Mathematics Fall 2011 PFUGS

MATH 499 (Sec 1): Singularities of Plane Curves
Contact: Evan Bullock, evanmb@rice.edu
Organizational Meeting: Wednesday, August 31 12:00pm HB 423
This seminar will tackle some basic problems in algebraic geometry using computational methods, but without a heavy emphasis on theory. The goal is to introduce students to some of the problems and methods of algebraic geometry without necessarily requiring them to understand some of the more technical details involved. In particular, we will be looking at plane curves and plane curve singularities, and performing calculations to classify various types of plane curves and their singularities. This will give students an introduction to some of the basic techniques of algebraic geometry, should they wish to continue in this area. Also, it will show how computational and algorithmic methods can be applied to mathematical research.

MATH 499 (Sec 3): Symbolic Dynamics
Contact: James Tanis, jtanis@rice.edu
Organizational Meeting: Friday, August 26 12:00pm HBH 423
Symbolic dynamics studies the behavior of symbolic systems as time moves forward or backward. It can be used to model physical phenomena such as magnetism and thermodynamic pressure, and it is also useful in coding. In our research project we will explore dynamical properties such as chaos, entropy, and the existence of periodic and dense orbits in some representative examples of symbolic systems.
The suggested prerequisite is Math 211. This is a two credit course that meets two hours per week. We will decide a schedule at the organizational.

Math 499 (Sec 4): Computing the Jones polynomial
Contact: Prudence Heck, ph6@rice.edu
Meeting: Wednesdays, 12:00-12:50pm HBH 423
The Jones polynomial has had a striking impact on knot theory. Yet its very definition makes it hard to compute, and no good geometric interpretation exists. S. Bigelow, following work of R. Lawrence, gave a new definition of the Jones polynomial, hinting at a topological interpretation. This PFUG seeks a method for computing the Jones polynomial from Lawrence and Bigelow?s definition. The only prerequisite is Math 212, although students should be comfortable working with matrices.

Mathematics Spring 2011 PFUGS

MATH 499 (Sec 1): Singularities of Plane Curves
Contact: Evan Bullock, evanmb@rice.edu, x4829

Math 499 (Sec 2): Geometric Calculus of Variations
Contact: Leobardo Rosales, lrosales@rice.edu, x4829
Course Page: http://www.owlnet.rice.edu/~lr7/VIGRE/
Description: The goal of the Fall VIGRE Geometric Calculus of Variations group is to study the boundary-weighted area functional. Part of the course will involve an introduction to the classical area minimization problem: given a boundary curve, find the surface which minimizes area amongst all surfaces with that given boundary. In the boundary-weighted area problem we require that our surfaces have boundaries consisting of at least a fixed curve, and we penalize by introducing a weight given by the size of the boundary.
This course can be taken for three units. We shall meet once a week for two hours, and a report shall be due at the end of the semester.
Prerequisites: Math 212, Math 211 recommended

Math 499 (Sec 4): Shadow Links II
Contact: Prudence Heck, Prudence.Heck@rice.edu, x4829
Description: Several important invariants of knots in 3-dimensional Euclidean space may be calculated via projections of knots in the plane with marked crossings. While general 3-manifolds do not project to a plane, we can, in some cases, still obtain certain knot diagrams called shadow knots. In this PFUG we will continue our attempt to recover certain invariants of knots in Euclidean space from their shadow diagrams and, if time permits, generalize these invariants to knots in other 3-manifolds. This PFUG should be accessible to students who did not participate in the Fall semester.

Math 699 (Sec 1): Current Topics in Topology
Contacts: Shelly Harvey and Tim Cochran
Seminar Page: http://math.rice.edu/~shelly/top_working//
Class Time and Location: Tuesdays 1:00-2:00pm

Math 699 (Sec 2): Research Methods in Algebraic Geometry
Contacts: Brendan Hassett and Zhiyuan Li
Seminar Page:http://math.rice.edu/~zl3/ags2010_fall/index.html
Class Time and Location: Tuesdays and Thursdays 10:50am - 12:05am
Description: In Fall 2010, this seminar has two components: Tuesday meetings are devoted to foundational results and key examples of Geometric Invariant Theory, with lectures by Brendan Hassett. The Algebraic Geometry Working seminar meets Thursdays and is coordinated by Zhiyuan Li. It features student lectures on the Hodge theory of compact Kaehler manifolds.

Math 699 (Sec 3): Spectral Theory Brown Bag Seminar
Contacts: David Damanik
Seminar Page: http://www.ruf.rice.edu/~dtd3/Brown-Bag-Seminar.htm
Class Time and Location: Wednesdays 12:05-1:00pm

Math 699 (Sec 4): Multiple_Valued Functions
Contact: Robert Hardt, hardt@rice.edu, x3280
Schedule: http://math.rice.edu/~zz1/math699/
Class Time and Location: Monday and Wednesdays 12:00-12:50pm in HB453
Description: A multiple-valued energy minimizing function is a generalization of an ordinary single-valued harmonic function. They are used for variational problems and to approximate mass minimizing chains of arbitrary dimension. We will derive their elementary analytic and geometric properties, following the book of Almgren and recent papers by De Lellis and Spandora.

