UIUC Dept. of Mathematics Seminar Calendar

       May 2012              June 2012              July 2012      
 Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
        1  2  3  4  5                   1  2    1  2  3  4  5  6  7
  6  7  8  9 10 11 12    3  4  5  6  7  8  9    8  9 10 11 12 13 14
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 20 21 22 23 24 25 26   17 18 19 20 21 22 23   22 23 24 25 26 27 28
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Thursday, January 19, 2012

Number Theory Seminar
11:00 am in 241 Altgeld Hall, Thursday, January 19, 2012

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Kevin Ford (UIUC Math)
Values of Euler's function not divisible by a given prime, and the distribution of Euler-Kronecker constants for cyclotomic fields
Abstract: For a give prime $q>2$, we investigate the first and second order terms in the asymptotic series for the counting function of $n$ with $q\nmid \phi(n)$. Part of the analysis involves the Euler-Kronecker constant $EK(q)$ for the cyclotomic field $Q(e^{2 \pi i/q})$. One of our theorems gives a (conditional on the prime $k$-tuples conjecture) disproof of conjectures of Ihara concerning the distribution of $EK(q)$. The distribution of primes in the arithmetic progression $1\mod q$ plays a central role in the results. This is joint work with Florian Luca and Pieter Moree.

Thursday, January 26, 2012

Thursday, February 2, 2012

Number Theory Seminar
11:00 am in 241 Altgeld Hall, Thursday, February 2, 2012

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Daniel Fiorilli (IAS Princeton)
On how the first term of an arithmetic progression can influence the distribution of an arithmetic sequence
Abstract: We will show that many arithmetic sequences have asymmetries in their distribution amongst the progressions mod q. The general phenomenon is that if we fix an integer a having some arithmetic properties (these properties depend on the sequence), then the progressions a mod q will tend to contain fewer elements of the arithmetic sequence than other progressions a mod q, on average over q. The observed phenomenon is for quite small arithmetic progressions, and the maximal size of the progressions is fixed by the nature of the sequence. Examples of sequences falling in our range of application are the sequence of primes, the sequence of integers which can be represented as the sum of two squares (or more generally by a fixed positive definite binary quadratic form) (with or without multiplicity), the sequence of twin primes (under Hardy-Littlewood) and the sequence of integers free of small prime factors. We will focus on these examples as they are quite fun and enlightening.

Thursday, February 9, 2012

Number Theory Seminar
11:00 am in 241 Altgeld Hall, Thursday, February 9, 2012

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Jimmy Tseng (UIUC Math)
Bounded Luroth expansions
Abstract: Luroth series expansions are in a family of various expansions of the real numbers, a family which includes continued fractions. Like for continued fractions, every real number can be expressed as a Luroth expansion. Also like for continued fractions, the digits of a Luroth expansion are generated by a self-map, the Luroth map. The digits are, of course, an encoding of the map and give us a geometric way of looking at the expansion. I will give a sketch of the proof of the following result (joint with B. Mance): the set B of numbers with bounded Luroth expansion, bounded continued fraction expansion, and bounded n-ary expansion for every integer n > 1 is a dense set of full Hausdorff dimension. (Each of these conditions on B would form a superset of zero Lebesgue measure.) The proof is based on applying a technique developed by W. Schmidt (or a later variant made by C. McMullen) to the Luroth map,which has infinite distortion, and is adapted from techniques that I developed in 2009 for cases of bounded distortion.

Thursday, February 16, 2012

Number Theory Seminar
11:00 am in 243 Altgeld Hall, Thursday, February 16, 2012

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Youness Lamzouri (UIUC Math)
Conditional bounds for the least quadratic non-residue and the values of L-functions at 1
Abstract: We study explicit and theoretical bounds for several interesting quantities in number theory, conditionally on the Generalized Riemann Hypothesis. Specifically, we improve the existing explicit bounds for the least quadratic non-residue and the least prime in an arithmetic progression. We also refine the classical conditional bounds of Littlewood for $L$-functions at $s=1$. In particular, we derive explicit upper and lower bounds for $L(1,\chi)$ and $\zeta(1+it)$, and deduce explicit bounds for the class number of imaginary quadratic fields. This is a joint work with X. Li and K. Soundararajan.

