We introduce a newinvariant describing the structure of sets of lengths in atomicmonoids and domains. For an atomic monoid $H$, let ${{\Delta }_{\rho }}\left( H \right)$ be the set of all positive integers d that occur as differences of arbitrarily long arithmetical progressions contained in sets of lengths havingmaximal elasticity $\rho \left( H \right)$. We study ${{\Delta }_{\rho }}\left( H \right)$ for transfer Krull monoids of finite type (including commutative Krull domains with finite class group) with methods from additive combinatorics, and also for a class of weakly Krull domains (including orders in algebraic number fields) for which we use ideal theoretic methods.

We characterize the Local Langlands Correspondence $\left( \text{LLC} \right)$ for inner forms of $\text{G}{{\text{L}}_{n}}$ via the Jacquet–Langlands Correspondence $\left( \text{JLC} \right)$ and compatibility with the Langlands Classification. We show that $\text{LLC}$ satisfies a natural compatibility with parabolic induction and characterize $\text{LLC}$ for inner forms as a unique family of bijections $\prod \left( \text{G}{{\text{L}}_{r}}\left( D \right) \right)\,\to \,\Phi \left( \text{G}{{\text{L}}_{r}}\left( D \right) \right)$ for each $r$, (for a fixed $D$), satisfying certain properties. We construct a surjective map of Bernstein centers $\mathfrak{Z}\left( \text{G}{{\text{L}}_{n}}\left( F \right) \right)\,\to \,\mathfrak{Z}\left( \text{G}{{\text{L}}_{r}}\left( D \right) \right)$ and show this produces pairs of matching distributions in the sense of Haines. Finally, we construct explicit Iwahori-biinvariant matching functions for unit elements in the parahoric Hecke algebras of $\text{G}{{\text{L}}_{r}}\left( D \right)$, and thereby produce many explicit pairs of matching functions.

We provide general inequalities that compare the surface area $S(K)$ of a convex body $K$ in ${{\mathbb{R}}^{n}}$ to the minimal, average, or maximal surface area of its hyperplane or lower dimensional projections. We discuss the same questions for all the quermassintegrals of $K$. We examine separately the dependence of the constants on the dimension in the case where $K$ is in some of the classical positions or $K$ is a projection body. Our results are in the spirit of the hyperplane problem, with sections replaced by projections and volume by surface area.

Culler and Vogtmann defined a simplicial space $O\left( g \right)$, called outer space, to study the outer automorphism group of the free group ${{F}_{g}}$. Using representation theoretic methods, we give an embedding of $O\left( g \right)$ into the analytification of $x\left( {{F}_{g}}\,,\,S{{L}_{2}}\left( \mathbb{C} \right) \right)$, the $S{{L}_{2}}\left( \mathbb{C} \right)$ character variety of ${{F}_{g}}$, reproving a result of Morgan and Shalen. Then we show that every point $v$ contained in a maximal cell of $O\left( g \right)$ defines a flat degeneration of $x\left( {{F}_{g}}\,,\,S{{L}_{2}}\left( \mathbb{C} \right) \right)$ to a toric variety $X\left( {{P}_{\Gamma }} \right)$. We relate $x\left( {{F}_{g}}\,,\,S{{L}_{2}}\left( \mathbb{C} \right) \right)$ and $X\left( v \right)$ topologically by showing that there is a surjective, continuous, proper map ${{\Xi }_{v}}\,:\,x\left( {{F}_{g}}\,,\,S{{L}_{2}}\,\left( \mathbb{C} \right) \right)\,\to \,X\left( v \right)$. We then show that this map is a symplectomorphism on a dense open subset of $x\left( {{F}_{g}}\,,\,S{{L}_{2}}\left( \mathbb{C} \right) \right)$ with respect to natural symplectic structures on $x\left( {{F}_{g}}\,,\,S{{L}_{2}}\left( \mathbb{C} \right) \right)$ and $X\left( v \right)$. In this way, we construct an integrable Hamiltonian system in $x\left( {{F}_{g}}\,,\,S{{L}_{2}}\left( \mathbb{C} \right) \right)$ for each point in a maximal cell of $O\left( g \right)$, and we show that each $v$ defines a topological decomposition of $x\left( {{F}_{g}}\,,\,S{{L}_{2}}\left( \mathbb{C} \right) \right)$ derived from the decomposition of $X\left( {{P}_{\Gamma }} \right)$ by its torus orbits. Finally, we show that the valuations coming from the closure of a maximal cell in $O\left( g \right)$ all arise as divisorial valuations built from an associated projective compactification of $x\left( {{F}_{g}}\,,\,S{{L}_{2}}\left( \mathbb{C} \right) \right)$.

We prove a new bound on collinear triples in subgroups of prime finite fields and use it to give some new bounds on exponential sums with trinomials.

We prove many simultaneous congruences mod 2 for elliptic and Hilbert modular forms among forms with different Atkin–Lehner eigenvalues. The proofs involve the notion of quaternionic $S$-ideal classes and the distribution of Atkin–Lehner signs among newforms.

Consider a finite sequence of linear contractions ${{S}_{j}}(x)\,=\,\rho x\,+{{d}_{j}}$ and probabilities $pj\,>\,0$ with $\Sigma \,{{p}_{j}}\,=\,1$. We are interested in the self-similar measure $\mu \,=\,\Sigma \,{{p}_{j}}\,\mu \,{}^\circ \,s_{j}^{-1}$, of finite type. In this paper we study the multi-fractal analysis of such measures, extending the theory to measures arising from non-regular probabilities and whose support is not necessarily an interval.

Under some mild technical assumptions, we prove that there exists a subset of supp $\mu $ of full $\mu $ and Hausdorff measure, called the truly essential class, for which the set of (upper or lower) local dimensions is a closed interval. Within the truly essential class we show that there exists a point with local dimension exactly equal to the dimension of the support. We give an example where the set of local dimensions is a two element set, with all the elements of the truly essential class giving the same local dimension. We give general criteria for these measures to be absolutely continuous with respect to the associated Hausdorff measure of their support, and we show that the dimension of the support can be computed using only information about the essential class.

To conclude, we present a detailed study of three examples. First, we show that the set of local dimensions of the biased Bernoulli convolution with contraction ratio the inverse of a simple Pisot number always admits an isolated point. We give a precise description of the essential class of a generalized Cantor set of finite type, and show that the $k$-th convolution of the associated Cantor measure has local dimension at $x\,\in \,(0,\,1)$ tending to 1 as $k$ tends to infinity. Lastly, we show that within a maximal loop class that is not truly essential, the set of upper local dimensions need not be an interval. This is in contrast to the case for finite type measures with regular probabilities and full interval support.

We introduce the tracial Rokhlin property for a conditional expectation for an inclusion of unital ${{\text{C}}^{*}}$-algebras $P\,\subset \,A$ with index finite, and show that an action $\alpha$ from a finite group $G$ on a simple unital ${{\text{C}}^{*}}$- algebra $A$ has the tracial Rokhlin property in the sense of N. C. Phillips if and only if the canonical conditional expectation $E:\,A\,\to \,{{A}^{G}}\,$ has the tracial Rokhlin property. Let $\mathcal{C}$ be a class of infinite dimensional stably finite separable unital ${{\text{C}}^{*}}$-algebras that is closed under the following conditions:

(1) If $A\,\in \,\mathcal{C}$ and $B\,\cong \,A$, then $B\,\in \,\mathcal{C}$.

(2) If $A\,\in \,\mathcal{C}$ and $n\,\in \,\mathbb{N}$, then ${{M}_{n}}\left( A \right)\,\in \,\mathcal{C}$.

(3) If $A\,\in \,\mathcal{C}$ and $p\,\in \,A$ is a nonzero projection, then $pAp\,\in \,\mathcal{C}$.

Suppose that any ${{\text{C}}^{*}}$-algebra in $\mathcal{C}$ is weakly semiprojective. We prove that if $A$ is a local tracial ${{\text{C}}^{*}}$-algebra in the sense of Fan and Fang and a conditional expectation $E:\,A\,\to \,P$ is of index-finite type with the tracial Rokhlin property, then $P$ is a unital local tracial $\mathcal{C}$-algebra.

