Graduate Student Seminar, Department of Mathematics, University of Notre Dame, 2013-2014 Graduate Student Seminar, Department of Mathematics, University of Notre Dame, 2013-2014 Graduate Student Seminar, Department of Mathematics, University of Notre Dame, 2013-2014 Graduate Student Seminar, Department of Mathematics, University of Notre Dame, 2013-2014 Graduate Student Seminar, Department of Mathematics, University of Notre Dame, 2013-2014 Graduate Student Seminar, Department of Mathematics, University of Notre Dame, 2013-2014

# Notre Dame Math Graduate Student Seminar, 2013-2014

The Graduate Student Seminar is put on by the Mathematics Graduate Student Association . GSS meets approximately every other Monday.

All talks are at 4:15 in HH229 (Fall)/ HH229 (Spring) unless otherwise noted.

Previous Semesters

To volunteer to give a talk, or for anything else regarding the seminar, contact Alexander Diaz Lopez.

## Schedule Fall 2013

Date Speaker Title
Monday, September 2 Brian Hall Classical and Semiclassical Limits in Quantum Mechanics
Monday, September 16 Lien-Yung Kao A Weil-Petersson Type (Pressure) Metric of Metric Graphs
Monday, September 30 Renato Ghini Bettiol Basic Bifurcation Theory and Geometric Variational Problems
Monday, October 14 Peter Ulrickson Poincare, His Duality Theorem, and the Beginnings of Algebraic Topology
Monday, November 4 Edward Burkard From Classical Mechanics to Symplectic Geometry
Monday, December 9 Jennifer Garbett Minkowski's Theorem

## Schedule Spring 2014

Date Speaker Title
Monday, February 3 Eric Wawerczyk Universal Quadratic Forms and Arithmetic in Quaternion Algebras
Monday, February 10 Ben Lewis Constructing a Quasi-Steady State Using Hamiltonian Flow
Monday, March 3 Luis Saumell Hilbert Scheme of Points of the Affine Space
Monday, March 17 Augusto Stoffel The Pontryagin-Thom Construction
Monday, March 31 Dominic Culver Generalized (co)homology theories and Brown's Representability Theorem
Monday, April 7 Panel Discussion Job Related Topics. Panelists: Nicole Kroeger, David Cook II, Brian Shourd and Yueh-Ju Lin
Monday, April 14 Xumin Jiang Boundary expansion for Singular Yamabe problem
Monday, April 28 Amy Buchmann Flow Induced by Bacterial Carpets and Transport of Microscale Loads

## Abstracts

### September 2, 2013

Speaker
Brian Hall
Title
Classical and Semiclassical Limits in Quantum Mechanics
Abstract

On the surface, quantum mechanics and classical mechanics appear totally different, with a classical particle being described by a trajectory in R^2n and a quantum particle by a "wavefunction" on R^n. Nevertheless, there is a widespread belief that in the limit as Planck's constant (which is a basic parameter in quantum mechanics) tends to zero, one should somehow recover classical mechanics. I will talk about how this idea works out for the energy spectrum of a particle moving in the real line. A simple result shows that the quantum energy spectrum fills in the entire range of the classical energy function for small values of Planck's constant; this is a form of the "classical limit." A deeper result is "semiclassical": the individual energy levels of the quantum system (which are inherently quantum mechanical quantities) can be determined approximately by a condition on the underlying classical trajectory.

No prior knowledge of quantum mechanics will be assumed (or required), and there will be plenty of pictures.

### September 16, 2013

Speaker
Lien-Yung Kao
Title
A Weil-Petersson Type (Pressure) Metric of Metric Graphs
Abstract

On the fertile ground of the Teichmüller-Thurston theory, we are fascinated by the classical Weil-Petersson metric. Following the pioneer works of S. Wolpert[7], M. Bridgeman[1, 2] and C. McMullen[4], we have a new dynamical, thermodynamic formalism, interpretation of the W.-P. metric on the Teichmüller space. In this talk, following the work of M. Pollicott and R. Sharp[5], we are going to discuss the simplest analogy on graphs and see how the whole machinery goes. And further explore some different properties from the classic W-P metric. If time permit, I will talk about some new results on Higher Teichmüller theory[3].

