtain an estimate for an average of shifted convolution sums of $GL_3$ Fourier coefficients.

We study a generalized Dedekind sum $S_{\chi_1,\chi_2}(a,c)$ attached to newform Eisenstein series $E_{\chi_1,\chi_2}(z,s)$. Our work shows the Dedekind sum is rarely substantially larger than $\log^3 c$. The method of proof first relates the size of the Dedekind sum to continued fractions. A result of Hensley from 1991 then controls the average size of the maximal partial quotient in the continued fraction expansion of $a/c$. We complement this result by computing approximate values of the Dedekind sum in some special cases, which in particular produces examples of large values of the Dedekind sum., Comment: 9 pages

We develop representation theoretic techniques to construct three dimensional non-semisimple topological quantum field theories which model homologically truncated topological B-twists of abelian Gaiotto-Witten theory with linear matter. Our constructions are based on relative modular structures on the category of weight modules over an unrolled quantization of a Lie superalgebra. The Lie superalgebra, originally defined by Gaiotto and Witten, is associated to a complex symplectic representation of a metric abelian Lie algebra. The physical theories we model admit alternative realizations as Chern-Simons-Rozansky-Witten theories and supergroup Chern-Simons theories and include as particular examples global forms of $\mathfrak{gl}(1 \vert 1)$-Chern-Simons theory and toral Chern-Simons theory. Fundamental to our approach is the systematic incorporation of non-genuine line operators which source flat connections for the topological flavour symmetry of the theory., Comment: 46 pages

We prove an asymptotic formula for the second moment of central values of Dirichlet $L$-functions restricted to a coset. More specifically, consider a coset of the subgroup of characters modulo $d$ inside the full group of characters modulo $q$. Suppose that $\nu_p(d) \geq \nu_p(q)/2$ for all primes $p$ dividing $q$. In this range, we obtain an asymptotic formula with a power-saving error term; curiously, there is a secondary main term of rough size $q^{1/2}$ here which is not predicted by the integral moments conjecture of Conrey, Farmer, Keating, Rubinstein, and Snaith. The lower-order main term does not appear in the second moment of the Riemann zeta function, so this feature is not anticipated from the analogous archimedean moment problem. We also obtain an asymptotic result for smaller $d$, with $\nu_p(q)/3 \leq \nu_p(d) \leq \nu_p(q)/2$, with a power-saving error term for $d$ larger than $q^{2/5}$. In this more difficult range, the secondary main term somewhat changes its form and may have size roughly $d$, which is only slightly smaller than the diagonal main term., Comment: 18 pages

We construct a two dimensional unoriented open/closed topological field theory from a finite graded group $\pi:\hat{G} \twoheadrightarrow \{1,-1\}$, a $\pi$-twisted $2$-cocycle $\hat{\theta}$ on $B \hat{G}$ and a character $\lambda: \hat{G} \rightarrow U(1)$. The underlying oriented theory is a twisted Dijkgraaf-Witten theory. The construction is based in the $(\hat{G}, \hat{\theta},\lambda)$-twisted Real representation theory of $\ker \pi$. In particular, twisted Real representations are boundary conditions and the generalized Frobenius-Schur element is its crosscap state., Comment: 24 pages

We present an updated design of the 220 GHz microwave kinetic inductance detector (MKID) pixel for SPT-3G+, the next-generation camera for the South Pole Telescope. We show results of the dark testing of a 63-pixel array with mean inductor quality factor $Q_i = 4.8 \times 10^5$, aluminum inductor transition temperature $T_c = 1.19$ K, and kinetic inductance fraction $\alpha_k = 0.32$. We optically characterize both the microstrip-coupled and CPW-coupled resonators, and find both have a spectral response close to prediction with an optical efficiency of $\eta \sim 70\%$. However, we find slightly lower optical response on the lower edge of the band than predicted, with neighboring dark detectors showing more response in this region, though at level consistent with less than 5\% frequency shift relative to the optical detectors. The detectors show polarized response consistent with expectations, with a cross-polar response of $\sim 10\%$ for both detector orientations., Comment: 6 pages, 5 figures, ASC 2022 proceedings

This paper is a self-contained introduction to the theory of renormalized Reshetikhin-Turaev invariants of links defined by Geer, Patureau-Mirand and Turaev. Whereas the standard Reshetikhin-Turaev theory of a $\mathbb{C}$-linear ribbon category assigns the trivial invariant to any link with a component colored by a simple object of vanishing quantum dimension, the renormalized theory does not. We give a streamlined development of the renormalized Reshetikhin-Turaev theory of links for the category of weight modules over the restricted unrolled quantum group of $\mathfrak{sl}_2(\mathbb{C})$ at an even root of unity., Comment: 30 pages

Let $E$ be an elliptic curve over $\mathbf{Q}$. We conjecture asymptotic estimates for the number of vanishings of $L(E,1,\chi)$ as $\chi$ varies over all primitive Dirichlet characters of orders 4 and 6, subject to a mild hypothesis on $E$. Our conjectures about these families come from conjectures about random unitary matrices as predicted by the philosophy of Katz-Sarnak. We support our conjectures with numerical evidence. Earlier work by David, Fearnley and Kisilevsky formulates analogous conjectures for characters of any odd prime order. In the composite order case, however, we need to justify our use of random matrix theory heuristics by analyzing the equidistribution of the squares of normalized Gauss sums. Along the way we introduce the notion of totally order $\ell$ characters to quantify how quickly quartic and sextic Gauss sums become equidistributed. Surprisingly, the rate of equidistribution in the full family of quartic (sextic, resp.) characters is much slower than in the sub-family of totally quartic (sextic, resp.) characters. A conceptual explanation for this phenomenon is that the full family of order $\ell$ twisted elliptic curve $L$-functions, with $\ell$ even and composite, is a mixed family with both unitary and orthogonal aspects., Comment: 31 pages, 5 figures

In this paper we construct twisted Real quasi-elliptic cohomology as the twisted KR-theory of loop groupoids. The theory systematically incorporates loop rotation and reflection. After establishing basic properties of the theory, we construct twisted elliptic Pontryagin characters and, without twists, Real analogues of the string power operation of quasi-elliptic cohomology. We also explore the relation of the theory to the Tate curve., Comment: 46 pages

We introduce an unrolled quantization $U_q^E(\mathfrak{gl}(1 \vert 1))$ of the complex Lie superalgebra $\mathfrak{gl}(1 \vert 1)$ and use its categories of weight modules to construct and study new three dimensional non-semisimple topological quantum field theories. These theories are defined on categories of cobordisms which are decorated by ribbon graphs and cohomology classes and take values in categories of graded super vector spaces. Computations in these theories are enabled by a detailed study of the representation theory of $U_q^E(\mathfrak{gl}(1 \vert 1))$, both for generic and root of unity $q$. We argue that by restricting to subcategories of integral weight modules we obtain topological quantum field theories which are mathematical models of Chern--Simons theories with gauge supergroups $\mathfrak{psl}(1 \vert 1)$ and $\mathfrak{gl}(1 \vert 1)$ coupled to background flat $\mathbb{C}^{\times}$-connections, as studied in the physics literature by Rozansky--Saleur and Mikhaylov. In particular, we match Verlinde formulae and mapping class group actions on state spaces of non-generic tori with results in the physics literature. We also obtain explicit descriptions of state spaces of generic surfaces, including their graded dimensions, which go beyond results in the physics literature., Comment: 54 pages. v2: Results strengthened to give a complete description of state spaces of non-generic tori for arbitrary q. Other minor improvements throughout