The main result is that if $A$ is simple, separable, unital nuclear, Jiang–Su absorbing and $E:\,A\,\to \,P$ has the tracial Rokhlin property, then $P$ is Jiang–Su absorbing. As an application, when an action α from a finite group $G$ on a simple unital ${{\text{C}}^{*}}$-algebra $A$ has the tracial Rokhlin property, then for any subgroup $H$ of $G$ the fixed point algebra ${{A}^{H}}$ and the crossed product algebra $A{{\rtimes }_{{{\alpha }_{|H}}}}$$H$ is Jiang–Su absorbing. We also show that the strict comparison property for a Cuntz semigroup $W\left( A \right)$ is hereditary to $W\left( P \right)$ if $A$ is simple, separable, exact, unital, and $E:\,A\,\to \,P$ has the tracial Rokhlin property.

This paper introduces a class of abstract linear representations on Banach convolution function algebras over homogeneous spaces of compact groups. Let $G$ be a compact group and $H$ a closed subgroup of $G$. Let $\mu $ be the normalized $G$-invariant measure over the compact homogeneous space $G/H$ associated with Weil's formula and $1\,\le \,p\,<\,\infty $. We then present a structured class of abstract linear representations of the Banach convolution function algebras ${{L}^{p}}\left( G/H,\,\mu \right)$.

We investigate the smoothing problem of limit linear series of rank one on an enrichment of the notions of nodal curves and metrized complexes called saturated metrized complexes. We give a finitely verifiable full criterion for smoothability of a limit linear series of rank one on saturated metrized complexes, characterize the space of all such smoothings, and extend the criterion to metrized complexes. As applications, we prove that all limit linear series of rank one are smoothable on saturated metrized complexes corresponding to curves of compact-type, and we prove an analogue for saturated metrized complexes of a theorem of Harris and Mumford on the characterization of nodal curves contained in a given gonality stratum. In addition, we give a full combinatorial criterion for smoothable limit linear series of rank one on saturated metrized complexes corresponding to nodal curves whose dual graphs are made of separate loops.

We prove that digital sequences modulo $m$ along squares are normal, which covers some prominent sequences, such as the sum of digits in base $q$ modulo $m$, the Rudin–Shapiro sequence, and some generalizations. This gives, for any base, a class of explicit normal numbers that can be efficiently generated.

We provide an explicit construction of representations in the discrete spectrum of two $p$-adic symmetric spaces. We consider $\text{G}{{\text{L}}_{n}}\left( F \right)\,\times \,\text{G}{{\text{L}}_{n}}\left( F \right)\backslash \text{G}{{\text{L}}_{2n}}\left( F \right)$ and $\text{G}{{\text{L}}_{n}}\left( F \right)\,\backslash \text{G}{{\text{L}}_{n}}\left( E \right)$, where $E$ is a quadratic Galois extension of a nonarchimedean local field $F$ of characteristic zero and odd residual characteristic. The proof of the main result involves an application of a symmetric space version of Casselman’s Criterion for square integrability due to Kato and Takano.

We study a relation between distinction and special values of local invariants for representations of the general linear group over a quadratic extension of $p$-adic fields. We show that the local Rankin–Selberg root number of any pair of distinguished representation is trivial, and as a corollary we obtain an analogue for the global root number of any pair of distinguished cuspidal representations. We further study the extent to which the gamma factor at 1/2 is trivial for distinguished representations as well as the converse problem.

Let ${{\mathbf{F}}_{q}}[T]$ be the ring of polynomials over the finite field of $q$ elements and $Y$ a large integer. We say a polynomial in ${{\mathbf{F}}_{q}}[T]$ is $Y$-smooth if all of its irreducible factors are of degree at most $Y$. We show that a ternary additive equation $a\,+\,b\,=\,c$ over $Y$-smooth polynomials has many solutions. As an application, if $S$ is the set of first $s$ primes in ${{\mathbf{F}}_{q}}[T]$ and $s$ is large, we prove that the $S$-unit equation $u\,+\,v\,=\,1$ has at least $\text{exp}\left( {{s}^{1/6-\in }}\,\text{log}\,\text{q} \right)$ solutions.

