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Tambara functors are the analogue of commutative rings in equivariant algebra. Nakaoka defined ideals in Tambara functors, leading to the definition of the Nakaoka spectrum of prime ideals in a Tambara functor. In this work, we continue the study of the Nakoaka spectra of Tambara functors. We describe, in terms of the Zariski spectra of ordinary commutative rings, the Nakaoka spectra of many Tambara functors. In particular: we identify the Nakaoka spectrum of the fixed point Tambara functor of any $G$-ring with the GIT quotient of its classical Zariski spectrum; we describe the Nakaoka spectrum of the complex representation ring Tambara functor over a cyclic group of prime order $p$; we describe the affine line (the Nakaoka spectra of free Tambara functors on one generator) over a cyclic group of prime order $p$ in terms of the Zariski spectra of $\mathbb{Z}[x]$, $\mathbb{Z}[x,y]$, and the ring of cyclic polynomials $\mathbb{Z}[x_0,\ldots,x_{p-1}]^{C_p}$. To obtain these results, we introduce a "ghost construction" which produces an integral extension of any $C_p$-Tambara functor, the Nakaoka spectrum of which is describable. To relate the Nakaoka spectrum of a Tambara functor to that of its ghost, we prove several new results in equivariant commutative algebra, including a weak form of the Hilbert basis theorem, going up, lying over, and levelwise radicality of prime ideals in Tambara functors. These results also allow us to compute the Krull dimensions of many Tambara functors., Comment: 52 pages. Comments welcome! v2: Improved exposition in section 3.2; updated remark 7.8

We compute Mackey functor-valued Tor over certain free incomplete Tambara functors, generalizing the computation of Tor over a polynomial ring on one generator. In contrast with the classical situation where the resulting Tor groups vanish above degree one, we present examples where Tor is nonvanishing in almost every degree. We also discuss a 2-primary analogue of the odd-primary Koszul complexes defined in our other work \cite{MQS24a}., Comment: 16 Pages, 3 Figures. Comments welcome!

In equivariant algebra, Mackey functors replace abelian groups and incomplete Tambara functors replace commutative rings. In this context, we prove that equivariant Hochschild homology can sometimes be computed using Mackey functor-valued Tor. To compute these Tor Mackey functors for odd primes $p$, we define cyclic-$p$-group-equivariant analogues of the Koszul resolution which resolve the Burnside Mackey functor (the analogue of the integers) as a module over free incomplete Tambara functors (the analogue of polynomial rings). We apply these Koszul resolutions to compute Mackey functor-valued Hochschild homology of free incomplete Tambara functors for cyclic groups of odd prime order and for the cyclic group of order 9.

We prove that the image of the total power operation for Burnside rings $A(G) \to A(G\wr\Sigma_n)$ lies inside a relatively small, combinatorial subring $\mathring A(G,n) \subseteq A(G \wr \Sigma_n)$. As $n$ varies, the subrings $\mathring A(G,n)$ assemble into a commutative graded ring $\mathring A(G)$ with a universal property: $\mathring A(G)$ carries the universal family of power operations out of $A(G)$. We construct character maps for $\mathring A(G,n)$ and give a formula for the character of the total power operation. Using $\mathring A(G)$, we extend the Frobenius--Wielandt homomorphism of Dress--Siebeneicher--Yoshida to wreath products compatibly with the total power operation. Finally, we prove a generalization of Burnside's orbit counting lemma that describes the transfer map $A(G \wr \Sigma_n) \to A(\Sigma_n)$ on the subring $\mathring A(G,n)$.

We compute the homotopy Mackey functors of the $KU_G$-local equivariant sphere spectrum when $G$ is a finite $q$-group for an odd prime $q$, building on the degree zero case from arXiv:2204.03797., Comment: 15 pages. Comments welcome!

