Thursdays 3:30-4:30 – Room M-102
Hui Xue (firstname.lastname@example.org)
[toggle_item title=”Jim Coykendall – Clemson University ” active=”false”]
Some generalizations of integrality[divider]
The notion of integrality is central in commutative algebra (and one could also say that it is the heartbeat of algebraic number theory). We say that an element is integral over the domain R if it is a root of a monic polynomial with coefficients in R. Integral extensions are of much important theoretical and practical importance in commutative algebra. An integral extension of a domain R is intimately related to R, and domains that are integrally closed are surprisingly well behaved. In this talk, we will begin in the realm of things integral and generalize to a number of newer notions that generalize this concept. There will be a number of (hopefully) illuminating examples given along the way to help build intuition.
[toggle_item title=”Sean Sather-Wagstaff – Clemson University ” active=”false”]
A gentle introduction to Gorenstein rings[divider]Two outside seminar speakers this semester will be speaking on Gorenstein rings. The purpose of this talk is to give motivation for, definition of, and examples of these objects.[/toggle_item]
[toggle_item title=”Adela Vraciu – University of South Carolina ” active=”false”]
Constructions of totally reflexive modules[divider]Let R be a commutative local Noetherian ring. Totally reflexive modules are a generalization of free modules, with nice homological properties.
It is known that if R is not Gorenstein, then either every totally reflexive module is free, or else there are infinitely many isomorphism classes of indecomposable totally reflexive modules. For a given ring R, it is not known in general how to decide which of the above holds, and, in the latter case, how to construct the infinitely many modules that are known to exist.
We describe concrete constructions of totally reflexive modules for some special classes of rings.
This is joint work with Cameron Atkins.
[toggle_item title=”Jintai Ding – University of Cincinnati ” active=”false”]
The LWE-based key exchange[divider]
Public key cryptosystems (PKC) are critical part of the foundation of modern communication systems, in particular, Internet. However Shor’s algorithm shows that the existing PKC like Diffie-Hellmann key exchange, RSA and ECC can be broken by a quantum computer. To prepare for the coming age of quantum computing, we need to build new public key cryptosystems that could resist quantum computer attacks. In this lecture, we present a practical and provably secure ( authenticated) key exchange protocol based on the learning with errors problems, which is conceptually simple and has strong provable security properties. This new constructions was established in 2011-2012. These protocols are indeed practical. We will explain that all the existing LWE based key exchanges are variants of this fundamental design.
In addition, we will explain how to use the signal function invented for KE for authentication schemes.
[toggle_item title=”Joseph Gunthier – City University of New York” active=”false”]Slicing the stars[divider]
How many algebraic numbers are there? Easy: infinitely many. But if we bound their height and degree, we get an interesting counting problem. One can parametrize algebraic numbers by their minimal polynomials, and then this becomes a question of counting lattice points in certain “star bodies.” We’ll explain how to asymptotically count algebraic numbers, integers, units and more by carefully counting lattice points in slices of these star bodies. Furthermore, by proving explicit error terms, we can talk meaningfully about “random” algebraic numbers. This is joint work with Robert Grizzard (University of Wisconsin-Madison).
[toggle_item title=”Brittney Falahola – University of Nebraska-Lincoln ” active=”true”]
Characterizing Gorenstein Rings of Prime Characteristic[divider]
Using the Frobenius ring map and various tools one can develop from it, I will give several characterizations of Gorenstein rings of prime characteristic p>0. This is joint work with Tom Marley.