Tuesday, February 21, 2012

Thursday, March 1, 2012

Number Theory Seminar
11:00 am in 243 Altgeld Hall, Thursday, March 1, 2012

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Xiannan Li (Department of Mathematics, University of Illinois at Urbana-Champaign)
The least prime that does not split in a number field
Abstract: I will begin by describing some classical work on unconditionally bounding the least quadratic non-residue dating back to Vinogradov and Burgess. A generalization of this problem is to bound the least prime that does not split completely in a number field, which was studied by K. Murty and then by Vaaler and Voloch. I will describe two different approaches, one based on zeros of L-functions, and the other on the theory of multiplicative functions, which give the best known bounds here when the degree of the number field is larger than 2.

Tuesday, March 27, 2012

Number Theory Seminar
11:00 am in 241 Altgeld Hall, Tuesday, March 27, 2012

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Vorrapan Chandee (CRM, University of Montreal)
Bounding S(t) via extremal functions
Abstract: Assuming the Riemann hypothesis we consider the argument function of the Riemann zeta function, $S(t)$. We will prove that for large $t$, $$|S(t)| \leq ( 1/4 + o(1) ) \log t /\log \log t,$$ which is an improvement of the previous work of Goldston and Gonek by a factor of 2. The result may reasonably be thought of as having attained the limit of existing methods of bounding $S(t)$ under RH. Two different approaches to improve a bound for $S(t)$ will be presented in the talk. Both methods rely on the solution of the Beurling-Selberg extremal problem from recent works by Carneiro, Littmann and Vaaler.

Thursday, March 29, 2012

Number Theory Seminar
11:00 am in 243 Altgeld Hall, Thursday, March 29, 2012

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Florian Luca (Univ. Morelia, Mexico)
On the local behavior of the Ramanujan tau function
Abstract: Let $\tau(n)$ be the Ramanujan function given by $$\sum_{n\ge 1} \tau(n) q^n = q\prod_{i\ge 1} (1-q^i)^{24}.$$ Lehmer conjectured that $\tau(n)\ne 0$ for all n, and this conjecture has been verified for all $n \le 22798241520242687999$: In my talk, I will present the main ideas of the following theorem obtained jointly with Yuri Bilu (Bordeaux): If $k$ is a positive integer such that $\tau(m) \ne 0$ for all $m \le k$, then for every permutation $\sigma$ of the first $k$ positive integers, there exist infinitely many positive integers $n$ such that $\tau(n + \sigma(1)) < \cdots < \tau(n + \sigma(k))$: In other words, the absolute value of the Ramanujan-function may behave in all possible ways (increasing, decreasing, etc.) on intervals of length $k$ provided of course that it does not vanish with a certain periodicity inside such intervals. The proof is quite elementary in essence (Chinese Remainder Theorem) although we appeal to lower bounds for linear forms in logarithms of algebraic numbers to justify certain estimates and to sieves and the truth of the Sato-Tate conjecture for $\tau(n)$ to avoid certain primes.

Tuesday, April 3, 2012

Number Theory Seminar
11:00 am in 241 Altgeld Hall, Tuesday, April 3, 2012

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Benjamin Smith (INRIA Saclay and Ecole Polytechnique Paris)
Point counting on genus 2 curves with real multplication
Abstract: Point counting -- that is, computing zeta functions of curves over finite fields --- is a fundamental problem in algorithmic number theory and cryptography. In this talk, we present an accelerated Schoof-type point-counting algorithm for curves of genus 2 equipped with an efficiently computable real multiplication endomorphism. Using our new algorithm, we can compute the zeta function of an explicit RM genus 2 curve over $\mathbb{F}_q$ in $O(\log^5 q)$ bit operations (vs. $O(\log^8 q)$ for the classical algorithm). This, together with a number of other practical improvements, yields a dramatic speedup for cryptographic-sized Jacobians over prime fields, as well as some record-breaking computations. (Joint work with D. Kohel and P. Gaudry)

