# GANITA Seminar

### From TorontoMathWiki

This page contains information about the GANITA Seminar at the University of Toronto. The seminar usually meets on Thursdays in the GANITA Lab, located at HU1012, 215 Huron Street. For more information, please contact Aaron Chow at aaron.chow@utoronto.ca.

# 2012 Fall

## November 01, **Sadoughian**, 13:00 @HU1012

Reza Sadoughian (McMaster University) | GANITA Seminar
| Thursday | November 01 | 13:00 | HU1012 | ||||||||

Title: Rankin L-functions and the twisted Birch and Swinnerton-Dyer Conjecture
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Abstract: We use Rankin's method to evaluate the central critical value of the L-series attached to an elliptic curve E over Q and certain odd irreducible 2-dimensional Artin representations. As an application we use the result to study the twisted Birch and Swinnerton-Dyer conjecture.
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[ arXiv] | 2012_11_01_Sadoughian_Notes | 2012_11_01 |

## October 25, **George**, 14:00 @HU1012

William George (University of Toronto) | GANITA Seminar
| Thursday | October 25 | 14:00 | HU1012 | ||||||||

Title: A Surface of Qi Cheng and Ming-Deh Huang With Applications to the ECDLP
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Abstract: I will investigate properties of an elliptic surface for which, by work of Qi Cheng and Ming-Deh Huang, if a positive density of the fibers have rank one then one can relate the ECDLP to certain lifting problems. I will show how by Rosen and Silverman's proof of certain cases of Nagao's Conjecture one can compute the generic rank of this surface and then I will look at some work of Helfgott towards computing the average root number of this surface.
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[ arXiv] | 2012_10_25_George_Notes | 2012_10_25 |

## September 27, **Hanson**, 14:00 @HU1012

Brandon Hanson (University of Toronto) | GANITA Seminar
| Thursday | September 27 | 14:00 | HU1012 | ||||||||

Title: Monochromatic Sums and Products in Finite Fields
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Abstract:
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[ arXiv] | 2012_09_27_Hanson_Notes | 2012_09_27 |

## September 27, **Mourtada**, 13:00 @HU1012

Mariam Mourtada (University of Toronto) | GANITA Seminar
| Thursday | September 27 | 13:00 | HU1012 | ||||||||

Title: TBA
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Abstract:
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[ arXiv] | 2012_09_27_Mourtada_Notes | 2012_09_27 |

## September 20, **Burko**, 14:00 @HU1012

Robby Burko (University of Toronto) | GANITA Seminar
| Thursday | September 20 | 14:00 | HU1012 | ||||||||

Title: Precision Estimates for Kedlaya's Point-Counting Algorithm Applied to Certain Singular Curves over Finite Fields
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Abstract:
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[ arXiv] | 2012_09_20_Burko_Notes | 2012_09_20 |

## September 13, **Seraj**, 13:00 @HU1012

Samer Seraj (University of Toronto) | GANITA Seminar
| Thursday | September 13 | 13:00 | HU1012 | ||||||||

Title: Sum of Cubes is Square of Sum
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Abstract: An intriguing mathematical fact about the first n natural numbers is that the sum of their cubes is the same as the square of their sum. We explore other sets with the same property by establishing definite answers to the question of finitude vs. infinitude of solutions for each n, and end by asking open problems. The material is from a paper written by Edward Barbeau and Samer Seraj.
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[ arXiv] | 2012_09_13_Seraj_Notes | 2012_09_13 |

## September 06, **Lim**, 14:00 @HU1012

Meng Fai Lim (University of Toronto) | GANITA Seminar
| Thursday | September 06 | 14:00 | HU1012 | ||||||||

Title: Some remarks on the BSD conjecture and (p-)parity conjecture
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Abstract: We present some (minor) improvement to results on BSD
conjecture and (p-)parity conjecture. Namely, we improve certain relativity results of Dokchitser-Dokchitser on the p-parity conjecture. We will also present a statement on the BSD conjecture which is a mild improvement of results of Kolyvagin, Kato and Abkary-Murty. The method used in is mainly group-theoretical. | |||||||||||||

[ arXiv] | 2012_09_06_Lim_Notes | 2012_09_06 |

# 2012 Summer

## August 30, **Zong**, 14:00 @HU1012

Ying Zong (University of Toronto) | GANITA Seminar
| Thursday | August 30 | 14:00 | HU1012 | ||||||||

Title: p-adic simpson correspondence, d'apres Faltings
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Abstract: I give an expose of Faltings' paper "a p-adic simpson correspondence".
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[ arXiv] | 2012_08_30_Zong_Notes | 2012_08_30 |

## August 23, **Eskandari**, 14:00 @HU1012

Payman Eskandari (University of Toronto) | GANITA Seminar
| Thursday | August 23 | 14:00 | HU1012 | ||||||||

Title: A deformation of the free associative algebra on a set
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Abstract:
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[ arXiv] | 2012_08_23_Eskandari_Notes | 2012_08_23 |

## July 26, **George**, 14:00 @HU1012

William George (University of Toronto) | GANITA Seminar
| Thursday | July 26 | 14:00 | HU1012 | ||||||||

