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Thursday, September 27, 2018 - 13:30 ,
Location: Skiles 006 ,
Stephen McKean ,
Georgia Tech ,
Organizer: Trevor Gunn

Bézout’s Theorem is the classical statement that generic curves of degree c and d intersect in cd points. However, this theorem requires that we work over an algebraically closed field. Using some tools from A^1-algebraic topology, we will give an arithmetic generalization of Bézout’s Theorem. We will also describe the geometric implications of this generalization over the reals.

Thursday, September 20, 2018 - 13:30 ,
Location: Skiles 006 ,
Trevor Gunn ,
Georgia Tech ,
Organizer: Trevor Gunn

We will give a brief introduction to matroids with a focus on representable matroids. We will also discuss the Plücker embedding of the Grassmannian.

Thursday, September 13, 2018 - 13:30 ,
Location: Skiles 006 ,
Trevor Gunn ,
Georgia Tech ,
Organizer: Trevor Gunn

Tropical geometry is a blend of algebraic geometry and polyhedral combinatorics that arises when one looks at algebraic varieties over a valued field. I will give a 50 minute introduction to the subject to highlight some of the key themes.

Thursday, September 6, 2018 - 13:30 ,
Location: Skiles 006 ,
Jaewoo Jung ,
Georgia Tech ,
Organizer: Trevor Gunn

One way to analyze a (finitely generated) module over a ring is to consider its minimal free resolution and look at its Betti table. The Betti table would be obtained by algebraic computations in general, but in case of the ideal (consists of relations) is generated by monomial quadratics, we can obtain Betti numbers (which are entries of the Betti table) by looking at the corresponding graphs and its associated simplicial complex. In this talk, we will introduce the Stanley-Reisner ideal which is the ideal generated by monomial quadratics and Hochster’s formula. Also, we will deal with some theorems and corollaries which are related to our topic.

Friday, April 20, 2018 - 10:00 ,
Location: Skiles 006 ,
Jose Acevedo ,
Georgia Tech ,
Organizer: Kisun Lee

In this talk we show how to obtain some (sometimes sharp) inequalities between subgraph densities which are valid asymptotically on any sequence of finite simple graphs with an increasing number of vertices. In order to do this we codify a simple graph with its edge monomial and establish a nice graphical notation that will allow us to play around with these densities.

Friday, April 13, 2018 - 10:00 ,
Location: Skiles 006 ,
Tim Duff ,
Georgia Tech ,
Organizer: Kisun Lee

The fundamental data structures for numerical methods in algebraic geometry are called "witness sets." The term "trace test" refers to certain numerical methods which verify the completeness of such witness
sets. It is natural to ask questions about the complexity of such a test and in what sense its output may be regarded as "proof." I will give a basic exposition of the trace test(s) with a view towards these questions

Friday, April 6, 2018 - 10:00 ,
Location: Skiles 006 ,
Jaewoo Jung ,
Georgia Tech ,
jaewoojung@gatech.edu ,
Organizer: Kisun Lee

H. Dao, C. Huneke, and J. Schweig provided a bound of the regularity of edge-ideals in their paper “Bounds on the regularity and projective dimension of ideals associated to graphs”. In this talk, we introduced their result briefly and talk about a bound of the regularity of Stanley-Reisner ideals using similar approach.

Friday, March 30, 2018 - 10:00 ,
Location: Skiles 006 ,
Jaewoo Jung ,
Georgia Tech ,
Organizer: Kisun Lee

One way to analyze a module is to consider its minimal free resolution and take a look its Betti numbers. In general, computing minimal free resolution is not so easy, but in case of some certain modules, computing the Betti numbers become relatively easy by using a Hochster's formula (with the associated simplicial complex. Besides, Mumford introduced Castelnuovo-Mumford regularity. The regularity controls when the Hilbert function of the variety becomes a polynomial. (In other words, the regularity represents how much the module is irregular). We can define the regularity in terms of Betti numbers and we may see some properties for some certain ideals using its associated simplicial complex and homology. In this talk, I will review the Stanley-Reisner ideals, the (graded) betti-numbers, and Hochster's formula. Also, I am going to introduce the Castelnuovo-Mumford regularity in terms of Betti numbers and then talk about a useful technics to analyze the Betti-table (using the Hochster's formula and Mayer-Vietories sequence).

Friday, March 16, 2018 - 10:00 ,
Location: Skiles 006 ,
Kisun Lee ,
Georgia Tech ,
klee669@gatech.edu ,
Organizer: Kisun Lee

Expanding the topic we discussed on last week, we consider the way to certify roots for system of equations with D-finite functions. In order to do this, we will first introduce the notion of D-finite functions, and observe the property of them. We also suggest two different ways to certify this, that is, alpha-theory and the Krawczyk method. We use the concept of majorant series for D-finite functions to apply above two methods for certification. After considering concepts about alpha-theory and the Krawczyk method, we finish the talk with suggesting some open problems about these.

Friday, March 9, 2018 - 10:00 ,
Location: Skiles 006 ,
Kisun Lee ,
Georgia Tech ,
klee669@gatech.edu ,
Organizer: Kisun Lee

This is an intoductory talk for the currently using methods for certifying roots for system of equations. First we discuss about alpha-theory which was constructed by Smale and Shub, and explain how this theory could be modified in order to apply in actual problems. In this step, we point out that alpha theory is still restricted only into polynomial systems and polynomial-exponential systems. After that as a remedy for this problem, we will introduce an interval arithmetic, and the Krawczyk method. We will end the talk with a discussion about how these current methods could be used in more general setting.