Department of Mathematics

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On various Gauss-Manin connections

by Robby Burko | University of Toronto
Time: 14:00  (Thursday, Mar. 03, 2011)
Location: HU1012, 215 Huron St
Abstract:
No Abstract

Details

Small gaps between primes: A result of Goldstein,Pintz and Yildirim

by Payman Eskandari | University of Toronto
Time: 13:00  (Thursday, Mar. 03, 2011)
Location: HU1012, 215 Huron St
Abstract:
The difference between two consecutive primes has been interesting mathematicians since the ancient times. The most famous unsolved problem in this direction is the twin prime conjecture, which can be stated as $\liminf_{r} p_{r+1}-p_r =2$, where $p_r$ denotes the rth prime. Let $N$ be a large number. By the prime number theorem, there are about $N/\log N$ primes between $N$ and $2N$, so the average gap between two consecutive primes in this range is $\log N$, which is almost $log p$ for any prime in the range. In this sense, the gap $p_{r+1}-p_r$ is in average about $\log p_r$. This motivates the study of $\liminf_{r}\frac{p_{r+1}-p_r}{\log p_r}$. It was first Erdos that proved (unconditionally) that this $\liminf$ is less than $1$, and several mathematicians including Ricci, Bombieri and Davenport improved the upper bound for this $\liminf$. In 2006, finally it was proved, by Goldston, Pintz and Yildirim that this $\liminf$ is indeed zero. My goal is to give a sketch of the proof of GPY result and discuss some of its ideas. Time permitting, I might also briefly discuss some parts of Soundararajan's review paper and Tao's blog article on the result.

Dates in this series

· Thursday, Jan. 14, 2010: On the Large Sieve Method in Algebraic Number Fields (Nataliya Laptyeva)

· Thursday, Jan. 21, 2010: About the Euler-Kronecker constant of a cyclotomic field  (Mariam Mourtada)

· Thursday, Feb. 04, 2010: On Side Channel Attacks of Product Ciphers (Aaron Chow)

· Thursday, Feb. 18, 2010: Logarithmic Derivatives of L-functions (Kumar Murty)

· Thursday, Mar. 11, 2010: On the Euler -Kronecker constant of a number field (Maryam Mourtada)

· Thursday, Apr. 15, 2010: Tate cycles on Abelian varieties (Vijay Patankar)

· Thursday, Apr. 15, 2010: Zeta functions of varieties (Kumar Murty)

· Thursday, Apr. 29, 2010: Generalized Brauer-Siegel Theorem (Maryam Mourtada)

· Tuesday, May. 18, 2010: Identity-Based Encryption (Ken Giuliani)

· Wednesday, May. 26, 2010: TBA (Maryam Mourtada)

· Wednesday, May. 26, 2010: Construction of Opaque Predicates (Vijay Patankar)

· Tuesday, Jun. 01, 2010: Mazur's Conjecture and its Applications to Generalizations of Hilbert's Tenth Problem (William George)

· Wednesday, Jun. 09, 2010: TBA (Robby Burko)

· Thursday, Jun. 10, 2010: Certain Classes of Reducible Polynomials over Finite Fields (Brandon Hanson)

· Wednesday, Jun. 16, 2010: TBA (Payman Eskandari)

· Thursday, Jun. 17, 2010: Introduction to Hyperelliptic Curve Cryptography (Aaron Chow)

· Wednesday, Jul. 07, 2010: On Omega Theorem for $\frac{Ll'}{L}(1,\chi_D)$ (Mariam Mourtada)

· Thursday, Sep. 16, 2010: Poitou-Tate duality over p-adic Lie extension of global fields (Meng Fai Lim)

· Thursday, Sep. 23, 2010: Hilbert's Tenth Problem over Large Subrings of Q and an Interesting Conjecture of Davis (William George)

· Thursday, Sep. 23, 2010: A variant of Lehmer's conjecture in the CM-case (Nataliya Laptyeva)