[1] M. Bridgeman, "Hausdorff dimension and the Weil-Petersson metric to quasi-Fuchsian space," Geom. and Top. 14(2010), 799-831.
[2] M. Bridgeman and E. Taylor, "An extension of the Weil-Petersson metric to quasi-Fuchsian space," Math. Ann. 341(2008), 927-943.
[3] M. Bridgeman, D. Canary, F. Labourie and A. Sambarino, "The pressure metric for convex presentations," Preprint (2013), 1-96.
[4] C. McMullen, "Thermodynamics, dimension and the Weil-Petersson metric," Invent. Math. 173(2008), 365-425.
[5] M. Pollicott and R. Sharp, "A Weil-Petersson metric of metric graphs," Preprint (2012), 1-13.
[6] W. Parry and M. Pollicott, "Zeta functions and the periodic orbit structure of hyperbolic dynamics," Astérisque, 187-188(1990).
[7] S. Wolpert, "Thurston's Riemannian metric for Teichmüller space," J. Diff. Geom. 23(1986), 143-174.

### September 30, 2013

Speaker
Renato Ghini Bettiol
Title
Basic Bifurcation Theory and Geometric Variational Problems
Abstract

Broadly speaking, bifurcation occurs when the Implicit Function Theorem fails. In the first half of this talk, I will make this statement precise and give some examples (the only prerequisite to understanding this part is multi-variable calculus). In the second half of this talk, I will discuss how bifurcation can be used to produce new solutions of PDEs and geometric variational problems, surveying on some recent results in [1], [2] and [3].

[1] R. B., P. Piccione, Delaunay type hypersurfaces in cohomogeneity one manifolds, preprint, arXiv:1306.6043 [math.DG].
[2] R. B., P. Piccione, Bifurcation and local rigidity of homogeneous solutions to the Yamabe problem on spheres, Calc. Var. Partial Differential Equations 47 (2013), no. 3-4, 789-807, arXiv:1107.5335 [math.DG].
[3] R. B., P. Piccione, Multiplicity of solutions to the Yamabe problem on collapsing Riemannian submersions, to appear in Pacific J. Math., arXiv:1304.5510 [math.DG].

### October 14, 2013

Speaker
Peter Ulrickson
Title
Poincare, His Duality Theorem, and the Beginnings of Algebraic Topology
Abstract

Henri Poincare founded the field of algebraic topology toward the end of the 19th century. We'll survey some definitions and results of Poincare's 1895 paper Analysis Situs and its five supplements. One goal of the talk is to answer the question - how did Poincare think about Poincare duality? The talk won't assume much familiarity with topology (don't worry if you aren't familiar with Poincare duality), and will include some historical remarks in addition to the mathematical content.

### November 4, 2013

Speaker
Edward Burkard
Title
From Classical Mechanics to Symplectic Geometry
Abstract

Starting with Newton's Second Law of Motion, we will talk about how to evolve classical mechanics into symplectic geometry. After that, we will talk about two well known items in symplectic topology: the nonsqueezing theorem and Arnold's conjecture on fixed points. This talk is intended to be at a fairly basic level, and more of an overview than anything else. Depending on the audience, some knowledge of manifolds may be assumed (at the level of the Basic Topology course).

[1] D. McDuff & M. Salamon, "Introduction to Symplectic Topology." 2nd edition. Oxford Mathematical Monographs, Oxford Science Publications, 1998
[2] L. Polterovich, "The geometry of the group of symplectic diffeomorphisms." ETH Lectures in Mathematics, Birkhauser, 2001

### December 2, 2013

Speaker
Jennifer Garbett
Title
Minkowski's Theorem
Abstract

Minkowski's Theorem, proved by Hermann Minkowski in 1889 became the foundation of the Geometry of Numbers, which Minkowski introduced to solve problems in number theory. In this talk we introduce Minkowski's Theorem and prove a special case. We then discuss a few applications of the theorem and use it to solve Polya's Orchard Problem. This talk will be very basic and should be widely accessible, as any required background material will be included.