Title: Root Numbers of Fibers of Elliptic Surface
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Abstract: I will look at some work of David Rohrlich concerning the behavior of the root number of fibers of certain elliptic surfaces. For certain surfaces, he shows that the variation of the root number is related to topological properties of the surface. The talk will focus on gaining a sense of for what one requires on a surface to have this kind of behavior.
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[ arXiv] | 2012_07_26_George_Notes | 2012_07_26 |

## July 19, **Sastry**, 15:00 @HU1012

Pramathanath Sastry (Chennai Mathematical Institute) | GANITA Seminar
| Thursday | July 19 | 15:00 | HU1012 | ||||||||

Title: Abelian varieties and cryptogrpahy
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Abstract:
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[ arXiv] | 2012_07_19_Sastry_Notes | 2012_07_19 |

## July 19, **Anghel**, 14:00 @HU1012

Catalina Anghel (University of Toronto) | GANITA Seminar
| Thursday | July 19 | 14:00 | HU1012 | ||||||||

Title: Results related to the self-power map
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Abstract:
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[ arXiv] | 2012_07_19_Anghel_Notes | 2012_07_19 |

## July 12, **Sastry**, 15:00 @HU1012

Pramathanath Sastry (Chennai Mathematical Institute) | GANITA Seminar
| Thursday | July 12 | 15:00 | HU1012 | ||||||||

Title: Abelian varieties and cryptogrpahy
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Abstract:
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[ arXiv] | 2012_07_12_Sastry_Notes | 2012_07_12 |

## July 12, **Burko**, 14:00 @HU1012

Robby Burko (University of Toronto) | GANITA Seminar
| Thursday | July 12 | 14:00 | HU1012 | ||||||||

Title: p-adic methods in computing zeta functions of singular curves over finite fields
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Abstract:
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[ arXiv] | 2012_07_12_Burko_Notes | 2012_07_12 |

## July 05, **Marie**, 15:00 @HU1012

Vincent Marie (ENSICAEN) | GANITA Seminar
| Thursday | July 05 | 15:00 | HU1012 | ||||||||

Title: Implementation of a payment protocol preserving privacy
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Abstract:
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[ arXiv] | 2012_07_05_Marie_Notes | 2012_07_05 |

## July 05, **Khachlouf**, 14:00 @HU1012

Mejdi Khachlouf (ENSICAEN) | GANITA Seminar
| Thursday | July 05 | 14:00 | HU1012 | ||||||||

Title: Enhancing privacy in the e-health information system
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Abstract: Numerous country-specific infrastructures for medical records are emerging. The critical impact of these sensitive data on the privacy of patients gives rise to many standardization activities. Security of such medical information systems has attracted a major attention due to the large amount of data shared by several actors, such as doctors or nurses in different institutions. However, the privacy preserving issue is at most partially treated. Concepts of data minimisation and data sovereignty for medical data are often neglected. The proposed architecture presents an e-health infrastructure intended to minimise personal data disclosure and data aggregation, whereas data sovereignty principle is realized in accordance with medical constraints.
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[ arXiv] | 2012_07_05_Khachlouf_Notes | 2012_07_05 |

## June 28, **Murty**, 15:00 @HU1012

Kumar Murty (University of Toronto) | GANITA Seminar
| Thursday | June 28 | 15:00 | HU1012 | ||||||||

Title: On Bounded Generation
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Abstract:
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[ arXiv] | 2012_06_28_Murty_Notes | 2012_06_28 |

## June 28, **Lim**, 14:00 @HU1012

Meng Fai Lim (University of Toronto) | GANITA Seminar
| Thursday | June 28 | 14:00 | HU1012 | ||||||||

Title: Growth of p-rank of Selmer groups of elliptic curves with split multiplicative reduction
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Abstract: This is a continuation of the talk given in Ganita seminar on 2012.04.26. Let p be a fixed prime. Suppose that E is an elliptic curve with split multiplicative reduction at a prime, and that the Tamagawa factor at the split multiplicative prime is divisible by p. In this talk, we present some results on the growth of Selmer p-rank of the above class of elliptic curves in certain p-adic Lie extension. This is a joint work with Prof. V. Kumar Murty.
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[ arXiv] | 2012_06_28_Lim_Notes | 2012_06_28 |

## June 21, **Mitra**, 15:00 @HU1012

Mahan Mitra (Ramakrishna Mission Vivekananda University) | GANITA Seminar
| Thursday | June 21 | 15:00 | HU1012 | ||||||||

Title: Unbounded quasi-morphisms and bounded generation (part 2)
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Abstract:
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[ arXiv] | 2012_06_21_Mitra_Notes | 2012_06_21 |

## June 21, **Murty**, 14:00 @HU1012

Kumar Murty (University of Toronto) | GANITA Seminar
| Thursday | June 21 | 14:00 | HU1012 | ||||||||

Title: The error term in the Sato-Tate conjecture (part 2)
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Abstract:
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[ arXiv] | 2012_06_21_Murty_Notes | 2012_06_21 |

## June 14, **Eskandari**, 16:00 @HU1012

Payman Eskandari (University of Toronto) | GANITA Seminar
| Thursday | June 14 | 16:00 | HU1012 | ||||||||

Title: TBA
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Abstract:
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[ arXiv] | 2012_06_14_Eskandari_Notes | 2012_06_14 |

## June 14, **Murty**, 15:20 @HU1012

Kumar Murty (University of Toronto) | GANITA Seminar
| Thursday | June 14 | 15:20 | HU1012 | ||||||||

