Akbulut and Ruberman shows how to construct absolutely exotic smooth structures
on compact 4-manifolds with boundary in their paper (Absolutely Exotic Compact 4-Manifolds). I will be talking about this construction.

In spin systems, the existence of a spectral gap has far-reaching consequences. "Frustration-free" spin systems form a subclass that is special enough to make the spectral gap problem amenable and, at the same time, broad enough to include physically relevant examples. We discuss "finite-size criteria", which allow to bound the spectral gap of the infinite system by the spectral gap of finite subsystems. We focus on the connection between spectral gaps and boundary conditions.

Speaker: Meera Mainkar, Central Michigan University (Visiting MSU this term)

Title: Anosov diffeomorphisms on nilmanifolds and algebraic units in number fields

Date: 11/02/2017

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

Anosov diffeomorphisms on nilmanifolds play an important and beautiful role in dynamics as the notion represents the most perfect kind of global hyperbolic behaviour, giving examples of structurally stable dynamical systems. Anosov diffeomorphisms on nilmanifolds give rise to special type of algebraic units in number fields. We will explore the properties of these algebraic units to understand the structure of the Lie algebra of the simply connected cover of the nilmanifolds admitting Anosov diffeomorphism.

Many of the most naturally occurring spaces have a surprisingly simple cell structure, having cells in only even dimensions. Complex Grassmanians, Thom spaces, and loops on a Lie group all have this property. Dually, one can consider generalized cohomology theories like complex K-theory with only even homotopy groups. Evaluating these cohomology theories on spaces with only even cells is extremely simple, and this interplay drives much of modern algebraic topology. I'll describe some classical work in this area before exploring some of the more recent, equivariant generalizations.

I will speak about both abstract Schur-Weyl duality from a categorical viewpoint as well as the concrete case for the symmetric groups and general linear groups. The concrete case says roughly that there is a one to one correspondence between irreducible finite dimensional representations of the symmetric groups and irreducible finite dimensional polynomial representations of the general linear groups. Time admitting, I may point out some connections with symmetric functions and polynomials.

Title: Math Ed Colloquium: The blending of academic and social support through apoyo and consejos in the mathematical success of three undergraduate Latinx engineering students

Date: 11/07/2017

Time: 1:15 PM - 2:45 PM

Place: 252 EH

Analyses of academic success among Latinxs in undergraduate STEM education have shed light on gendered disparities between Latinx women and Latinx men. In the United States, Latinx women have higher grade point averages and degree completion rates than Latinx men even though they also report lower levels of confidence. Despite the masculinization of engineering and mathematics spaces perpetuated through women’s underrepresentation and norms of engagement, Latinx women earn over half of science and engineering degrees conferred to their racial group. With undergraduate mathematics serving as a gatekeeper for advanced STEM coursework among historically marginalized populations, qualitative analyses of Latinx undergraduate men’s experiences of mathematical success can illuminate ways to inform more socially affirming STEM postsecondary educational opportunities for them. This presentation shares findings from a phenomenological study of mathematical success as a socially exclusionary experience among three Latinx men pursuing engineering majors at a large predominantly white, four-year University. A cross-case analysis of the three men’s mathematics experiences documented using multiple data sources reveals how academically and socially supportive opportunities, likened to apoyo (moral support) and consejos (narratives of advice) from Latinx culture, shaped their high school and undergraduate mathematical success. Implications are raised to inform change in mathematics education and STEM support programs to better meet Latinx men’s academic and social needs across the K-16 STEM pipeline.

Title: A slice filtration for certain twisted Eilenberg-Mac Lane spectra

Date: 11/09/2017

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

A space X can be described by its Postnikov tower, whose stages have only the homotopy groups of X in a range. Equivariantly, there is an analogue of the Postnikov filtration called the slice filtration. After reviewing some previously known examples, I will describe the slice filtration for twisted Eilenberg-Mac Lane spectra when the group of equivariance is K_4, the Klein four group. This is joint work with C. Yarnall.

The Euclidean plane consists of ordered pairs of real numbers p=(x,y) equipped with the well-known distance function d(p_1,p_2)=((x_1-x_2)^2+(y_1-y_2)^2)^{1/2}. An isometry is a self-map of the plane that preserves all distances between pairs of points.
After briefly classifying the isometries of the plane, I'll prove the following surprising result: If a self-map of the plane carries pairs of points that are distance one apart to pairs of points that are distance one apart, then the map is an isometry. That is, maps that preserve distance one preserve all distances!
Time permitting, I'll discuss the relationship of this problem to the more combinatorial problem of determining the chromatic number of the plane and some open problems.

