Algebra and Number Theory
See here for our Algebra and Number Theory Seminar Schedule.
|C.F. Gauß||E. Noether||D. Hilbert||F.J. MacWilliams|
|James Beidleman||Group Theory|
|Alberto Corso||Commutative Algebra|
|Paul Eakin||Commutative Algebra|
|Edgar Enochs||Homological Algebra|
|Heide Gluesing-Luerssen||Coding Theory, Systems and Control Theory|
|Kenneth Kubota||Number Theory|
|David Leep||Quadratic Forms|
|Uwe Nagel||Commutative Algebra and Algebraic Geometry|
|Avinash Sathaye||Algebraic Geometry|
|Current Doctoral Students|
|Nicholas Armenoff (Nagel)|
|Sema Gunturkun (Nagel)|
|Ray Kremer (Enochs)|
|Theodoros Kyriopoulos (Enochs)|
|Casey Monday (Enochs)|
|Furuzan Ozbek (Enochs)|
|Bill Robinson (Nagel)|
|Stephen Sturgeon (Nagel)|
|Carolyn Troha (Gluesing-Luerssen)|
|Emeriti and Former Members|
|Donald Coleman (Emeritus)||Group Rings|
|Cornelia Yuen(PostDoc)||Commutative Algebra and Algebraic Geometry|
|Bonnie Smith (PostDoc)||Commutative Algebra|
In Fall 2012 the seminar takes place Wednesdays, 2:00 - 3:00 pm, in POT 745. Coordinator: Uwe Nagel
Graduate Courses in Algebra
- MA 561 - Modern Algebra I - A. Corso
- MA 565 - Linear Algebra - D. Leep
- Ma 765 - Topics in Coding Theory - H. Gluesing-Luerssen
- MA 661 - Modern Algebra II - H. Gluesing-Luerssen
- MA 764 - Selected Topics in Algebra - E. Enochs
- Ma 765 - Schubert Varieties - U. Nagel
- MA 561 - Modern Algebra I - H. Gluesing-Luerssen
- MA 565 - Linear Algebra - A. Corso
- Ma 765 - Topics in Number Theory - D. Leep
- MA 661 - Modern Algebra II - E. Enochs
- MA 764 - Introduction to Quadratic Number Fields - D. Leep
- Ma 765 - Cohen-Macaulay Rings - A. Corso
- MA 561 - Modern Algebra I - E. Enochs
- MA 565 - Linear Algebra - H. Gluesing-Luerssen
- Ma 667 - Theory of Groups - J. Beidleman
- Ma 764 - Algebraic Geometry - U. Nagel
- MA 661 - Modern Algebra II - U. Nagel
- Ma 764 - Homological Algebra - E. Enochs
- MA 765 - Introduction to Coding Theory - H. Gluesing-Luerssen
- MA 561 - Modern Algebra I - U. Nagel
- MA 565 - Linear Algebra - H. Gluesing-Luerssen
- Ma 764 - Selected Topics in Algebra: Commutative Algebra - A. Corso
Current and Former Events
- PASI: Commutative Algebra and its interactions to Geometry, Olinda (Brazil), August 3-14, 2009
- 2nd Bluegrass Algebra Conference, March 7-8, 2009. Organized by Alberto Corso and Uwe Nagel.
- Math Club talk given by David Cox: Tangents to four unit spheres, March 5, 2009.
- Three Challenges of Claude Shannon, Eighth Annual Hayden-Howard Lecture presented by Joachim Rosenthal, University of Zürich (Switzerland) , April 2, 2008. This talk is suitable for advanced undergraduates!
Abstract: In 1948/1949 Claude Shannon wrote two papers [Sha48,Sha49] which became the foundation of modern information theory. The papers showed that information can be compressed up to the `entropy', that data can be transmitted error free at a rate below the capacity and that there exist provable secure cryptographic systems. These were all fundamental theoretical result. The challenge remained to build practical systems which came close to the theoretical optimal systems predicted by Shannon.
In this overview talk we will explain how the first two challenges concerning coding theory have resulted in practical solutions which are very close to optimal. Then we explain why the gap between the practical implementation of cryptographic protocols with the theoretical result of Shannon is largest.
