# Graduate Colloquium Spring 2019

McHenry Library Room 4130

Refreshments served at 3:30 in room 4161

For further information, please contact Graduate Student John McHugh or call 831-459-2969

**Thursday April 11, 2019**

**Christy Hightower, Science & Engineering Distinguished Librarian, UCSC**

** Scholarly Communication Trends, Tools and Tips: Five Things Every Mathematics Graduate Student Should Know**Scholarly communication is the system through which the written or oral results of research and other scholarly writings are created, evaluated for quality, distributed, located by other researchers, obtained in full text (or other media) for reading or consumption, and preserved for future use. The system includes both formal means of communication, such as publication in peer-reviewed books or journals, and informal channels, such as the communication or discussion of scholarship through email and social media. Libraries and librarians support researchers throughout the entire cycle of scholarly communication.

As you can imagine, the topic of scholarly communication is a large one! This talk will focus on five trends, tools or tips that every math graduate student should be aware of. Topics covered will include some tips for locating and obtaining books and articles, the use of citation management software for organizing your notes about the research articles, books, blogs, and websites that you are reading and downloading, as well as copyright basics for authors.

**Thursday April 18, 2019**

No Seminar

**Thursday April 25, 2019**

**Alejandro Bravo Doddoli, UCSC****From Euler-Lagrange Equation to Hamel Equation**

Euler-Lagrange equations are very well know for being a powerful way to attack variational problems, and specially, in the context of mechanics, to give a covariant definition of Newton equations.

Given a manifold $M$ with local coordinates $(q_1. \dots, q_n)$, there is a natural way to give coordinates to the tangent bundle as follow, $(q_1, \dots, q_n, v_1, \dots, v_n)$ and a vector $v \in TM$ at a point $(q_1, \dots, q_n,)$ is given by $ v_1 \frac{\partial }{\partial q_1} + \dots + v_n \frac{\partial }{\partial q_n}$. The Euler-Lagrange approach is taking these coordinates as a local coordinates. However, the set $\{ \frac{partial }{ \partial q_i} \}$ is not the unique base for the tangent space of $M$. we can consider a set of $n$ lineal independent vectors field $\{ X_i \}$ and write the vector $v$ in this base as $v = \sum_{i=1}^n u_i X_i$, where $u_i$ are called quasi-velocities, this approach is more general since the vector fields may not commute, hence they cannot be generated by a local coordinates. Hamel equations are Euler Lagrange equations written in the local coordinates $(q_1, \dots, q_n, u_1, \dots, u_n)$. This new approch can be very helpful for systems with symmetries.We will review some examples such as the rigid body, pendulum and chaplying slight.

**Thursday May 2, 2019**

**Alejandro Bravo Doddoli, UCSC****Introduction to Geometry of Goursat Distribution and Bifurcation Theory**

A Goursat flag is a chain $D_s \subset D_{s-1} \subset \dots \subset D_{1} \subset D_0 = TM$. of subbundles of the tagent bundle $TM$ such that $conrank D_i = i$ and $D_{i-1}$ kis generated by the vector fields in $D_i$ and their Lie brackets. Engel, Goursat, and Cartan studied these flags and establish a normal form them, valid at generic point $M$. After that Kumpera, ruiz and Mormil discovered a Goursat flags with singularities, and the number of these grows exponentially with the conrank $s$. Finally Montgomery classify all the singularities.

The distribution associated with problem of $n$-trailers, is a example of the Goursat's flag with all the singularities. Then we can fiend a critical vector fields, which flow is a periodic orbit that foliated the singularity. Since my master degree I wanted to find consequences in the dynamics, crated by the singularity. I will try to explain how much I have found about the persistent of those periodic orbits.

**Thursday May 9, 2019**

No Seminar

**Thursday May 16, 2019**

**TBA**

**Thursday May 23, 2019**

**TBA**

**Thursday May 30, 2019**

**TBA**

**Thursday June 6, 2019 **

No Seminar due to special Math Tea honoring the 2019 Graduating Mathematics MA and PhD students! Please join us in celebrating: McHenry 3rd Floor Breezeway, 3:00pm - 4:30pm