The Arithmetic Seminar [Department of Mathematics and Statistics, Binghamton University]

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=====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, Arithmetic Topology, etc.

**PLACE and TIME**: This semester the seminar meets primarily on Tuesdays at 4:00 pm, with possible special lectures at other days and times. The in-house talks will be in-person, while visitors outside of Binghamton area will be in-person or by Zoom: [[https://binghamton.zoom.us/j/92745369515?pwd=gg9R8gOQrFpFOwe4T3c6nUbUcNrLPq.1|Zoom link]]\\ 

**ORGANIZERS**: \\    **Regular Faculy:** [[:people:borisov:|Alexander Borisov]], [[:people:mazur:|Marcin Mazur]], [[:people:adrian:|Adrian Vasiu]]. \\     **Post-Docs:** [[:people:hdang2::|Huy Dang]]

**Current Ph.D. students:** [[:people:grads:hari:|Hari Asokan]], [[:people:grads:mithunp:|Mithun Padinhare Veettil]].


**Graduated Ph.D. students** (in number theory and related topics): [[:people:grads:snopce:|Ilir Snopce]] (Dec. 2009), [[:people:grads:xiao:|Xiao Xiao]] (May 2011), [[:people:grads:jinghao:|Jinghao Li]] (May 2015), [[:people:grads:ding:|Ding Ding]] (Dec. 2015),
[[:people:grads:milano:|Patrick Milano]] (May 2018), [[:people:grads:zhou:|Changwei Zhou]] (May 2019), Patrick Carney (May 2023), [[:people:grads:lamoureux:|Sarah Lamoureux]] (Sep. 2023),  [[:people:grads:sengupta:|Sayak Sengupta]] (May 2024).

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**SEMINAR ANNOUNCEMENTS**: To receive announcements of seminar talks by email, please join our [[http://www1.math.binghamton.edu/mailman/listinfo/Arithmeticsem|mailing list]].

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

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===Previous Arithmetic Seminar Talks===
  * ------------------------- [[seminars:arit:arit_fall2025]]
  * [[seminars:arit:arit_spring2025]]  -------- [[seminars:arit:arit_fall2024]]
  * [[seminars:arit:arit_spring2024]]  -------- [[seminars:arit:arit_fall2023]]
  * [[seminars:arit:arit_spring2023]]  -------- [[seminars:arit:arit_fall2022]]
  * [[seminars:arit:arit_spring2022]]  -------- [[seminars:arit:arit_fall2021]]
  * [[seminars:arit:arit_spring2021]]  -------- [[seminars:arit:arit_fall2020]]
  * [[seminars:arit:arit_spring2020]]  -------- [[seminars:arit:arit_fall2019]]
  * [[seminars:arit:arit_spring2019]]  -------- [[seminars:arit:arit_fall2018]]
  * [[seminars:arit:arit_spring2018]]  -------- [[seminars:arit:arit_fall2017]]
  * [[seminars:arit:arit_spring2017]]  -------- [[seminars:arit:arit_fall2016]]
  * [[seminars:arit:arit_spring2016]]  -------- [[seminars:arit:arit_fall2015]]
  * [[seminars:arit:arit_spring2015]]  -------- [[seminars:arit:arit_fall2014]]

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===Spring 2026===

   * **January 27**  \\  **//Speaker//**: NA \\      **//Title//**: Organizational Meeting    \\ **//Abstract//**: \\

   * **February 3 (2:45-3:45 pm, cross-listed from Algebra Seminar) ** \\    **//Speaker//**: Tim Riley (Cornell University) \\      **//Title//**: Conjugator length \\ **//Abstract//**: The conjugacy problem for a finitely generated group $G$ asks for an algorithm which, on input a pair of words u and v, declares whether or not they represent conjugate elements of $G$. The conjugator length function $CL$ is its most direct quantification: $CL(n)$ is the minimal $N$ such that if $u$ and $v$ represent conjugate elements of $G$ and the sum of their lengths is at most $n$, then there is a word $w$ of length at most $N$ such that $uw=wv$ in $G$.  I will talk about why this function is interesting and how it can behave, and I will highlight some open questions.  En route I will talk about results variously with Martin Bridson, Conan Gillis, and Andrew Sale, as well as recent advances by Conan Gillis and Francis Wagner.\\

   * **February 10**  \\    **//Speaker//**: Alexander Borisov (Binghamton) \\      **//Title//**: A structure sheaf for Kirch topology, an update \\ **//Abstract//**:  Kirch topology on $\mathbb N$ goes back to a 1969 paper of Kirch. It can be defined by a basis of open sets that consists of all infinite arithmetic progressions $a+d\mathbb N_0$, such that $gcd(a,d)=1$ and $d$ is square-free. It is Hausdorff, connected, and locally connected. I will give an update on my current work on a natural presheaf of functions on this topological space: locally integer polynomial functions. In particular, I will discuss when the sheafification is equal to the presheaf, and when it is bigger. I will also discuss (Cech) cohomology. In particular, I will give examples with trivial and nontrivial H^1. No prior knowledge of the topic is assumed. This talk will also serve as an introduction to Mithun's talk next week. \\

