Title: Algebra in Topology and Topology in Algebra

Date: 04/02/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

How do we quantify the difference between the surface of a basketball and the surface of a doughnut? Algebraic objects, such as numbers, can be used to study objects in topology called spaces. But the tools of topology can also be used to study objects in algebra. In this talk we will explore the fascinating interplay between algebra and topology and see how it is manifested in a tool called Algebraic K-theory.
This talk will be accessible to both undergraduate and graduate students.

We will define and study hypergraphic polytopes. These polytopes make up a proper subset of all generalized permutahedra and include all graphic zonotopes. We will show how the normal fan of hypergraphic polytopes can be understood in terms of acyclic orientations of hypergraphs. This will provide additional understanding of the antipode of the hypergraphic Hopf algebra from last week.

Speaker: Armin Schikorra, University of Pittsburgh

Title: On free boundary problems for conformally invariant variational functions

Date: 04/04/2018

Time: 4:10 PM - 5:00 PM

Place: C517 Wells Hall

I will present a regularity result at the free boundary for critical
points of a large class of conformally invariant variational
functionals. The main argument is that the Euler-Lagrange equation can
be interpreted as a coupled system, one of local nature and one of
nonlocal nature, and that both systems (and their coupling) exhibit an
antisymmetric structure which leads to regularity estimates.

The pentagram map is a discrete dynamical system introduced by Richard Schwartz, which acts on the space of all planar polygons. More generally, the map is defined on the space of all "twisted polygons". In this talk, we will define twisted polygons, and then construct a coordinate system on the space of all twisted polygons, and write a formula for the pentagram map in these coordinates. If there is time, we will discuss a Poisson structure on the space of polygons which can be used to show that the pentagram map is a completely integrable system (in the sense of Liouville).

Speaker: Juanita Pinzon-Calcedo, North Carolina State

Title: Gauge Theory and Knot Concordance

Date: 04/05/2018

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

Knot concordance can be regarded as the study of knots as boundaries of surfaces embedded in spaces of dimension 4. Specifically, two knots K_0 and K_1 are said to be smoothly concordant if there is a smooth embedding of the 2--dimensional annulus S^1 × [0, 1] into the 4--dimensional cylinder S^3 × [0, 1] that restricts to the given knots at each end. Smooth concordance is an equivalence relation, and the set of smooth concordance classes of knots, C, is an abelian group with connected sum as the binary operation. The algebraic structure of C, the concordance class of the unknot, and the set of knots that are topologically slice but not smoothly slice are much studied objects in low-dimensional topology. Gauge theoretical results on the nonexistence of certain definite smooth 4-manifolds can be used to better understand these objects. In this talk I will explain how the study of instantons can be used to shown that (1) the group of topologically slice knots up to smooth concordance contains a subgroup isomorphic to Z^\infty, and (2) satellite operations that are similar to cables are not homomorphisms on C.

Historically, it is geometry which led to important developments in several areas of mathematics including number theory. But recently there have been several instances of number theory being applied to settle important questions in geometry. I will talk about two problems in whose solution number theory has been used in a crucial way.
The first one, settled in collaboration with Sai-Kee Yeung, is classification of fake projective planes and their higher dimensional analogs. (I recall that fake projective planes are smooth projective complex surfaces with same Betti numbers as the complex projective plane, but which are not isomorphic to the complex projective plane. The first such surface was constructed by David Mumford.)
The second problem concerns compact Riemannian manifolds and it has the following very interesting formulation due to Mark Kac: “Can one hear the shape of a drum?”. In precise mathematical terms, the question asks whether two compact Riemannian manifolds with same spectrum (i.e., the set of eigenvalues counted with multiplicities) are isometric. The answer is in general “no”. However, Andrei Rapinchuk and I investigated Kac’s question, using number theoretic results and tools, for a particularly nice class of manifolds, namely locally symmetric spaces. The answer turned out to be very interesting and has led to several other developments which, if time permits, I will mention.

A familiar idea in math and computing has recently made a big splash in redistricting lawsuits: if you want to understand a large, complicated space with mysterious structure, you should just drop yourself down in the space and walk around randomly for a long time. What you see when you explore may produce a good representative sample of the space, even if your exploration is way shorter than the time it would take to see everything. This idea is gaining traction in trying to understand whether a congressional redistricting plan is reasonable or not, by comparing it to a huge ensemble of other possibilities found by random walk in the space of plans. I'll overview some of these ideas and tell you how they've played out in Wisconsin, North Carolina, and Pennsylvania.

