Mathematics Spring 2013 PFUGS

Math 499 (Sec 1): Geometry of Harmonic Tones in Drums
Contact: Sinan Ariturk, ariturk@rice.edu
Course Page: Math 499 (Section 001)
Meeting: Wednesdays 12:00-12:50pm, TBA
The harmonic tones of a drum have a beautiful and complicated geometric structure that we will seek to understand. There is a profound relationship between the shape of the drum and the harmonic tones that the drum produces. One geometric object of particular importance is the nodal set, which is the set of points that remain stationary while the rest of the drum vibrates. The goal of our research project will be to understand what kind of shapes can and cannot occur as nodal sets.

Math 499 (Sec 3): Symbolic Dynamics
Contact: James Tanis, jtanis@rice.edu
Meeting: Fridays 12:00-12:50pm, TBA
We will learn the fundamentals of symbolic dynamics, and there are no prerequisites. Students are encouraged to be creative by coming up with their own exercises and problems and then solve them.
Those who are new to the subject will draw material from a textbook for the first half of the semester and, with the instructor's guidance, will pursue their own interests in the subject during the second half of the semester. Those who took the course in Fall 2012 are encouraged to try to obtain results or partial results on some open problems. We will discuss results from recent professional papers that are relevant or of interest.

Fall 2012 PFUGS

Math 499 (Sec 1): Geometry of Harmonic Tones in Drums
Contact: Sinan Ariturk, ariturk@rice.edu
Meeting: Mondays 12:00-12:50pm, HBH423
The harmonic tones of a drum have a beautiful and complicated geometric structure that we will seek to understand. There is a profound relationship between the shape of the drum and the harmonic tones that the drum produces. One geometric object of particular importance is the nodal set, which is the set of points that remain stationary while the rest of the drum vibrates. The goal of our research project will be to understand what kind of shapes can and cannot occur as nodal sets.

Math 499 (Sec 3): Symbolic Dynamics
Contact: James Tanis,jtanis@rice.edu
Meeting: Fridays 12:00-12:50pm, HBH423
One can model a physical system using symbolic dynamics by associating symbols to physical states and discretizing time. The symbolic system now evolves by shifting one symbol to another in discrete time intervals, in accordance with the evolution of the physical system in continuous time. Symbolic dynamics has found applications to thermodynamics and celestial mechanics in physics and to coding and linear algebra in mathematics, for example. We will study the fundamentals of this subject.

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: Fridays 3:00-4:30pm, HBH427

Math 699 (Sec 003): Derived categories and moduli
Contact: Brendan Hassett, hassett@rice.edu
Course Page: http://www.math.rice.edu/~hassett/teaching/699fall12/699main.html
This course will introduce the elements of derived categories with a view toward applications in moduli theory. Bridgeland stability conditions for moduli of complexes on the projective plane and K3 surfaces will be the main goal.

Math 699 (Sec 004): Seminar on Minimal Surfaces
Contact: Robert Hardt,hardt@rice.edu, Mike Wolf, mwolf@rice.edu
We are organizing a working seminar on "Minimal Surfaces". We plan for everyone to take turns working through sections of the recent AMS text by Colding and Minicozzi. This book gives a good introduction to many approaches and results from the classical roots, emphasizing the relation to complex analysis, to the authors' own recent powerful results about limits of sequences of minimal surfaces.

Math 699 (Sec 005): Spectral Theory Learning Seminar
Contact: Milivoje Lukic, milivoje.lukic@rice.edu

Spring 2012 PFUGS

MATH 499 (Sec 1): Introduction to p-adic Numbers
Contact: Evan Bullock, evanmb@rice.edu
Meeting: Tuesdays and Thursdays, 10:50-11:30am HB 453

Math 499 (Sec 4): Computing the Jones polynomial
Contact: Prudence Heck, ph6@rice.edu
Meeting: Wednesdays, 12:00-12:50pm HBH 453

MATH 499 (Sec 3): Low Dimensional Dynamics
Contact: James Tanis, jtanis@rice.edu
Meeting: Mondays, Wednesdays, Fridays, 12:00-12:50pm in HBH 453

Math 699 (Sec 1): Current Topics in Topology
Contacts: Tim Cochran, cochran@rice.edu; Shelly Harvey, shelly@rice.edu
Meeting: Tuesdays, 1:00-2:10pm HBH 453

Math 699: Toric Varieties
Contacts: Evan Bullock, Brian Lehmann, Tony Varilly-Alvarado

Math 699 (Sec 3): Spectral Theory
Contact: David Damanik, damanik @rice.edu
Meeting: Wednesdays, 12:00-12:50pm
We discuss recent work of ourselves and others in mathematical domains related to spectral theory and dynamical systems.

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.

Math 699 (Sec 1): Current Topics in Topology
Contacts: Tim Cochran, cochran@rice.edu; Shelly Harvey, shelly@rice.edu
Meeting: Tuesdays, 1:00-2:00pm HBH 423

Math 699 (Sec 4): Topics in Geometric Measure Theory
Contact: Robert Hardt, hardt@rice.edu, x3280
Meeting: Wednesdays, 12:00-12:50pm

Math 699 (Sec 5): Spectral Theory
Contact: David Damanik, damanik @rice.edu
Meeting: Wednesdays, 12:00-12:50pm
We discuss recent work of ourselves and others in mathematical domains related to spectral theory and dynamical systems.

Mathematics Summer 2011 PFUGS

Math 499: Geometric Calculus of Variations
Contact: Leobardo Rosales, lrosales@rice.edu, x4829

Math 499: Topics in Geometric Measure Theory
Contacts: Robert Hardt, hardt@rice.edu; Mike Wolf, mwolf@rice.edu

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