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Session A-3: Three-Act Math Tasks, Lindsey Herlehy 2017 Illinois Mathematics and Science Academy

Session A-3: Three-Act Math Tasks, Lindsey Herlehy

Professional Learning Day

Participants will engage in a Three-Act Math task highlighting the application of properties of geometrical figures. Developed by Dan Meyer, an innovative and highly regarded mathematics instructor, Three-Act Math tasks utilize pedagogical skills that elicit student curiosity, collaboration and questioning. By posing a mathematical problem through active storytelling, this instructional approach redefines real-world mathematics and clarifies the role that a student plays in the learning process. Participants will be given multiple resources where they can access Three-Act Math tasks appropriate for upper elementary grades through Algebra and Geometry courses.


Drawing A Triangle On The Thurston Model Of Hyperbolic Space, Curtis D. Bennett, Blake Mellor, Patrick D. Shanahan 2017 Loyola Marymount University

Drawing A Triangle On The Thurston Model Of Hyperbolic Space, Curtis D. Bennett, Blake Mellor, Patrick D. Shanahan

Blake Mellor

In looking at a common physical model of the hyperbolic plane, the authors encountered surprising difficulties in drawing a large triangle. Understanding these difficulties leads to an intriguing exploration of the geometry of the Thurston model of the hyperbolic plane. In this exploration we encounter topics ranging from combinatorics and Pick’s Theorem to differential geometry and the Gauss-Bonnet Theorem.


Computation Of Real Radical Ideals By Semidefinite Programming And Iterative Methods, Fei Wang 2016 The University of Western Ontario

Computation Of Real Radical Ideals By Semidefinite Programming And Iterative Methods, Fei Wang

Electronic Thesis and Dissertation Repository

Systems of polynomial equations with approximate real coefficients arise frequently as models in applications in science and engineering. In the case of a system with finitely many real solutions (the $0$ dimensional case), an equivalent system generates the so-called real radical ideal of the system. In this case the equivalent real radical system has only real (i.e., no non-real) roots and no multiple roots. Such systems have obvious advantages in applications, including not having to deal with a potentially large number of non-physical complex roots, or with the ill-conditioning associated with roots with multiplicity. There is a corresponding, but ...


On The Perfect Reconstruction Of The Structure Of Dynamic Networks, Alan Veliz-Cuba 2016 University of Dayton

On The Perfect Reconstruction Of The Structure Of Dynamic Networks, Alan Veliz-Cuba

Annual Symposium on Biomathematics and Ecology: Education and Research

No abstract provided.


Non-Commutative Automorphisms Of Bounded Non-Commutative Domains, John E. McCarthy, Richard M. Timoney 2016 Washington University in St Louis

Non-Commutative Automorphisms Of Bounded Non-Commutative Domains, John E. Mccarthy, Richard M. Timoney

Mathematics Faculty Publications

We establish rigidity (or uniqueness) theorems for non-commutative (NC) automorphisms that are natural extensions of classical results of H. Cartan and are improvements of recent results. We apply our results to NC domains consisting of unit balls of rectangular matrices.


On The Free And G-Saturated Weight Monoids Of Smooth Affine Spherical Varieties For G=Sl(N), Won Geun Kim 2016 The Graduate Center, City University of New York

On The Free And G-Saturated Weight Monoids Of Smooth Affine Spherical Varieties For G=Sl(N), Won Geun Kim

All Graduate Works by Year: Dissertations, Theses, and Capstone Projects

Let $X$ be an affine algebraic variety over $\mathbb{C}$ equipped with an action of a connected reductive group $G$. The weight monoid $\Gamma(X)$ of $X$ is the set of isomorphism classes of irreducible representations of $G$ that occur in the coordinate ring $\mathbb{C}[X]$ of $X$. Losev has shown that if $X$ is a smooth affine spherical variety, that is, if $X$ is smooth and $\mathbb{C}[X]$ is multiplicity-free as a representation of $G$, then $\Gamma(X)$ determines $X$ up to equivariant automorphism.

Pezzini and Van Steirteghem have recently obtained a combinatorial characterization of the weight ...


K-Theory Of Root Stacks And Its Application To Equivariant K-Theory, Ivan Kobyzev 2016 The University of Western Ontario

K-Theory Of Root Stacks And Its Application To Equivariant K-Theory, Ivan Kobyzev

Electronic Thesis and Dissertation Repository

We give a definition of a root stack and describe its most basic properties. Then we recall the necessary background (Abhyankar’s lemma, Chevalley-Shephard-Todd theorem, Luna’s etale slice theorem) and prove that under some conditions a quotient stack is a root stack. Then we compute G-theory and K-theory of a root stack. These results are used to formulate the theorem on equivariant algebraic K-theory of schemes.