Free algebras are always free as modules over the base ring in classical algebra. In equivariant algebra, free incomplete Tambara functors play the role of free algebras and Mackey functors play the role of modules. Surprisingly, free incomplete Tambara functors often fail to be free as Mackey functors. In this paper, we determine for all finite groups conditions under which a free incomplete Tambara functor is free as a Mackey functor. For solvable groups, we show that a free incomplete Tambara functor is flat as a Mackey functor precisely when these conditions hold. Our results imply that free incomplete Tambara functors are almost never flat as Mackey functors. However, we show that after suitable localizations, free incomplete Tambara functors are always free as Mackey functors., Comment: V2: added section 6.1 on the restriction functor V3: Updated according to referee report. Accepted version, to appear in IMRN

We present 2,241 exoplanet candidates identified with data from the Transiting Exoplanet Survey Satellite (TESS) during its two-year prime mission. We list these candidates in the TESS Objects of Interest (TOI) Catalog, which includes both new planet candidates found by TESS and previously-known planets recovered by TESS observations. We describe the process used to identify TOIs and investigate the characteristics of the new planet candidates, and discuss some notable TESS planet discoveries. The TOI Catalog includes an unprecedented number of small planet candidates around nearby bright stars, which are well-suited for detailed follow-up observations. The TESS data products for the Prime Mission (Sectors 1-26), including the TOI Catalog, light curves, full-frame images, and target pixel files, are publicly available on the Mikulski Archive for Space Telescopes., Comment: 39 pages, 16 figures. The Prime Mission TOI Catalog is included in the ancillary data as a CSV. For the most up-to-date catalog, refer to https://tess.mit.edu/toi-releases/

Planets occur most frequently around cool dwarfs, but only a handful of specific examples are known to orbit the latest-type M stars. Using TESS photometry, we report the discovery of two planets transiting the low-mass star called LP 791-18 (identified by TESS as TOI 736). This star has spectral type M6V, effective temperature 2960 K, and radius 0.17 R_Sun, making it the third-coolest star known to host planets. The two planets straddle the radius gap seen for smaller exoplanets; they include a 1.1 R_Earth planet on a 0.95 day orbit and a 2.3 R_Earth planet on a 5 day orbit. Because the host star is small the loss of light during these planets' transits is fairly large (0.4% and 1.7%). This has allowed us to detect both planets' transits from ground-based photometry, refining their radii and orbital ephemerides. In the future, radial velocity observations and transmission spectroscopy can both probe these planets' bulk interior and atmospheric compositions, and additional photometric monitoring would be sensitive to even smaller transiting planets., Comment: 20 pages, 7 figures, 2 tables, 2 planets, 1 cool star. Submitted to AAS Journals

We produce light curves for all ~34,000 targets observed with K2 in Campaign 17 (C17), identifying 34 planet candidates, 184 eclipsing binaries, and 222 other periodic variables. The location of the C17 field means follow-up can begin immediately now that the campaign has concluded and interesting targets have been identified. The C17 field has a large overlap with C6, so this latest campaign also offers a rare opportunity to study a large number of targets already observed in a previous K2 campaign. The timing of the C17 data release, shortly before science operations begin with the Transiting Exoplanet Survey Satellite (TESS), also lets us exercise some of the tools and methods developed for identification and dissemination of planet candidates from TESS. We find excellent agreement between these results and those identified using only K2-based tools. Among our planet candidates are several planet candidates with sizes < 4 R_E and orbiting stars with KepMag < 10 (indicating good RV targets of the sort TESS hopes to find) and a Jupiter-sized single-transit event around a star already hosting a 6 d planet candidate., Comment: Submitted. 19 pages, 5 figures, 5 tables, 34 planet candidates, 184 EBs, 34,000 lightcurves

We generalize a construction of families of moderate rank elliptic curves over $\mathbb{Q}$ to number fields $K/\mathbb{Q}$. The construction, originally due to Steven J. Miller, \'Alvaro Lozano-Robledo and Scott Arms, invokes a theorem of Rosen and Silverman to show that computing the rank of these curves can be done by controlling the average of the traces of Frobenius, the construction for number fields proceeds in essentially the same way. One novelty of this method is that we can construct families of moderate rank without having to explicitly determine points and calculating determinants of height matrices., Comment: Version 1.0, 4 pages, sequel to arXiv:math/0406579

