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The stable reduction theorem was proved by Grothendieck for Abelian varieties and subsequently by Deligne and Mumford for projective curves.
This talk first solves explicitly a Riemann-Hilbert problem of confluent hypergeometric systems. It then conjectures and proves a special case that the WKB approximation of the monodromy data of confl...
In the last decade, there has been a growing interest in the study of rational points on homogeneous spaces of linear algebraic groups defined over fields having good arithmetical properties other tha...
Equidistribution is an important theme in number theory. The Sato-Tate conjecture, which was established by Richard Taylor et.al. in 2008, asserts that given an elliptic curve over Q without complex m...
For a smooth projective variety over a number field, the Beilinson—Bloch conjecture predicts that Chow groups of algebraic cycles modulo rational equivalence can be determined using cohomological data...
Let $k$ be a field of characteristic 0 and let $C$ be the a smooth affine curve over $k$. We will first discuss applications of strong approximation to the theory of integral quadratic forms over $C$....
Let $\{a(t):t \in \mathbb{R}\}$ be a diagonalizable subgroup of $SL(d,\mathbb{R})$ for which the expanded horosphere $U$ is abelian. By the Birkhoff ergodic theorem, for any point $x \in SL(d,\mathbb{...
Growth curves of the Morada Nova sheep males and females were described using nonlinear models and the relationships between body weight and thoracic circumference were evaluated. Altogether 1516 repe...
Can a cell sense its own shape? Working in the MBL’s Whitman Center research community, scientists from Dartmouth College developed an ingenious experiment to ask this question. Their concl...
For smooth families of projective algebraic curves, we extend the notion of intersection pairing of metrized line bundles to a pairing on line bundles with flat relative connections. In this setting, ...
In this paper we consider questions of the following type. Let k be a base eld and K=k be a eld extension. Given a geometric object X over a eld K (e.g. a smooth curve of genus g) what is the le...
The classical principle of least action says that orbits of mechanical systems extremize action; an important subclass are those orbits that minimize action. In this paper we utilize this principle al...
Given an elliptic curve E over a number field k, the Galois action on the torsion points of E induces a Galois representation, ρE : Gal(k/k) → GL2(Z b). For a fixed number field k, we describe the ima...
Let E be an elliptic curve over the rationals that does not have complex multiplication. For each prime `, the action of the absolute Galois group on the `-torsion points of E can be given in terms of...

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