Mathematics Fall 2010 PFUGS

MATH 499 (Sec 1): Singularities of Plane Curves
Contact: Evan Bullock, evanmb@rice.edu, x4829
Course Page: http://math.rice.edu/~evanmb/math499fall10/
Class Time and Location: Thursdays, 2:30-3:20pm in HBH453
Description: In this course we will study singularities of plane curves (the solution sets to polynomial equations f(x,y)=0 in two variables) using techniques of computational algebraic geometry.
Prerequisites: Math 212/221 and 355/354

Math 499 (Sec 2): Surfaces Minimizing Boundary
Contact: Leobardo Rosales, lrosales@rice.edu, x4829
Course Page: http://www.owlnet.rice.edu/~lr7/VIGRE/
Class Time and Location: Tuesdays 4:00-6:00pm in HB453
Description: The goal of the Fall VIGRE Geometric Calculus of Variations group is to study the "boundary-weighted area functional." Part of the course will involve an introduction to the classical area minimization problem: given a boundary curve, find the surface which minimizes area amongst all surfaces with that given boundary. In the boundary-weighted area problem we require that our surfaces have boundaries consisting of at least a fixed curve, and we penalize by introducing a weight given by the size of the boundary .

Math 499 (Sec 3): Semiclassical limit in Quantum Mechanics and Nonlinear Waves
Contact: Sergey Belov, belov@rice.edu, x4829
Course Page: http://math.rice.edu/~smb4/Math499Fall2010/
Class Time and Location: Tuesdays 12:10-12:50pm in HB453
Description: The semiclassical limit is taken in Quantum mechanics to connect with the classical mechanics. Curiously, the zero-dispersion limit in Nonlinear waves leads in many cases to similar expressions as in the semiclassical limit. We will investigate these relations and discuss some examples of nonlinear equations in the semiclassical limit (NLS, KdV).
Prerequisites: Math 211 and some complex variables; familiarity with proofs (Math 221), Complex analysis (Math 382), PDEs (Math 381) and Matlab is helpful but not required

Math 499 (Sec 4): Shadow Links
Contact: Prudence Heck, Prudence.Heck@rice.edu, x4829
Class Time and Location: Mondays 12:00-12:50pm in HB427
Course Page: http://math.rice.edu/~ph6/VIGRE/
Description: Shadow links are diagrams of links in certain 3-manifolds. That is, links in spaces that locally look like 3-dimensional Euclidean space, but globally are quite different. We will attempt to recover certain invariants of knots in 3-dimensional Euclidean space via shadow links and, if time permits, generalize these invariants to knots in other 3-manifolds.

Math 499 (Sec 5): Spatial Graphs
Contact: Danielle O'Donnol, odonnol@rice.edu, x4829
Class Time and Location: Fridays 12:00-12:50pm in HB453
Description: The study of spatial graphs combines aspects of combinatorics and knot theory. In this class we will learn about spatial graphs and tackle some open problems concerning spatial graphs.
Prerequisites: One of MATH 221 Honors Calculus, or any 300+ math class (so that you have experience with proof based homework assignments).

Math 699 (Sec 001): Current Topics in Topology
Contacts: Shelly Harvey and Tim Cochran
Seminar Page: http://math.rice.edu/~shelly/top_working
Class Time and Location: Tuesdays 1:00-2:00pm in HB453

Mathematics Spring 2010 PFUGS

MATH 499 (Sec 1): Singularities of Plane Curves and Groebner Bases
Contact: Evan Bullock, evanmb@rice.edu, x2372.
Course Page: http://math.rice.edu/~evanmb/math499spring10/

MATH 499 (Sec 2): A Minimization Model of Time Step Cell Network Evolution
Contact: Leobardo Rosales, lrosales@rice.edu, x4896
Course Page: http://www.owlnet.rice.edu/~lr7/VIGRE/

MATH 499 (Sec 4): Knots and 3 Manifolds
Contact: Prudence Heck, Prudence.Heck@rice.edu, x3283

Math 699 (Sec 001): Current Topics in Topology
Contacts: Shelly Harvey and Tim Cochran
Seminar Page:http://math.rice.edu/~shelly/top_working
Class Time and Location: Tuesdays 1:00-2:00pm in HB453

Math 699 (Sec 3): Brown Bag Seminars - We discuss recent work of ourselves and others in mathematical domains related to spectral theory and dynamical systems.

Math 699 (Sec 4): This seminar will again meet twice a week. The Tuesday meeting (2:30 HB227), led by Leo Rosales, will involve the study of Allard's regularity theorem for varifolds following "Lectures on Geometric Measure Theory " by L.Simon. The Thursday meeting (12:00 HB453), led by Robert Hardt, will involve multiple talks by the participants on rectifiable currents and their regularity as solutions of geometric variational problems.