Title: Seiberg-Witten monopoles with multiple spinors on a surface times a circle

Date: 04/10/2018

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

I will discuss a generalisation of the 3-dimensional Seiberg-Witten equations which was studied by Haydys and Walpuski in relation to Yang-Mills theory on manifolds with special holonomy. The main difference from the classical setting is the non-compactness of the moduli space of solutions. I will explain how to tackle this problem and count the solutions in the special case when the underlying 3-manifold is the product of a Riemann surface and a circle. The main ingredient is a holomorphic description of the moduli space of solutions and its compactification. It allows us to relate our problem to classical results on holomorphic vector bundles on Riemann surfaces.

Speaker: Daniel Johnston, Grand Valley State University

Title: On Rainbow Turán Numbers

Date: 04/10/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

For a fixed graph F, we consider the maximum number of edges in a properly edge-colored graph on n vertices which does not contain a rainbow copy of F, that is, a copy of F all of whose edges receive a different color. This maximum, denoted by ex^*(n; F), is the rainbow Turán number of F, and its systematic study was initiated by Keevash, Mubayi, Sudakov and Verstr\"ate [Combinatorics, Probability and Computing 16 (2007)]. In this talk, we look ex^*(n; F) when F is a forest of stars, and consider bounds on ex^*(n; F) when F is a path with m edges, disproving a conjecture in the aforementioned paper for m = 4. This is based on joint work with Cory Palmer, Puck Rombach, and Amites Sarkar.

Title: Spin Geometry, Bochner’s Method, and Vanishing Theorems

Date: 04/11/2018

Time: 4:10 PM - 5:00 PM

Place: C204A Wells Hall

On a compact Riemannian manifold X, we can give two different Laplace operators, namely the Dirac Laplacian and the Bochner Laplacian. Their difference is of order zero, and can be expressed in terms of the curvature tensor of X. Using harmonic theory, Bochner was able to conclude that the vanishing of certain Betti numbers of X under appropriate positivity assumptions on the curvature tensor.

Title: Thick morphisms of (super)manifolds, non-linear pullbacks, and homotopy algebras

Date: 04/12/2018

Time: 11:00 AM - 12:00 PM

Place: C304 Wells Hall

Abstract: I will speak about the notion of a "thick morphism", which generalizes ordinary smooth maps. Like ordinary maps, a thick morphism induces an action on functions (pullback), but unlike the familiar case, such pullbacks are, in general, non-linear transformations. They have the form of formal non-linear differential operators and are constructed by some perturbative procedure. (Thick morphisms themselves are defined as formal canonical relations between the cotangent bundles specified by generating functions of particular type.) Being non-linear, these pullbacks cannot be algebra homomorphisms; however, their derivatives at each point turn out to be homomorphisms.
The non-linearity is a feature essential for application to homotopy bracket structures on manifolds. Roughly, "non-linearity" = "homotopy". A thick morphism intertwining odd master Hamiltonians of two S-infinity structures (which is practically described by a Hamilton-Jacobi type equation for the generating function) induces an L-infinity morphism of the corresponding homotopy Poisson algebras. Application to homotopy Poisson structures was our primary motivation; but there are also applications to vector bundles and Lie algebroids.
There are two parallel versions: "bosonic" (for even functions) and "fermionic" (for odd functions). The bosonic version has a quantum counterpart. "Quantum pullbacks" have the form of particular Fourier integral operators. There is also an application to "quantum brackets" induced by BV-type operators.

Title: Link homology and Floer homology in pictures

Date: 04/12/2018

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

There are no fewer than eight link homology theories which admit spectral sequences from Khovanov homology. These theories have very different origins -- representation theory, gauge theory, symplectic topology -- so it's natural to ask for some kind of unifying theory. I will attempt to describe this theory using Bar-Natan's pictorial formulation of link homology. This strengthens a result of Baldwin, Hedden, and Lobb and proves new functoriality results for several link homology theories. It may also be useful for studying mutation. (I won't assume much specific knowledge of these link homology theories!)

Speaker: Maria Gualdani, George Washington University

Title: The Landau equation: old and new

Date: 04/12/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Kinetic equations are used to describe evolution of interacting particles. The most famous kinetic equation is the Boltzmann equation: formulated by Ludwig Boltzmann in 1872, this equation describes motion of a large class of gases. Later, in 1936 Lev Landau derived from the Boltzmann equation a new mathematical model for motion of plasma. This latter equation was named the Landau equation. One of the main features of the Landau and Boltzmann equations is nonlocality, meaning that particles interact at large, non-infinitesimal length scales. The Boltzmann and Landau equations present integro-differential operators that are highly nonlinear, singular and with degenerating coefficients. Despite the fact that many mathematicians and physicists have been working on these equations, many important questions are still unanswered due to their mathematical complexity. In this talk we concentrate on the mathematical results of the Landau equation. We will first review existing results and open problems and in the second part of the talk we will focus on recent developments of well-posedness and regularity theory.

Speaker: Brian Wetton, University of British Columbia

Title: Asymptotic Analysis of Implicit Time Stepping for Allen Cahn Dynamics

Date: 04/13/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

There is a growing awareness that fully implicit time stepping methods are needed to accurately compute phase field models with metastable dynamics, such as energy gradient flows of Allen-Cahn and Cahn-Hillard. The superior accuracy of fully implicit time stepping compared to energy stable schemes is shown in a number of ways. The criticisms of fully implicit time stepping in the literature have been that the resulting nonlinear system has multiple solutions; that even when there is a correct local solution the system is difficult to solve numerically; and that this solution may not decrease the energy. Using the asymptotic structure of metastable solutions, it is shown that when time steps are chosen appropriately to the dynamics, locally unique solutions to the fully implicit problem exist that decrease energy. In addition, there is a simple preconditioner that gives a condition number independent of spatial resolution and order parameter for a conjugate gradient solve for Newton iterates for the nonlinear system. The asymptotic results are confirmed in numerical experiments, part of a larger computational benchmark project. This is joint work with Xinyu Cheng, Dong Li, and Keith Promislow. Some recent, related work by Jinchao Xu will also be discussed.

Title: A snorkeling tour into Calculus student WeBWorK data.

Date: 04/16/2018

Time: 4:10 PM - 5:00 PM

Place: C109 Wells Hall

In Fall of 2017, the MTH133 students generated around 300,000 records through their use of WeBWorK; a lot of the information gathered is either not presented in the grading reports on WeBWorK, or are difficult to collect in one place. I'll start by giving a guided, not-very-deep dive into these records, highlighting some obvious and some not-so-obvious facets. This will be followed by audience participation in speculating on the correct interpretation of the data as well as brainstorming on how we can inform our curricular design with these kinds of feedback. Suggestions for directions to pursue the analysis further will be welcomed.

We describe a new algorithm which determines if the intersection of a quasiconvex subgroup of a negatively curved group with any of its conjugates is infinite. The algorithm is based on the concepts of a coset graph and a weakly Nielsen generating set of a subgroup. We also give a new proof of decidability of a membership problem for quasiconvex subgroups of negatively curved groups.

This talk will be a review of open questions and some recent results on the
(classical and quantum) Uniform Electron Gas (UEG), which is a fundamental
model in quantum chemistry. The UEG is a gas of electrons placed in such a way
that their density is constant everywhere in space.
We will in particular compare this model with the one-component plasma
(Jellium), which has a constant positive background, but no special constraint
on the electronic density. Jellium is believed to play a central role for
random matrices and random Schrödinger operators. In some cases Jellium is the
same as the UEG, but in some other cases, these could be different.
Results in collaboration with Elliott H. Lieb (Princeton) and Robert Seiringer
(Vienna).

Speaker: Ekaterina Rapinchuk, Michigan State University

Title: An Auction Dynamics Approach to Semi-Supervised Data Classification

Date: 04/17/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

We reinterpret the semi-supervised data classification problem using an auction dynamics framework inspired by real life auctions. This novel forward and reverse auction procedure for data classification requires remarkably little training/labeled data and readily incorporates volume/class size constraints. We prove that the algorithm always terminates with the right properties for any choice of parameters and derive its computational complexity. Experimental results on benchmark machine learning datasets show that our approach exceeds the performance of current state-of-the-art methods, while requiring a fraction of the computational time. This is joint work with Matt Jacobs and Selim Esedoglu.