Topological And Hq Equivalence Of Prime Cyclic P-Gonal Actions On Riemann Surfaces, Sean A. Broughton 2016 Rose-Hulman Institute of Technology

Topological And Hq Equivalence Of Prime Cyclic P-Gonal Actions On Riemann Surfaces, Sean A. Broughton

Mathematical Sciences Technical Reports (MSTR)

Two Riemann surfaces S1 and S2 with conformal G-actions have topologically equivalent actions if there is a homeomorphism h : S1 -> S2 which intertwines the actions. A weaker equivalence may be defined by comparing the representations of G on the spaces of holomorphic q-differentials Hq(S1) and Hq(S2). In this note we study the differences between topological equivalence and Hq equivalence of prime cyclic actions, where S1/G and S2/G have genus zero.


The Implicit Function Theorem And Free Algebraic Sets, Jim Agler, John E. McCarthy 2016 Washington University in St Louis

The Implicit Function Theorem And Free Algebraic Sets, Jim Agler, John E. Mccarthy

Mathematics Faculty Publications

We prove an implicit function theorem for non-commutative functions. We use this to show that if p ( X;Y ) is a generic non-commuting polynomial in two variables, and X is a generic matrix, then all solutions Y of p ( X;Y ) = 0 will commute with X


Aspects Of Non-Commutative Function Theory, Jim Agler, John E. McCarthy 2016 Washington University in St Louis

Aspects Of Non-Commutative Function Theory, Jim Agler, John E. Mccarthy

Mathematics Faculty Publications

We discuss non commutative functions, which naturally arise when dealing with functions of more than one matrix variable.


The Log-Exponential Smoothing Technique And Nesterov’S Accelerated Gradient Method For Generalized Sylvester Problems, N. T. An, Daniel J. Giles, Nguyen Mau Nam, R. Blake Rector 2016 Thua Thien Hue College of Education

The Log-Exponential Smoothing Technique And Nesterov’S Accelerated Gradient Method For Generalized Sylvester Problems, N. T. An, Daniel J. Giles, Nguyen Mau Nam, R. Blake Rector

Mathematics and Statistics Faculty Publications and Presentations

The Sylvester or smallest enclosing circle problem involves finding the smallest circle enclosing a finite number of points in the plane. We consider generalized versions of the Sylvester problem in which the points are replaced by sets. Based on the log-exponential smoothing technique and Nesterov’s accelerated gradient method, we present an effective numerical algorithm for solving these problems.


Minimizing Differences Of Convex Functions With Applications To Facility Location And Clustering, Mau Nam Nguyen, R. Blake Rector, Daniel J. Giles 2016 Portland State University

Minimizing Differences Of Convex Functions With Applications To Facility Location And Clustering, Mau Nam Nguyen, R. Blake Rector, Daniel J. Giles

Mathematics and Statistics Faculty Publications and Presentations

In this paper we develop algorithms to solve generalized Fermat-Torricelli problems with both positive and negative weights and multifacility location problems involving distances generated by Minkowski gauges. We also introduce a new model of clustering based on squared distances to convex sets. Using the Nesterov smoothing technique and an algorithm for minimizing differences of convex functions called the DCA introduced by Tao and An, we develop effective algorithms for solving these problems. We demonstrate the algorithms with a variety of numerical examples.


Spherical Tropicalization, Anastasios Vogiannou 2016 University of Massachusetts Amherst

Spherical Tropicalization, Anastasios Vogiannou

Doctoral Dissertations May 2014 - current

In this thesis, I extend tropicalization of subvarieties of algebraic tori over a trivially valued algebraically closed field to subvarieties of spherical homogeneous spaces. I show the existence of tropical compactifications in a general setting. Given a tropical compactification of a closed subvariety of a spherical homogeneous space, I show that the support of the colored fan of the ambient spherical variety agrees with the tropicalization of the closed subvariety. I provide examples of tropicalization of subvarieties of GL(n), SL(n), and PGL(n).


Algebraicity Of Rational Hodge Isometries Of K3 Surfaces, Nikolay Buskin 2016 University of Massachusetts Amherst

Algebraicity Of Rational Hodge Isometries Of K3 Surfaces, Nikolay Buskin

Doctoral Dissertations May 2014 - current

Consider any rational Hodge isometry

$\psi:H^2(S_1,\QQ)\rightarrow H^2(S_2,\QQ)$ between any two K\"ahler $K3$

surfaces $S_1$ and $S_2$. We prove that the cohomology class of $\psi$ in $H^{2,2}(S_1\times S_2)$

is a polynomial in Chern classes of coherent analytic sheaves

over $S_1 \times S_2$. Consequently, the cohomology class of $\psi$ is algebraic

whenever $S_1$ and $S_2$ are algebraic.