We prove an essentially optimal large sieve inequality for self-dual Eisenstein series of varying levels. This bound can alternatively be interpreted as a large sieve inequality for rationals ordered by height. The method of proof is recursive, and has some elements in common with Heath-Brown's quadratic large sieve, and the asymptotic large sieve of Conrey, Iwaniec, and Soundararajan., Comment: 40 pages. v2: refereed version implementing various corrections. Accepted for publication by Acta Arithmetica, special issue in honor of Henryk Iwaniec

We improve on the spectral large sieve inequality for symmetric-squares. We also prove a lower bound showing that the most optimistic upper bound is not true for this family., Comment: 16 pages

Motivated by periodicity theorems for Real $K$-theory and Grothendieck--Witt theory and, separately, work of Hori-Walcher on the physics of Landau-Ginzburg orientifolds, we introduce and study categories of Real matrix factorizations. Our main results are generalizations of Kn\"{o}rrer periodicity to categories of Real matrix factorizations. These generalizations are structurally similar to $(1,1)$-periodicity for $KR$-theory and $4$-periodicity for Grothendieck-Witt theory. We use techniques from Real categorical representation theory which allow us to incorporate into our main results equivariance for a finite group and discrete torsion twists., Comment: 31 pages. v2: Typos fixed, results slightly strengthened

How self-organization leads to the emergence of structure in social populations remains a fascinating and open question in the study of complex systems. One frequently observed structure that emerges again and again across systems is that of self-similar community, i.e., homophily. We use a game theoretic perspective to explore a case in which individuals choose affiliation partnerships based on only two factors: the value they place on having social contacts, and their risk tolerance for exposure to threat derived from social contact (e.g., infectious disease, threatening ideas, etc.). We show how diversity along just these two influences are sufficient to cause the emergence of self-organizing homophily in the population. We further consider a case in which extrinsic social factors influence the desire to maintain particular social ties, and show the robustness of emergent homophilic patterns to these additional influences. These results demonstrate how observable population-level homophily may arise out of individual behaviors that balance the value of social contacts against the potential risks associated with those contacts. We present and discuss these results in the context of outbreaks of infectious disease in human populations. Complementing the standard narrative about how social division alters epidemiological risk, we here show how epidemiological risk may deepen social divisions in human populations., Comment: 17 pages, 4 figures

We use the action of Atkin-Lehner operators to generate a family of reciprocity formulas for newform Dedekind sums. This family of reciprocity formulas provides symmetries which we use to investigate the kernel of these Dedekind sums., Comment: 9 pages, 2 figures

We prove an improved spectral large sieve inequality for the family of $SL_3(\mathbb{Z})$ Hecke-Maass cusp forms. The method of proof uses duality and its structure reveals unexpected connections to Heath-Brown's large sieve for cubic characters., Comment: 11 pages

This work presents results from simulations of the 14 July 2000 ("Bastille Day") solar proton event. We used the Energetic Particle Radiation Environment Model (EPREM) and the CORona-HELiosphere (CORHEL) software suite within the SPE Threat Assessment Tool (STAT) framework to model proton acceleration to GeV energies due to the passage of a CME through the low solar corona, and compared the model results to GOES-08 observations. The coupled simulation models particle acceleration from 1 to 20 $R_\odot$, after which it models only particle transport. The simulation roughly reproduces the peak event fluxes, and timing and spatial location of the energetic particle event. While peak fluxes and overall variation within the first few hours of the simulation agree well with observations, the modeled CME moves beyond the inner simulation boundary after several hours. The model therefore accurately describes the acceleration processes in the low corona and resolves the sites of most rapid acceleration close to the Sun. Plots of integral flux envelopes from multiple simulated observers near Earth further improve the comparison to observations and increase potential for predicting solar particle events. Broken-power-law fits to fluence spectra agree with diffusive acceleration theory over the low energy range. Over the high energy range, they demonstrate the variability in acceleration rate and mirror the inter-event variability observed solar-cycle 23 GLEs. We discuss ways to improve STAT predictions, including using corrected GOES energy bins and computing fits to the seed spectrum. This paper demonstrates a predictive tool for simulating low-coronal SEP acceleration., Comment: 18 pages; 9 figures

We study the algebraic $K$-theory and Grothendieck-Witt theory of proto-exact categories, with a particular focus on classes of examples of $\mathbb{F}_1$-linear nature. Our main results are analogues of theorems of Quillen and Schlichting, relating the $K$-theory or Grothendieck-Witt theories of proto-exact categories defined using the (hermitian) $Q$-construction and group completion.