Thursday, April 5, 2012

Number Theory Seminar
11:00 am in 243 Altgeld Hall, Thursday, April 5, 2012

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Jingjing Huang (Penn State Math)
Sums of unit fractions
Abstract: The results presented in this talk are joint work of Robert Vaughan with myself. We are mainly concerned with the Diophantine equation $$\frac{a}{n}=\frac1{x_1}+\frac1{x_2}+\cdots+\frac1{x_k}$$ and its number of positive integer solutions $R_k(n;a)$. Now the distribution of the function $R_2(n;a)$ is well understood. More precisely, by averaging over $n$, the first moment and second moment behaviors of $R_2(n;a)$ have been established. Furthermore, we have shown that, after normalisation, $R_2(n;a)$ satisfies Gaussian distribution, which is an analog of the classical theorem of Erdos and Kac. Now, in this talk, I will mainly talk about the following result. Let $E_a(N)$ denote the number of $n\le N$ such that $R_2(n;a)=0$. It is established that when $a\ge3$ we have $$E_a(N)\sim C(a) \frac{N(\log\log N)^{2^{m-1}-1}}{(\log N)^{1-1/2^m}},$$ with $m$ defined in the talk. This result significantly improves a result of Hoffmeister and Stoll. I will explain how to prove this theorem. The next project would be to study the ternary case $k=3$. While the conjecture, by Erd\H{o}s, Straus and Schinzel, that for fixed $a\ge 4$, we have $R_3(n;a)>0$ when $n$ is sufficiently large, is still wide open, here I will talk about some partial results on the mean value $\sum_{n\le N}R_3(n;a)$ if time permits.

Tuesday, April 10, 2012

Tuesday, April 17, 2012

Thursday, April 19, 2012

Tuesday, April 24, 2012

Joint Ergodic Theory/Number Theory Seminar
11:00 am in 347 Altgeld Hall, Tuesday, April 24, 2012

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Francesco Cellarosi (IAS/MSRI)
Ergodic Properties of Square-Free Numbers
Abstract: We study binary and multiple correlations for the set of square-free numbers and we construct a dynamical systems naturally associated to them. We prove that such dynamical system has pure point spectrum and it is therefore isomorphic to a translation on a compact abelian group. In particular, the system is ergodic but not weakly mixing, and it has zero metric entropy. The latter results were announced recently by Peter Sarnak and our approach provides an alternative approach. Joint work with Yakov Sinai.

Thursday, April 26, 2012

Number Theory Seminar
11:00 am in 243Altgeld Hall, Thursday, April 26, 2012

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Youness Lamzouri (Department of Mathematics, University of Illinois at Urbana-Champaign)
Discrepancy bounds for the distribution of the Riemann zeta function
Abstract: In 1930 Bohr and Jessen proved that for any $1/2<\sigma\leq 1$, $\log \zeta(\sigma+it)$ has a continuous limiting distribution in the complex plane. As a consequence they deduced that the set of values of $\log \zeta(\sigma+it)$ is everywhere dense in $\mathbb{C}$. Harman and Matsumoto obtained a quantitative version of the Bohr-Jessen Theorem using Fourier analysis on a multidimensional torus. In this talk we shall present a different approach which leads to uniform discrepancy bounds for the distribution of $\log \zeta(\sigma+it)$ that improve the Matsumoto-Harman estimates.

