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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.

Friday, March 2, 2018 - 10:00 ,
Location: Skiles 254 ,
Marcel Celaya ,
Georgia Tech ,
mcelaya@gatech.edu ,
Organizer: Kisun Lee

In this talk we will discuss the paper of Adiprasito, Huh, and Katz titled "Hodge Theory for Combinatorial Geometries," which establishes the log-concavity of the characteristic polynomial of a matroid.

Friday, February 23, 2018 - 10:00 ,
Location: Skiles 006 ,
Tim Duff ,
Georgia Tech ,
tduff3@gatech.edu ,
Organizer: Kisun Lee

TBA

Friday, February 23, 2018 - 10:00 ,
Location: Skiles 006 ,
Tim Duff ,
Georgia Tech ,
tduff3@gatech.edu ,
Organizer: Kisun Lee

Polyhedral homotopy methods solve a sparse, square polynomial system by deforming it into a collection of square "binomial start systems." Computing a complete set of start systems is generally a difficult combinatorial problem, despite the successes of several software packages. On the other hand, computing a single start system is a special case of the matroid intersection problem, which may be solved by a simple combinatorial algorithm. I will give an introduction to polyhedral homotopy and the matroid intersection algorithm, with a view towards possible heuristics that may be useful for polynomial system solving in practice.

Friday, February 16, 2018 - 10:10 ,
Location: Skiles 006 ,
Libby Taylor ,
Georgia Tech ,
libbyrtaylor@gmail.com ,
Organizer: Kisun Lee

Algebraic geometry has a plethora of cohomology theories, including the derived functor, de Rham, Cech, Galois, and étale cohomologies. We will give a brief overview of some of these theories and explain how they are unified by the theory of motives. A motive is constructed to be a “universal object” through which all cohomology theories factor. We will motivate the theory using the more familiar examples of Jacobians of curves and Eilenberg-Maclane spaces, and describe how motives generalize these constructions to give categories which encode all the cohomology of various algebro-geometric objects. The emphasis of this talk will be on the motivation and intuition behind these objects, rather than on formal constructions.

Friday, February 9, 2018 - 10:10 ,
Location: Skiles 254 ,
Marc Härkönen ,
Georgia Tech ,
harkonen@gatech.edu ,
Organizer: Kisun Lee

As a continuation to last week's talk, we introduce the ring D of differential operators with complex coefficients, or the Weyl algebra. As we saw last week, the theory of the ring R, the ring of differential operators with rational function coefficients, is in many ways almost the same as the regular polynomial ring. The ring D however will look slightly different as its structure is much finer. We will look at filtrations, graded rings and Gröbner bases induced by weight vectors. Finally we will present an overview on the integration algorithm of holonomic D-modules and mention some applications.

Friday, February 2, 2018 - 10:10 ,
Location: Skiles 254 ,
Marc Härkönen ,
Georgia Tech ,
harkonen@gatech.edu ,
Organizer: Kisun Lee

Differential operator rings can be described as polynomial rings over differential operators. We will study two of them: first the relatively simple ring of differential operators R with rational function coefficients, and then the more complicated ring D with polynomial coefficients, or the Weyl algebra. It turns out that these rings are non-commutative because of the way differential operators act on smooth functions. Despite this, with a bit of work we can show properties similar to the regular polynomial rings, such as division, the existence of Gröbner bases, and Macaulay's theorem. As an example application, we will describe the holonomic gradient descent algorithm, and show how it can be used to efficiently solve computationally heavy problems in statistics.

Friday, January 26, 2018 - 10:00 ,
Location: Skiles 254 ,
Trevor Gunn ,
Georgia Tech ,
tgunn@gatech.edu ,
Organizer: Kisun Lee

We will first give a quick introduction to automatic sequences. We will then outine an algebro-geometric proof of Christol's theorem discovered by David Speyer. Christol's theorem states that a formal power series f(t) over GF(p) is algebraic over GF(p)(t) if and only if there is some finite state automaton such that the n-th coefficent of f(t) is obtained by feeding in the base-p representation of n into the automaton. Time permitting, we will explain how to use the Riemann-Roch theorem to obtain bounds on the number of states in the automaton in terms of the degree, height and genus of f(t).

Friday, November 17, 2017 - 10:00 ,
Location: Skiles 114 ,
Timothy Duff ,
GA Tech ,
Organizer: Timothy Duff

Motivated by the general problem of polynomial system solving, we state and sketch a proof Kushnirenko's theorem. This is the simplest in a series of results which relate the number of solutions of a "generic" square polynomial system to an invariant of some associated convex bodies. For systems with certain structure (here, sparse coefficients), these refinements may provide less pessimistic estimates than the exponential bounds given by Bezout's theorem.

Friday, November 3, 2017 - 10:00 ,
Location: Skiles 114 ,
Jaewoo Jung ,
GA Tech ,
Organizer: Timothy Duff

We continue our discussion of free resolutions and Stanley-Reisner ideals. We introduce Hochster's formula and state results on the behavior of Betti tables under clique-sums.