We study the algebraic $K$-theory and Grothendieck-Witt theory of proto-exact categories of vector bundles over monoid schemes. Our main results are the complete description of the algebraic $K$-theory space of an integral monoid scheme $X$ in terms of its Picard group $\operatorname{Pic}(X)$ and pointed monoid of regular functions $\Gamma(X, \mathcal{O}_X)$ and a description of the Grothendieck-Witt space of $X$ in terms of an additional involution on $\operatorname{Pic}(X)$. We also prove space-level projective bundle formulae in both settings.

We establish sharp bounds for the second moment of symmetric-square $L$-functions attached to Hecke Maass cusp forms $u_j$ with spectral parameter $t_j$, where the second moment is a sum over $t_j$ in a short interval. At the central point $s=1/2$ of the $L$-function, our interval is smaller than previous known results. More specifically, for $|t_j|$ of size $T$, our interval is of size $T^{1/5}$, while the previous best was $T^{1/3}$ from work of Lam. A little higher up on the critical line, our second moment yields a subconvexity bound for the symmetric-square $L$-function. More specifically, we get subconvexity at $s=1/2+it$ provided $|t_j|^{6/7+\delta}\le |t| \le (2-\delta)|t_j|$ for any fixed $\delta>0$. Since $|t|$ can be taken significantly smaller than $|t_j|$, this may be viewed as an approximation to the notorious subconvexity problem for the symmetric-square $L$-function in the spectral aspect at $s=1/2$.

Newform Dedekind sums are a class of crossed homomorphisms that arise from newform Eisenstein series. We initiate a study of the kernel of these newform Dedekind sums. Our results can be loosely described as showing that these kernels are neither "too big" nor "too small." We conclude with an observation about the Galois action on Dedekind sums that allows for significant computational efficiency in the numerical calculation of Dedekind sums., Comment: 8+epsilon pages, 2 figures

Interplanetary shocks are one of the proposed sources of suprathermal ion populations (i.e., ions with energies of a few times the solar wind energy). Here, we present results from a series of three-dimensional hybrid simulations of collisionless shocks in the solar wind. We focus on the influence of the shock-normal angle, $\theta_{Bn}$, and the shock speed, $V_s$, on producing protons with energies a few to hundreds of times the thermal energy of the upstream plasma. The combined effects of $\theta_{Bn}$ and $V_s$ result in shocks with Alfv\'en Mach numbers in the range 3.0 to 6.0 and fast magnetosonic Mach numbers in the range 2.5 to 5.0, representing moderate to strong interplanetary shocks. We find that $\theta_{Bn}$ largely organizes the shape of proton energy spectra while shock speed controls acceleration efficiency. All shocks accelerate protons at the shock front but the spectral evolution depends on $\theta_{Bn}$. Shocks with $\theta_{Bn} \geq 60^\circ$ produce isolated bursts of suprathermal protons at the shock front while shocks with $\theta_{Bn} \leq 45^\circ$ create suprathermal beams upstream of the shock. Downstream proton energy spectra have exponential or smoothed broken power-law forms when $\theta_{Bn} \geq 45^\circ$, and a single power-law form when $\theta_{Bn} \leq 30^\circ$. Protons downstream of the strongest shocks have energies at least 100 times the upstream thermal energy, with $\theta_{Bn} \leq 30^\circ$ shocks producing the highest energy protons and $\theta_{Bn} \geq 60^\circ$ shocks producing the largest number of protons with energies at least a few times the thermal energy., Comment: 20 pages; 10 figures