seminars:arit:arit

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), 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