Title:
766666666622222222222222229950000003333333333311488Speaker: Michael Filaseta
Abstract: We refer to a block of digits in a natural number n as a sequence of equal digits which cannot be extended to a longer sequence of equal digits. For example, 2211000111 has 4 blocks of digits, namely 22, 11, 000, and 111. Note that two blocks, 11 and 111, are formed using the same digit. We show that the number of blocks of digits of a^n as n goes to infinity tends to infinity except when it shouldn't. The number in the title is the largest number divisible by 2^52, relatively prime to 5, which consists of blocks where no two of the blocks are formed using the same digit. There is no natural number divisible by 2^53 and relatively prime to 5 consisting of blocks with no two formed using the same digit. This is past joint work with Richard Blecksmith and Charles Nicol.
Title:
Baker's Method 101Speaker: Michael Filaseta
Abstract: Despite the title, this is not a talk about cooking. I will first explain how Baker's method can be used to show that the largest prime factor of n(n+1) tends to infinity with n. Then I will finish off the arguments from the previous lecture showing that the number of blocks of digits in base b of a^n tends to infinity whenever log(a)/log(b) is irrational.
Title:
Exponential sums associated to prehomogeneous vector spaces over finite fieldsSpeaker: Frank Thorne
Abstract: I will discuss ongoing work of myself and Taniguchi (and which is being further extended by Dan Kamenetsky). We are developing a mathod for evaluating exponential sums associated to prehomogeneous vector spaces over finite fields, and for using these to obtain "level of distribution" statements for various arithmetic sequences. I will discuss how we are doing this and why anyone might care.
Title:
On a constant associated with the Prouhet-Tarry-Escott ProblemSpeaker: Maria Markovich
Abstract: For n a positive integer, the Prouhet-Tarry-Escott Problem asks for two different sets of n positive integers for which the sum of the kth powers of the elements of one set is equal to the sum of the kth powers of the elements of the second set for each positive integer k < n. For n > 12, it is not known whether such sets exist. I will give some background on this problem and then show how Newton polygons can be used to determine information on the size of a certain constant associated with the problem.
Title:
Exponential sums associated to prehomogeneous vector spaces over finite fields, Part IISpeaker: Frank Thorne
Abstract: This is a continuation of the previous talk by the speaker with promises of explaining the material on the prehomogeneous vector space part of the title.
Title:
Exponential sums associated to prehomogeneous vector spaces over finite fields, Part IIISpeaker: Frank Thorne
Abstract: Further progress is made on the subject of the last talk.
Title:
The 1729 K_3 SurfaceSpeaker: Sarah Trebat-Leder (Emory University)
Abstract:
Pretalk: The first part of the talk will be an overview of elliptic curves, geared towards BSD and the distribution of ranks, and the second part will explain a result of Stewart and Top about the number of cubic twists of x^3 + y^3 = 1 with rank at least 2 or 3.
Research Talk: We revisit the mathematics that Ramanujan developed in connection with the famous "taxi-cab" number 1729. A study of his writings reveals that he had been studying Euler's diophantine equation a^3+b^3=c^3+d^3. It turns out that Ramanujan's work anticipated deep structures and phenomena which have become fundamental objects in arithmetic geometry and number theory. We find that he discovered a K3 surface with Picard number 18, one which can be used to obtain infinitely many cubic twists over Q with rank at least 2.
Title:
Roots of polynomials with integer coefficientsSpeaker: Michael Filaseta
Abstract: We begin with the one-line proof that if f(x) is a monic non-constant polynomial with complex coefficients and with |f(0)| at least 1, then it must have a root on or outside the unit circle. We will then discuss some mathematics that developed from this simple idea (for example, Mahler measure). Then we will turn to the basic question of what else can be said in the original result if instead we consider f(x) to have integer coefficients. Finally, we discuss applications of these more recent investigations.
Title:
Some applications of two theorems of CapelliSpeaker: Michael Filaseta
Abstract: The main theorem of Capelli that we discuss describes the factorization of a composition of two polynomials. This theorem has a number of interesting consequences, and I will describe a couple of them before focusing on a recent application connected to trace fields of hyperbolic 3-manifolds under Dehn fillings. This is joint work with Stavros Garoufalidis and Josh Harrington.