Tuesday, May 1, 2012

Number Theory Seminar
11:00 am in 241 Altgeld Hall, Tuesday, May 1, 2012

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Alina Cojocaru (Univ. Illinois Chicago)
Frobenius fields for elliptic curves
Abstract: Let E be an elliptic curve defined over $\mathbb{Q}$. For a prime p of good reduction for E, let $\pi_p$ be the $p$-Weil root of E and $\mathbb{Q}(\pi_p)$ the associated imaginary quadratic field generated by $\pi_p$. In 1976, Serge Lang and Hale Trotter formulated a conjectural asymptotic formula for the number of primes $p < x$ for which $\mathbb{Q}(\pi_p)$ is isomorphic to a fixed imaginary quadratic field. I will discuss progress on this conjecture, in particular an average result confirming the predicted asymptotic formula. This is joint work with Henryk Iwaniec and Nathan Jones.

Tuesday, August 28, 2012

Thursday, September 6, 2012

Number Theory Seminar
11:00 am in 241 Altgeld Hall, Thursday, September 6, 2012

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Bruce Berndt (UIUC Math)
Unpublished Manuscripts Published with Ramanujan's Lost Notebook
Abstract: Published with Ramanujan's lost notebook are several partial manuscripts. Some evidently were intended to be portions of papers that he had published. Others are partial manuscripts of papers that were never completed. In this lecture, we discuss examples of both types. For the former, we offer speculation on why Ramanujan never included the results in his published papers. The manuscripts are over a broad range of topics, including classical analysis, analytic number theory, diophantine approximation, and elementary mathematics.

Thursday, September 13, 2012

Number Theory Seminar
11:00 am in 241 Altgeld Hall, Thursday, September 13, 2012

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Dermot McCarthy (Texas A&M)
Hypergeometric Functions over Finite Fields and Modular Forms
Abstract: Hypergeometric functions over finite fields were introduced by Greene in the 1980's as analogues of the classical hypergeometric function. His motivation was to `bring some order' to the area of character sums and their evaluations by appealing to the rich theory of the classical function, and, in particular, its transformation properties. Since then, these finite field hypergeometric functions have also exhibited interesting properties in other areas. In particular, special values of these functions have been related to the Fourier coefficients of certain elliptic modular forms. Relationships with Siegel modular forms of higher degree are also expected. We will outline recent work on proving an example of such a connection, whereby a special value of the hypergeometric function is related to an eigenvalue associated to a Siegel eigenform of degree 2. This is joint work with Matt Papanikolas.

Thursday, September 20, 2012

Thursday, September 27, 2012

Number Theory Seminar
11:00 am in 241 Altgeld Hall, Thursday, September 27, 2012

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Armin Straub (UIUC Math)
Arithmetic aspects of short random walks
Abstract: We revisit a classical problem: how far does a random walk travel in a given number of steps (of length 1, each taken along a uniformly random direction)? Although such random walks are asymptotically well understood, surprisingly little is known about the exact distribution of the distance after just a few steps. For instance, the average distance after two steps is (trivially) given by $4/\pi$; but what is the average distance after three steps? In this talk, we therefore focus on the arithmetic properties of short random walks and consider both the moments of the distribution of these distances as well as the corresponding density functions. It turns out that the even moments have a rich combinatorial structure which we exploit to obtain analytic information. In particular, we find that in the case of three and four steps, the density functions can be put in hypergeometric form and may be parametrized by modular functions. Much less is known for the density in case of five random steps, but using the modularity of the four-step case we are able to deduce its exact behaviour near zero.

Thursday, October 4, 2012

Number Theory Seminar
11:00 am in 241 Altgeld Hall, Thursday, October 4, 2012

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Ken Stolarsky (UIUC Math)
Distribution mod 1: three points of view
Abstract: Three areas that connect with distribution of sequences mod 1 are number theory, numerical integration, and approximation theory. The most important reference here is the classic book of Kuipers and Niederreiter. Now a "new" framework based on reproducing kernels may become prominent. We outline this, examine the one-dimensional setting, and briefly indicate how it impacts higher dimensional distributions. This material has been gleaned from papers of J. Brauchat, J. Dick, F. Hickernell, and F. Pillichshammer. This talk is intended to be accessible to first year graduate students.

