Sums of Squares in Extremal Combinatorics
I will explain how sums of squares in the gluing algebra of partially labelled graphs can be used to prove graph density inequalities in extremal combinatorics. This is the essence of semidefinite method of Lovasz and Szegedy, and also Cauchy-Schwartz calculus of Razborov. I will introduce the sums of squares method, do several examples of sums of squares proofs, and then present simple explicit inequalities that show strong limitations of the sums of squares method. No background in combinatorics is necessary. This is joint work with Annie Raymond, Mohit Singh and Rekha Thomas.
Code Constructions for Distributed Storage
Iwan Duursma — University of Illinois at Urbana-Champaign
(Joint ADM/RTG Seminar)
Storage solutions for storing data on a single disk or a combination of several disks do not perform well at the scale of data centers that employ thousands of servers. Over the last decade two new families of error-correcting codes have been introduced that better address the requirements for large scale distributed storage of data. We discuss the main parameters for these families and present a variety of algebraic and graph theoretic methods that can be used to construct codes with good trade offs among the main parameters.
Shuffle-Compatible Permutation Statistics
It has been observed since the early work of Richard Stanley that several well-known permutation statistics are “compatible” with the operation of shuffling permutations. In joint work with Ira Gessel, we formalize this notion of a shuffle-compatible permutation statistic and develop a unifying framework for studying shuffle-compatibility, which has close connections to the theory of P-partitions, quasisymmetric functions, and noncommutative symmetric functions. In this talk, I will survey the main results of our work as well as several new directions of research concerning shuffle-compatibility.
The power edge ideal of a finite simple graph
Every electric power system can be modeled by a graph G whose vertices represent electrical buses and whose edges represent power lines. A phasor measurement unit (PMU) is a monitor that can be placed at a bus to observe the voltage at that bus as well as the current and its phase through all incident power lines. The problem of monitoring the entire electric power system using the fewest number of PMUs is closely related to vertex covering and dominating set problems in graph theory.
In this talk, we will give an overview of the PMU placement problem and its connections to commutative ring theory. By defining the power edge ideal IPG of a graph G, we will show how to use graphs of electric power grids to generate polynomial rings with desired algebraic properties. In particular, we will classify the trees G for which IPG is Cohen-Macaulay and prove that every such ideal is also a complete intersection.
This project is joint work with Alan Hahn, Frank Moore, and Sean Sather-Wagstaff.
Mixed Integer Polynomial Optimization – Algorithms and Complexity for Linear and Non-Linear Cases when the Dimension is Fixed
(Joint AMD and OR Seminar)
Integer Programming (or Integer Optimization) is the problem if optimizing an objective function over a set of feasible points that have the restriction that some variables must take integer values. Integer linear programming (optimizing a linear function over linear constraints and integer constraints) is NP-Hard, while the continuous counterpart, Linear Programming, is polynomial time solvable. Since the linear case is NP-Hard, the non-linear case of integer programming is also NP-Hard. The story becomes much different when the dimension is considered a fixed number. For instance, what is the complexity of integer linear programming when there are only 3 variables? Lenstra proved in the 80’s that in fact, for any fixed dimension, integer linear programming can be solved in polynomial time. This result hinges on the lattice basis reduction algorithms such as the famous LLL algorithm. Surprisingly, the question of complexity in fixed dimension is very unclear even for quadratic integer programming. We will survey Lenstra-type algorithms for integer programming and show recent results on convex and non-convex mixed integer programming.
Random groups and cubulations
Yen Duong — Invited AWM speaker
Special cube complexes have been a hot topic for a few years now for their versatility and usefulness. Following a construction of Sageev we can cubulate all sorts of groups, and specifically I’ll explain Gromov’s density model of random groups, the square model of random groups, and examples of Sageev’s cubulation with random groups, and why we would want to do all this.
Factorization and the Half-Factorial Property
The notion of unique factorization is central in commutative algebra and number theory (with very important ramifications spilling over into almost every branch of mathematics). In general, “factorization” may be considered the study of the multiplicative decomposition(s) of elements in a ring (usually an integral domain for our purposes), and generalizations of unique factorization are of much interest.
One of the most infamous generalizations of unique factorization is the half-factorial property. Loosely speaking, a half-factorial domain is an integral domain in which elements may not factor uniquely, but any two decompositions of the same nonzero element into irreducibles will have the same number of irreducibles involved (counting possible multiplicities). The aim of this talk will be to give an overview of this generalization of unique factorization from its inception through some very recent results. Along the way, the talk will be seasoned and flavored with many examples comparing and contrasting the half-factorial property with the more familiar notion of unique factorization. Graduate students are especially encouraged to have their lives changed by this talk!
Efficient Fully Homomorphic Encryption Schemes
As cloud computing, internet of things (IoT) and blockchain technology become increasingly prevalent, there is an urgent need to protect the privacy of massive volumes of sensitive data collected or stored in distributed computer networks or cloud servers, as many of the networks or servers can be vulnerable to external and internal threats such as malicious hackers or curious insiders. The Holy-Grail of cryptography is to have practical fully homomorphic encryption (FHE) schemes that allow any third party (including cloud servers, hackers, miners or insiders) to perform searching or analytics of an arbitrary function on encrypted data without decryption, while no information on the original data is ever leaked. The breakthrough was made by Gentry in 2009 who discovered the first FHE scheme, and since then many improvements have been made on designing more efficient homomorphic encryption schemes. The main bottlenecks are in bootstrapping and large cipher expansion factor (the size ratio of ciphertexts over messages): the current best FHE schemes compute bootstrapping of one bit operation in 0.013 second and still have a cipher expansion factor of 10,000. In this talk, we present a compact FHE scheme whose bootstrapping speed is slightly slower but whose cipher expansion factor is only 6 under private-key encryption and 20 under public-key encryption, hence practical in term of storage.
Minimal Ramification Problem
Minimal ramification problem can be considered as a quantitative version of Inverse Galois Problem. The latter is a question that asks whether every finite group is a Galois group of a Galois extension of the field of rational numbers Q. It is one of the important problems in Field Arithmetic and Algebraic Number Theory. For a given finite group G, let Ram(G) be the minimal integer for which there exists a Galois extension of N/Q that ramifies exactly at Ram(G) primes. The task in minimal ramification problem is to compute or to bound Ram(G). In this talk, I will give an overview of the problem and some results, and then we will talk about current and future research.