Title: The error term in the Sato-Tate conjecture
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Abstract:
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[ arXiv] | 2012_06_14_Murty_Notes | 2012_06_14 |

## June 14, **Mitra**, 14:40 @HU1012

Mahan Mitra (Ramakrishna Mission Vivekananda University) | GANITA Seminar
| Thursday | June 14 | 14:40 | HU1012 | ||||||||

Title: Unbounded quasi-morphisms and bounded generation
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Abstract:
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[ arXiv] | 2012_06_14_Mitra_Notes | 2012_06_14 |

## May 03, **Lim**, 14:00 @HU1012

Meng Fai Lim (University of Toronto) | GANITA Seminar
| Thursday | May 03 | 14:00 | HU1012 | ||||||||

Title: Growth of p-rank of Selmer groups of elliptic curves with split multiplicative reduction
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Abstract: This is a joint work with Prof. V. Kumar Murty. We present a result on the growth of Selmer p-rank of elliptic curves with split multiplicative reduction in an infinite Hilbert p-class tower. We are namely interested in the case when the Tamagawa factor at the split multiplicativ prime is divisible by p. We will show that under certain assumptions, the p-rank is unbounded. This result will complement the case for CM-abelian varieties considered by us, and also the result of Matar. The idea of the proof revolves around the techniques occuring in the theme of the so-called "Mazur Control Theorem". We will use a slight variant of this as considered by Matsuno.
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[ arXiv] | 2012_05_03_Lim_Notes | 2012_05_03 |

## April 26, **George**, 14:00 @HU1012

William George (University of Toronto) | GANITA Seminar
| Thursday | April 26 | 14:00 | HU1012 | ||||||||

Title: Finding Quadratic Residue Fibres as Outputs in an Algorithm
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Abstract:
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[ arXiv] | 2012_04_26_George_Notes | 2012_04_26 |

# 2012 Winter

## March 08, **Burko**, 14:00 @HU1012

Robby Burko (University of Toronto) | GANITA Seminar
| Thursday | March 08 | 14:00 | HU1012 | ||||||||

Title: Vanishing of Log-Cohomology
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Abstract:
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[ arXiv] | 2012_03_08_Burko_Notes | 2012_03_08 |

## March 01, **Zaytseva**, 14:00 @HU1012

Anastasia Zaytseva (University of Toronto) | GANITA Seminar
| Thursday | March 01 | 14:00 | HU1012 | ||||||||

Title: Generalized Gamma function
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Abstract: "The Bohr-Mollerup theorem states that $\log{\Gamma(x)}$" is the only eventially convex function which satisfies the functional equation $f(x+1)-f(x)=\log{x}$ for $x>0$ with the additional condition $f(1)=0$. It is natural to consider higher analogues of this function, R_k(x), defined as the eventially convex solution of the equation $f(x+1)-f(x)=\log^k{x}$, for $x>0$, $k\in\mathbb{Z}^{+}$ with some initial condition. The case $k=2$ was studied by Deninger in 1983 in his paper "On the analogue of the formula of Chowla and Selberg". In particular, he showed connection between $R_2(x)$ and the Hurwitz zeta function and used it to obtain several relations for $R_2(x)$. In my talk I will discuss Deninger's work and further properties of the functions $R_k(x)$.
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[ arXiv] | 2012_03_01_Zaytseva_Notes | 2012_03_01 |

## February 16, **Eskandari**, 14:00 @HU1012

Payman Eskandari (University of Toronto) | GANITA Seminar
| Thursday | February 16 | 14:00 | HU1012 | ||||||||

Title: Motives as systems of realizations
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Abstract:
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[ arXiv] | 2012_02_16_Eskandari_Notes | 2012_02_16 |

## February 09, **George**, 14:00 @HU1012

William George (University of Toronto) | GANITA Seminar
| Thursday | February 09 | 14:00 | HU1012 | ||||||||

Title: Finding Rank One Lifts and the Elliptic Curve Discrete Log Problem
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Abstract: I will discuss my attempts to find desirable lifts of elliptic curves with the aim of fulfilling a hypothesis of Miri and Murty that would show the equivalence of ECDLP to a lifting problem. I will then compare this to related work done by Cheng and Huang in 2006.
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[ arXiv] | 2012_02_09_George_Notes | 2012_02_09 |

## January 26, **Chow**, 15:00 @HU1012

Aaron Chow (University of Toronto) | GANITA Seminar
| Thursday | January 26 | 15:00 | HU1012 | ||||||||

Title: Primality Testing via Modular Forms
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Abstract: This is an informal discussion about work in progress on primality testing using Fourier coefficients of modular forms.
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[ arXiv] | 2012_01_26_Chow_Notes | 2012_01_26 |

## January 26, **Burko**, 14:00 @HU1012

Robby Burko (University of Toronto) | GANITA Seminar
| Thursday | January 26 | 14:00 | HU1012 | ||||||||

Title: TBA
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Abstract:
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[ arXiv] | 2012_01_26_Burko_Notes | 2012_01_26 |

## January 19, **Zaytseva**, 14:00 @HU1012

Anastasia Zaytseva (Queen's University) | GANITA Seminar
| Thursday | January 19 | 14:00 | HU1012 | ||||||||

Title: On the analogue of a Kronecker Limit Formula
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Abstract: Let $Q(x,y)=ax^2+bxy+cy^2$ be a definite quadratic form such that $a,c>0, b^2-4ac=1$. The Epstein zeta-function is defined as
\[K(Q,s)=\sum_{m,n}\frac{1}{Q(m,n)^s}\] for $Re(s)>1$. The summation here is over all integers $(m,n)\neq (0,0)$. Kronecker showed that $K(Q,s)$ can be extended meromorphically to the neighborhood $s=1$ with the only pole at $s=1$. The Kronecker's limit formula gives the first two coefficients in the Laurent expansion of $K(Q,s)$ at $s=1$. This important formula can applied to study the properties of the Dedekind zeta function of quadratic fields. In 1975, Zagier proved the analogue of the Kronecker's Theorem for the function \[Z(Q,s)=\sum_{m>0,n>-1}\frac{1}{Q(m,n)^s},\] where $Q(x,y)=ax^2+bxy+cy^2$ is indefinite quadratic form with conditions $a,b,c>0 , b^2-4ac=1$. In this talk we will discuss the theorem's of Kronecker and Zagier and their connection with the Dedekind zeta function. | |||||||||||||

[ arXiv] | 2012_01_19_Zaytseva_Notes | 2012_01_19 |

# 2011 Fall

## November 24, **Shahrokhi Tehrani**, 14:00 @HU1012

Shervin Shahrokhi Tehrani (University of Toronto) | GANITA Seminar
| Thursday | November 24 | 14:00 | HU1012 | ||||||||

Title: The Geometry of Siegel Modular Forms
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Abstract: Local system, on the moduli space of abelian surfaces, are locally constant sheaves that obtained by action of GSp(4) on a finite dimensional vector spaces. We will consider the cohomology of these kinds of bundles and make connection to theory of Siegel automorphic forms of genus two AND theta correspondence.
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[ arXiv] | 2011_11_24_Shahrokhi Tehrani_Notes | 2011_11_24 |

## November 10, **George**, 15:00 @HU1012

William George (University of Toronto) | GANITA Seminar
| Thursday | November 10 | 15:00 | HU1012 | ||||||||

Title: Primes in Elliptic Divisibility Sequences
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Abstract: I will look at some results on the primes that appear in the denominator of the multiples of a generator for a rank one curve that Poonen uses to show topological properties of certain sets of S-integers points on the curve. I will then look at a refinement of some of these ideas developed by Eisentrager and Everest.
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[ arXiv] | 2011_11_10_George_Notes | 2011_11_10 |

## November 10, **Anghel**, 14:00 @HU1012

Catalina Anghel (University of Toronto) | GANITA Seminar
| Thursday | November 10 | 14:00 | HU1012 | ||||||||

Title: Lower Bounds for Discrete Logarithms and Related Problems
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Abstract:
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[ arXiv] | 2011_11_10_Anghel_Notes | 2011_11_10 |

## November 03, **Chow**, 15:00 @HU1012

Aaron Chow (University of Toronto) | GANITA Seminar
| Thursday | November 03 | 15:00 | HU1012 | ||||||||

Title: The Rank One Lifting Problem and the AKS Algorithm
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Abstract: This will be a survey of two papers. The first one is by Miri and Murty on the equivalence of the lifting problem for elliptic curves of rank one and the discrete logarithm problem. The second paper is by Agrawal, Kayal, and Saxena on their primality testing algorithm.
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[ arXiv] | 2011_11_03_Chow_Notes | 2011_11_03 |

## November 03, **Eskandari**, 14:00 @HU1012

Payman Eskandari (University of Toronto) | GANITA Seminar
| Thursday | November 03 | 14:00 | HU1012 | ||||||||

Title: The algebra of multiple zeta values
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Abstract: This will be an introductory talk on the algebra of multiple zeta values. The first part of the talk will be devoted to defining multiple zeta values and their algebra and some elementary observations such as writing them as integrals. In the second part I will give a conjectural description of the algebra of multiple zeta values.
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[ arXiv] | 2011_11_03_Eskandari_Notes | 2011_11_03 |

## October 27, **Venkatesan**, 14:00 @HU1012

Ramarathnam Venkatesan (Microsoft Research) | GANITA Seminar
| Thursday | October 27 | 14:00 | HU1012 | ||||||||

Title: Non Abelian Analogues of Lattice Rounding
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Abstract:
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[ arXiv] | 2011_10_27_Venkatesan_Notes | 2011_10_27 |

## October 13, **Burko**, 15:00 @HU1012

Robby Burko (University of Toronto) | GANITA Seminar
| Thursday | October 13 | 15:00 | HU1012 | ||||||||

Title: Limiting Frobenius Structures and the Legendre family of Elliptic curves
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Abstract:
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[ arXiv] | 2011_10_13_Burko_Notes | 2011_10_13 |

## October 13, **Zaytseva**, 14:00 @HU1012

Anastasia Zaytseva (Queen's University) | GANITA Seminar
| Thursday | October 13 | 14:00 | HU1012 | ||||||||

Title: On the arithmetic properties of the incomplete Gamma functions
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Abstract: The arithmetic properties of the Gamma function at rational points is an open problem in Number Theory. Apart from very few classical results, we still don't know whether \Gamma(p/q) is transcendental or even irrational. I will show how the theory of E-functions may be used to study the arithmetic nature of the incomplete Gamma functions, which are functions closely related to the Gamma function. This is a joint work with Ram Murty.
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[ arXiv] | 2011_10_13_Zaytseva_Notes | 2011_10_13 |

## October 06, **George**, 15:00 @HU1012

William George (University of Toronto) | GANITA Seminar
| Thursday | October 06 | 15:00 | HU1012 | ||||||||

Title: A Conjecture of Mazur for Elliptic Surfaces and a Decision Problem
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Abstract: I will look at a conjecture of Mazur regarding, for any given elliptic surface, the topology properties of the set of fibers with positive rank. In particular, I will look at some known special cases and the implications of the conjecture to relating certain decision problems.
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[ arXiv] | 2011_10_06_George_Notes | 2011_10_06 |

## October 06, **Tehrani**, 14:00 @HU1012

Shervin Shahrokhi Tehrani (University of Toronto) | GANITA Seminar
| Thursday | October 06 | 14:00 | HU1012 | ||||||||

Title: Local Systems
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Abstract: I am going to talk about two different kind of bundles on the moduli spaces of abelian varieties. The Local systems have important applications in arithmetic geometry.
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[ arXiv] | 2011_10_06_Tehrani_Notes | 2011_10_06 |

## September 29, **Anghel**, 15:00 @HU1012

Catalina Anghel (University of Toronto) | GANITA Seminar
| Thursday | September 29 | 15:00 | HU1012 | ||||||||

Title: '
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Abstract:
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[ arXiv] | 2011_09_29_Anghel_Notes | 2011_09_29 |

## September 29, **Burko**, 14:00 @HU1012

Robby Burko (University of Toronto) | GANITA Seminar
| Thursday | September 29 | 14:00 | HU1012 | ||||||||

Title: '
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Abstract:
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[ arXiv] | 2011_09_29_Burko_Notes | 2011_09_29 |

## September 22, **Lim**, 14:00 @HU1012

Meng Fai Lim (University of Toronto) | GANITA Seminar
| Thursday | September 22 | 14:00 | HU1012 | ||||||||

Title: '
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Abstract:
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[ arXiv] | 2011_09_22_Lim_Notes | 2011_09_22 |

## September 15, **Virag**, 13:00 @HU1012

Balint Virag (University of Toronto) | GANITA Seminar
| Thursday | September 15 | 13:00 | HU1012 | ||||||||

Title: Limits of random matrices and the Brownian Carousel
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Abstract: The Montgomery conjecture relates the eigenvalues of random matrices to zeros of the Riemann zeta function. I will describe a new representation of limits of random matrices in terms of a random self-adjoint operator (called the Brownian carousel) based on Brownian motion on $\text{SL}_2(\mathbb{R})$. This can be used to make conjectures about what the Riemann zeta function itself looks like near a random location on the critical line.
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[ arXiv] | 2011_09_15_Virag_Notes | 2011_09_15 |

# 2011 Summer

## August 23, **Chow**, 10:00 @HU1012

Aaron Chow (University of Toronto) | GANITA Seminar
| Tuesday | August 23 | 10:00 | HU1012 | ||||||||

Title: eHMAC: A side-channel resistant HMAC proposal
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Abstract: A keyed message authentication code (MAC) protects a messages integrity and authenticity with a secret key. The standard way of constructing a MAC from a cryptographic (block-based) hash function is the HMAC construction (RFC 2104). However, there are side-channel attacks that allow forgery of HMAC's based on SHA-1 and SHA-2, and more recently SHA-3 candidates. In this talk, HMAC is extended to resist these side-channel attacks, and its security is discussed.
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[ arXiv] | 2011_08_23_Chow_Notes | 2011_08_23 |

## August 16, **Milcak**, 11:00 @HU1012

Juraj Milcak (University of Toronto) | GANITA Seminar
| Tuesday | August 16 | 11:00 | HU1012 | ||||||||

Title: On the Lonely Runner Conjecture
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Abstract: The Lonely Runner Conjecture was first stated by J. M. Wills in 1967 and independently by T. W. Cusick in 1974. The conjecture is the very simple assertion that for any set $V$ of $k-1$ positive integers there exists some $\alpha \in \mathbb{R}$ such that for all $v \in V$, $\Vert \alpha v \Vert \geq \dfrac{1}{k}$, where $\Vert \cdot \Vert$ denotes the distance to the nearest integer. The conjecture has been proved for $k \le 7$, using a wide variety of methods. In this talk we will discuss some of the approaches taken to prove the above cases. The case $k=2$ is trivial; $k=3$ has been settled by Betke and Wills using Diophantine approximation methods, as well as by Cusick by solving a related "view-obstruction" problem. We explore a new constructive solution for this case. The proofs for $k=4,5,6$ follow a case by case approach and will not be discussed. We further investigate the most recent proof for $k=7$. Finally, we will show that the proposed bound is always tight for any $k$, and prove the conjecture when the set of integers $V$ forms some particular types of sequences. | |||||||||||||

[ arXiv] | 2011_08_16_Milcak_Notes | 2011_08_16 |

## August 09, **David**, 10:00 @HU1012

Chantal David http://www.mathstat.concordia.ca/faculty/cdavid/ (Concordia University) | GANITA Seminar
| Tuesday | August 09 | 10:00 | HU1012 | ||||||||

Title: Fluctuations in the number of points of curves over finite fields
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Abstract: We study in this talk the distribution of the number of points for two families of curves over a finite field with q elements: cyclic covers of $P^1$ and smooth plane curves. The Katz-Sarnak philosophy makes predictions about the statistics for such families in the large q limit when the genus is fixed. We are looking at the complementary statistics, when the genus varies, but the field of definition is fixed. In that case, one can obtain statistics for the distribution of the number of points by sieving the families of curves.This is joint work with A. Bucur, B. Feigon and M. Lalin. | |||||||||||||

[ arXiv] | 2011_08_09_David_Notes | 2011_08_09 |

## July 19, **Shakoori**, 10:00 @BA6180

Azar Shakoori http://azarshakoori.com/ (University of Cantabria) | GANITA Seminar
| Tuesday | July 19 | 10:00 | BA6180 | ||||||||

Title: Polynomial Algebra by Values and Its Applications
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Abstract: The polynomial root-finding problem has a long and rich history in computational math- ematics, mainly because of its frequent applications in engineering. It may be the case that the polynomial system as it appears naturally in the application formulation is not pre- sented in the familiar monomial basis. In this talk, I will present our root-finding algorithm, a hybrid of a resultant-type al- gorithm and eigenvalue techniques. Here, we use some new formulations for the so-called B ´ezout matrix and the companion matrix pencil associated with a pair of bivariate poly- nomial equations described by their values (i.e., in the Lagrange polynomial basis). The approach we take avoids conversion between bases. This is important because converting from the basis in which the polynomials are given to the monomial basis (or any other basis) is likely to degrade the conditioning of the problem. I will also describe some of the applications of our algorithm; one such application in Computer-Aided-Geometric-Design is to compute the topology of an algebraic curve whose parametric or implicit equations have rather high degrees, numerous terms and very large coefficients. Applying the currently available methods for computing the topology of the curve, with such polynomial equations, requires a big amount of memory space and a significant amount of computing time. This is one of the main reasons why, in this work, we assume that neither the implicit equation, nor the parametrization of the given curve are known. Instead, sufficient number of points on the curve and the corresponding values of the equations and derivatives are known, and the degree is also given or can be deduced. | |||||||||||||

[ arXiv] | 2011_07_19_Shakoori_Notes | 2011_07_19 |

## July 12, **Kahrobaei**, 10:00 @HU1012

Delaram Kahrobaei (City University of New York) | GANITA Seminar
| Tuesday | July 12 | 10:00 | HU1012 | ||||||||

Title: A decade of using non-commutative groups in cryptography
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Abstract:
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[ arXiv] | 2011_07_12_Kahrobaei_Notes | 2011_07_12 |

## July 05, **Guerreiro**, 10:00 @HU1012

Marinês Guerreiro [1] (Universidade Federal de Vicosa (Brasil)) | GANITA Seminar
| Tuesday | July 05 | 10:00 | HU1012 | ||||||||

Title: Computing weights in abelian group codes
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Abstract: In this talk we propose a technique to compute idempotents that generate minimal abelian codes in group algebras of abelian groups of exponent pq, for p and q distinct odd prime numbers over a field of characteristic two, under some restrictions. We also present a way to compute the weight of the minimal codes. The main results extend previous work done by C. Polcino Milies and R.A. Ferraz. This is a joint work with R.A. Ferraz, C. P. Milies and G. Chalom from IME-USP (São Paulo).
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[ arXiv] | 2011_07_05_Guerreiro_Notes | 2011_07_05 |

## June 23, **Anghel**, 11:00 @HU1012

Catalina Anghel [2] (University of Toronto) | GANITA Seminar
| Thursday | June 23 | 11:00 | HU1012 | ||||||||

Title: On the number of solutions of exponential congruences
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Abstract:
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[ arXiv] | 2011_06_23_Anghel_Notes | 2011_06_23 |

## June 23, **Migonney**, 10:00 @HU1012

Coline Migonney [3] (Ensicaen) | GANITA Seminar
| Thursday | June 23 | 10:00 | HU1012 | ||||||||

Title: Concepts for the development of a secure & privacy e-health platform
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Abstract:
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[ arXiv] | 2011_06_23_Migonney_Notes | 2011_06_23 |

## June 21, **Plateaux**, 13:00 @HU1012

Aude Plateaux [4] (Ensicaen) | GANITA Seminar
| Tuesday | June 21 | 13:00 | HU1012 | ||||||||

Title: Notion of privacy and security within the secure electronic transactions
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Abstract:
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[ arXiv] | 2011_06_21_Plateaux_Notes | 2011_06_21 |

## June 16, **Burko**, 11:00 @HU1012

Robby Burko [5] (University of Toronto) | GANITA Seminar
| Thursday | June 16 | 11:00 | HU1012 | ||||||||

Title: A Method for Computing Zeta Functions of Varieties over Finite Fields using Fibrations
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Abstract: In 2006 Lauder extended his p-adic deformation method to include fibrations of certain varieties. The idea is to treat an n-dimensional variety as a parametrized family of (n-1)-dimensional subvarieties, and compute the action of Frobenius on this family. This reduces complexity of the original deformation method, since it only deals with the (n-1) level cohomology, however there are some restrictions on the types of varieties which can be used in this method. These ideas were generalized in G.M. Walker's Thesis in 2009.
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[ arXiv] | 2011_06_16_Burko_Notes | 2011_06_16 |

