Seminars and Colloquia by Series

Hopf Algebras and Cohomology of Lie Groups

Series
Geometry Topology Student Seminar
Time
Wednesday, April 1, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tao YuGeorgia Tech

In 1941, Hopf gave a proof of the fact that the rational cohomology of a compact connected Lie group is isomorphic to the cohomology of a product of odd dimensional spheres. The proof is natural in the sense that instead of using the classification of Lie groups, it utilizes the extra algebraic structures, now known as Hopf algebras. In this talk, we will discuss the algebraic background and the proof of the theorem.

Linear and rational factorization of tropical polynomials

Series
Algebra Seminar
Time
Monday, March 30, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Bo LinGeorgia Tech

Already for bivariate tropical polynomials, factorization is an NP-Complete problem.In this talk, we will introduce a rich class of tropical polynomials in n variables, which admit factorization and rational factorization into well-behaved factors. We present efficient algorithms of their factorizations with examples. Special families of these polynomials have appeared in economics,discrete convex analysis, and combinatorics. Our theorems rely on an intrinsic characterization of regular mixed subdivisions of integral polytopes, and lead to open problems of interest in discrete geometry.

The talk will be held online via Bluejeans. Use the following link to join the meeting.

Cancelled due to COVID-19: Counting extensions revisited

Series
Combinatorics Seminar
Time
Friday, March 27, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Lutz WarnkeGeorgia Tech

We consider rooted subgraphs in random graphs, i.e., extension counts such as (i) the number of triangles containing a given vertex or (ii) the number of paths of length three connecting two given vertices. 
In 1989, Joel Spencer gave sufficient conditions for the event that, with high probability, these extension counts are asymptotically equal for all choices of the root vertices.  
For the important strictly balanced case, Spencer also raised the fundamental question whether these conditions are necessary. 
We answer this question by a careful second moment argument, and discuss some intriguing problems that remain open. 

Post-grazing dynamics of a vibro-impacting energy generator--Postponed

Series
SIAM Student Seminar
Time
Friday, March 27, 2020 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Larissa SerdukovaGT Math

The motion of a forced vibro-impacting inclined energy harvester is investigated in parameter regimes with asymmetry in the number of impacts on the bottom and top of the device. This motion occurs beyond a grazing bifurcation, at which alternating top and bottom impacts are supplemented by a zero velocity impact with the bottom of the device. For periodic forcing, we obtain semi-analytical expressions for the asymmetric periodic motion with a ratio of 2:1 for the impacts on the device bottom and top, respectively. These expressions are derived via a set of nonlinear maps between different pairs of impacts, combined with impact conditions that provide jump dis continuities in the velocity. Bifurcation diagrams for the analytical solutions are complemented by a linear stability analysis around the 2:1 asymmetric periodic solutions, and are validated numerically. For smaller incline angles, a second grazing bifurcation is numerically detected, leading to a 3:1 asymmetry. For larger incline angles, period doubling bifurcations precede this bifurcation. The converted electrical energy per impact is reduced for the asymmetric motions, and therefore less desirable under this metric.

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