I will discuss the question of when a contractible manifold fails to be a cork; this happens either if there is no exotic involution on its boundary, or it fails to be Stein. I will give examples of Stein non-corks, and non-Stein non-loose corks. I will discuss open questions of whether there are loose-corks that can not be corks, or if there are infinite order corks (joint with Danny Ruberman).

Title: Symplectic Geometry of the moduli space of Projective Structures on Riemann surfaces.

Date: 11/14/2017

Time: 3:00 PM - 4:00 PM

Place: C304 Wells Hall

The moduli space of quadratic differentials on Riemann surfaces can be viewed as the total space of the cotangent bundle to the moduli space of Riemann surfaces. By choosing a base projective connection which varies holomorphically in moduli, the moduli space of projective structures is identified with the moduli space of quadratic differentials. A projective connection defines, via the monodromy map, a representation of the fundamental group of the Riemann surface into PSL(2,C), i.e. a point in the character variety.
We study the symplectic geometry induced via these maps and show: The homological symplectic structure on the moduli space of quadratic differentials (defined explicitly in terms of Darboux coordinates which involve the double cover arising from a quadratic differential) is identified with the canonical symplectic structure on the cotangent bundle to the moduli space of Riemann surfaces. Choosing the base projective connection as Bergman, Schottky, and Wirtinger induces equivalent symplectic structures on the moduli space of projective connections. Finally we show that under the monodromy map with base Bergman projective connection, the homological symplectic structure induces the Goldman bracket on the character variety. A comparison with the work of Kawai is made. This is joint work with Marco Bertola and Dmitry Korotkin.

Speaker: Eliot Bongiovanni, Michigan State Universtiy

Title: Proving the Least-Area Tetrahedral Tile of Space

Date: 11/14/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

We prove the least-area, unit-volume, tetrahedral tile of Euclidean space, without the assumption that the tiling uses only orientation-preserving images of the tile. Using a graph-theoretical approach, we define a class of tetrahedra that potentially tile with less surface area than the orientation-preserving minimizer, the Sommerville No. 1. We find that without the assumption of orientation preservation, the winner remains the Sommerville No. 1.
The talk summarizes "The Least-Area Tetrahedral Tile of Space" by Eliot Bongiovanni, Alejandro Diaz, Arjun Kakkar, and Nat Sothanaphan, a product of the 2017 NSF Williams College SMALL REU. Preprint: https://arxiv.org/abs/1709.04139

Speaker: Vladimir Chernousov, University of Alberta

Title: Classification of torsors over Laurent polynomial rings

Date: 11/15/2017

Time: 3:00 PM - 4:00 PM

Place: C304 Wells Hall

We will talk about classification of torsors of reductive group schemes over Laurent polynomial rings and applications in infinite dimensional Lie algebras. Joint work with A. Pianzola and P. Gille.

In this talk after going through the statements of Hodge decomposition and sketch of proof, we will try to visit some applications of it in Riemannian Geometry and Complex Geometry.

Title: Cluster Algebras and Representation Theory Part I

Date: 11/16/2017

Time: 10:00 AM - 10:50 AM

Place: C304 Wells Hall

One of the main motivations for the study of cluster algebras is to provide an algebraic-geometric framework for studying problems arising from representation theory. In the first talk, we will focus on the cluster theory of configuration spaces of decorated flags. The tropical sets of the latter spaces parametrize top components of the affine Grassmannian convolution varieties. By the geometric Satake Correspondence, they parametrize bases in the tensor invariants of representations of the Langlands dual groups. If time permits, I will talk about their connections with Fock-Goncharov Duality Conjecture and Mirror Symmetry.

Title: Approximating Continuous Functions on Persistence Diagrams for Machine Learning Tasks

Date: 11/16/2017

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

Many machine learning tasks can be boiled down to the following idea: Approximate a continuous function defined on a topological space (the ``ground truth'') given the function values (or approximate function values) on some subset of the points. This formulation has been well studied Euclidean data; however, more work is necessary to extend these ideas to arbitrary topological spaces.
In this talk, we focus on the task of classification and regression on the space of persistence diagrams endowed with the bottleneck distance, (D,d_B). These objects arise in the field of Topological Data Analysis as a signature which gives insight into the underlying structure of a data set. The issue is that the structure of (D,d_B) is not directly amenable to the application of existing machine learning theories. In order to properly create this theory, we will give a full characterization of compact sets in D; provide simple, exemplar functions for vectorization of persistence diagrams; and show that, in practice, this method is quite successful in classification and regression tasks on several data sets of interest.