We consider three point positive masses moving on ${{S}^{2}}$ and ${{H}^{2}}$. An Eulerian-relative equilibrium is a relative equilibrium where the three masses are on the same geodesic. In this paper we analyze the spectral stability of these kind of orbits where the mass at the middle is arbitrary and the masses at the ends are equal and located at the same distance from the central mass. For the case of ${{S}^{2}}$, we found a positive measure set in the set of parameters where the relative equilibria are spectrally stable, and we give a complete classification of the spectral stability of these solutions, in the sense that, except on an algebraic curve in the space of parameters, we can determine if the corresponding relative equilibrium is spectrally stable or unstable. On ${{H}^{2}}$, in the elliptic case, we prove that generically all Eulerian-relative equilibria are unstable; in the particular degenerate case when the two equal masses are negligible, we get that the corresponding solutions are spectrally stable. For the hyperbolic case we consider the system where the mass in the middle is negligible; in this case the Eulerian-relative equilibria are unstable.

Let $k$ be a number field. We describe the category of Laumon 1-isomotives over $k$ as the universal category in the sense of M. Nori associated with a quiver representation built out of smooth proper $k$-curves with two disjoint effective divisors and a notion of $H_{\text{dR}}^{1}$ for such “curves with modulus”. This result extends and relies on a theorem of J. Ayoub and L. Barbieri-Viale that describes Deligne's category of 1-isomotives in terms of Nori's Abelian category of motives.

The $\text{SL}\left( 2,\mathbb{C} \right)$-representation varieties of punctured surfaces form natural families parameterized by monodromies at the punctures. In this paper, we compute the loci where these varieties are singular for the cases of one-holed and two-holed tori and the four-holed sphere. We then compute the de Rham cohomologies of these varieties of the one-holed torus and the four-holed sphere when the varieties are smooth via the Grothendieck theorem. Furthermore, we produce the explicit Gauß-Manin connection on the natural family of the smooth $\text{SL}\left( 2,\mathbb{C} \right)$-representation varieties of the one-holed torus.

In this paper, we prove the spherical fundamental lemma for metaplectic group $M{{p}_{2n}}$ based on the formalism of endoscopy theory by J. Adams, D. Renard, and W.-W. Li.

For a Shimura variety of Hodge type with hyperspecial level structure at a prime $p$, Vasiu and Kisin constructed a smooth integral model (namely the integral canonical model) uniquely determined by a certain extension property. We define and study the Ekedahl-Oort stratifications on the special fibers of those integral canonical models when $p\,>\,2$. This generalizes Ekedahl-Oort stratifications defined and studied by Oort on moduli spaces of principally polarized abelian varieties and those defined and studied by Moonen, Wedhorn, and Viehmann on good reductions of Shimura varieties of PEL type. We show that the Ekedahl-Oort strata are parameterized by certain elements $w$ in the Weyl group of the reductive group in the Shimura datum. We prove that the stratum corresponding to $w$ is smooth of dimension $l\left( w \right)$ (i.e., the length of $w$) if it is non-empty. We also determine the closure of each stratum.

For a finite abelian $p$-group $A$ of rank $d\,=\,\dim\,A/pA$, let ${{\mathbb{M}}_{A}}\,:=\,\text{lo}{{\text{g}}_{p}}\,{{\left| A \right|}^{1/d}}$ be its (logarithmic) mean exponent. We study the behavior of the mean exponent of $p$-class groups in pro-$p$ towers $\text{L/K}$ of number fields. Via a combination of results from analytic and algebraic number theory, we construct infinite tamely ramified pro-$p$ towers in which the mean exponent of $p$-class groups remains bounded. Several explicit examples are given with $p\,=\,2$. Turning to group theory, we introduce an invariant $\underline{\mathbb{M}}\left( G \right)$ attached to a finitely generated pro-$p$ group $G$; when $G\,=\,\text{Gal}\left( \text{L/K} \right)$, where $L$ is the Hilbert $p$-class field tower of a number field $K$, $\underline{\mathbb{M}}\left( G \right)$ measures the asymptotic behavior of the mean exponent of $p$-class groups inside $\text{L/K}$. We compare and contrast the behavior of this invariant in analytic versus non-analytic groups. We exploit the interplay of group-theoretical and number-theoretical perspectives on this invariant and explore some open questions that arise as a result, which may be of independent interest in group theory.