# Graduate Colloquium Winter 2016

McHenry Building - Room 4130

Refreshments served at 3:30 in room 4161

For further information, please contact Gabriel Martins or call 831-459-2969

**Wednesday January 20, 2016**

**Wednesday January 20, 2016**

*P-adic L-functions and Eisenstein Series*

#### Joseph Ferrara, University of California, Santa Cruz

**
** L-functions encode important arithmetic information about finite field extensions of the rational numbers. In this talk, I will define and discuss some properties of L-functions associated to abelian extensions of number fields in order to motivate p-adic L-functions. P-adic L-functions are continuous functions of a p-adic variable, which interpolate special values of classical L-functions. The main part of this talk will be used to explain the role of Eisenstein series in the definition of certain p-adic L-functions.

**Wednesday January 27, 2016**

*Hanging out in Neighborhoods of Infinity*

#### Connor Jackman, University of California, Santa Cruz

**
** In this talk I will describe a result in Newton's gravitational 3-body problem. The result is that upon fixing the masses, angular momentum and a negative energy, then at these levels there exists a neighborhood of infinity containing no entire orbit. The plan is to first explain the setting and motivation assuming no prior background in celestial mechanics. Then I will give some details for the result's proof.

**Wednesday February 3, 2016**

**Wednesday February 3, 2016**

*What is machine learning and why do people care?*

#### Alex Beloi, University of California, Santa Cruz

**
** In this talk I will give an introduction to what machine learning is and types of problems people try to address with machine learning algorithms. Specifically I'll discuss artificial neural networks, some classic results on their properties and their recent rise to prominence. We'll talk about some amazing things that can and have been done with machine learning, but also some apparent limitations and obstacles that still need to be overcome.

**Wednesday February 10, 2016**

**Wednesday February 10, 2016**

*Trace forms and Lattice-based Cryptography*

#### Richard Klevan, University of California, Santa Cruz

**
** In this talk we investigate the trace form of a number field as it arises in the theory of lattice-based cryptography. After specializing to the case of a cyclotomic field, our goal will be to prove a (new?) formula which reduces the form in fields of arbitrary conductor to the case where the conductor is square-free. After this I will prove an explicit formula for the trace form in the square-free conductor case and, as a corollary, derive a symmetric polynomial representation of the form in the case of prime power conductor. If time allows I will describe how the reduction formula obtained can be used to extend some recent results of Interlando related to the Shortest Vector Problem in lattice cryptography (as well as the associate sphere packing problem).

**Wednesday February 17, 2016 **

**Wednesday February 17, 2016***Magnetic traps in 2 dimensions*

#### Gabriel Martins, University of Califoria, Santa Cruz

**
** In this talk, I'll describe how one can use magnetic fields to trap charged particles in the plane to a given bounded region. In mathmatical terms, I'll show how we can explore the Hamiltonian structure of the Lorentz equation to prove that a magnetic field defined on a bounded region, which is not intergrable along normal rays emanating from its bountry must have a complete Hamiltonian flow. This problem can be interpreted as a planar toy model for the dynamic inside of a Tokamak, a magnetic confinement device used to generate energy through nuclear fusion.

**Wednesday February 24, 2016**

*An Introduction to Vector Valued Modular Forms*

#### Richard Gottesman, University of California, Santa Cruz

**
** I will give an introduction to modular forms and vector valued modular forms. I will explain the basic theory of integral and half integral weight vector valued modular forms. I will also state some of the major open problems in the field. No previous familiarity with modular forms or vector valued modular forms will be assumed.

**Wednesday March 2, 2016**

*The Conley-Zehnder Index*

#### Jeongmin Shon, University of California, Santa Cruz

**
** In this talk, I will introduce the Conley-Zehnder index for nondegenerate paths in the symplectic linear group $Sp(2n, \mathbb{R})$. I will start with the construction of acontinuous map $\rho: Sp(2n, \mathbb{R}) \rightarrow S^1$ and mention some properties of $\rho.$ Then I will talk about a decomposition of $Sp(2n, \mathbb{R})$ into two connected open subsets and a closed subset. Finally, I will define the Conley-Zehnder index.

**Thursday March 10, 2016**

*Units of the Burnside Ring*

#### Jamison Barsotti, University of California, Santa Cruz

**
** If G is a finite group, the Burnside Ring B(G) is defined as the Grothendieck ring of the semi-ring of isomorphism classes of finite G-sets, with addition given by disjoint union and multiplication by cartesian product. There exists an embedding of B(G) into the ring of superclass functions on G, which implies that the unit group, B(G)^\times, is an elementary abelian 2-group. However, in general even knowing the rank of B(G)^\times becomes a difficult question. In particular, Bouc has answered this question for B(P)^\times, when P is a p-group. I will also discuss a connection between B(G)^\times and Feit-Thompson's Odd Order theorem.

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