Speaker: Gavish Nir, Technion Israel Institute of Technology

Title: A model of highly concentrated electrolyte solutions

Date: 11/17/2017

Time: 4:10 PM - 5:00 PM

Place: C100 Wells Hall

The Poisson-Nernst-Planck (PNP) theory is one of the most widely used analytical methods to describe electrokinetic phenomena for electrolytes. The model, however, considers isolated charges and thus is valid only for dilute ion concentrations. The key importance of concentrated electrolytes in applications has led to the development of a large family of generalized PNP models. However, the wide family of generalized PNP models fails to capture key phenomena recently observed in experiments and simulations, such as self-assembly, multiple-time relaxation, and under-screening in concentrated electrolytes.
In this talk, we derive a thermodynamically consistent mean-field model for concentrated solutions that goes beyond the PNP framework. The result is a modeling framework that contains the essential ingredients for describing electrolytes over the whole range of concentrations - from dilute electrolyte solutions to highly concentrated solution, such as ionic liquids. The model describes self-assembly, multiple-time relaxation, and under-screening, and reveals a mechanism of under-screening. Furthermore, the model predicts distinct transport properties which are not governed by Einstein-Stokes relations, but are rather effected by inter-diffusion and emergence of nano-structure.
Joint work with Doron Elad and Arik Yochelis.

Title: On the minimal compactification of the Cayley Grassmannian

Date: 11/20/2017

Time: 4:10 PM - 5:30 PM

Place: C304 Wells Hall

A 3-fold cross product operation exists only in dimensions 4
and 8. A 4-dimensional subspace of an 8 dimensional vector space is
called a Cayley plane if it is closed under the 3-fold cross product
operation. The Cayley Grassmannian is the space of all Cayley planes. It
is naturally a homogeneous space with an algebraic torus action, and it
resides in Gr(4,8). Over complex numbers, this space is not compact. In
this talk, after I talk about the necessary background, I will explain
some of the results I obtained on the minimal compactification of the
Cayley Grassmannian.

Speaker: Rachel Domagalski, Michigan State University

Title: Tree Cover Number and Maximum Semidefinite Nullity of Graphs Part 1

Date: 11/21/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

For a given multigraph G, we can associate a complex Hermitian matrix A=[a_{ij}] as follows: a_{ij}=0 if i, j are distinct and v_i,v_j are nonadjacent, a_{ij} is nonzero if i, j are distinct and v_i,v_j are ends of a single edge, a_{ij} is a complex number if i=j or if v_i,v_j are joined by parallel edges. In this talk we consider the collection of positive semidefinite complex Hermitian matrices associated with a given multigraph G, denoted S_+(G). The minimum rank of A in S_+(G) is called the minimum semidefinite rank, denoted mr_+(G). The corresponding maximum semidefinite nullity, denoted by M_+(G), satisfies mr_+(G) + M_+(G)=|G| where |G| is the number of vertices in G.
In order to compute M_+(G) for a given graph G, certain graph parameters have been developed. One such graph parameter is called the tree cover number of G, denoted T(G). This is the minimum number of vertex disjoint simple trees occurring as induced subgraphs of G that cover all vertices of G. This parameter is easier to compute than M_+(G). Also, it has been conjectured that for any multigraph G, T(G) <= M_+(G).
In this talk, I will build the background knowledge to understanding the maximum semidefinite nullity of a graph and the tree cover number parameter.

Building on a talk delivered with Anita Wager at NCTM, I will use the notion of a zeroth law in physics -- a principle that comes before all others -- to argue that joy is foundational to both the content and practice standards in mathematics. Attending to joy provides a powerful signpost for making decisions about both research and teaching. I will unpack the concept of joy by examining a variety of classroom interactions, demonstrating that joy comes not only from engaging in mathematics in conjunction with other pleasurable activities, such as talk and play, but also from the work of mathematics itself.

Speaker: Rachel Domagalski, Michigan State University

Title: Tree Cover Number and Maximum Semidefinite Nullity of Graphs Part 2

Date: 11/28/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

In this talk, we continue our discussion of the tree cover number of a graph and its relationship to maximum semidefinite nullity. We find the tree cover number of line graphs, shadow graphs, corona of two graphs, cartesian product of two graphs, and a few more. In these cases, we verify the conjecture: the tree cover number of a graph is less than or equal to the maximum semidefinite nullity of its associated complex Hermitian matrix.

