# TTU Algebra Seminar

## Next seminar

Friday 20 November, 3–4 pm  in MA 111

### Ming Zhao (U. Houston)   "Mixing sets: Complexity and Applications"

Abstract.  Mixing sets arise as an important substructure in general mixed-integer programming. In this talk, I will present two mixing sets. For the first one, I will give an algebraic characterization of all the extreme points. Although the number of extreme points grows exponentially in the worst case, we can still find the optimal solution within polynomial time. The other mixing set is an essential substructure underlying chance-constrained programming problems with finite discrete distributions. I will show families of strong inequalities that can tighten the formulation and improve the computational time in branch-and-cut algorithm.

TTU Department
of Mathematics
and Statistics

## Fall 2015

### Ming Zhao (U. Houston)   "Mixing sets: Complexity and Applications"

Abstract. Mixing sets arise as an important substructure in general mixed-integer programming. In this talk, I will present two mixing sets. For the first one, I will give an algebraic characterization of all the extreme points. Although the number of extreme points grows exponentially in the worst case, we can still find the optimal solution within polynomial time. The other mixing set is an essential substructure underlying chance-constrained programming problems with finite discrete distributions. I will show families of strong inequalities that can tighten the formulation and improve the computational time in branch-and-cut algorithm.

### James Staff   "The Compact Lie Groups $$SU(3)$$ and $$G_2$$"

Abstract. The group of automorphisms of the real octonion algebra $$\mathbb{O}$$ is the compact form of the exceptional Lie group $$G_2$$. We can view the group $$SU(3)$$ as the stabilizer of a point in the sphere $$S^6$$ of imaginary units in $$\mathbb{O}$$, which exhibits $$G_2$$ as an $$SU(3)$$-principal bundle over $$S^6$$. These well-known facts already paint a nice picture of the structure of $$G_2$$, and the explicit embeddings obtained in this way should be useful to describe the moduli space of flat principal connections on a $$G_2$$-principal bundle over a standard torus $$\Sigma \cong S^1 \times S^1$$. Such moduli spaces appear prominently in Chern-Simons theory. It is a work in progress to explicitly compute the Weyl quantization of this moduli space for the gauge group $$SU(3)$$. If time permits, we will discuss how these results can be used to obtain the quantization for the gauge group $$G_2$$.

### Odin Jesse   "Real Radical Ideals with Singular"

Abstract. This talk will serve as a casual follow-up to some of the questions that arose during Dr. de Farias' talk last week. We'll discuss the moment matrix approach to the computation of real radical ideals based on Lasserre and Laurent's 2009 work. We'll do some small examples by hand then turn to software for more interesting cases. We'll talk about the "realrad" package in Singular and use it to compute the real radicals of ideals that can assist in optimization problems. We'll also take a look at some seemingly small ideals for which the real radical algorithm can't produce a result in a realistic amount of time on average desktop hardware.

### Ismael Regis de Farias Jr.   "Challenging problems in computational polynomial optimization"

Abstract. The problem of minimizing a polynomial function over a semi-algebraic set is one of the most important in optimization, both in theory as well as in practice. It turns out that polynomial optimization is a robust model, and it includes as a special case, such problems as 0-1 mixed-integer quadratic programming. The use of algebraic methods, both theory and computational, is clearly possible. However, this is a largely unexplored approach. I will present a few of these problems and thoughts as to how they can benefit from theory and computational methods from algebraic geometry.

### Odin Jesse   "Theta Bodies and Classification of SOS Ideals"

Abstract. If a polynomial can be written as the sum of squares of other polynomials then it is certainly non-negative over the reals. More generally, if a polynomial is congruent to a sum of squares in a quotient ring for an ideal then it is non-negative over the real variety of the ideal. It turns out that the converse is true in certain nice cases. In this talk we'll motivate this study of non-negativity with some applications in optimization. We'll introduce some tools from real algebraic geometry including the Real Radical of an ideal, a real version of the Nullstellensatz, and the Theta Body hierarchy of an ideal. These can be used to study the SOS structure of polynomial ideals. An ideal is called $$(d,k)$$-SOS if every polynomial of degree $$\leq d$$ that is non-negative over the variety of the ideal can be written not only as a sum of squares, but as a sum of squares of polynomials of degree $$\leq k$$. Ideals with this property, especially the $$(1,1)$$-SOS ideals, are cooperative with a number of optimization techniques that use Semi-Definite Programming algorithms. A classification of ideals by $$(d,k)$$-SOS properties is an open problem. We'll look at some applications that demonstrate why the classification is interesting and we'll conclude with a discussion of the situations in which a classification is known.

### Ismael Regis de Farias Jr.   "Theta bodies for polynomial ideals"

Abstract. I will review the recent work of Gouveia, Parrilo, and Thomas on a semidefinite programming hierarchy of relaxations for polynomial optimization. I will define and motivate Theta bodies, and give some of its main properties. In the end I will present a few open problems on polynomial optimization related to Theta bodies.

### Ismael Regis de Farias Jr.   "Connections between algebra and optimization"

Abstract. This is the first of n (where n is a positive integer) presentations that will explore the virtually unexplored synergism between algebra and optimization. In this one, I will give a gentle introduction to the field of optimization and will hint a few possibilities of interaction with algebra.

### Chris Monico   "Root-finding over finite fields"

Abstract. In this elementary talk, aimed at graduate students, we will present a well-known technique for finding roots of univariate polynomials over finite fields.

### Lars Winther Christensen  "A variation on Baer's criterion II"

Abstract. Baer's criterion says, when one phrases it in cohomological terms, that injectivity of a module can be detected by vanishing of Ext against cyclic modules. I will discuss a recent result that says that one can replace cyclic modules with fields; it comes out of joint work with Srikanth Iyengar.

### Lars Winther Christensen  "A variation on Baer's criterion I"

Abstract. Baer's criterion says, when one phrases it in cohomological terms, that injectivity of a module can be detected by vanishing of Ext against cyclic modules. I will discuss a recent result that says that one can replace cyclic modules with fields; it comes out of joint work with Srikanth Iyengar.

Organizer:  Lars Winther Christensen