seminars:arit
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| * **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 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//**: TBA \\ **//Abstract//**: TBA \\ | + | * **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 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//**: TBA \\ **//Abstract//**: TBA \\ | + | * **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 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 $p$. \\ | ||
seminars/arit.1771525073.txt · Last modified: 2026/02/19 13:17 by borisov