Working Seminar on Arithmetic Geometry
This seminar is mostly focused in topics of arithmetic geometry. More specifically on elliptic, hyperelliptic, and superelliptic curves, Jacobians, torsion points, Selmer, Shafarevich-Tate groups, splitting of Jacobians, descent, and related topics.
Occasionally we have invited speakers on group theory, algebraic geometry, number theory, computational algebra, etc.
If you are a student in mathematics or computer science, a colleague from the surrounding universities and want to attend, please feel free to contact us.
Speaker: Hung Ngoc Nguyen
University of Akron
Title: Character degrees of finite groups
Date: Nov. 18, 2014
Abstract: A representation of degree $n$ of a group $G$ is a way to represent elements in $G$ by $n\times n$ invertible matrices in such a way that the rule of group operation corresponds to matrix multiplication. The character afforded by a representation is a function on the group which associates to each group element the trace of the corresponding matrix and therefore it carries the essential information about the representation in a more condensed form. The degree of a character is exactly the degree of the representation affording it. There is no doubt that degree is the most important piece of information in a character and, therefore, character degrees are key tools to study the structure of finite groups.
In character theory of finite groups, a natural and important question is: what can the character degrees of a finite group say about the structure of the group? or in other words, how much information regarding the structure of a group can be determined from its character degrees? We will present some recent results in this line of research. In particular, we will discuss the connection between character degrees and several important characteristics of finite groups such as (quasi)simplicity, solvability, or nilpotency.
Speaker: Esengul Salturk,
The University of Scranton, PA
Title: Coding Theory and The Counting Problem
Day: October 14, 2014
Room: SEB 364
Abstract One of the most important fields in the application of algebra is the theory of codes which deals with the reliable communication of information from one point to another. The theory of error correcting codes has been studied since 1948, beginning with the seminal work of Shannon and Hamming. While in early stages codes were defined over finite fields, in 1994, there was a significant change. After the paper of Hammons and his collaborates, a complete study of codes over finite rings began. Within the study of codes, determining the number of codes with given parameters has been one of the most important problem of combinatorial coding theory. Since 1948, number of sub-codes of a linear code has been considered. This problem was completely solved for codes over finite fields by the well known Gaussian coefficients. Recently, the counting problem over finite chain rings and finite principal ideal rings has been studied. Additionally the number of additive Z2Z4 codes has also been determined. This talk will describe the history and solution of the counting problem in the study of error-correcting codes.
Speaker: T. Shaska
Title: Reduction theory for binary forms
Abstract: We will go over some of the reduction techniques for binary forms and how they can be used to determine equations for superelliptic curves with minimal height. Papers of J. Cremona and M. Stoll and M. Stoll will be discussed in detail.
1) M. Stoll, J. Cremona, On the reduction theory of binary forms
2) M. Stoll, Reduction theory of point clusters in projective space, arXiv:0909.2808 [math.NT]
3) M. Barghava, A. Yang, On the number of integral binary n-ic forms having bounded Julia invariant, arXiv:1312.7339 [math.NT]
Title: Hedge fund crowds and mispricing
Speaker: Blerina Reca, University of Toledo
Abstract: Recent models and the popular press suggest that hedge funds follow similar strategies resulting in crowded equity positions that destabilize markets. Inconsistent with this assertion, we find that hedge fund equity portfolios are remarkably independent. Moreover, when hedge funds do buy and sell the same stocks, their demand shocks drive prices toward, rather than away from, fundamental values. Even in periods of extreme market stress, we find no evidence that hedge funds exert negative externalities on security prices due to their crowded trades. Our results have important implications for the ongoing debate regarding hedge fund regulation.
Speaker: J. Gutierrez, University of Cantabria, Spain
Prof. Gutierrez of University of Cantabria (Spain) will be visiting our group during the month of July and will hold two lectures on Lattice reduction techniques and their applications.
Day: July 17 and July 22, 2014
Room: SEB 364
Abstract: Historically, lattices were investigated since the late 18th century by mathematicians such as Lagrange and Gauss. In the 19th century, important results due to Minkowski motivated the use of lattice theory in the theory and geometry of numbers. More recently, lattices have become a topic of active research in mathematics and computer science. They are used as an algorithmic tool to solve a wide variety of problems; they have many applications in cryptography and cryptanalysis; and they have some unique properties from a computational complexity point of view. These
are the topics that we will see in this talk.
In the first part we give some basic background on lattices including the lattice basis technique with emphasis to LLL reduction and the corresponding algorithm. Then we will present applications to:
- Recovering zeros of polynomials over finite fields
- Noisy polynomial interpolation
- Predicting the linear congruential generator on elliptic curves
- Computing the linear complexity of sequences.
Title: Moduli height of curves
Speaker: Lubjana Beshaj
Abstract: We define the moduli height of curves and compare that to the Mahler measure, l_2, and l_inf of the defining polynomial. Some results will be proven for curves of given height defined over the ring of integers of a Dedekind domain.
Day: May 6, 2014
Room: SEB 364
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