Topology Of The Affine Springer Fiber In Type A, Tobias Wilson 2016 University of Massachusetts - Amherst

Topology Of The Affine Springer Fiber In Type A, Tobias Wilson

Doctoral Dissertations May 2014 - current

We develop algorithms for describing elements of the affine Springer fiber in type

A for certain 2 g(C[[t]]). For these , which are equivalued, integral, and regular,

it is known that the affine Springer fiber, X, has a paving by affines resulting from

the intersection of Schubert cells with X. Our description of the elements of Xallow

us to understand these affine spaces and write down explicit dimension formulae. We

also explore some closure relations between the affine spaces and begin to describe the

moment map for the both the regular and extended torus action.


Skein Theory And Algebraic Geometry For The Two-Variable Kauffman Invariant Of Links, Thomas Shelly 2016 University of Massachusetts - Amherst

Skein Theory And Algebraic Geometry For The Two-Variable Kauffman Invariant Of Links, Thomas Shelly

Doctoral Dissertations May 2014 - current

We conjecture a relationship between the Hilbert schemes of points on a singular plane curve and the Kauffman invariant of the link associated to the singularity. Specifcally, we conjecture that the generating function of certain weighted Euler characteristics of the Hilbert schemes is given by a normalized specialization of the difference between the Kauffman and HOMFLY polynomials of the link. We prove the conjecture for torus knots. We also develop some skein theory for computing the Kauffman polynomial of links associated to singular points on plane curves.


Quasi-Platonic Psl2(Q)-Actions On Closed Riemann Surfaces, Sean A. Broughton 2015 Rose-Hulman Institute of Technology

Quasi-Platonic Psl2(Q)-Actions On Closed Riemann Surfaces, Sean A. Broughton

Mathematical Sciences Technical Reports (MSTR)

This paper is the first of two papers whose combined goal is to explore the dessins d'enfant and symmetries of quasi-platonic actions of PSL2(q). A quasi-platonic action of a group G on a closed Riemann S surface is a conformal action for which S/G is a sphere and S->S/G is branched over {0, 1,infinity}. The unit interval in S/G may be lifted to a dessin d'enfant D, an embedded bipartite graph in S. The dessin forms the edges and vertices of a tiling on S by dihedrally symmetric polygons, generalizing the ...


Quasi-Platonic Psl2(Q)-Actions On Closed Riemann Surfaces, Sean Broughton 2015 Rose-Hulman Institute of Technology

Quasi-Platonic Psl2(Q)-Actions On Closed Riemann Surfaces, Sean Broughton

S. Allen Broughton

This paper is the first of two papers whose combined goal is to explore the dessins d'enfant and symmetries of quasi-platonic actions of PSL2(q). A quasi-platonic action of a group G on a closed Riemann S surface is a conformal action for which S/G is a sphere and S->S/G is branched over {0, 1,infinity}. The unit interval in S/G may be lifted to a dessin d'enfant D, an embedded bipartite graph in S. The dessin forms the edges and vertices of a tiling on S by dihedrally symmetric polygons, generalizing the ...


Integrability And Regularity Of Rational Functions, Greg Knese 2015 Washington University in St. Louis

Integrability And Regularity Of Rational Functions, Greg Knese

Mathematics Faculty Publications

Motivated by recent work in the mathematics and engineering literature, we study integrability and non-tangential regularity on the two-torus for rational functions that are holomorphic on the bidisk. One way to study such rational functions is to fix the denominator and look at the ideal of polynomials in the numerator such that the rational function is square integrable. A concrete list of generators is given for this ideal as well as a precise count of the dimension of the subspace of numerators with a specified bound on bidegree. The dimension count is accomplished by constructing a natural pair of commuting ...


Quasi-Platonic Psl(2,Q)-Actions On Closed Riemann Surfaces, Sean Broughton 2015 Rose-Hulman Institute of Technology

Quasi-Platonic Psl(2,Q)-Actions On Closed Riemann Surfaces, Sean Broughton

S. Allen Broughton

This paper is the first of two papers whose combined goal is to explore the dessins d'enfant and symmetries of quasi-platonic actions of PSL(2,q). A quasi-platonic action of a group G on a closed Riemann S surface is a conformal action for which S/G is a sphere and S -> S/G is
branched over {0,1,infinity}. The unit interval in S/G may be lifted to a dessin d'enfant D, an embedded bipartite graph in S.The dessin forms the edges and vertices of a tiling on S by dihedrally symmetric polygons, generalizing the ...


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