### February 3, 2014

Speaker
Eric Wawerczyk
Title
Universal Quadratic Forms and Arithmetic in Quaternion Algebras
Abstract

The object of study will be integral discrete subrings of division quaternion algebras over the rational numbers such that the Norm Form restricted to the subring is universal. Lagranges Theorem states that every natural number, n, can be represented as a sum of four squares (n=a^2 + b^2 + c^2 + d^2). This means that the quadratic form (w^2 + x^2 + y^2 + z^2) is universal over the integers, i.e. it represents all natural numbers. This quadratic form can be realized as the Norm form of the Hamilton quaternion algebra (-1, -1, R). In general, given a field F and two non-zero elements, a and b, of F, we can construct a quaternion algebra denoted (a,b / F). In these quaternion algebras we can find systems of arithmetic (discrete subrings). The goal of the talk will be to answer the question: How many different universal quaternionic integer systems are there and where do they live?

### February 10, 2014

Speaker
Ben Lewis
Title
Constructing a Quasi-Steady State Using Hamiltonian Flow
Abstract

The WKB method classically approximates the wavefunctions of quantum mechanical systems using only spacial zones and gluing. We will present a time and space dependent approach to this problem which does not require gluing. In some sense this approach is more canonical, because, unlike the traditional WKB method, our method treats position and momentum equally. This new approach provides a novel and simple solution to a century old problem. Based on the paper "Bohr-Sommerfeld quantization rules in the semiclassical limit" by George A Hagedorn and Sam L Robinson.

### March 3, 2014

Speaker
Luis Saumell
Title
Hilbert Scheme of Points of the Affine Space
Abstract

Abstract.

### March 17, 2014

Speaker
Augusto Stoffel
Title
The Pontryagin-Thom construction
Abstract

The Pontryagin-Thom construction provides a bridge between the study of manifolds and homotopy theory. I will describe two applications of this construction: Pontryagin's calculation of certain homotopy groups of spheres through the study of framed cobordism, and Thom's translation of the problem of classification of manifolds up to cobordism into a purely homotopy-theoretical (and approachable) problem.

### March 31, 2014

Speaker
Dominic Culver
Title
Generalized (co)homology theories and Brown's representability theorem
Abstract

Singular (co)homology is a very useful invariant of a topological space satisfying nice formal properties, and it can be shown that these formal properties uniquely determine singular (co)homology. These properties are called the Eilenberg-Steenrod axioms. Omitting the dimension axiom'' one arrives at the definition of a \emph{generalized (co)homology theory}, examples of which are cobordism and $K$-theory. Brown's representability theorem relates these generalized (co)homology theories to stable homotopy theory, namely the theorem says that any generalized cohomology theory can be represented by an $\Omega$-spectrum. In this talk, I will discuss generalized cohomology theories and sketch a proof of Brown's Representability Theorem. I will also give an application to geometry.

### April 14, 2014

Speaker
Xumin Juang
Title
Boundary expansion for Singular Yamabe problem
Abstract

I will take Singular Yamabe problem as an example to show how we construct boundary expansion for its solution. This expansion is on the distance function to the boundary, and since the solution is singular up to boundary, this expansion shows optimal regularity of the solution, and completely describes how a singular solution behaves near boundary. Similar expansion holds for minimal graphs in hyperbolic space, which is my recent joint work with Professor Qing Han, and a complex Monge-Ampere related to constructing Kahler-Einstein metric on pseudoconvex domain.

### April 28, 2014

Speaker
Amy Buchmann
Title
Flow Induced by Bacterial Carpets and Transport of Microscale Loads
Abstract

Microfluidics devices carry very small volumes of liquid though channels and have been used in many biological applications including drug discovery and development. In many microfluidic experiments, it would be useful to mix the fluid within the chamber. However, the traditional methods of mixing and pumping at large length scales don't work at small length scales. Recent experimental work has suggested that the flagella of bacteria may be used as motors in microfluidics devices by creating a bacterial carpet [1]. Mathematical modeling can be used to investigate this idea and to quantify flow induced by bacterial carpets. I will introduce the method of regularized stokeslets [2] and show how this can be implemented to model fluid flow above bacterial carpets and the transport of microscale loads. Model validation and preliminary results will be presented.
[1] N. Darnton, L. Turner, K. Breuer, and H. Berg, Moving fluid with bacterial carpets, Biophys. J., 86 (2004), pp. 1863-1870.
[2] R. Cortez, The method of regularized stokeslets, SIAM J. Sci. Comput., 23 (2001), p. 1204.

## Previous Years

MGSA - Math Department - University of Notre Dame =