Algebraic Geometry

Torsion of Rational Elliptic Curves over the Cyclotomic Extensions of ℚ

Speaker: 
Omer Avci
Date: 
Thu, Oct 31, 2024
Location: 
PIMS, University of Calgary
Conference: 
UCalgary Algebra and Number Theory Seminar
Abstract: 

Let E be an elliptic curve defined over ℚ. Let p > 3 be a prime such that p - 1 is not divisible by 3, 4, 5, 7, 11. In this article, we classify the groups that can arise as E(ℚ(ζp))tors up to isomorphism. The method illustrates techniques for eliminating possible structures that can appear as a subgroup of E(ℚab)tors.

Class: 

Orienteering with One Endomorphism

Speaker: 
Renate Scheidler
Date: 
Thu, Oct 24, 2024
Location: 
PIMS, University of Calgary
Conference: 
UCalgary Algebra and Number Theory Seminar
Abstract: 

Given two elliptic curves, the path finding problem asks to find an isogeny (i.e. a group homomorphism) between them, subject to certain degree restrictions. Path finding has uses in number theory as well as applications to cryptography. For supersingular curves, this problem is known to be easy when one small endomorphism or the entire endomorphism ring are known. Unfortunately, computing the endomorphism ring, or even just finding one small endomorphism, is hard. How difficult is path finding in the presence of one (not necessarily small) endomorphism? We use the volcano structure of the oriented supersingular isogeny graph to answer this question. We give a classical algorithm for path finding that is subexponential in the degree of the endomorphism and linear in a certain class number, and a quantum algorithm for finding a smooth isogeny (and hence also a path) that is subexponential in the discriminant of the endomorphism. A crucial tool for navigating supersingular oriented isogeny volcanoes is a certain class group action on oriented elliptic curves which generalizes the well-known class group action in the setting of ordinary elliptic curves.

Class: 

Parametrization of rings of finite rank - a geometric approach and their use in counting number fields

Speaker: 
Gaurav Patil
Date: 
Thu, Oct 17, 2024
Location: 
PIMS, University of Calgary
Conference: 
UCalgary Algebra and Number Theory Seminar
Abstract: 

We describe parametrizations of rings that generalize the notions of monogenic rings and binary rings. We use these parametrizations to give better lower bounds on the number of number fields of degree n and bounded discriminant.

Class: 

Understanding arithmetic and geometry through cutting and pasting

Speaker: 
Ravi Vakil
Date: 
Thu, Sep 21, 2023
Location: 
Online
Conference: 
PIMS Network Wide Colloquium
Abstract: 

Euler’s famous formula tells us that (with appropriate caveats), a map on the sphere with f countries (faces), e borders (edges), and v border-ends (vertices) will satisfy v-e+f=2. And more generally, for a map on a surface with g holes, v-e+f=2-2g. Thus we can figure out the genus of a surface by cutting it into pieces (faces, edges, vertices), and just counting the pieces appropriately. This is an example of the topological maxim “think globally, act locally”. A starting point for modern algebraic geometry can be understood as the realization that when geometric objects are actually algebraic, then cutting and pasting tells you far more than it does in “usual” geometry. I will describe some easy-to-understand statements (with hard-to-understand proofs), as well as easy-to-understand conjectures (some with very clever counterexamples, by M. Larsen, V. Lunts, L. Borisov, and others). I may also discuss some joint work with Melanie Matchett Wood.

Speaker biography:

Ravi Vakil is a Professor of Mathematics and the Robert K. Packard University Fellow at Stanford University, and was the David Huntington Faculty Scholar. He received the Dean's Award for Distinguished Teaching, an American Mathematical Society Centennial Fellowship, a Frederick E. Terman fellowship, an Alfred P. Sloan Research Fellowship, a National Science Foundation CAREER grant, the presidential award PECASE, and the Brown Faculty Fellowship. Vakil also received the Coxeter-James Prize from the Canadian Mathematical Society, and the André-Aisenstadt Prize from the CRM in Montréal. He was the 2009 Earle Raymond Hedrick Lecturer at Mathfest, and a Mathematical Association of America's Pólya Lecturer 2012-2014. The article based on this lecture has won the Lester R. Ford Award in 2012 and the Chauvenet Prize in 2014. In 2013, he was a Simons Fellow in Mathematics.

Class: 

Non-realizability of polytopes via linear programming

Speaker: 
Amy Wiebe
Date: 
Wed, Apr 20, 2022
Location: 
Online
Conference: 
Emergent Research: The PIMS Postdoctoral Fellow Seminar
Abstract: 

A classical question in polytope theory is whether an abstract polytope can be realized as a concrete convex object. Beyond dimension 3, there seems to be no concise answer to this question in general. In specific instances, answering the question in the negative is often done via “final polynomials” introduced by Bokowski and Sturmfels. This method involves finding a polynomial which, based on the structure of a polytope if realizable, must be simultaneously zero and positive, a clear contradiction. The search space for these polynomials is ideal of Grassmann-Plücker relations, which quickly becomes too large to efficiently search, and in most instances where this technique is used, additional assumptions on the structure of the desired polynomial are necessary.

In this talk, I will describe how by changing the search space, we are able to use linear programming to exhaustively search for similar polynomial certificates of non-realizability without any assumed structure. We will see that, perhaps surprisingly, this elementary strategy yields results that are competitive with more elaborate alternatives and allows us to prove non-realizability of several interesting polytopes.

Class: 

Skeleta for Monomial Quiver Relations

Speaker: 
Jesse Huang
Date: 
Wed, Dec 1, 2021
Location: 
PIMS, University of Alberta
Zoom
Online
Conference: 
Emergent Research: The PIMS Postdoctoral Fellow Seminar
Abstract: 

I will introduce a skeleton obtained directly from monomial relations in a finite quiver without cycles, and relate the construction to some classical examples in mirror symmetry. This is work in progress with David Favero.

Class: 

Differential Equations and Algebraic Geometry - 5

Speaker: 
Andreas Malmendier
Date: 
Mon, Nov 15, 2021
Location: 
PIMS, University of Alberta
Zoom
Online
Conference: 
PIMS Network Courses
Differential Equations and Algebraic Geometry
Abstract: 

This is a guest lecture in the PIMS Network Wide Graduate Course in Differential Equations in Algebraic Geometry.

Class: 

Differential Equations and Algebraic Geometry - 4

Speaker: 
Matt Kerr
Date: 
Fri, Nov 5, 2021
Location: 
PIMS, University of Alberta
Zoom
Online
Conference: 
PIMS Network Courses
Differential Equations and Algebraic Geometry
Abstract: 

This is a guest lecture in the PIMS Network Wide Graduate Course in Differential Equations in Algebraic Geometry.

Class: 

Differential Equations and Algebraic Geometry - 3

Speaker: 
Adrian Clingher
Date: 
Wed, Nov 3, 2021
Location: 
PIMS, University of Alberta
Zoom
Online
Conference: 
PIMS Network Courses
Differential Equations and Algebraic Geometry
Abstract: 

This is a guest lecture in the PIMS Network Wide Graduate Course in Differential Equations in Algebraic Geometry.

Class: 

Differential Equations and Algebraic Geometry - 2

Speaker: 
Hossein Movasati
Date: 
Sat, Oct 30, 2021
Location: 
PIMS, University of Alberta
Zoom
Conference: 
PIMS Network Courses
Differential Equations and Algebraic Geometry
Abstract: 

This is a guest lecture in the PIMS Network Wide Graduate Course in Differential Equations in Algebraic Geometry.

Class: 

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