Baker’s conjecture and Eremenko’s conjecture for functions with negative zeros

We introduce a new technique that allows us to make progress on two long standing conjectures in transcendental dynamics: Baker's conjecture that a transcendental entire function of order less than 1/2 has no unbounded Fatou components, and Eremenko's conjecture that all the components of the escaping set of an entire function are unbounded. We show that both conjectures hold for many transcendental entire functions whose zeros all lie on the negative real axis, in particular those of order less than 1/2. Our proofs use a classical distortion theorem based on contraction of the hyperbolic metric, together with new results which show that the images of certain curves must wind many times round the origin.     

Serre’s conjecture and its consequences

After some generalities about the absolute Galois group of \mathbb Q\mathbb Q, we present the historical context in which Serre made his modularity conjecture. This was recently proved by Wintenberger and the author ([22], [23]), with an input of Kisin ([24]). The focus of these notes is on the applications of the conjecture. Some of the applications are based on the methods used in the proof.     

Boyle’s conjecture and perfect localizations

ABSTRACT

In this article we study the behavior of left QI-rings under perfect localizations. We show that a perfect localization of a left QI-ring is a left QI-ring. We prove that Boyle’s conjecture is true for left QI-rings with finite Gabriel dimension such that every hereditary torsion theory in the Gabriel filtration is perfect. As corollary, we get that Boyle’s conjecture is true for left QI-rings which satisfy the restricted left socle condition, a result proved by Faith in [6 Faith, C. (1976). On hereditary rings and Boyle’s conjecture. Arch. Math. 27(1):113–119.[Crossref] , [Google Scholar]].     

Graph norms and Sidorenko’s conjecture

Let H and G be two finite graphs. Define h _H (G) to be the number of homomorphisms from H to G. The function h _H (·) extends in a natural way to a function from the set of symmetric matrices to ℝ such that for A _G , the adjacency matrix of a graph G, we have h _H (A _G ) = h _H (G). Let m be the number of edges of H. It is easy to see that when H is the cycle of length 2n, then h _H (·)^1/m is the 2n-th Schatten-von Neumann norm. We investigate a question of Lovász that asks for a characterization of graphs H for which the function h _H (·)^1/m is a norm.     

Weak closure and Oliver’s p-group conjecture

To date almost all verifications of Oliver’s p-group conjecture have proceeded by verifying a stronger conjecture about weakly closed quadratic subgroups. We construct a group of order 3⁴⁹ which refutes the weakly closed conjecture but satisfies Oliver’s conjecture.     

Real normalized differentials and Arbarello’s conjecture

Using meromorphic differentials with real periods, we prove Arbarello’s conjecture that any compact complex cycle of dimension g - n in the moduli space M _g of smooth algebraic curves of genus g must intersect the locus of curves having a Weierstrass point of order at most n.     

On Hua-Tuan’s conjecture

Let G be a finite group and |G| = pn, p be a prime. For 0 m n, sm(G) denotes the number of subgroups of of order pm of G. Loo-Keng Hua and Hsio-Fu Tuan have ever conjectured: for an arbitrary finite p-group G, if p > 2, then sm(G) ≡ 1, 1 + p, 1 + p + p2 or 1 + p + 2p2 (mod p3). In this paper, we investigate the conjecture, and give some p-groups in which the conjecture holds and some examples in which the conjecture does not hold.     

On Olsson’s conjecture for blocks with metacyclic defect groups of odd order

Let G be a finite group, and let B be a p-block of G with defect group D. Let k ₀(B) denote the number of ordinary irreducible characters of height 0 in B. In 1984 Olsson proposed a conjecture: k₀(B)\leqq |D:D￠|{k_{0}(B)\leqq |D:D'|}. In this paper, we will verify Olsson’s conjecture in the case that D is metacyclic and p is odd.     

On Sarnak’s conjecture and Veech’s question for interval exchanges

Using a criterion due to Bourgain [10] and the generalization of the self-dual induction defined in [19], for each primitive permutation we build a large family of k-interval exchanges satisfying Sarnak’s conjecture, and, for at least one permutation in each Rauzy class, smaller families for which we have weak mixing, which implies a prime number theorem, and simplicity in the sense of Veech.     

On Jordan’s inequality

We present sharp upper and lower bounds for the function \(\sin (x)/x\). Our bounds are polynomials of degree 2n, where n is any nonnegative integer.     

On Jordan’s theorem

In the present paper, we give some remarks on the well-known Jordan theorem and Hamiltonians.     

Goldbach’s conjecture in max-algebra

The Goldbach conjecture is one of the best known open problems in number theory. It claims that every even integer greater than 2 can be written as the sum of two primes. The present paper formulates a max-algebraic claim that is equivalent to Goldbach’s conjecture. The max-algebraic analogue allows examination of the conjecture by the methods of max-algebra. A max-algebra is an algebraic structure in which classical addition \(+\) and multiplication \(\times \) are replaced by the operations maximum \(\oplus \) and addition \(\otimes \), in other words \(a\oplus b=\max \{a,b\}\) and \(a\otimes b=a+b\).     

On De Jong’s conjecture

Let X be a smooth projective curve over a finite field F _q . Let ρ be a continuous representation π(X) → GL _n (F), where F = F _l ((t)) with F _l being another finite field of order prime to q. Assume that is irreducible. De Jong’s conjecture says that in this case is finite. As was shown in the original paper of de Jong, this conjecture follows from an existence of an F-valued automorphic form corresponding to ρ is the sense of Langlands. The latter follows, in turn, from a version of the Geometric Langlands conjecture. In this paper we sketch a proof of the required version of the geometric conjecture, assuming that char(F) ≠ 2, thereby proving de Jong’s conjecture in this case.     

Remarks on Harada’s conjecture

An open conjecture by Harada from 1981 gives an easy characterization of the p-blocks of a finite group in terms of the ordinary character table. Kiyota and Okuyama have shown that the conjecture holds for p-solvable groups. In the present work we extend this result using a criterion on the decomposition matrix. In this way we prove Harada’s Conjecture for several new families of defect groups and for all blocks of sporadic simple groups. In the second part of the paper we present a dual approach to Harada’s Conjecture.     

Counterexamples to Mercat’s conjecture

In this paper, we provide counterexamples to Mercat’s conjecture on vector bundles on algebraic curves. For any \({n \geq 4}\), we provide examples of curves lying on K3 surfaces and vector bundles on those curves which invalidate Mercat’s conjecture in rank n.     

On Ueno’s conjecture K

We show that if X is a smooth complex projective variety with Kodaira dimension 0 then the Kodaira dimension of a general fiber of its Albanese map is at most . J. A. Chen was partially supported by NCTS, TIMS, and NSC of Taiwan. C. D. Hacon was partially supported by NSF research grant no: 0456363 and an AMS Centennial Scholarship. We would like to thank J. Kollár, R. Lazarsfeld, C.-H. Liu, M. Popa, P. Roberts, and A. Singh for many useful comments on the contents of this paper.     

On Hua-Tuan’s conjecture Ⅱ

Let G be a group of order pn, p a prime. For 0 m n, sm(G) denotes the number of subgroups of order pm of G. Loo-Keng Hua and Hsio-Fu Tuan had ever conjectured: for an arbitrary finite p-group G, if p > 2, then sm(G) ≡ 1, 1+p, 1+p+p2 or 1+p+2p2(mod p3). The conjecture has a negative answer. In this paper, we further investigate the conjecture and propose two new conjectures.