Title:
Large gaps between consecutive prime numbersSpeaker: Kevin Ford (University of Illinois)
Abstract: In 1938, Rankin showed that that maximal gap between consecutive prime numbers less than x is at least c log x log_2 x log_4 x/(log_3 x)^2 for some constant c, where log_k is the k-th iterate of log. Since then, there have been improvements to the constant c and it has been conjectured (Erdos $10,000 problem) that the result holds for ANY c. This conjecture was recently proved by the speaker in joint work with Ben Green, Sergei Konyagin and Terence Tao (and at about the same time, independently by James Maynard). Later, with Green, Konyagin, Maynard and Tao we improved the bound by a factor log_3 x. We will describe the main ideas of the proof, emphasizing how tools from various areas come into play, such as sieve methods, primes in arithmetic progressions, probabilistic methods, and combinatorial methods. We will describe some related problems and applications, including chains of large prime gaps, long strings of composites containing a perfect k-th power, and application to the least prime in an arithmetic progression.
Title:
Torsion in odd degreeSpeaker: Abbey Bourdon (University of Georgia)
Abstract: Let E be an elliptic curve defined over a number field F. It is a classical theorem of Mordell and Weil that the collection of points of E with coordinates in F form a finitely generated abelian group. We seek to understand the subgroup of points with finite order. In particular, given a positive integer d, we would like to know precisely which abelian groups arise as the torsion subgroup of an elliptic curve defined over a number field of degree d. I will discuss recent progress on this problem for the special class of elliptic curves with complex multiplication (CM). In particular, if d is odd, we now have a complete classification of the groups that arise as the torsion subgroup of a CM elliptic curve defined over a number field of degree d. This is joint work with Paul Pollack.
Title:
Torsion of CM elliptic curves: analytic aspectsSpeaker: Paul Pollack (University of Georgia)
Abstract: For each positive integer d, let T(d) denote the supremum of all orders of groups E(F)[tors] appearing for an elliptic curve E defined over a degree d number field F. A celebrated theorem of Merel asserts that T(d) < infinity for all d. However, the known quantitative results in this direction are far from the conjectured truth. Let T_{CM}(d) be defined the same way as T(d), but with the restriction to CM elliptic curves. I will discuss some recent statistical results concerning T_{CM}(d) and related functions. Perhaps surprisingly, the "anatomy of integers" (as pioneered by Paul Erdos) plays a key role in the proofs. Joint work with Abbey Bourdon and Pete L. Clark.
Title:
The modified Mobius functionSpeaker: Harsh Mehta
Abstract: We will modify the Mobius function and carry out some general analytic number theory to prove results like (messy sum involving Omega(n) and i) = pi^2/105 where Omega(n) counts, with multiplicity, the number of prime factors of n. Should time permit I will talk about analytic properties of corresponding Zeta functions.
Title:
Density of polynomials with squarefree discriminantSpeaker: Jerry Wang (Princeton)
Abstract: The problem of the density of squarefree discriminant polynomials is an old one, being considered by many people, and the density being conjectured by Lenstra. A proof has been out of question for a long time. The reason it was desired is that a squarefree discriminant polynomial f immediately gives the ring of integers of Q[x]/f(x) and its Galois group. In recent joint work with Manjul Bhargava and Arul Shankar, we counted the number of odd degree polynomials with squarefree discriminant and proved the conjecture of Lenstra. In this talk, I will explain the general strategy of the squarefree sieve and the specific strategy to deal with discriminants which in turn leads to counting integral orbits for a representation of a non-reductive group.
Title:
Extremal covering systemsSpeaker: Ognian Trifonov
Abstract: A distinct covering system is a finite collection of congruences x = a_i (mod m_i), for 1 < m_1 < m_2 < \cdots < m_k, such that every integer satisfies at least one of them. Bob Hough recently proved that the least modulus of any distinct covering system is no more than 10^(18). Suppose m is a positive integer such that there is a distinct covering with least modulus m. We are interested in finding the least positive integer L(m) such that there is a distinct covering system with least modulus m and such that all moduli divide L(m). It is well known that L(2)=12. We show that L(3)=120 and L(4) = 360.
Title:
Modular Galois representations with small imagesSpeaker: Matt Boylan
Abstract: I will briefly discuss how congruences for modular form coefficients modulo a prime p arise when the mod p Galois representation attached to the modular form has small image, where small means that it does not contain SL(2, Z/pZ). When this happens, we say that p is exceptional for the form f. I will present work that I did in the (distant) past which identifies precisely which primes are exceptional for forms in a natural family, the eta-quotient newforms.
Title:
The minimum modulus problem for covering systemsSpeaker: Bob Hough (Oxford)
Abstract:
A distinct covering system of congruences is a finite collection of arithmetic progressions to distinct moduli
a_i mod m_i, 1 < m_1 < m_2 < ... < m_k
whose union is the integers. Answering a question of Erdos, I have shown that the least modulus m_1 of a distinct covering system of congruences is at most 10^{16}. I will describe aspects of the proof, which involves the theory of smooth numbers and a relative form of the Lovasz local lemma.