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Series: Combinatorics Seminar

Wednesday, December 5, 2018 - 14:00 ,
Location: Skiles 006 ,
Monica Flamann ,
Georgia Tech ,
Organizer: Sudipta Kolay

Series: Analysis Seminar

Series: Algebra Seminar

One variation of the Waring problem is to ask for the shortest non-trivial equations of the form f_1^d + ... + f_r^d = 0, under various conditions on r, d and where f_j is a binary form. In this talk I'll limit myself to quadratic forms, and show all solutions for r=4 and d=3,4,5. I'll also give tools for you to find such equations on your own. The talk will touch on topics from algebra, analysis, number theory, combinatorics and algebraic geometry and name-check such notables as Euler, Sylvester and Ramanujan, but be basically self-contained. To whet your appetite: (x^2 + xy - y^2)^3 + (x^2 - xy - y^2)^3 = 2x^6 - 2y^6.

Series: Geometry Topology Seminar

Monday, December 3, 2018 - 13:00 ,
Location: Skiles 006 ,
Oleg Lazarev ,
Columbia ,
Organizer: John Etnyre

Series: High Dimensional Seminar

Note the special time!

TBA

Series: Algebra Seminar

In this talk we will discuss an arithmetic analogue of the gonality of a nice curve over a number field: the smallest positive integer e such that the points of residue degree bounded by e are infinite. By work of Faltings, Harris--Silverman and Abramovich--Harris, it is understood when this invariant is 1, 2, or 3; by work of Debarre-Fahlaoui these criteria do not generalize. We will focus on scenarios under which we can guarantee that this invariant is actually equal to the gonality using the auxiliary geometry of a surface containing the curve. This is joint work with Geoffrey Smith.

Series: Stochastics Seminar

Heavy tailed distributions have been shown to be
consistent with data in a variety of systems with multiple time
scales. Recently, increasing attention has appeared in different
phenomena related to climate. For example, correlated additive and
multiplicative (CAM) Gaussian noise, with infinite variance or heavy
tails in certain parameter regimes, has received increased attention in
the context of atmosphere and ocean dynamics. We discuss how CAM noise
can appear generically in many reduced models. Then we show how reduced
models for systems driven by fast linear CAM noise processes can be
connected with the stochastic averaging for multiple scales systems
driven by alpha-stable processes. We identify the conditions under
which the approximation of a CAM noise process is valid in the averaged
system, and illustrate methods using effectively equivalent fast,
infinite-variance processes. These applications motivate new
stochastic averaging results for systems with fast processes driven by
heavy-tailed noise. We develop these results for the case of
alpha-stable noise, and discuss open problems for identifying
appropriate heavy tailed distributions for these multiple scale systems.
This is joint work with Prof. Adam Monahan (U Victoria) and Dr. Will
Thompson (UBC/NMi Metrology and Gaming).

Series: Other Talks

Oral Comprehensive Exam

The purpose of this work is approximation of generic Hamiltonian dynamical systems by those with a finite number of islands. In this work, we will consider a Lemon billiard as our Hamiltonian dynamical system apparently with an infinitely many islands. Then, we try to construct a Hamiltonian dynamical system by deforming the boundary of our lemon billiard to have a finite number of islands which are the same or sub-islands of our original system. Moreover, we want to show elsewhere in the phase space of the constructed billiard is a chaotic sea. In this way, we will have a dynamical system which preserves some properties of our lemon billiards while it has much simpler structure.