Anthony Várilly-Alvarado– December 1

Rational surfaces and locally recoverable codes

Anthony Várilly-AlvaradoRice University

 Motivated by large-scale storage problems around data loss, a budding branch of coding theory has surfaced in the last decade or so, centered around locally recoverable codes.  These codes have the property that individual symbols in a codeword are functions of other symbols in the same word.  If a symbol is lost (as opposed to corrupted), it can be recomputed, and hence a code word can be repaired.  Algebraic geometry has a role to play in the design of codes with locality properties. In this talk I will explain how to use algebraic surfaces to both reinterpret constructions of optimal codes already found in the literature, and to find new locally recoverable codes, many of which are optimal (in a suitable sense).  This is joint work with Cecília Salgado and Felipe Voloch.


Justin Lyle- November 12

Recent progress on the Auslander-Reiten and Huneke-Weigand Conjectures

Justin LyleUniversity of Arkansas

Various conjectures have emerged as far back as the 1950’s concerning conditions on the vanishing of Ext/Tor. Among the most celebrated are the Auslander-Reiten and Huneke-Wiegand conjectures, which pose that a finitely-generated module over a commutative Noetherian ring should be free in certain scenarios. We will discuss some recent progress on these conjectures. In particular, we will journey through some surprising connections between the Huneke-Wiegand conjecture and several seemingly disparate theories. Using these ideas we will provide conditions, some necessary and some sufficient, for the Huneke-Wiegand conjecture to hold. As a culmination, we will discuss a recent proof of the speaker of the Auslander-Reiten conjecture for graded Gorenstein domains.


Shi Bai-November 10

 Enumeration-based Lattice Reduction

Shi Bai Florida Atlantic University

Lattice reduction algorithms have received much attention in recent years due to their relevance in cryptography. In this talk, we will discuss some of the recent developments on enumeration-based lattice reduction algorithms.

First, we will discuss a lattice reduction algorithm that achieves root Hermite factor k^(1/(2k)) in time k^(k/8 + o(k)) and polynomial memory. This improves the previously best known enumeration-based algorithms which achieve the same quality, but in time k^(k/(2e) +o(k)). The main idea is to conduct the preprocessing in a larger dimension than the enumerate dimension. Second, we discuss the usage of approximate enumeration oracles in BKZ, together with extended preprocessing ideas. In the end, we will illustrate some simulated results to demonstrate their practical behavior.

This talk is based on the following joint work:

M. R. Albrecht, S. Bai, P. A. Fouque, P. Kirchner, D. Stehlé, W. Wen: Faster Enumeration-Based Lattice Reduction: Root Hermite Factor k^(1/(2k)) Time k^(k/8+o(k)). CRYPTO (2) 2020: 186-212.

M. R. Albrecht, S. Bai, J. Li, J. Rowell: Lattice Reduction with Approximate Enumeration Oracles – Practical Algorithms and Concrete Performance. CRYPTO (2) 2021: 732-759.


Rafael D’Oliveira-November 3

Secure Distributed Matrix Multiplication

Rafael D’OliveiraClemson University

Matrix multiplication is, oftentimes, the most expensive computational task in an algorithm. It is the computational bottleneck for training many of the now well-celebrated learning algorithms, for example. To speed up the algorithm, the data can be distributed on many machines to perform the computations in parallel. This sharing of the data, however, raises security concerns when the data is sensitive and has to remain private, such as financial or medical data. Secure distributed matrix multiplication (SDMM) studies how to parallelize matrix multiplication while keeping the data secure.

In this talk, we present a combinatorial tool, called the degree table, and show how to utilize it to construct codes for SDMM which are currently the best performing for their parameters. I will also show lower bounds for this technique and characterize the total time complexity for SDMM codes, showing that if the parameters of the code are not chosen carefully, the total time might be larger than simply performing the computation locally.


Joseph Skelton- October 8

Symbolic Powers of Cover Ideals and the Koszul Property

Joseph SkeltonClemson University

Let G be a finite simple graph. This talk investigates the properties of symbolic powers of cover ideals associated to G. The results begin with combinatorially changing a given graph by whiskering (introducing a degree one vertex) a cycle cover of the graph. We show that this process forces all symbolic powers of the cover ideal J(G) to be Koszul or, equivalently, componentwise linear. We follow this result with a method of gluing two graphs, with Koszul cover ideals, together to obtain a new graph whose cover ideal, and all its symbolic powers, are Koszul. 

We then move to studying a spanning bipartite subgraph, B_G, of G. We first give conditions on J(G) which give that symbolic power J(G)^{(s)} is not componentwise linear for s>2. We then use this result and  B_G to give necessary and sufficient conditions on J(G) such that J(G)^{(s)} is componentwise linear for all s>1. In particular when G is a graph such that G\ N[A] has a simplicial vertex for any independent set A, or when G is a bipartite graph. 

This is joint work with Yan Gu, Tài Hà, and S. Selvaraja.


Selvi Kara- September 29

Blow-Up Algebras of Strongly Stable Ideals

Selvi KaraiUniversity of Utah

Let A be a polynomial ring and I1,…, Ir be a collection of ideals in A. The multi-Rees algebra R( I1,…, Ir) of this collection of ideals encode many algebraic properties of these ideals, their products, and powers. Additionally, the multi-Rees algebra  R( I1,…, Ir) arise in successive blowing up of Spec A at the subschemes defined by I1,…,Ir. Due to this connection, Rees and multi-Rees algebras are also called blow-up algebras in the literature.

In this talk, we will focus on Rees and multi-Rees algebras of strongly stable ideals. In particular, we will discuss the Koszulness of these algebras through a systematic study of these objects via three parameters: the number of ideals in the collection, the number of Borel generators of each ideal, and the degrees of Borel generators. In our study, we utilize combinatorial objects such as fiber graphs to detect Gröbner bases and Koszulness of these algebras. This talk is based on a joint work with Kuei-Nuan Lin and Gabriel Sosa.

Recording link: