Schedule
Spring 2024
All seminars are scheduled for 11:30--12:30 in the seminar room.
February 2: Robert Fraser
Title: Hausdorff dimension and Fourier dimension
Abstract: The Fourier transform is an important tool for studying the geometry of fractal sets. We will discuss Frostman’s lemma for determining the Hausdorff dimension of a set, and describe the definition of Fourier dimension. We will compute the Fourier dimension for some examples.
February 9: Catherine Searle
Title: On an extension and a generalization of a theorem of Montgomery and Yang
Abstract: Montgomery and Yang showed that a smooth homotopy $n$-sphere admitting a smooth circle action with a codimension two fixed point is $S^1$-equivariantly diffeomorphic to $S^n$. We extend this result to homotopy complex projective space. This is joint work in progress with Christine Escher.
February 16: Mark Meyer
Title: The Schneider non-convexity index
Abstract: It was shown in 2021 that Lebesgue measure has a strong additivity property called fractional superadditivity, and an initial attempt was made at completely characterizing volumes of sumsets under Minkowski addition. In my talk I will define a set function called the Schneider non-convexity index, then show that analogous to Lebesgue measure, the Schneider non-convexity index is fractionally subadditive. If time permits, I will present an initial attempt at completely characterizing the convexity index of sumsets.
February 23: Aubrey Wolfe
Title: The Four Vertex Theorem
Abstract: We are going to explore The Four Vertex Theorem. We will look at a proof of The Four Vertex Theorem with some examples. Slides are available here.
March 1: Thalia Jeffres
Title: Adjoint Operators, Part One: Basic Elements
Abstract: This is intended to be Part One of two talks. In this first one, I will review quickly the finite-dimensional case, making contact with phenomena, such as existence of a basis of eigenvectors, that have direct analogues in certain infinite dimensional settings. In the context of differential operators, I will calculate some elementary examples demonstrating how to find the adjoint operator, with indications of some of the issues that arise in more general settings.
March 8: No Seminar
March 15: Spring Break
March 22: Yueh-Ju Lin
Title: Ambient space and conformally compact Einstein manifolds
Abstract: In this talk, I will first introduce Fefferman-Graham ambient space, which is a powerful tool to construct local scalar conformal invariants. Such conformal invariants are not easy to come by, in contrast to Riemannian invariants via Weyl invariant theory. As a special example of Poincare manifolds (which can be viewed as "hyperboloids" in the ambient space), I will talk about conformally compact Einstein (CCE) manifolds and define the renormalized volume. If time permits, I will talk about the Gauss-Bonnet-type formula for the renormalized volume on CCE manifolds.
March 29: No seminar
April 5: Thanh Nguyen
Title: Brief Introduction Into Diophantine Approximation
Abstract: Diophantine Approximation is the study of the density of Q in R and how well real numbers are approximated by rationals. In this talk I will introduce
some basic definitions, notations and important results. Two fundamental re sults in this field are Dirichlet approximation theorem and Kintchine’s theorem. I will give a proof of the former and a partial proof of the latter.
April 12: Samuel Bartel
Title: Almost Non-negative Curvature and Maximal Symmetry Rank in Low Dimensions
Abstract: In this talk we discuss a result classifying closed, simply connected, almost non-negatively curved n-manifolds, with 4 ≤ n ≤ 6, admitting an isometric and effective torus action of maximal symmetry rank.
April 19: Xiaolong Li
April 26: Farida Ghazawneh
Fall 2023
All seminars are scheduled for 11:30--12:30 in the seminar room.
September 1: Catherine Searle:
Title: Positive Curvature and Discrete Abelian Symmetries
Abstract: Positively curved manifolds are notoriously hard to classify. In the early 90's, based on the observation that the few
known examples are all highly symmetric, Karsten Grove proposed his "Symmetry Program" which suggests trying to classify such manifolds with the additional hypothesis of symmetry. Much work has been focused on the case of continuous symmetries. In this talk, I'll discuss the case when the isometry group is finite and present some recent results when it is an elementary abelian two-group. This is joint work with Lee Kennard and Elahe Khalili Samani.
September 8: Dan Grady
Title: Supersymmetric field theories and topological modular forms.
Abstract: I will survey an open conjecture by Stolz and Teichner relating 2|1 Euclidean supersymmetric field theories to the cohomology theory known as topological modular forms (TMF). If time permits, I will discuss a possible plan of attack for this conjecture, using joint work with Dmitri Pavlov on the geometric cobordism hypothesis.
September 15: TBA
September 22: Sef Altakarli
September 29: Xiaolong Li
Title: Einstein four-manifolds and sectional curvature.
Abstract: I will talk about four-dimensional Einstein manifolds with positive scalar curvature and present a classification result under a pinching condition on the sectional curvatures. This is joint work with Prof. Xiaodong Cao at Cornell University.
October 6: Elton Bowman
October 13: Yueh-Ju Lin
October 20: Canceled
October 27: Robert Fraser
November 3: Samuel Bartel
November 10: Mark Meyer
November 17: Sien Gong (University of Kansas): TBA
November 24: Thanksgiving Break
December 1: Farida Ghazawneh
Spring 2023
All seminars are scheduled for 3:30--4:30 in the seminar room.
January 23: Daniel Grady
Title: Differential cohomology operations
Abstract: In this talk, I will discuss a geometric refinement of a cohomology theory called differential cohomology. If time permits, I will discuss some calculations of differential cohomology operations, analogous to power operations in ordinary cohomology.
January 30: Xiaolong Li
Title: What is a viscosity solution?
Abstract: The viscosity solution concept was introduced in the early 1980s by Pierre-Louis Lions and Michael G. Crandall as a generalization of the classical concept of what is meant by a 'solution' to a partial differential equation (PDE). It has been found that the viscosity solution is the natural solution concept to use in many applications of PDE's, including for example first order equations arising in dynamic programming (the Hamilton–Jacobi–Bellman equation), differential games (the Hamilton–Jacobi–Isaacs equation) or front evolution problems, as well as second-order equations such as the ones arising in stochastic optimal control or stochastic differential games. In this talk, I will explain what a viscosity solution is and summarize some of its basic properties.
February 6: Robert Fraser
February 13: Yueh-Ju Lin
February 20: Elton Bowman
February 27: Thanh Nguyen
Title: Asymptotic Behavior of Oscillatory Integrals of the First Kind
March 6: Mark Meyer
Title: The convexification effect of Minkowski summation
March 13: Spring Break, no seminar
March 20: Thalia Jeffres
Title: Fixed Point Methods and Iteration Schemes, Part Three
Abstract: This will be the third talk in a survey of fixed point methods and iterations schemes as methods for solving problems in differential equations and differential geometry. I will finish up two illustrations of the Arzela-Ascoli theorems: The Perron process, and an application to the Monge-Amp\'{e}re eigenvalue problem. Then I will describe the continuity method.
The talk will be a continuation of the theme of the earlier two talks, but it's totally self-contained and independent. It will be accessible to students, and this means all students, not just advanced students already undertaking thesis work in related areas.
March 27:
April 3: Aubrey Wolfe
Title: The Isoperimetric Inequality.
Abstract: The isoperimetric inequality is a geometric inequality relating the perimeter of a set with the area of the set. We will look at a proof of the inequality and if time permits some applications of the inequality.
April 10: Thalia Jeffres
Title: Fixed Point Methods and Iteration Schemes, Part Four
Abstract: In Part Four, I will discuss the Continuity Method, giving statements of the relevant theorems, and two illustrations of the method. Although it's a continuation of earlier talks, it will be self-contained. I will also include some background material on the implicit function theorem, including brief mention of this theorem in the infinite dimensional setting. It will be accessible to everyone.
April 17: Dan Grady
Title: An Atiyah-Hirzebruch spectral sequence for differential K-theory and applications to type II string theory
Abstract: It has been a long standing conjecture that the background fields for type II string theory are described by a mathematical object called twisted differential K-theory. This talk will survey work that provides a partial resolution to this conjecture. I will begin with a discussion of the physics motivation, then I will discuss K-theory and its differential refinement.
I will continue with a survey of joint work with H. Sati, which constructs an AHSS for differential K-theory. Finally, I will explain how the differentials in the spectral sequence lead to quantization conditions on the RR-fields, which agree with the quantization conditions that exist in the physics literature.
April 24: Farida Ghazawneh
Title: Error correcting codes and symmetry
Abstract: In this talk, we will introduce error correcting codes and show how error correcting codes can be used as tool in the classification of positively curved manifolds with symmetry.
May 1: Samuel Bartel
Fall 2022
All seminars are scheduled for 11:30 --12:30 in the seminar room.
September 9: Daniel Grady
Title: Deformation classes of invertible field theories and the Freed--Hopkins conjecture, part I
Abstract: In this talk, I will discuss a recent result which provides an affirmative answer to a conjecture by Freed and Hopkins. The conjecture concerns a classification of reflection positive invertible field theories. I will begin by reviewing motivation and background on reflection positive theories. Then I will state the conjecture and sketch of the proof.
September 16: Daniel Grady
Title: Deformation classes of invertible field theories and the Freed--Hopkins conjecture, part II
Abstract: In this talk, I will discuss a recent result which provides an affirmative answer to a conjecture by Freed and Hopkins. The conjecture concerns a classification of reflection positive invertible field theories. I will begin by reviewing motivation and background on reflection positive theories. Then I will state the conjecture and sketch of the proof.
September 23: Xiaolong Li
Title: The curvature operator of the second kind
Abstract: In this talk, I will first introduce the notion of the curvature operator of the second kind, and then I will compute the eigenvalues of the curvature operator of the second kind on some geometric examples such as spheres and product of spheres. The talk only uses (advanced) linear algebra and is accessible to graduate students.
September 30: Farida Ghazawneh
Title: The Connectedness Lemma
Abstract: Wilking's Connectedness Lemma states that for a compact, positively curved n-manifold with a totally geodesic, embedded, compact submanifold of codimension k, the inclusion of the submanifold is (n-2k+1)-connected in M. In this talk, I will present a sketch of the proof and mention some important applications for positively curved manifolds with symmetries.
October 7: Md. Ibrahim Kholil
Title: A uniqueness theorem for inverse problems in quasilinear anisotropic media
Abstract: We study the question of whether one can uniquely determine a scalar quasilinear conductivity in an anisotropic medium by making voltage and current measurements at the boundary.
October 14: Robert Fraser
Title: The uncertainty principle in Harmonic Analysis
Abstract: The uncertainty principle is a guiding heuristic that is essential to all of modern harmonic analysis. In this talk, we will discuss the uncertainty principle in one dimension and the Shannon-Nyquist sampling theorem. Time permitting, we will discuss higher-dimensional versions of the uncertainty principle and the wavepacket decomposition for the Schrodinger equation.
October 21: Dmitri Pavlov, Texas Tech University
Title: The geometric cobordism hypothesis
Abstract: I will explain my recent work with Daniel Grady on locality of functorial field theories (arXiv:2011.01208) and the geometric cobordism hypothesis (arXiv:2111.01095). The latter generalizes the Baez–Dolan cobordism hypothesis to nontopological field theories, in which bordisms can be equipped with geometric structures, such as smooth maps to a fixed target manifold, Riemannian metrics, conformal structures, principal bundles with connection, or geometric string structures. Applications include a generalization of the Galatius–Madsen–Tillmann–Weiss theorem, a solution to a conjecture of Stolz and Teichner on representability of concordance classes of functorial field theories, and a construction of power operations on the level of field theories (extending the recent work of Barthel-Berwick-Evans-Stapleton).
October 28: Catherine Searle
Title: When is an Alexandrov space smoothable?
Abstract: In this talk, I will discuss the problem of when an Alexandrov space is smoothable.
