Let $A_f$ be the abelian variety attached by Shimura to a normalized newform $f\in S_2(\Gamma_1(N))^{\operatorname{new}}$. We prove that for any integer $n > 1$ the set of pairs of endomorphism algebras $\big( \operatorname{End}_{\overline{\mathbb{Q}}}(A_f) \otimes \mathbb{Q}, \operatorname{End}_\mathbb{Q}(A_f) \otimes \mathbb{Q} \big)$ obtained from all normalized newforms $f$ with complex multiplication such that $\dim A_f=n$ is finite. We determine that this set has exactly 83 pairs for the particular case $n=2$ and show all of them. We also discuss a conjecture related to the finiteness of the set of number fields $\operatorname{End}_\mathbb{Q}(A_f) \otimes \mathbb{Q}$ for the non-CM case.

We establish a partial regularity result for weak solutions of nonsingular parabolic systems with subquadratic growth of the type $$ \partial_t u - \mathrm{div} a(x,t,u,Du) = B(x,t,u,Du), $$ where the structure function $a$ satisfies ellipticity and growth conditions with growth rate $\frac{2n}{n+2} < p < 2$. We prove Hölder continuity of the spatial gradient of solutions away from a negligible set. The proof is based on a variant of a harmonic type approximation lemma adapted to parabolic systems with subquadratic growth.

We consider Smoluchowski's equation with a homogeneous kernel of the form $a(x,y) = x^\alpha y ^\beta + x^\beta y^\alpha$ with $-1 < \alpha \leq \beta < 1$ and $\lambda := \alpha + \beta \in (-1,1)$. We first show that self-similar solutions of this equation are infinitely differentiable and prove sharp results on the behavior of self-similar profiles at $y = 0$ in the case $\alpha < 0$. We also give some partial uniqueness results for self-similar profiles: in the case $\alpha = 0$ we prove that two profiles with the same mass and moment of order $\lambda$ are necessarily equal, while in the case $\alpha < 0$ we prove that two profiles with the same moments of order $\alpha$ and $\beta$, and which are asymptotic at $y = 0$, are equal. Our methods include a new representation of the coagulation operator, and estimates of its regularity using derivatives of fractional order.

We show that on an NTA domain if each tangent measure to harmonic measure at a point is a polynomial harmonic measure then the associated polynomials are homogeneous. Geometric information for solutions of a two-phase free boundary problem studied by Kenig and Toro is derived.

Inspired by recent works on rings satisfying Auslander's conjecture, we study invariants, called Auslander bounds, and prove that they have strong relations to some homological conjectures.

Let $M$ be a rotationally symmetric Riemannian manifold, and $\Delta$ be the Laplace-Beltrami operator on $M$. We establish a necessary and sufficient condition for the constant function 1 to be a semismall perturbation of $-\Delta +1$ on $M$, and give optimal sufficient conditions for uniqueness of nonnegative solutions of the Cauchy problem to the heat equation. As an application, we determine the structure of all nonnegative solutions to the heat equation on $M\times(0,T)$.

We provide very general symmetrization theorems in arbitrary dimension and codimension, in products, warped products, and certain fiber bundles such as lens spaces, including Steiner, Schwarz, and spherical symmetrization and admitting density.

We obtain some results in both Lorentz and Finsler geometries, by using a correspondence between the conformal structure (Causality) of standard stationary spacetimes on $M = \mathbb{R} \times S$ and Randers metrics on $S$. In particular: (1) For stationary spacetimes: we give a simple characterization of when $\mathbb{R} \times S$ is causally continuous or globally hyperbolic (including in the latter case, when $S$ is a Cauchy hypersurface), in terms of an associated Randers metric. Consequences for the computability of Cauchy developments are also derived. (2) For Finsler geometry: Causality suggests that the role of completeness in many results of Riemannian Geometry (geodesic connectedness by minimizing geodesics, Bonnet-Myers, Synge theorems) is played by the compactness of symmetrized closed balls in Finslerian Geometry. Moreover, under this condition we show that for any Randers metric $R$ there exists another Randers metric $\tilde R$ with the same pregeodesics and geodesically complete. Even more, results on the differentiability of Cauchy horizons in spacetimes yield consequences for the differentiability of the Randers distance to a subset, and vice versa.

The theory of Muckenhoupt's weight functions arises in many areas of analysis, for example in connection with bounds for singular integrals and maximal functions on weighted spaces. We prove that a certain averaging process gives a method for constructing $A_p$ weights from a measurably varying family of dyadic $A_p$ weights. This averaging process is suggested by the relationship between the $A_p$ weight class and the space of functions of bounded mean oscillation. The same averaging process also constructs weights satisfying reverse Hölder ($RH_p$) conditions from families of dyadic $RH_p$ weights, and extends to the polydisc as well.

Using Fréchet algebraic technique, we show the existence of a nuclear Fréchet space without basis, thus providing yet another proof (of a different flavor) of a negative answer to a well known problem of Grothendieck from 1955. Using Fefferman's construction (which is based on complex-variable technique) of a $C^\infty$-function on the unit circle with certain properties, we give much simpler, transparent, and "natural" examples of restriction spaces without bases of nuclear Fréchet spaces of $C^\infty$-functions; these latter spaces, being classical objects of study, have attracted some attention because of their relevance to the theories of PDE and complex dynamical systems, and harmonic analysis. In particular, the restriction space $A^\infty(E)$, being a quotient algebra of the algebra $A^\infty(\Gamma)$, is the central one to other examples; the algebras $A^\infty$ had played a crucial role in solving a well-known problem of Kahane and Katznelson in the negative.

The first aim of the present paper is to compare various sub-Riemannian structures over the three dimensional sphere $S^3$ originating from different constructions. Namely, we describe the sub-Riemannian geometry of $S^3$ arising through its right action as a Lie group over itself, the one inherited from the natural complex structure of the open unit ball in $\mathbb{C}^2$ and the geometry that appears when it is considered as a principal $S^1$-bundle via the Hopf map. The main result of this comparison is that in fact those three structures coincide. We present two bracket generating distributions for the seven dimensional sphere $S^7$ of step 2 with ranks 6 and 4. The second one yields to a sub-Riemannian structure for $S^7$ that is not widely present in the literature until now. One of the distributions can be obtained by considering the CR geometry of $S^7$ inherited from the natural complex structure of the open unit ball in $\mathbb{C}^4$. The other one originates from the quaternionic analogous of the Hopf map.

We establish $L^p$-boundedness for a class of operators that are given by convolution with product kernels adapted to curves in the space. The $L^p$ bounds follow from the decomposition of the adapted kernel into a sum of two kernels with singularities concentrated respectively on a coordinate plane and along the curve. The proof of the $L^p$-estimates for the two corresponding operators involves Fourier analysis techniques and some algebraic tools, namely the Bernstein-Sato polynomials. As an application, we show that these bounds can be exploited in the study of $L^p-L^q$ estimates for analytic families of fractional operators along curves in the space.

We characterize (up to endpoints) the $k$-tuples $(p_1,\ldots,p_k)$ for which certain $k$-linear generalized Radon transforms map the product $L^{p_1} \times \cdots \times L^{p_k}$ boundedly into $\mathbb{R}$. This generalizes a result of Tao and Wright.