This talk will be an introduction to train tracks and the Nielsen-Thurston Classification of elements in the mapping class group of a surface. We will then discuss some properties of pseudo-Anosov mapping classes and methods for constructing and understanding them.

Speaker: Yu Deng, Courant Institute of Mathematical Sciences, NYU

Title: Spacetime resonance and asymptotic behavior for wave equations

Date: 11/29/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

The method of spacetime resonance was developed by Germain-Masmoudi-Shatah to study small data long time problems for nonlinear dispersive equations. In this talk we will consider the wave equation □u=uΔu in 3+1 dimensions. Global existence for this equation was proved by Lindblad and Alinhac. By applying the method of spacetime resonance, we are able to obtain the precise asymptotics for small solutions, which turns out to exhibit a subtle nonlinear behavior. Our work also justifies the ideas behind a conjecture of Lindblad and Rodnianski. This is joint work with F. Pusateri.

Title: Cluster algebras and representation theory Part II

Date: 11/30/2017

Time: 10:00 AM - 10:50 AM

Place: C304 Wells Hall

In the second part, we will further study the configuration spaces of decorated flags and their connections with the geometric Satake Correspondence, Fock-Goncharov Duality Conjecture, and Mirror Symmetry. We will present a natural construction of the cluster coordinates of configuration spaces and generalize them to the moduli space of decorated G-local systems.

In this talk, I will discuss a technique for studying the stable homotopy groups of spheres called the Mahowald invariant. This technique takes an element in the stable homotopy groups of spheres and produces a nontrivial element in a higher stable homotopy group. In the first part of the talk, I will review some classical computations in stable homotopy theory, introduce the Mahowald invariant, and state Mahowald and Ravenel’s computation of the Mahowald invariants of 2^i for i ≥ 1.
In the second part of the talk, I will discuss a generalization of this technique to motivic stable homotopy theory, or stable homotopy theory for schemes. I will give a brief introduction to motivic homotopy theory, explain some of the differences between the classical and motivic Mahowald invariants, and conclude with some motivic Mahowald invariants. Time permitting, I will discuss a part of the computation which uses a new motivic cohomology theory.

Title: Entropy stable high order discontinuous Galerkin methods for hyperbolic conservation laws

Date: 11/30/2017

Time: 3:05 PM - 3:55 PM

Place: C304 Wells Hall

It is well known that semi-discrete high order discontinuous Galerkin (DG) methods satisfy cell entropy inequalities for the square entropy for both scalar conservation laws and symmetric hyperbolic systems, in any space dimension and for any triangulations. However, this property holds only for the square entropy and the integrations in the DG methods must be exact. It is significantly more difficult to design DG methods to satisfy entropy inequalities for a non-square convex entropy, and / or when the integration is approximated by a numerical quadrature. In this talk, we report on our recent development of a unified framework for designing high order DG methods which will satisfy entropy inequalities for any given single convex entropy, through suitable numerical quadrature which is specific to this given entropy. Our framework applies from one-dimensional scalar cases all the way to multi-dimensional systems of conservation laws. For the one-dimensional case, our numerical quadrature is based on the methodology established in the literature, with the main ingredients being summation-by-parts (SBP) operators derived from Legendre Gauss-Lobatto quadrature, the entropy stable flux within elements, and the entropy stable flux at element interfaces. We then generalize the scheme to two-dimensional triangular meshes by constructing SBP operators on triangles based on a special quadrature rule. A local discontinuous Galerkin (LDG) type treatment is also incorporated to achieve the generalization to convection-diffusion equations. Numerical experiments will be reported to validate the accuracy and shock capturing efficacy of these entropy stable DG methods.This is a joint work with Tianheng Chen.

Title: Elliptic curves, Euler systems, and Iwasawa theory

Date: 11/30/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Some of the most fascinating pieces of mathematics, such as Dirichlet’s class number formula and the Birch and Swinnerton-Dyer conjecture, build a bridge between the distant worlds of arithmetic and analysis. Euler systems and Iwasawa theory provide an intermediate step between the two, and both have been at the source of much of the progress to date on the BSD conjecture and its many generalizations. In this talk, I will expand on some of these ideas, including a brief overview of some of the recent developments in the area.