## Graduate Number Theory Seminar/ Learning Seminar

Co-Organized by Louis Gaudet (louis.gaudet@rutgers.edu) and Brooke Ogrodnik (bl481@math.rutgers.edu, website)

Co-Sponsored by the generosity of the Math Department Academic Initiatives Program (AIP) and the Math Graduate Student Organization (GSO).

If you are interested in giving a talk or would just like to learn please do not hesitate to email the organizer.

The Graduate Number Theory Seminar and Learining Seminar are two seminars that are held periodically throughout the semester.

The Graduate Number Theory Seminar (GNTS) aimed at giving graduate students and postdocs a place to tell others about what they are working on or find interesting, as well as giving them a space and audience to practice giving talks.

The Learning Seminar (LS) currently is a potpourri of topics that interest the speaker. If you have a recommendation for a theme for the Fall semester, please email the organizer. However, overall it is similar to GNTS except faculty are welcome to attend and/or give talks.

#### Next Talk

Date: | 4-6-2021 |
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Speaker: | Nathan Mehlhop |

Title: | Waring's Problem and the Circle Method |

Abstract: | In 1770, Lagrange showed that every natural number can be written as the sum of four perfect squares. Is there always a finite number g(k) of terms needed when we instead consider positive kth powers? This is Waring's problem, and it was proven in 1909 by Hilbert, but the method gave no useful numerical bounds. In 1919, Hardy and Littlewood introduced the circle method and gave asymptotics for the number of representations of N as the sum of s many positive kth powers. This method is widely applicable to questions of additive number theory. I will introduce the circle method in the context of Waring's problem. |

##### Seminars take place Tuesdays at 4pm ET online. Email the organizer for the links.

#### Future Talks

- 4-13 David Herrera
- 4-20 Postdoc Panel

#### Spring 20201- Tuesdays at 4pm ET via Zoom

March 30th (LS): | Claire Burrin, "Modular forms of half-integer weight and applications to equidistribution" |
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March 23rd (GNTS): | Max Weinreich, "The pentagram map and arithmetic dynamics" |

March 9th (LS): | Chris Lutsko, "Pair Correlation of the sequence $\alpha n^\theta \mod 1$" |

March 2nd (GNTS): | Edna Jones, " An Asymptotic Local-Global Principle for Integral Kleinian Sphere Packings" |

February 23rd (LS): | Forrest Thurman, "The Reproducing Kernel for the Hilbert Space of Modular Forms" |

February 16th (GNTS): | Kevin Kwan, "Zeros of Riemann Zeta Function, III" |

February 9th (LS): | Alex Walker, " Topics in Analytic Continuation" |

February 2nd (GNTS): | Louis Gaudet, "Critical Zeros of the Riemann Zeta Function" |

Janurary 26th (LS): | Kim Klinger-Logan, "Pair Correlation for GL(Z) L-Functions" |

#### Fall 2020- Wednesdays 1:30pm EDT via Zoom

November 18th (GNTS): | Louis Gaudet, "Evaluating a mollified Möbius sum" |
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November 11th(LS): | Brooke Ogronik, "Dirichlet's Class Number Formula" |

November 4th(GNTS): | Nick Backes, "On The Theory of p-adic Manifolds" |

October 28th (GNTS): | Andre Hernandez-Espiet, "The Rankin-Selberg Method" |

October 21st (LS): | Kim Klinger-Logan, "Low Energy String Theory and Eisenstein Series" |

October 14th (LS): | Alex Karlovitz, "Crystallographic Sphere Packings and Local-Global Problems} |

October 7th (GNTS): | Edna Jones, "Möbius Transformations and the Bends and Centers of Generalized Circles, Spheres, and Hyperspheres" |

September 30th (LS): | Louis Gaudet, "Hecke Characters and Applications" |

September 23rd (GNTS): | George Hauser, "Shimura's "On the Holomorphy of Certain Dirichlet Series" " |

September 16th (LS): | Alex Walker, "Sets of Large Harmonic Sum Containing no Long Arithmetic Progressions" |

#### Summer 2020- Wednesdays 11am via Zoom

August 26th (LS): | George Hauser, "Fourier Coefficients of an Eisenstein Distribution on the Universal Cover of SL(2,R)." |
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August 12th (LS): | Edna Jones, "The Local-Global Principle for Integral Soddy Sphere Packings" |

July 29th (GNTS): | Brooke Ogrodnik, "On the Local-Global Conjecture for Commutator Traces." |

July 15th (GNTS): | Doyon Kim, "Automorphic distributions and the zeros on the critical line pt.2" |

July 1st (GNTS): | Alex Karlovitz, "Three Local-Global Problems for Thin Orbits" |

June 17th (LS): | Surya Gavva, "Prime Number Theorem by Convolution method" |

#### Spring 2020- Wednesdays 9:20-10:20 in Hill 423

May 6th (LS): | Doyon Kim, Automorphic distributions and the zeros on the critical line" |
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April 29nd (LS): | Max Weinreich, "Automorphism groups of dynamical systems over finite fields" |

April 22nd (LS): | Louis Gaudet, "Vinogradovs theorem on sums of three primes, Part 2," via online seminar |

April 15th (LS): | Louis Gaudet, "Vinogradovs theorem on sums of three primes, Part 1," VIa Online Seminar |

April 8th (GNTS): | Andre Hernandez-Espiet, "Closed Geodesics and Binary Quadratic Forms," Via Online Seminar |

April 1st (LS): | Alex Karlovitz, "A Theorem on the Base Eigenvalues of Hecke Groups," Via Online Seminar |

March 11th (LS): | Brooke Ogrodnik, "Local-Global Conjecture for Thin Groups" |

March 4th (GNTS): | Claire Burrin, "Elkies-McMullen" |

February 26th (LS): | Alex Kontorovich, "Part 2 of Ratner's Theorems" |

February 19th (GNTS): | Edna Jones, "Local Densities of Diagonal Integral Ternary Quadratic Forms at Odd Primes." |

February 12th (LS): | Alex Kontorovich, "Ratner's Theorems Part 1" |

February 5th (GNTS): | Alex Karlovitz, "Harmonic Analysis on the Hyperbolic Plane" |

January 29th (LS): | Alex Kontorovich, "A Review on Ratner's Theorems" |

#### Fall 2019- Wednesdays 10:30-11:30 in Hill 425

December 11th (GNTS): | Forrest Thurman, "Modularity Property of the Theta Functions of Lattices" |
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December 4th (LS): | Alex Karlovitz, "Continuation of Chapter 2 of Ratner's Theorems" |

November 20th (LS): | George Hauser, "Chapter 2 of Ratner's Theorems" |

November 13th (LS): | Alex Kontorovich, "Ratner's Theorems" |

November 6th (GNTS): | Doyon Kim, "Congruent numbers and elliptic curves" |

October 30th (LS): | Alex Walker, "Arithmetic Applications of Ratner's Theorems" |

October 23rd (GNTS): | Edna Jones, "Siegel's Formula" |

October 16th (GNTS): | Louis Gaudet, "A zero-density theorem for the Riemann zeta function" |

October 9th (GNTS): | Alex Karlovitz, "An Introduction to Maass Forms and their Fourier Expansions" |

October 2nd (GNTS): | George Hauser, "The Dedekind Eta Function and Its Multiplier System" |

September 25th (LS): | Matthew Welsh, "The Oppenhiem conjecture and Ratner's theorems" |

September 18th (LS): | Alex Kontorovich, "Introduction to Ratner's theorems" |

September 11th (GNTS): | Brooke Ogrodnik, "Reciprocal Geodesics" |