**Problem of the Week**

**Math Club**

**BUGCAT 2020**

**Zassenhaus Conference**

**Hilton Memorial Lecture**

**BingAWM**

seminars:arit

**TOPICS**: Arithmetic in the broadest sense that includes Number Theory (Elementary Arithmetic, Algebraic, Analytic, Combinatorial, etc.), Algebraic Geometry, Representation Theory, Lie Groups and Lie Algebras, Diophantine Geometry, Geometry of Numbers, Tropical Geometry, Arithmetic Dynamics, etc.

**PLACE and TIME**: This semester the seminar meets primarily on Tuesdays at 4:15 p.m. on Zoom, with possible special lectures on Mondays at 3:30 or other days. Zoom link

**ORGANIZERS**:

**Regular Faculy:** Alexander Borisov, Marcin Mazur, Adrian Vasiu,

**Post-Docs:** Vaidehee Thatte,
Fikreab Solomon Admasu.

**Current Ph.D. students:** Patrick Carney, Andrew Lamoureux, Micah Loverro, and Sayak Sengupta.

**Graduated Ph.D. students** (in number theory and related topics): Ilir Snopce (Dec. 2009), Xiao Xiao (May 2011), Jinghao Li (May 2015), Ding Ding (Dec. 2015),
Patrick Milano (May 2018), Changwei Zhou (May 2019).

**SEMINAR ANNOUNCEMENTS**: To receive announcements of seminar talks by email, please join our mailing list.

**Related seminar**: Upstate New York Online Number Theory Colloquium (bi-weekly, online):
http://people.math.binghamton.edu/borisov/UpstateNYOnline/Colloquium.html

**Related seminar**: The student/postdoc “No Theory Seminar”:
https://sites.google.com/view/vaideheethatte/service-outreach/nts-bu

- ————————- Fall 2020
- Spring 2020 ——– Fall 2019
- Spring 2019 ——– Fall 2018
- Spring 2018 ——– Fall 2017
- Spring 2017 ——– Fall 2016
- Spring 2016 ——– Fall 2015
- Spring 2015 ——– Fall 2014

**February 16**

: N/A*Speaker*

: Organizational Meeting*Title*

: We will discuss plans for this semester*Abstract*

**February 23**

: Fikreab Solomon Admasu (Binghamton)*Speaker*

: Bhargava's composition law for binary cubic forms*Title*

: The Delone-Fadeev correspondence shows that binary cubic forms with integer coefficients parametrize orders in cubic fields. With this result in mind, Bhargava constructs a binary cubic form from 2x3x3 boxes of integers and proves that there is a natural composition law for the boxes of integers. The group resulting from this law is then shown to be isomorphic to the class group of a corresponding cubic order. This is a cubic analogue of Gauss's theory of composition for binary quadratic forms and its relation to ideal classes of quadratic orders. The talk is based on Bhargava's “Higher composition laws II: On cubic analogues of Gauss composition.”*Abstract*

**March 2**

: Hyuk Jun Kweon (MIT)*Speaker*

: Bounds on the Torsion Subgroups of Néron-Severi Group Schemes*Title*

: Let $X \hookrightarrow \mathbb{P}^r$ be a smooth projective variety defined by homogeneous polynomials of degree $\leq d$ over an algebraically closed field $k$. Let $\mathbf{Pic}\, X$ be the Picard scheme of $X$, and $\mathbf{Pic}\, ^0 X$ be the identity component of $\mathbf{Pic}\, X$. The N\'eron–Severi group scheme of $X$ is defined by $\mathbf{NS} X = (\mathbf{Pic}\, X)/(\mathbf{Pic}\, ^0 X)_{\mathrm{red}}$, and the N\'eron–Severi group of $X$ is defined by $\mathrm{NS}\, X = (\mathbf{NS} X)(k)$. We give an explicit upper bound on the order of the finite group $(\mathrm{NS}\, X)_{{\mathrm{tor}}}$ and the finite group scheme $(\mathbf{NS} X)_{{\mathrm{tor}}}$ in terms of $d$ and $r$. As a corollary, we give an upper bound on the order of the torsion subgroup of second cohomology groups of $X$ and the finite group $\pi^1_{et}(X,x_0)^{\mathrm{ab}}_{\mathrm{tor}}$. We also show that $(\mathrm{NS}\, X)_{\mathrm{tor}}$ is generated by $(\deg X -1)(\deg X - 2)$ elements.*Abstract*

**March 9**

: Thomas Morrill (Trine)*Speaker*

: TBA*Title*

: TBA*Abstract*

**March 16**

: Fikreab Solomon Admasu (Binghamton)*Speaker*

: Bhargava's composition law for binary cubic forms 2*Title*

: TBA*Abstract*

**March 23**

: TBA*Speaker*

: TBA*Title*

: TBA*Abstract*

**March 30**

: TBA*Speaker*

: TBA*Title*

: TBA*Abstract*

**April 6**

: TBA*Speaker*

: TBA*Title*

: TBA*Abstract*

**April 13**

: TBA*Speaker*

: TBA*Title*

: TBA*Abstract*

**April 20**

: TBA*Speaker*

: TBA*Title*

: TBA*Abstract*

**April 27**

: TBA*Speaker*

: TBA*Title*

: TBA*Abstract*

**May 4**

: TBA*Speaker*

: TBA*Title*

: TBA*Abstract*

**May 11**

: TBA*Speaker*

: TBA*Title*

: TBA*Abstract*

**May 18**

: TBA*Speaker*

: TBA*Title*

: TBA*Abstract*

seminars/arit.txt · Last modified: 2021/02/24 08:28 by borisov

Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Noncommercial-Share Alike 3.0 Unported