We establish a truncated non-integrated defect relation for meromorphic mappings from M into P n C intersecting hypersurfaces in subgeneral position. This is joint work with S. Quang, N. Phuong, and L. Abstract: This talk contains two parts. In the first part, we give some applications of Nevanlinna theory to normal families and normal functions. In the second part, we present a second main theorem for holomorphic curves intersecting one hypersurface in projective space.

## Symbolic Asymptotics: Functions of Two Variables, Implicit Functions

Abstract: In , Corvaja and Zannier obtained a new proof of Siegel's theorem on integral points on curves based on Schmidt's celebrated Subspace Theorem. Soon after that and based on earlier work , Evertse and Ferretti applied Schmidt's theorem to give diophantine results for homogeneous polynomials of higher degree on a projective variety in P n. This has led to further work of A. Levin, P. Autissier, M. Ru, G. The bounds come from a detailed analysis of the kernel of the Cartier operator. This is joint work with John B.

A classical topic connecting geometry and diphantine equations is the study of Heronian triangles, that is, triangles with sides of integer length and with integer area.

I will show that this sequence, and a closely related 3rd-order linear recurrence, occur in an unexpected way in the study of certain factorials. The question of when this order can be a power of 2 leads to the new concept of a "Gauss prime", the study of which is closely related to the sequences mentioned above.

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- Heavy Metals in the Aquatic Environment. Proceedings of the International Conference Held in Nashville, Tennessee, December 1973;
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Apart from explaining these various connections, I will derive some divisibility properties of the sequences in question. The techniques include the geometry of numbers, representation theory, and polynomial invariants of finite groups. Time permitting, I will also discuss some related counting questions involving variants on the classical discriminant.

Quantized number theory. Riemann hypothesis, fractal cohomology and spectra of Polya—Hilbert operators via regularized determinants. We plan to discuss two approaches to " Quantized number theory " QNT developed by the author and his collaborators. A first approach, using a suitable version of the infinitesimal shift of the real line and the associated spectral operator, which is a version of the quantized Riemann zeta function or of a more general quantized L -function.

In that setting, several reformulations of the Riemann hypothesis RH are obtained in terms of the invertibility of the spectral operator, and motivated in part by the earlier work of the author and his collaborators on the theory of complex fractal dimensions and associated reformulations of RH as inverse spectral problems for fractal strings. It can be represented by a quantized Dirichlet series and a quantized Euler product which converge in a suitable sense even within the critical interval 0,1 i.

This work is developed in detail in the forthcoming research monograph by M. Lapidus and H. Royal Soc. A, A second approach, based on the use of the infinitesimal shift in the complex plane on an appropriately weighted Bergman space of entire functions and the use of suitable regularized spectral determinants. In that work, we construct a generalized "Polya—Hilbert operator" the spectrum of which is precisely the divisor i. Accordingly, as in the Weil conjectures, and at least in the case of number fields as hoped for in later work by a number of mathematicians including C.

Deninger , the Riemann zeta function, the zeta functions of varieties over finite fields i. The latter Frobenius operator is the unbounded linear operator induced by the Polya—Hilbert operator on the total "fractal cohomology space" associated with the underlying spectrum of fractal complex dimensions i. This work is carried out in a paper by M. Lapidus and T. Montgomery et al. In the latter book are also proposed and explored several possible geometric or topological interpretations of this work, expressed, in particular, in terms of suitable sheaves and toposes.

However, a lot remains to be done in this direction in order to uncover the true geometric meaning of the above analytic and algebraic but fully rigorous constructions, or of appropriate modifications thereof. We then prove that all of the pairs occur in one tree, and demonstrate properties of the tree. The corresponding Galois representation was discovered by J. In this talk we give it an interpretation in terms of Anderson motives.

## Abstracts - HINT - Hawaiʻi Number Theory conference

We give an explicit inequality between its Faltings height and its primes of bad reduction, and discuss various applications. This is a joint work with Lucia Di Vizio. Then, in a sense that will be made more precise in the talk, Euler's gamma function essentially allows to classify the abelian equations mentioned in the title, where the word 'essentially' hides here that there are important and deep differences when we compare with Hayes's theory. Although Berkovich spaces usually appear in a non-archimedean setting, their general definition actually allows arbitrary Banach rings as base rings, e.

The global functions on those spaces are typically convergent arithmetic power series, i. We will explain how the geometric setting can help prove nice properties of rings of convergent arithmetic power series such as noetherianity in several variables, thus extending a result of D. Harbater in one variable. In this talk we give some of the first results when the underlying group is not cyclic.

We review this structure and describe some of the aspects required for its proof. This is a joint work with David Burns. Coleman made a conjecture on the module of circular distributions around 's. This conjecture is a natural globalized version of the fact that all norm-compatible families of units in towers of local cyclotomic fields arise by evaluating a "Coleman power series" at roots of unity, as had been proved in Inventiones The conjecture is also related in sprit with the archimedean characterization of cyclotomic units that Coleman had given in Crelle's journal In this talk, we introduce recent improvements of the earlier works on the conjecture using the arguments of Euler systems together with theorems on Iwasawa modules.

We we will also outline recent results applying our methods to trinomials of more general degree. The set of smooth rational curves on an Enriques surface therefore gives a configuration of spheres in 8 dimensions.

If an Enriques surface has a smooth rational curve, but is otherwise generic, then this configuration of spheres is an Apollonian packing. An amazing theorem of Christol states that a power series y over a finite field is an algebraic function if and only if its coefficient sequence can be produced by a finite automaton, which is a limited model of a computer with no memory. The proof uses combinatorics and linear algebra, but hidden in the theorem there is geometric information about a curve that contains y in its function field.

I make this explicit by demonstrating a precise link between the complexity of the automaton and the geometry of the curve. We discuss some recent progress in techniques for establishing effective results for unlikely intersections. The methods are tied to equidistribution results and thus are naturally applicable to problems from arithmetic dynamics. The focus of the talk will be on the basic analytic techniques involved with particular emphasis on the challenges faced in establishing effective constants when moving from polynomial to rational dynamics.

In studying the equi distribution of shapes of quartic number fields, one relies heavily on Bhargava's parametrizations which brings with it a notion of resolvent ring.

## Forum Mathematicum

Maximal rings have unique resolvent rings so it is possible to live a long and healthy life without understanding what they are. The authors have decided, however, to forsake such bliss and look into what ever are these rings and what happens if we consider their shapes along with our initial number fields. What is the geometric intuition for Euler characteristics? MO: Intuition behind Riemann's bilinear relations.

MO: What does the Jacobian of a curve tell us about the curve? MO: How do you see the genus of a curve, just looking at its function field? See Tsimerman generals. See Donaldson p. Equidistribution results Weyl style using this?

See MO. How does this differ from van der Corput method? Or the ergodic method Dylan mentioned? Complex Analysis and Riemann Surfaces.

### Number Theory Seminar

Explanations 6. Victor Wang. Uniformization and quotients MSE for complex plane Donaldson p. Related Lessons. Philosophy of Alternative Representations. Measure and Integration Theory. Fourier and Harmonic Analysis.