As quantum computers become more and more advanced, currently standard cryptographic protocol becomes weaker and weaker. Supersingular elliptic curve isogeny graphs are a promising new direction for post-quantum cryptographic protocol. The supersingular elliptic curve isogeny graph has vertices which are isomorphism classes of supersingular elliptic curves over an algebraically closed field of characteristic p, and edges which are degree-l isogenies up to equivalence. In this thesis, we study supersingular elliptic curve isogeny graphs from three directions. First, we describe the relationship between the supersingular elliptic curve isogeny graph over an algebraically closed field of characteristic p and a supersingular isogeny graph defined over the prime field. We provide a deep study and heuristics on how the supersingular elliptic curves which are defined over the prime field or which are p-power Frobenius conjugate sit inside of the supersingular elliptic curve isogeny graph. Second, we attach level structure to supersingular elliptic curves and study the endomorphism rings of these objects. We provide an equivalence of categories with objects in a quaternion algebra, and study the l-isogeny graph of supersingular elliptic curves with level structure. Third, we study oriented supersingular elliptic curve isogeny graphs, where the orientation is a choice of embedding of a quadratic field into the endomorphism ring of a supersingular elliptic curve. We study the ``volcano" and ``cordillera" structure of these oriented isogeny graphs, and we provide algorithms for walking the supersingular l-isogeny graph via paths on the oriented l-isogeny graph.
The study of Maltsev conditions is a significant part of universal algebra, with classical characterizations of families of varieties (congruence permutable, distributive, modular...) and recent advanced results by Hobby, McKenzie, Kearnes, Kiss, among others. In particular, the interplay between distinct Maltsev conditions for congruence modular varieties has led to a refined theory for such varieties.
Recall that a Maltsev condition is, roughly, a statement of the form "there are some n and terms t1,...,tn such that a certain finite set of equations hold". As we mentioned, many deep and sophisticated results are known about Maltsev conditions. On the other hand, when two conditions are compared, really little is known about the exact value of the smallest n as above. For example, a simple observation by A. Day asserts that if some variety V has k Jónsson terms witnessing congruence distributivity, then V has 2k-1 Day terms witnessing congruence modularity. About fifty years ago Day asked whether this result is best possible, but, to the best of our knowledge, an exact solution is not yet known.
A deeper problem (asked by Lakser, Taylor, Tschantz in 1985) concerns the relative lengths of sequences of Day and Gumm terms characterizing congruence modularity. More recently, Kazda, Kozik, McKenzie, Moore provided still another characterization of congruence distributive and modular varieties by means of "directed" terms. Again, the exact relationships between the lengths of the sequences of terms is not known. A solution of the above problems is supposed to provide either interesting exotic examples of congruence modular and distributive varieties, or more refined structure theorems.
We shall present recent results about the above Day, LTT and KKMM problems, with an unexpected application to congruence distributive varieties.
Relative lengths of Maltsev conditions
Mar. 29, 2022 3pm (MATH 350)
Topology
Juan Moreno (CU Boulder)
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This talk will outline the TVBW models which use a spherical fusion category and combinatorial structures on manifolds to produce oriented (2+1)D TQFTs .