· Thursday, Sep. 30, 2010: A variant of Lehmer's conjecture in the CM-case (part 2) (Nataliya Laptyeva)

· Thursday, Sep. 30, 2010: Class number one problem for real quadratic fields of certain type (Kostadinka Lapkova)

· Thursday, Oct. 07, 2010: Incomplete character sums (Catalina Anghel)

· Thursday, Oct. 07, 2010: A Side Channel on Hash Functions (Aaron Chow)

· Thursday, Oct. 14, 2010: TBA (Nataliya Laptyeva)

· Thursday, Oct. 21, 2010: TBA (Payman Eskandari)

· Thursday, Oct. 21, 2010: Applications of Recurrence to Number Theory (Brandon Hanson)

· Thursday, Oct. 28, 2010: A variant of Lehmer's conjecture in the CM-case (part 4) (Nataliya Laptyeva)

· Thursday, Nov. 04, 2010: TBA (Robby Burko)

· Thursday, Nov. 04, 2010: TBA (Hamid Usefi)

· Thursday, Nov. 11, 2010: TBA (Catalina Anghel)

· Thursday, Nov. 11, 2010: TBA (Kostadinka Lapkova)

· Thursday, Nov. 18, 2010: Duality over p-adic Lie extensions of global fields (Meng Fai Lim)

· Thursday, Nov. 25, 2010: Selmer Ranks of Twists of Elliptic Curves (William George)

· Thursday, Dec. 02, 2010: A variant of Lehmer's conjecture in the CM-case (continued) (Nataliya Laptyeva)

· Thursday, Jan. 20, 2011: Algebraic Eraser and Non-Abelian Group-Based Cryptography (Hamid Usefi)

· Thursday, Jan. 20, 2011: Proof of one of Bourgain's theorems on exponential sums, (Catalina Anghel)

· Thursday, Jan. 27, 2011: Side-Channel Attacks on SHA-3 Candidates (Aaron Chow)

· Thursday, Feb. 03, 2011: Nonvanishing of quadraric Dirichlet L-functions at $s=\frac{1}{2}$ (Mariam Mourtada)

· Thursday, Feb. 10, 2011: On the Brun-Titchmarch Theorem (Nataliya Laptyeva)

· Thursday, Mar. 03, 2011: On various Gauss-Manin connections (Robby Burko)

· Thursday, Mar. 03, 2011: Small gaps between primes: A result of Goldstein,Pintz and Yildirim (Payman Eskandari)

· Thursday, Mar. 10, 2011: The Bivariate Polynomial Injection Problem (William George)

· Thursday, Mar. 17, 2011: Probabilistic reasoning and the number of prime divisors (Brandon Hanson)

· Thursday, Mar. 24, 2011: The growth of the Tate-Shafarevich group in p-Hilbert class field towers (Meng Fai Lim )

· Thursday, Mar. 31, 2011: Side-Channel Attacks on SHA-3 Candidates (Part 2) (Aaron Chow)

· Thursday, Apr. 14, 2011: Elliptic Cartan subalgebra (Ying Zong)

· Thursday, Apr. 21, 2011: Digital Signature Scheme with a generator kept secret (Nikolajs Volkovs)

· Thursday, May. 26, 2011: Concepts for the development of a secure and private e-health platform (Coline Migonney)

· Thursday, May. 26, 2011: Zero-knowledge identification protocols (Sabrina Mameri)

· Thursday, Jun. 02, 2011: Some Consequences of Mazur's Conjecture (William George)

· Thursday, Jun. 09, 2011: Average over fundamental discriminants of $L'/L(\sigma,\chi_D) ~ , 1/2<\sigma <1$ (Mariam Mourtada)

· Thursday, Jun. 16, 2011: A Method for Computing Zeta Functions of Varieties over Finite Fields using Fibrations (Robby Burko)