De Bruijn and Newman introduced a deformation of the completed Riemann zeta function $\zeta$, and proved there is a real constant $\Lambda$ which encodes the movement of the nontrivial zeros of $\zeta$ under the deformation. The Riemann hypothesis is equivalent to the assertion that $\Lambda\leq 0$. Newman, however, conjectured that $\Lambda\geq 0$, remarking, "the new conjecture is a quantitative version of the dictum that the Riemann hypothesis, if true, is only barely so." Andrade, Chang and Miller extended the machinery developed by Newman and Polya to $L$-functions for function fields. In this setting we must consider a modified Newman's conjecture: $\sup_{f\in\mathcal{F}} \Lambda_f \geq 0$, for $\mathcal{F}$ a family of $L$-functions. We extend their results by proving this modified Newman's conjecture for several families of $L$-functions. In contrast with previous work, we are able to exhibit specific $L$-functions for which $\Lambda_D = 0$, and thereby prove a stronger statement: $\max_{L\in\mathcal{F}} \Lambda_L = 0$. Using geometric techniques, we show a certain deformed $L$-function must have a double root, which implies $\Lambda = 0$. For a different family, we construct particular elliptic curves with $p + 1$ points over $\mathbb{F}_p$. By the Weil conjectures, this has either the maximum or minimum possible number of points over $\mathbb{F}_{p^{2n}}$. The fact that $#E(\mathbb{F}_{p^{2n}})$ attains the bound tells us that the associated $L$-function satisfies $\Lambda = 0$., Comment: Version 1.1, 12 pages

The Cosmology Large Angular Scale Surveyor (CLASS) is an experiment to measure the signature of a gravita-tional-wave background from inflation in the polarization of the cosmic microwave background (CMB). CLASS is a multi-frequency array of four telescopes operating from a high-altitude site in the Atacama Desert in Chile. CLASS will survey 70\% of the sky in four frequency bands centered at 38, 93, 148, and 217 GHz, which are chosen to straddle the Galactic-foreground minimum while avoiding strong atmospheric emission lines. This broad frequency coverage ensures that CLASS can distinguish Galactic emission from the CMB. The sky fraction of the CLASS survey will allow the full shape of the primordial B-mode power spectrum to be characterized, including the signal from reionization at low $\ell$. Its unique combination of large sky coverage, control of systematic errors, and high sensitivity will allow CLASS to measure or place upper limits on the tensor-to-scalar ratio at a level of $r=0.01$ and make a cosmic-variance-limited measurement of the optical depth to the surface of last scattering, $\tau$., Comment: 23 pages, 10 figures, Presented at SPIE Astronomical Telescopes and Instrumentation 2014: Millimeter, Submillimeter, and Far-Infrared Detectors and Instrumentation for Astronomy VII. To be published in Proceedings of SPIE Volume 9153

The Cosmology Large Angular Scale Surveyor (CLASS) experiment aims to map the polarization of the Cosmic Microwave Background (CMB) at angular scales larger than a few degrees. Operating from Cerro Toco in the Atacama Desert of Chile, it will observe over 65% of the sky at 38, 93, 148, and 217 GHz. In this paper we discuss the design, construction, and characterization of the CLASS 38 GHz detector focal plane, the first ever Q-band bolometric polarimeter array., Comment: 15 pages, 4 figures. Presented at SPIE Astronomical Telescopes and Instrumentation 2014: Millimeter, Submillimeter, and Far-Infrared Detectors and Instrumentation for Astronomy VII. To be published in Proceedings of SPIE Volume 9153

Modularity maximization has been one of the most widely used approaches in the last decade for discovering community structure in networks of practical interest in biology, computing, social science, statistical mechanics, and more. Modularity is a quality function that measures the difference between the number of edges found within clusters minus the number of edges one would statistically expect to find based on random chance. We present a natural generalization of modularity based on the difference between the actual and expected number of walks within clusters, which we call walk-modularity. Walk-modularity can be expressed in matrix form, and community detection can be performed by finding leading eigenvectors of the walk-modularity matrix. We demonstrate community detection on both synthetic and real-world networks and find that walk-modularity maximization returns significantly improved results compared to traditional modularity maximization., Comment: 13 pages, 6 figures, 1 table