Thursday, October 11, 2012

Tuesday, October 16, 2012

Thursday, October 18, 2012

Number Theory Seminar
11:00 am in 241 Altgeld Hall, Thursday, October 18, 2012

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Michael Filaseta (Univ. South Carolina Math.)
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Abstract: If $p$ is a prime with decimal representation $d_{n} d_{n-1} \dots d_{1} d_{0}$, then a theorem of A. Cohn implies that the polynomial $f(x) = d_{n} x^{n} + d_{n-1}x^{n-1} + \cdots + d_{1}x + d_{0}$ is irreducible. One can view this result as following from the fact that if $g(x) \in \mathbb Z[x]$ with $g(0) = 1$, then $g(x)$ has a root in the disk $D = \{ z \in \mathbb C: |z| \le 1 \}$. On the other hand, that such a $g(x)$ has a root in $D$ has little to do with $g(x)$ having integer coefficients. In this talk, we discuss a perhaps surprising result about the location of a zero of such a $g(x)$ that makes use of its coefficients being in $\mathbb Z$ and discuss the implications this has on generalizations of Cohn's theorem. A variety of open problems will be presented. This research is joint work with a former student, Sam Gross.

Thursday, November 8, 2012

Tuesday, November 13, 2012

Number Theory Seminar
11:00 am in 241 Altgeld Hall, Tuesday, November 13, 2012

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David Zywina (Institute For Advanced Study)
Elliptic surfaces and the Inverse Galois Problem
Abstract: The Inverse Galois Problem asks whether every finite group $G$ occurs as the Galois group of some extension of $\mathbb{Q}$, i.e., whether there is a Galois extension $K/\mathbb{Q}$ such that $Gal(K/\mathbb{Q})$ is isomorphic to $G$. This problem is still wide open, even in the special case of simple groups. By studying the Galois action on the \'etale cohomology of some well-chosen elliptic surfaces, we will prove many new cases of the Inverse Galois problem. In particular, we will explain why the simple groups $PSL_2(\mathbb{F}_p)$ and $PSp_4(\mathbb{F}_p)$ both occur as Galois extensions of $\mathbb{Q}$ for all sufficiently large primes $p$. An important role will be played by the Birch and Swinnerton-Dyer conjecture for certain elliptic curves.

Thursday, November 15, 2012

Number Theory Seminar
11:00 am in 241 Altgeld Hall, Thursday, November 15, 2012

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Jeff Vaaler (Univ. Texas - Austin)
Bounds for small generators of number fields
Abstract: Let $k$ be an algebraic number field with degree $d$ over the rational field $\mathbb{Q}$, and discriminant $\Delta_k$ . If $k$ has a real embedding then we prove that $k$ has a generator $\alpha$ such that $H(\alpha) \le |\Delta_k|^{1/2d}$, where $H(\alpha)$ is the absolute multiplicative Weil height. This verifies a conjecture of W. Ruppert. If $k$ has no real embedding the situation is more complicated, and we are only able to obtain a conditional result. In this case we prove a similar bound on the height of a generator for a number fields $k$, but we must assume that the Dedekind zeta-function associated to the Galois closure of $k/\mathbb{Q}$ satisfies the generalized Riemann hypothesis. This is joint work with Martin Widmer.

Tuesday, November 27, 2012

Thursday, November 29, 2012

Tuesday, December 4, 2012

Thursday, December 6, 2012

Number Theory Seminar
11:00 am in 241 Altgeld Hall, Thursday, December 6, 2012

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Shaoshi Chen (North Carolina State Univ. )
On the Summability of Bivariate Rational Functions
Abstract: This talk contains two parts: First, I will give a brief introduction to Zeilberger's method of creative telescoping. Second, I will talk about a recent work, in which we present criteria for deciding whether a bivariate rational function can be written as a sum of two (q)-difference of bivariate rational functions. Using these criteria, we show how certain double sums can be evaluated, first, in terms of single sums and, finally, in terms of values of special functions. (This is a joint work with Michael F. Singer)