## June 16, **Stein**, 10:00 @HU1012

Andreas Stein [6] (University of Oldenburg) | GANITA Seminar
| Thursday | June 16 | 10:00 | HU1012 | ||||||||

Title: Ellipic Curves Public-Key Cryptography - Applications and Attacks
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Abstract: In recent years, elliptic curves have become objects of intense investigation because of their significance to public-key cryptography. The major advantage of ECC is that the cryptographic security is in general believed to grow exponentially with the length of the input parameters. This implies short parameters, short digital signatures, and fast computations. We provide a survey of elliptic curves over finite fields and their interactions with algorithmic number theory. Our main focus will be the discussion of various interesting attacks to the so-called elliptic curve discrete logarithm problem (ECDLP) and their mathematical background as well as their important impact on public-key cryptography. For several attacks, results on algebraic curves, especially hyperelliptic curves, are needed.
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[ arXiv] | 2011_06_16_Stein_Notes | 2011_06_16 |

## June 09, **Mourtada**, 10:00 @HU1012

Mariam Mourtada [7] (University of Toronto) | GANITA Seminar
| Thursday | June 09 | 10:00 | HU1012 | ||||||||

Title: Average over fundamental discriminants of $L'/L(\sigma,\chi_D), 1/2 < \sigma < 1$.
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Abstract: This is a subsequent work following the average over fundamental discriminants of $L'/L(1,\chi_D)$. We will show under GRH that this average is a constant for all $\sigma$ . Then we will look at a specific set of $\sigma$ that has a positive measure, and show that the average is still a constant, unconditionally.
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[ arXiv] | 2011_06_09_Mourtada_Notes | 2011_06_09 |

## June 02, **George**, 13:00 @HU1012

William George [8] (University of Toronto) | GANITA Seminar
| Thursday | June 02 | 13:00 | HU1012 | ||||||||

Title: Some Consequences of Mazur's Conjecture
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Abstract: I will explore some of the ideas related to Mazur's Conjecture on the topological closure (in the real topology) of the rational points on varieties. Particularly, I will look at its implications for Diophantine decidability and how it affects the specialized ranks on elliptic surfaces.
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[ arXiv] | 2011_06_02_George_Notes | 2011_06_02 |

## May 26, **Mameri**, 13:00 @HU1012

Sabrina Mameri [9] (Ensicaen) | GANITA Seminar
| Thursday | May 26 | 13:00 | HU1012 | ||||||||

Title: Zero-knowledge identification protocols
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Abstract:
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[ arXiv] | 2011_05_26_Mameri_Notes | 2011_05_26 |

# 2011 Winter

## April 21, **Volkovs**, 13:00 @HU1012

Nikolajs Volkovs (PRATA) | GANITA Seminar
| Thursday | April 21 | 13:00 | HU1012 | ||||||||

Title: Digital Signature Scheme with a generator kept secret
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Abstract: We present the digital signature algorithm where a generator should not be used for verifying a signature, and therefore it can be kept secret. Furthermore, we show that the problem of extracting a private key from a public key is a new problem, different from the DLP.
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[ arXiv] | 2011_04_21_Volkovs_Notes | 2011_04_21 |

## April 14, **Zong**, 13:00 @HU1012

Ying Zong [10] (University of Toronto) | GANITA Seminar
| Thursday | April 14 | 13:00 | HU1012 | ||||||||

Title: Elliptic Cartan subalgebra
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Abstract: Given a semi-simple Lie algebra s and an irreducible representation E of s, over a field k, does s have a Cartan sub-algebra acting irreducibly on E? With Kumar Murty, we give a solution to this question when k is a p-adic field, and apply it to a problem of splitting abelian varieties.
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[ arXiv] | 2011_04_14_Zong_Notes | 2011_04_14 |

## March 31, **Chow**, 13:00 @HU1012

Aaron Chow [11] (University of Toronto) | GANITA Seminar
| Thursday | March 31 | 13:00 | HU1012 | ||||||||

Title: Side-Channel Attacks on SHA-3 Candidates (Part 2)
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Abstract: This talk is a follow up on a side-channel attack using CPA against HMACs based on the current SHA-3 candidates.
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[ arXiv] | 2011_03_31_Chow_Notes | 2011_03_31 |

## March 24, **Lim**, 13:00 @HU1012

Meng Fai Lim [12] (University of Toronto) | GANITA Seminar
| Thursday | March 24 | 13:00 | HU1012 | ||||||||

Title: The growth of the Tate-Shafarevich group in p-Hilbert class field towers
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Abstract: The existence of an infinite p-Hilbert class field tower was first established by Golod and Shaferevich in 1964. Stark then asked whether the p-class rank of the layers in an infinite p-class field tower tends to infinity. It turns out that this should be the case by a conjecture of Fontaine and Mazur. In a paper of Murty and Ouyang, they consider an analog of Stark's question and show that for a certain class of elliptic curves with complex multiplication, its p-rank of the Selmer group in the p-class field tower tends to infinity. In this talk, we will consider another analog of the above problems, namely, we will show that, under certain assumptions on p, the p-rank of the Tate-Shafarevich group of an abelian variety with complex multiplication in the infinite p-class field tower tends to infinity. This is a joint work with Prof. Kumar Murty.
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[ arXiv] | 2011_03_24_Lim_Notes | 2011_03_24 |