- UIC-Purdue Workshop, December 2-3, 2006.
- The History of Imaginary Numbers presented by Robin Hartshorne, University of California (Berkeley). April 6, 2006. This talk is suitable for all undergraduates!
- Midwest Algebra, Geometry and their Interactions Conference (MAGIC 05), October 7-11, 2005.
- Lipman-Fest, May 17-21, 2004.
- Bluegrass Algebra Conference and Hayden-Howard Lectureship, April 11-13, 2003.
Colloquia in Algebra and Geometry
- Duality Aspects in Coding Theory, Heide Gluesing-Luerssen, University of Kentucky , March 3, 2011.
- The canonical model of a singular curve, Steven L. Kleiman, Massachusetts Institute of Technology , March 6, 2009.
- The History of Imaginary Numbers, Robin Hartshorne, University of California (Berkeley), April 6, 2006.
- Algebraic Description and Effective Computation of Certain Structures in Algebraic Geometry, Aron Simis, Universidade Federal de Pernambuco (Brazil), April 4, 2006.
- On the Core of Ideals, Claudia Polini, University of Notre Dame, April 3, 2006.
- h-vectors of Gorenstein Polytopes, Tim Römer, University of Osnabrück (Germany), March 9, 2006.
- Design and Analysis of Convolutional Codes, Heide Gluesing-Luerssen, University of Groningen (The Netherlands), January 31, 2006.
- How to detect finiteness of Gorenstein homological dimension, Lars Winther Christensen, University of Nebraska, November 10, 2005.
- Rationality of the Zeta function of a finite graph, Hyman Bass, University of Michigan, September 23, 2005.
- Some expected properties of algebras, Uwe Nagel, University of Kentucky, November 14, 2002.
- Some things Ramanujan may have had up his sleeve, George Andrews, Penn State, March 4, 2002.
- Aspects of Liaison Theory, Uwe Nagel, University of Paderborn (Germany), February 15, 2002.
- Intersection multiplicities, Anurag Singh, University of Utah, February 4, 2002.
- Gorenstein Artin algebras, Hema Srinivasan, University of Missouri, November 20, 2001.
- Simultaneous resolutions, Dale Cutkosky, University of Missouri, November 19, 2001.
- Zero cycles, Euler class and existence of unimodular elements, Shrikant M. Bhatwadekar, Tata Institute of Fundamental Research, November 1, 2001.
Sample Qualifying Coursework for Doctoral Students
- MA 565 - Linear Algebra
Vector spaces: Basic definitions, dimension, matrices and linear transformations.
- MA 561 - Modern Algebra I
Groups: Basic definitions, isomorphism theorems, permutation groups, structure of finitely generated abelian groups, groups acting on sets, the Sylow theorems, solvable groups.
Rings: Basic definitions, ideals, prime and maximal ideals, quotient rings, Euclidean rings, PID's and UFD's, field of fractions, polynomial rings, irreducibility criteria.
- MA 661 - Modern Algebra II
Fields: Algebraic extensions, splitting fields, separable extensions, finite fields.
Galois Theory: Fundamental Theorem of Galois Theory, Galois group of polynomials, solvability of polynomial equations, symmetric polynomials.
1) Abstract Algebra (3rd edition), by D. Dummit and R. Foote
Preliminaries; Ch. 1; Ch. 2; Ch. 3; Ch. 4; Ch. 6 (sect. 1); Ch. 7; Ch. 8; Ch. 9; Ch. 13; Ch. 14
2) Algebra, by T. Hungerford
Ch. 1 (sec. 2-6); Ch. 2 (sec. 1, 2, 4-8); Ch. 3; Ch. 4 (sec. 1, 2, 6); Ch. 5 (sec. 1-6, 9); Ch. 8 (sec. 1-3)
3) Algebra (2nd edition), by S. Lang
Ch. 1 (sec. 1-6, 10); Ch. 2; Ch. 3 (sec. 1,2, 5); Ch. 5; Ch. 6 (sec. 1-5); Ch. 7; Ch. 8 (sec. 1-3, 7); Ch. 15 (sec. 2)
Here are some old prelim exams:
Maintained by Heide Gluesing-Luerssen.
Last update: April 19, 2012