   * **February 17**  \\    **//Speaker//**: Mithun Veettil (Binghamton) \\      **//Title//**: Some results on the  Locally LIP functions \\ **//Abstract//**: Locally LIP functions are obtained as a result of sheafification of the presheaf LIP on some infinite subset $X$ of $N={1,2,3,...}$, with a prescribed topology. Often we work with Kirch topology on $N$ that makes $N$ a connected, locally connected, and Hausdorff space. \\ If the set $X$ is a union of non-connected open sets, then we can easily define a locally LIP function on $X$ that is not a LIP function globally. In fact, even if the space $X$ is connected, a locally LIP function on $X$  need not be a LIP function on $X$. In this talk, we will look at $X=N$\ $6N$, which is connected, and construct a locally LIP function that is not LIP on $X$. Also, we will show that this is not the case if one works with $\mathbb{Z}[1/2]$ instead of $\mathbb{Z}$ for the above set $X$.\\

   * **February 24**  \\    **//Speaker//**: Hari Asokan (Binghamton) \\      **//Title//**: Variation of Geometric Invariant Theory \\ **//Abstract//**:  Geometric Invariant Theory is used to construct quotients of group actions on varieties, but the outcome depends on a choice of linearization. Variation of Geometric Invariant Theory (VGIT) studies the different quotients resulting from changing this choice. In this talk I will give an informal introduction to VGIT, focusing on how stability changes as linearization varies. \\

   * **March 3**  \\    **//Speaker//**: Connor Stewart (CUNY)  \\      **//Title//**: Conductor–Discriminant Inequality for Tamely Ramified Cyclic Covers \\ **//Abstract//**: We consider $\mathbb{Z}/n$-covers $X\to\mathbb{P}^1$ defined over discretely valued fields $K$ with excellent valuation ring $\mathcal{O}_K$ and perfect residue field of characteristic not dividing $n$. Two standard measures of bad reduction for such a curve $X$ are the Artin conductor of its minimal regular model over $\mathcal{O}_K$ and the valuation of the discriminant of a Weierstrass equation for $X$. We prove an inequality relating these two measures. Specifically, if $X$ is given by an affine equation $y^n = f(x)$ with $f(x) \in \mathcal{O}_K[x]$, and if $\mathcal{X}$ is its minimal regular model over $\mathcal{O}_K$, then the negative of the Artin conductor of $\mathcal{X}$ is bounded above by $(n-1)v_K(\textrm{disc (rad}\ f))$. This extends previous work of Ogg, Saito, Liu, Srinivisan, and Obus-Srinivasan on elliptic and hyperelliptic curves. (Joint work with Andrew Obus and Padmavathi Srinivasan.) \\

   * **March 10**  \\    **//Speaker//**: Eric Yin (Binghamton) \\      **//Title//**:  Generating abelian extensions with elliptic curves with complex multiplication. \\ **//Abstract//**: An elliptic curve with complex multiplication is one with extra endomorphisms, ones that are not simply given by multiplication-by-m maps. In this talk we discuss how this extra structure allows us to find an analogy to Kronecker-Weber, generating abelian extensions of imaginary quadratic fields through torsion points on elliptic curves with CM. In addition, we discuss the role the ideal class group plays in both measuring ramification and classifying elliptic curves with a given endomorphism ring. This lets us describe both the Hilbert class field, the maximal unramified abelian extension, and the maximal abelian extension of any imaginary quadratic field.\\

   * **March 17**  \\    **//Speaker//**: Anitha Srinivasan (Comillas University, Madrid), [[https://binghamton.zoom.us/j/92745369515?pwd=gg9R8gOQrFpFOwe4T3c6nUbUcNrLPq.12|by Zoom]] \\      **//Title//**: The generalized  Markoff equation \\ **//Abstract//**: The talk will look at  various aspects of the generalized Markoff equation $a^2+b^2+c^2=3abc+m$ ($m\ge 0$), giving an overview of all the exciting work in the area.  A few examples of topics that will be mentioned are: the classification of solution triples $(a, b, c)$ that come from $k$-Fibonacci sequences,  open conjectures (which $m's$ have no solutions?), counting algorithms for the number of solutions (trees) and the Markoff equation mod $p$.  \\

   * **March 24**  \\    **//Speaker//**: Jauing Jun (SUNY New Paltz)  \\      **//Title//**: Categorical approach to stability of tropical toric vector bundles \\ **//Abstract//**: Developing a suitable notion of vector bundles in tropical geometry has recently attracted considerable interest. In a recent work, Khan and Maclagan introduced tropical vector bundles using matroids, inspired by Klyachko’s classification of toric vector bundles, and studied their stability properties. In this talk, we reinterpret their notion of stability through the framework of the categorical approach to stability proposed by André. This perspective clarifies the structure underlying their results and places them in a broader conceptual setting. This is joint work with Alex Sistko and Cameron Wright. \\

   * **April 14**  \\    **//Speaker//**: Anubhav Nanavaty (Cornell University)  \\      **//Title//**: TBA \\ **//Abstract//**: TBA \\

   * **April 21**  \\    **//Speaker//**: Joe Kramer-Miller (Lehigh University)  \\      **//Title//**: TBA \\ **//Abstract//**: TBA \\

   * **April 28**  \\    **//Speaker//**: TBA  \\      **//Title//**: TBA \\ **//Abstract//**: TBA \\