unQVNTS (Vermont)

We survey the theory of zeta functions of varieties over finite fields and the development of crystalline cohomology as a tool for studying them. We then introduce ideas from higher category theory and algebraic topology to define Topological Hochschild Homology, and discuss its relation to crystalline cohomology and zeta functions.

We discuss the theory of Witt vectors with the p-adic integers as a motivating example. We define the Witt polynomials and the associated ring operations to define Witt vectors for any commutative ring. Finally, we discuss the three important operators on Witt vectors known as the Verschiebung, restriction, and Frobenius.

The shape of a number field is an invariant which refines the discriminant. In this talk we present Piper H's PhD thesis on the equidistribution of shapes of S_3-, S_4-, and S_5-number fields when they are ordered by discriminant. We then discuss joint work that is in progress on the joint distribution of the shapes of a field and its resolvent. This talk will be accessible to anyone who knows some Galois theory as well as the definition of a number field and its ring of integers.

Fully Homomorphic Encryption (FHE) allows us to perform arbitrary computations over encrypted data, without the decryption keys or knowing anything about the data itself. FHE has numerous applications where privacy is of utmost concern. For example, it enables outsourcing the storage of private genomic data to a commercial cloud server, while also enabling private queries to match a set of encrypted biomarkers to encrypted genomes to calculate the probability of genomic diseases. Ever since Gentry’s first construction based on hard problems in lattices, there have been several FHE schemes proposed based on different hard problems. All existing schemes have a common trait; all fresh ciphertexts have a small noise that is added to the plaintext during encryption. Homomorphic evaluations on ciphertexts cause this noise to grow and result in a decryption error if it passes a certain threshold. The challenge is to find an efficient technique to refresh the evaluated ciphertext so that the noise is reduced to the initial levels of a fresh ciphertext. I will start this talk with a description of Regev’s Learning with Errors (LWE) problem and its ring variant R-LWE. I will next talk about how they are used in homomorphic encryption, particularly by presenting BGV [Brakerski, Gentry, Vaikuntanathan] and FV [Fan, Vercauteren] FHE schemes. These two schemes are the basis for two widely used FHE libraries HElib of IBM Research and SEAL of Microsoft Research. I will conclude my talk by demonstrating real world examples of how these schemes and libraries can be used in real life to protect the privacy of sensitive data while providing efficiency to its users.

Security schemes based on RSA or ECC are known to be vulnerable under quantum attacks. There has been an increasing interest in the study of cryptosystems based on the hardness of computing the endomorphism rings or isogenies of supersingular elliptic curves. Both of these problems are equivalent and it is not known if they can be broken by quantum computers. Here we study the hardness of the problem of generating the endomorphism ring of a supersingular elliptic curve, which corresponds to a maximal order in a quaternion algebra, by looking at cycles of the ell-isogeny graph. This strategy was first explored by Kohel (1996). We present two main results: One is a condition for two cycles to be linearly independent. The second one is a criterion under which two cycles do not generate a maximal order. This is joint work with Efrat Bank, Kirsten Eisentraeger, Travis Morrison and Jennifer Park.

(Joint work with Kyle Hammer, Thomas Mattman, and Daniel Vallieres of CSU Chico)
We consider an unramified Galois covering of a graph X by a graph Y, and denote the group of automorphisms of Y over X by G. For the graph Y, the Jacobian J(Y) is a finite group with a variety of other names whose order is the tree-number of Y. In our situation, J(Y) becomes a module over the group ring Z[G]. Using L-functions, we define an element in this group ring and show that it annihilates the group J(Y). This is an analog of the classical Stickelberger theorem for cyclotomic fields.

(Joint work with Kyle Hammer, Thomas Mattman, and Daniel Vallieres of CSU Chico)
We consider an unramified Galois covering of a graph X by a graph Y, and assume that the group G of automorphisms of Y over X is abelian. Then L-functions are defined for the characters of G, leading to the definition of a Stickelberger element and a Stickelberger ideal St in the integral group ring Z[G] of G . We prove a formula relating the index of this ideal in the augmentation ideal to the order of the Jacobian J(Y) of the graph Y.

The class number is among the most important invariants associated to a number field. Conjecturally, its behavior (at the “good” primes) is governed by the heuristics of Cohen-Lenstra-Martinet. By contrast, the genus number is supported at the “bad” primes, but is easier to understand. It is very natural to ask about the density of genus number one fields among all number fields of a fixed degree and signature. We prove that approximately 96.23% of cubic fields, ordered by discriminant, have genus number one. Finally, we show that a positive proportion of totally real cubic fields with genus number one fail to be norm-Euclidean. Time permitting, I will further discuss norm-Euclidean questions and the case of quintic fields. This is joint work with Amanda Tucker.

Error correcting codes are systems for incorporating redundancy into stored or transmitted data, so that errors can be identified and even corrected. A good error correcting code is efficient and can correct many errors relative to its efficiency. These codes are ubiquitous in the digital age, and many excellent codes arise from algebraic constructions. The increasing importance of cloud computing and storage has created a need for codes that protect against server failure in large computing facilities. One way of approaching this problem is to ask for local recovery. An error correcting code is said to be locally recoverable if any symbol in a code word can be recovered by accessing a subset of the other symbols. This subset is known as the helper or recovery set for the given symbol. It may be desirable to have many disjoint recovery sets for each symbol, in case of multiple server failures or to provide many options for recovery. Barg, Tamo, and Vladut recently constructed LRCs with one and two disjoint recovery sets from algebraic curves. This talk presents a generalization of this construction to three or more recovery sets, using fiber products curves over finite fields. This is joint work with Kathryn Haymaker and Gretchen Matthews.

We start by giving preliminaries on ramification, cyclotomic fields, and ideal class groups. We then introduce some basic, useful theorems in class field theory. We set up the machinery to present a reflection theorem about class numbers. We use these results together with Vandiver's conjecture and Stickelberger's Theorem to generalize that result, with the goal of identifying the class group of a cyclotomic field as a quotient of a group ring.

For a positive integer m, with zeta_m denoting a primitive mth root of unity, each unit of the real cyclotomic field K_m^+ = QQ(zeta_m + zeta_m^-1) has an associated "signature" indicating the sign (positive or negative) of each of its phi(m)/2 real embeddings. The collection of such unit signatures is an elementary abelian 2-group whose rank measures how many different possible signs arise from units of K_m^+. I will discuss some recent results (joint with D. Dummit and H. Kisilevsky) on signatures of circular units in these fields: we show that the (circular) unit signature rank tends to infinity with m, and that the difference between the unit signature rank and its maximum possible value is bounded in certain vertical towers but can become arbitrarily large in general.