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Friday, February 23, 2018 - 15:00 ,
Location: Skiles 271 ,
Jiaqi Yang ,
GT Math ,
Organizer: Jiaqi Yang

We will present a rigorous proof of non-existence of homotopically non-trivial invariant circles for standard map:x_{n+1}=x_n+y_{n+1}; y_{n+1}=y_n+\frac{k}{2\pi}\sin(2\pi x_n).This a work by J. Mather in 1984.

Friday, February 23, 2018 - 13:55 ,
Location: Skiles 269 ,
Prof. Justin Kakeu ,
Morehouse University ,
Organizer: Sung Ha Kang

We use a stochastic dynamic programming approach to address the following question: Can a homogenous resource extraction model (one without extraction costs, without new discoveries, and without technical progress) generate non-increasing resource prices? The traditional answer to that question contends that prices should exhibit an increasing trend as the exhaustible resource is being depleted over time (The Hotelling rule). In contrast, we will show that injecting concerns for temporal resolution of uncertainty in a resource extraction problem can generate a non-increasing trend in the resource price. Indeed, the expected rate of change of the price can become negative if the premium for temporal resolution of uncertainty is negative and outweighs both the positive discount rate and the short-run risk premium. Numerical examples are provided for illustration.

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.

Series: School of Mathematics Colloquium

A distinct covering system of congruences is a finite collection of arithmetic progressions $$a_i \bmod m_i, \qquad 1 < m_1 < m_2 < \cdots < m_k.$$Erdős asked whether the least modulus of a distinct covering system of congruences can be arbitrarily large. I will discuss my proof that minimum modulus is at most $10^{16}$, and recent joint work with Pace Nielsen, in which it is proven that every distinct covering system of congruences has a modulus divisible by either 2 or 3.

Wednesday, February 21, 2018 - 14:00 ,
Location: Skiles 006 ,
Kevin Kodrek ,
GaTech ,
Organizer: Anubhav Mukherjee

There are a number of ways to define the braid group. The traditional definition involves equivalence classes of braids, but it can also be defined in terms of mapping class groups, in terms of configuration spaces, or purely algebraically with an explicit presentation. My goal is to give an informal overview of this group and some of its subgroups, comparing and contrasting the various incarnations along the way.

Series: Analysis Seminar

I will speak how to ``dualize'' certain martingale estimates related to the dyadic square function to obtain estimates on the Hamming and vice versa. As an application of this duality approach, I will illustrate how to dualize an estimate of Davis to improve a result of Naor--Schechtman on the real line. If time allows we will consider one more example where an improvement of Beckner's estimate will be given.

Series: PDE Seminar

We prove the existence and the linear stability of Cantor families of small amplitude time quasi-periodic standing water wave solutions - i.e. periodic and even in the space variable x - of a bi-dimensional ocean with finite depth under the action of pure gravity. Such a result holds for all the values of the depth parameter in a Borel set of asymptotically full measure. This is a small divisor problem. The main difficulties are the quasi-linear nature of the gravity water waves equations and the fact that the linear frequencies grow just in a sublinear way at infinity. To overcome these problems, we first reduce the linearized operators obtained at each approximate quasi-periodic solution along the Nash-Moser iteration to constant coefficients up to smoothing operators, using pseudo-differential changes of variables that are quasi-periodic in time. Then we apply a KAM reducibility scheme that requires very weak Melnikov non-resonance conditions (which lose derivatives both in time and space), which we are able to verify for most values of the depth parameter using degenerate KAM theory arguments. This is a joint work with P. Baldi, M. Berti and R. Montalto.

Series: Geometry Topology Seminar

Now that the geometrization conjecture has been proven, and the virtual Haken conjecture has been proven, what is left in
3-manifold topology? One remaining topic is the computational complexity of geometric topology problems. How difficult is it to
distinguish the unknot? Or 3-manifolds from each other? The right approach to these questions is not just to consider quantitative
complexity, i.e., how much work they take for a computer; but also qualitative complexity, whether there are efficient algorithms with
one or another kind of help. I will discuss various results on this theme, such as that knottedness and unknottedness are both in NP; and
I will discuss high-dimensional questions for context.