seminars:arit

## The Arithmetic Seminar

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 on Mondays at 3:30 p.m. or on Tuesdays at 4:15 p.m. in WH 100E, with possible special lectures at other days. Before the talks, there will be refreshments in WH-102.

ORGANIZERS:
Regular Faculy: Alexander Borisov, Marcin Mazur, Adrian Vasiu,
Post-Docs: Vaidehee Thatte.

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

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).

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#### Fall 2018

• August 27 (Monday)
Speaker: N/A
Title: Organizational Meeting
Abstract: We will discuss schedule and speakers for this semester
• September 18 (Tuesday)
Speaker: Vaidehee Thatte (Binghamton)
Title: Ramification Theory for Degree $p$ extensions of Arbitrary Valuation Rings in Positive Characteristic, part 1
• Abstract: In classical ramification theory, we consider extensions of complete discrete valuation rings with perfect residue fields. We would like to study arbitrary valuation rings with possibly imperfect residue fields and possibly non-discrete valuations of rank $\geq 1$, since many interesting complications arise for such rings. In particular, defect may occur (i.e. we can have a non-trivial extension, such that there is no extension of the residue field or the value group) when the characteristic is positive. We will discuss some new results in the equal characteristic case, similar results are true in the mixed characteristic $(0,p)$ case.
We will begin with a few examples of Artin-Schreier extensions of valuation fields and explicitly compute some invariants of ramification theory in each case. Time permitting, we will also discuss a generalization of the classical Swan conductor.
• September 24/25
Speaker: Alexander Borisov (Binghamton)
Title: Singular Fano Varietes
• Abstract: TBA
• October 2 (Tuesday)
Speaker: Vaidehee Thatte (Binghamton)
Title: Ramification Theory for Degree $p$ extensions of Arbitrary Valuation Rings in Positive Characteristic, part 2
• Abstract(tentative): We will continue discussing Artin-Schreier extensions of valuation fields in positive characteristic. We will present some results that relate the “higher ramification ideal” of the extension with the ideal generated by the inverses of Artin-Schreier generators via the norm map. We will also introduce a generalization and further refinement of Kato's refined Swan conductor for such extensions.
• October 8 (Monday)
Speaker: TBA
Title: TBA
• October 15 (Monday)
Speaker: Viji Thomas (Cleveland State)
Title: TBA
• October 23 (Tuesday)
Speaker: Vaidehee Thatte (Binghamton)
Title: Ramification Theory for Degree $p$ extensions of Arbitrary Valuation Rings in Positive Characteristic, part 3
• Abstract(tentative): Let $K$ be a valued field of characteristic $p > 0$ with henselian valuation ring $A$. Let $L$ be a non-trivial Artin-Schreier extension of $K$ with $B$ as the integral closure of $A$ in $L$. In the classical theory, $B$ is generated as an $A$-algebra by a single element. This is not true when there is defect. We will discuss a result that allows us to write $B$ as a “filtered union over $A$”, in such cases.
We will conclude the series with some remarks on how to obtain analogous results in the mixed characteristic $(0, p)$ case.
• November 6 (Tuesday)
Speaker: Marie Langlois (Cornell)
Title: TBA
• November 13 (Tuesday)
Speaker: Andrew Obus (Baruch COllege)
Title: TBA
• November 19 (Monday)
Speaker: TBA
Title: TBA
• November 26 (Monday)
Speaker: Isabel Leal (Courant)
Title: TBA
• December 3/4
Speaker: Sayak Sengupta (Binghamton)
Title: TBA