· Thursday, Jun. 16, 2011: Ellipic Curves Public-Key Cryptography - Applications and Attacks (Prof. Dr. Andreas Stein)

· Tuesday, Jun. 21, 2011: Notion of privacy and security within the secure electronic transactions (Aude Plateaux)

· Thursday, Jun. 23, 2011: Concepts for the development of a secure & privacy e-health platform (Coline Migonney)

· Thursday, Jun. 23, 2011: On the number of solutions of exponential congruences (Catalina Anghel)

· Tuesday, Jul. 05, 2011: Computing weights in abelian group codes (Marinês Guerreiro)

· Tuesday, Jul. 12, 2011: A decade of using non-commutative groups in cryptography (Delaram Kahrobaei)

· Tuesday, Jul. 19, 2011: Polynomial Algebra by Values and Its Applications (Azar Shakoori)

· Tuesday, Aug. 09, 2011: Fluctuations in the number of points of curves over finite fields (Chantal David)

· Tuesday, Aug. 16, 2011: On the Lonely Runner Conjecture (Juraj Milcak)

· Tuesday, Aug. 23, 2011: eHMAC: A side-channel resistant HMAC proposal (Aaron Chow)

· Thursday, Sep. 15, 2011: Limits of random matrices and the Brownian Carousel (Balint Virag)

· Thursday, Sep. 22, 2011: TBA (Meng Fai Lim)

· Thursday, Sep. 29, 2011: TBA (R. Venkatesan)

· Thursday, Sep. 29, 2011: Pseudo-randomness of subsets related to primitive roots (Catalina Anghel)

· Thursday, Oct. 06, 2011: Local Systems (Shervin Shahrokhi Tehrani)

· Thursday, Oct. 06, 2011: A Conjecture of Mazur for Elliptic Surfaces and a Decision Problem (William George)

· Thursday, Oct. 13, 2011: Limiting Frobenius Structures and the Legendre family of Elliptic curves (Robby Burko)

· Thursday, Oct. 13, 2011: On the arithmetic properties of the incomplete Gamma functions (Anastasia Zaytseva)

· Thursday, Oct. 27, 2011: Non Abelian Analogues of Lattice Rounding (Ramarathnam Venkatesan)

· Thursday, Nov. 03, 2011: The algebra of multiple zeta values (Payman Eskandari)

· Thursday, Nov. 03, 2011: The Rank One Lifting Problem and the AKS Algorithm (Aaron Chow)

· Thursday, Nov. 10, 2011: Lower Bounds for Discrete Logarithms and Related Problems (Catalina Anghel)

· Thursday, Nov. 10, 2011: Primes in Elliptic Divisibility Sequences (William George)

· Thursday, Nov. 24, 2011: The Geometry of Siegel Modular Forms (Shervin Shahrokhi Tehrani)

· Thursday, Jan. 19, 2012: On the analogue of a Kronecker Limit Formula (Anastasia Zaytseva)

· Thursday, Mar. 01, 2012: Generalized Gamma function (Anastasia Zaytseva)

· Thursday, Apr. 26, 2012: TBA (William George)

· Thursday, Jun. 14, 2012: Unbounded quasi-morphisms and bounded generation  (Mahan Mitra)

· Thursday, Jun. 14, 2012: TBA (Payman Eskandari)

· Thursday, Jun. 14, 2012: The error term in the Sato-Tate conjecture  (Kumar Murty)

· Thursday, Jun. 28, 2012: Growth of p-rank of Selmer groups of elliptic curves with split multiplicative reduction  (Meng Fai Lim)

· Thursday, Aug. 30, 2012: p-adic simpson correspondence, d'apres Faltings  (Ying Zong)

· Thursday, Nov. 01, 2012: Rankin L-functions and the twisted Birch and Swinnerton-Dyer  (Reza Sadoughian)

· Thursday, Jan. 31, 2013: Density of Positive Rank Fibers with Composite Order mod p (William George)