We will review the history of this question and discuss a new result that partially answers it.
This is joint work with Pedro Solorzano and Fred Wilhelm.
November 4: Thalia Jeffres
Title: Fixed Point Theorems and Iteration Schemes, Part Two.
Abstract:
The statements of Fixed Point Theorems are abstract or functional-analytic, even topological. My emphasis is to show some ways to apply these theorems and to extract from them solution methods.
In the category of oldie but goodie, I will describe applications of the Arzela-Ascoli theorems. In Part One, we discussed two instances in which the Contraction Mapping Principle was applied directly, and in future seminars I will cover the Continuity Method, the Schauder Method, the method of sub- and supersolutions, and one other method based on being able somehow to order elements in an iteration. I will give two or three illustrations for each method.
The discussion will be accessible to everyone; the minimum to follow the discussion is Calc III.
November 11: No seminar
November 18: Yueh-Ju Lin
Title: Volume comparison theorem from Ricci to scalar, and Q-curvatures.
Abstract: In this talk, I will first review classical Bishop-Gromov volume comparison theorem (which has assumption on the Ricci curvature) and it's application. Then I will mention two conjectures regarding volume comparison for scalar curvature and some known partial results. In general, scalar curvature is too weak to control the volume but we can consider some special cases. This type of phenomenon can also be generalized to Q-curvature (a much weaker 4th-order curvature quantity). If time permits, I will briefly explain the idea of proof of local volume comparison for scalar and Q-curvatures.
November 25: Thanksgiving Break
December 2: Samuel Bartel
Title: Classifying 4-Manifolds with S1-Symmetry and Lower Curvature Conditions
Abstract:
In this talk I will discuss a classification theorem for 4-manifolds which admit an effective, isometric S1-action and are positively, non-negatively, or almost non-negatively curved.
Spring 2022
March 2: Xiaolong Li (online):
Title: Curvature operator of the second kind and proof of Nishikawa's conjecture
Abstract: In 1986, Nishikawa conjectured that a closed Riemannian manifold with positive curvature operator of the second kind is diffeomorphic to a spherical space form and a closed Riemannian manifold with nonnegative curvature operator of the second kind is diffeomorphic to a Riemannian locally symmetric space. Recently, the positive case of Nishikawa's conjecture was proved by Cao-Gursky-Tran and the nonnegative case was settled by myself. In this talk, I will first give an introduction about curvature operator of the second kind and then present a proof of Nishikawa's conjecture under weaker assumptions.
March 9: The week of MGC
March 16: Spring Break
March 23: No seminar
March 30: Farida Ghazawneh, Frankel's Theorem
April 6: No seminar
April 13: Yueh-Ju Lin, Deformation of Q-curvature and volume comparison
April 20: Catherine Searle, Positive curvature and Cohomogeneity two
April 27: No seminar
May 3: Thalia Jeffres, Fixed Point Theorems and Iteration Schemes
Fall 2021
September 1: Xiaolong Li, A Brief Introduction to the Ricci Flow ( slides of the talk )
September 8: Xiaolong Li, Ricci flow on Surfaces (slides of the talk )
September 15: No talk
September 22: Yueh-Ju Lin, Introduction to Yamabe Problem
September 29: Faiz Shabo, Conformal Change of Metric and some Differential Operators
October 6: Farida Ghazawneh, Isometric circle actions on positively curved closed 4-manifolds, Part I
October 13: Farida Ghazawneh, Isometric circle actions on positively curved closed 4-manifolds, Part II
October 20: No talk
October 27: Faiz Shabo, The existence and uniqueness of the solution for the evolution equation