MATH 699 (Sec 5): This seminar, led by Anthony V'arilly-Alvarado, will serve as a continuation of the algebraic stacks seminar running this semester, with an emphasis on the technicalities of category theory, descent, Grothendieck topologies, etc. This will meet twice a week on Tuesday and Thursday from 10:50-Noon.

MATH 699 (Sec 6): This seminar, led by Benjamin Waters, will cover Kleiman's paper, "Toward a numerical theory of ampleness" and transition into Debarre's book, "Higher-Dimensional Algebraic Geometry". Participants in this seminar will give talks to each other and share solutions to exercises in Debarre's book. It will meet twice a week on Monday and Friday from 4:00-5:00pm.

Mathematics Fall 2009 PFUGS

MATH 499 (Sec 1): Computational Algebraic Geometry
Contact: Evan Bullock, evanmb@rice.edu, x2372
Seminar Page: math.rice.edu/~evanmb/math499fall09/
Class Time and Location: Thursdays 12:10pm in HB423

MATH 499 (Sec 2): Geometric Calculus of Variations
Contact: Leobardo Rosales, lrosales@rice.edu, x4896
Seminar Page: www.owlnet.rice.edu/~lr7/vigre/
Class Time and Location: Thursdays 9:00-10:50am in HB438

MATH 499 (Sec 4): Legendrian Knots and Links
Contact: Prudence Heck, Prudence.Heck@rice.edu, x3283
Seminar Page: math.rice.edu/~ph6/VIGRE/
Class Time and Location: 12:00-12:50pm in HB427

Math 699 (Sec 1): Current Topics in Topology
Contact: Tim Cochran, cochran@rice.edu, x5265
              Shelly Harvey, shelly@rice.edu, x3659
Seminar Page: math.rice.edu/~shelly/top_working/
Class Time and Location: Tuesdays 1:00-2:00pm in HB453

Math 699 (Sec 3): Brown Bag Seminar
Contact: David Damanik, damanik@rice.edu, x3273
Seminar Page: www.ruf.rice.edu/~dtd3/Brown-Bag-Seminar.htm
Class Time and Location: Wednesdays 12:00-1:00pm in HB423

Math 699 (Sec 5): Algebraic Stacks
Contact: Evan Bullock, evanmb@rice.edu, x2372
             Anthony Várilly-Alvarado, av15@rice.edu, x4597
Seminar Page: math.rice.edu/~av15/AlgStacks.html
Class Time and Location: Mondays and/or Fridays 1:00-2:50pm in AL B209

Math 699 (Sec 001): Current Topics in Topology
Contacts: Shelly Harvey and Tim Cochran
Seminar Page: http://math.rice.edu/~shelly/top_working
Class Time and Location: Tuesdays 1:00-2:00pm in HB453

Mathematics Spring 2009 PFUGS

MATH 499 (Sec 2): Edge Length Minimizing Polyhedra
Contact: Leobardo Rosales, lrosales@rice.edu, x4896
Seminar Page: www.owlnet.rice.edu/~lr7/vigre/
Class Time and Location: Mondays and Wednesdays 12:10-12:50pm in Mart101
                                      Fridays 4:00-5:00pm in HB453

MATH 499 (Sec 6): Michell Trusses
Contact: Rolf Ryham, ryham@rice.edu, x2385
Seminar Page: www.owlnet.rice.edu/~rjr1/michell.html
Class Time and Location: Tuesdays 3:00-5:00pm in AL B209
                                      Fridays 4:00-5:00pm in AL B209

Math 699 (Sec 1): Current Topics in Topology
Contact: Tim Cochran, cochran@rice.edu, x5265
              Shelly Harvey, shelly@rice.edu, x3659
Seminar Page: math.rice.edu/~shelly/top_working/
Class Time and Location: Tuesdays 1:05-2:20pm in HB453

Math 699 (Sec 3): Brown Bag Seminar
Contact: David Damanik, damanik@rice.edu, x3273
Class Time and Location: Wednesdays 12:00-1:00pm in HB423

Math 699 (Sec 4): Geometric Measure Theory
Contact Leobardo Rosales, lrosales@rice.edu, x4896
Class Time and Location: Thursdays 10:50am-12:05pm in HB438

Math 699 (Sec 5): Representation Theory
Contact Brendan Hassett, hassett@rice.edu, x5261
            Elizabeth Dan-Cohen, edc@rice.edu, x5319
Class Time and Location: Mondays and Fridays 2:00-2:50pm in HB423

Math 699 (Sec 6): Seminar on Dynamical Systems
Contact Jonathan Fickenscher, jonfick@rice.edu, x2786
Class Time and Location: Tuesdays 12:05-1:00pm in HB453

Mathematics Fall 2008 PFUGS