## March 17, **Hanson**, 13:00 @HU1012

Brandon Hanson [13] (University of Toronto) | GANITA Seminar
| Thursday | March 17 | 13:00 | HU1012 | ||||||||

Title: Probabilistic reasoning and the number of prime divisors
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Abstract: The function $\omega(n)$ which counts the number of distinct prime divisors has been attacked in the past with some nice probabilistic arguments. The theorem of Hardy-Ramanujan and Turan gives an analog of Law of Large Numbers, that $\omega(n)$ has normal order $\log\log n$. Even more interesting, the theorem of Erdos-Kac tells us that $\omega(n)$ is $\log\log n$ + a normal random variable. We'll discuss these results.
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[ arXiv] | 2011_03_17_Hanson_Notes | 2011_03_17 |

## March 10, **George**, 13:00 @HU1012

William George [14] (University of Toronto) | GANITA Seminar
| Thursday | March 10 | 13:00 | HU1012 | ||||||||

Title: The Bivariate Polynomial Injection Problem
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Abstract: One can find examples of polynomials in two variables with integer coefficients that yield an injection from $\mathbb{Z}^2$ to $\mathbb{Z}$; however, it is surprisingly difficult to determine whether there is such a polynomial injection from $\mathbb{Q}^2$ to $\mathbb{Q}$. Indeed, while it is widely believed that such a polynomial should exist, the best results towards this question reduce the problem to an analogue of the abc conjecture or, via a recent result of Bjorn Poonen, to the Bombieri-Lang conjecture. We will look at these results and talk about how this problem relates to issues of Diophantine definability.
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[ arXiv] | 2011_03_10_George_Notes | 2011_03_10 |

## March 03, **Burko**, 14:00 @HU1012

Robby Burko [15] (University of Toronto) | GANITA Seminar
| Thursday | March 03 | 14:00 | HU1012 | ||||||||

Title: On various Gauss-Manin connections
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Abstract:
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[ arXiv] | 2011_03_03_Burko_Notes | 2011_03_03 |

## March 03, **Eskandari**, 13:00 @HU1012

Payman Eskandari [16] (University of Toronto) | GANITA Seminar
| Thursday | March 03 | 13:00 | HU1012 | ||||||||

Title: Small gaps between primes: A result of Goldstein, Pintz and Yildirim
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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.
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[ arXiv] | 2011_03_03_Eskandari_Notes | 2011_03_03 |

## February 10, **Laptyeva**, 13:00 @HU1012

Nataliya Laptyeva [17] (University of Toronto) | GANITA Seminar
| Thursday | February 10 | 13:00 | HU1012 | ||||||||

Title: On the Brun-Titchmarch Theorem
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Abstract:
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[ arXiv] | 2011_02_10_Laptyeva_Notes | 2011_02_10 |

## February 03, **Mourtada**, 13:00 @HU1012

Mariam Mourtada [18] (University of Toronto) | GANITA Seminar
| Thursday | February 03 | 13:00 | HU1012 | ||||||||

Title: Nonvanishing of quadraric Dirichlet L-functions at $s=\frac{1}{2}$
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Abstract: Soundararajan proved that for at least 87.5% of odd-squarefree integers $d$, we have $L(\frac{1}{2},\chi_{8d}) \neq 0$ . This striking proportion of nonvanishing has been proved through the introduction of a "Mollifier". Historically, this was used by Selberg when he proved that a positive proportion of zeros of the Riemann Zeta function lie on the critical line. In my talk, I will state and prove some of the lemmas involved, in order to conclude the theorem. One thing to know, is that the central point is not special here, a similar result can be proved for any point $\sigma+it$ in the critical strip, and for fundamental discriminants in any arithmetic progression $a \pmod{b}$.
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[ arXiv] | 2011_02_03_Mourtada_Notes | 2011_02_03 |

## January 27, **Chow**, 13:00 @HU1012

Aaron Chow [19] (University of Toronto) | GANITA Seminar
| Thursday | January 27 | 13:00 | HU1012 | ||||||||

Title: Side-Channel Attacks on SHA-3 Candidates
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Abstract: A message authentication code (MAC) is used to authenticate a message with a secret key. It protects both a message's integrity and authenticity. A hash-based message authentication code (HMAC) is one based on a hash function. In this talk, a side-channel attack using CPA on HMACs based on the current SHA-3 candidates will be discussed. This has an impact especially on hardware-based implementations.
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[ arXiv] | 2011_01_27_Chow_Notes | 2011_01_27 |

## January 20, **Anghel**, 14:00 @HU1012

Catalina Anghel [20] (University of Toronto) | GANITA Seminar
| Thursday | January 20 | 14:00 | HU1012 | ||||||||

Title: Proof of one of Bourgain's theorems on exponential sums
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Abstract:
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[ arXiv] | 2011_01_20_Anghel_Notes | 2011_01_20 |

## January 20, **Usefi**, 13:00 @HU1012

Hamid Usefi [21] (University of Toronto) | GANITA Seminar
| Thursday | January 20 | 13:00 | HU1012 | ||||||||

Title: Algebraic Eraser and Non-Abelian Group-Based Cryptography
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Abstract:
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[ arXiv] | 2011_01_20_Usefi_Notes | 2011_01_20 |