November 3: Catherine Searle, An Introduction to Alexandrov Spaces or 13 Ways to Say a Lower Curvature Bound
November 10:
November 17: Farida Ghazawneh
November 24: Thanksgiving Break
December 1:
Spring 2020
This semester we are running the seminar mainly on Alexandrov Geometry, reading the book A Course in Metric Geometry
February 5: Catherine Searle and Jacqueline Chan, "Math Origami and a Magic Trick"
February 12: Cassidy Lee, Chapter 1 from A Course in Metric Geometry
February 19: Cassidy Lee, continuation of Chapter 1 from A Course in Metric Geometry
February 26: Cassidy Lee and Aubrey Wolfe, continuation of Chapter 1 and Chapter 2 from A Course in Metric Geometry
March 4: Aubrey Wolfe, continuation of Chapter 2 from A Course in Metric Geometry
March 11: Aubrey Wolfe, continuation of Chapter 2 from A Course in Metric Geometry
March 18:
March 25: Spring Break
April 1:
April 8: Austin Bosgraaf, 4-manifolds with a lower curvature bound and continuous symmetry
April 15:
April 22:
April 29:
May 6:
Fall 2019
This semester we are once again running the seminar mainly on Topological Data Analysis, reading the book Computational Topology - An Introduction by Edelsbrunner and Harer
August 20: Organizational meeting
August 27: Adam Jaeger, "Probability Theory"
September 3: Adam Jaeger, "Graphs"
September 10: Austin Bosgraaf, "Point Set Topology"
September 17: Jason Clemens, "Graphs"
September 24: Yueh-Ju Lin "Surfaces, I"
October 1: Yueh-Ju Lin "Surfaces, II"
October 8: Catherine Searle "Almost non-negatively curved 4-manifolds with circle symmetry"
October 15: Fall Break
October 22: Justin Ryan, "Conformal Parametrization of Surfaces, Part I"
October 29: Austin Bosgraaf "Complexes"
November 5: Austin Bosgraaf, "4-manifolds of positive curvature with continuous symmetries"
November 12:
November 19: Justin Ryan, "Conformal Parametrization of Surfaces, Part II"
November 26:
December 3:
Previous Academic Years
Spring 2019
February 5: Austin Bosgraaf, An introduction to differentiable structures on manifolds
February 12: Aubrey Wolfe, An introduction to differentiable structures on manifolds, continued
February 19: Jason Clemens, Modulus on Graphs: Properties, computation, and related problems, I.
February 26: Jason Clemens, Modulus on Graphs: Properties, computation, and related problems, II.
March 5:
March 12: Spring Break
March 19:
March 26:
April 2:
April 9:
April 16:
April 23:
April 30:
Fall 2018
This semester we are running the seminar on Topological Data Analysis.
September 12: Adam Jaeger, "Persistent Homology, Persisence Diagrams and the bottleneck distance"
September 26: Jenny Pinkston, "Persistence Modules"
October 3: Aubrey Wolfe, "Persistence Modules, continued"
October 10: Austin Bosgraaf, "Persistence Modules, continued"
October 17: Yueh-Ju Lin, "Persistence Modules, continued"
October 24: Yueh-Ju Lin, "Persistence Modules, continued"
October 31: Yueh-Ju Lin, "Persistence Modules, continued"
November 7: Adam Jaeger, "Persistence Modules, continued"
November 14: Adam Jaeger, "Barcodes"
November 28: Austin Bosgraaf, "Barcodes, continued"
December 5: Jenny Pinkston, "Barcodes continued"
Fall 2017
September 22: Thalia Jeffres, WSU: "Self Adjoint Operators, I". Here are notes from the talk, and here is a
bibliography for the two talks.
October 6: Thalia Jeffres, WSU: "Self Adjoint Operators, II".
October 20: Justin Ryan, WSU: "Pseudo-Riemannian Lie algebras of modified H-type". Here are the slides from the talk.
November 3: James Reimer, WSU: "Invariant Polynomials Arising from the Monge Ampere Operator on Kahler-Einstein Manifolds".
November 20: Natasha Schlittenhardt, WSU: "Simple Hypersurface Singularities".
December 1: Catherine Searle, WSU: "Symmetries of manifolds with lower curvature bounds".
Spring 2017
February 23:
March 9:
March 23:
April 6:
April 20:
May 4: Mark Walsh, Maynooth University