Title: Kummer Theory on products of elliptic curves over a p-adic field

Date: 04/18/2018

Time: 3:00 PM - 4:00 PM

Place: C304 Wells Hall

In this talk I will present some very recent work, joint with I. Leal about zero-cycles on a product X=E_1 X E_2 of two elliptic curves over a p-adic field. In this work we prove that the cycle map to etale cohomology is injective for a large variety of cases, using a method introduced by Raskind and Spiess, namely using an analogue of the Milnor K-group of a field, defined by Kato and Somekawa. As an application, we obtain some new results about zero-cycles over local and global fields.
Throughout the talk I will only assume some basic familiarity with elliptic curves. Everything else will be self-contained and explained in the talk.

Speaker: Elizabeth de Freitas, Manchester Metropolitan University

Title: What is a Mathematical Concept?

Date: 04/18/2018

Time: 3:30 PM - 5:00 PM

Place: 252 EH

This presentation explores alternative approaches to the question: What is a mathematical concept? Philosophical and historical insights about the nature of mathematical concepts are discussed, with special attention to how concepts emerge and are established through particular mathematical practices. Such work shifts our attention to the material labour and onto generative nature of mathematical activity. New mathematical concepts emerge and old ones are creatively deformed when embodied practices redistribute what is considered sensible and perceptible. I discuss the pedagogical implications of this approach, and the important way such theoretical framing shifts our thinking about mathematics dis/ability. My aim is to rescue mathematical concepts from the staid curricular lists which entomb them, and to consider examples of how we might reanimate concepts in classroom activity.

Speaker: Ben Salisbury, Central Michigan University

Title: Crystal structure of certain PBW bases

Date: 04/18/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Lusztig's theory of PBW bases gives a way to realize the crystal $B(\infty)$ for any complex-simple Lie algebra where the underlying set consists of Kostant partitions. In fact, there are many different such realizations: one for each reduced expression of the longest element of the Weyl group. There is an explicit algorithm to calculate the actions of the crystal operators, but it can be quite complicated. In this talk, we will explain how, for certain reduced expressions, the crystal operators can also be described by a much simpler bracketing rule. Conditions describing these reduced expressions will be given in every type except $E_8$, $F_4$ and $G_2$ and several examples will be provided. This is joint work with Jackson Criswell, Peter Tingley, and Adam Schultze.

Title: Topological Hochschild homology and logarithmic geometry

Date: 04/19/2018

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

While a need to compute algebraic K-theory led topologists to consider spectrally-enriched versions of Hochschild homology, over the years topological Hochschild homology (THH) has emerged as an interesting invariant in its own right. In this talk, I will introduce some of these interesting properties and show how logarithmic geometry can help us shine light on THH's arithmetic structure.

: Consider 2 or 3-dimensional Brownian motion and the set of its cut points. In this talk, we will discuss about its relationship with the intersection exponent and prove the existence of its Minkowski content as a random Borel measure. If time permits, I will also explain how we identify the Minkowski content with the scaling limit of the counting measure of pivotal points for percolation on the triangular lattice in the 2-dimensional case. This is a joint project with Nina Holden, Greg Lawler and Xin Sun.

Speaker: Zinovy Reichstein, University of British Columbia

Title: Simplifying polynomials by Tschirnhaus transformations: old and new

Date: 04/19/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

I will revisit classical problems of simplifying polynomials in one variable by Tschirnhaus transformations. Surprisingly, many of the old questions are still open. I will restate them in geometric terms and discuss recent work in this area.

Title: Sensor fusion via two types of diffusion — with sleep dynamics and fetal health as examples.

Date: 04/20/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Quantifying the intrinsic structure from a given massive dataset, which is often nonlinear and complex, is a common challenge shared in almost all scientific fields, including data science. The problem is becoming more challenging when the data are from multiple sensors with heterogenous data types. The diffusion geometry is a flexible framework for this challenge that has led to several convincing results with solid theoretical backup. We will discuss how to apply the diffusion geometry, particularly the alternating diffusion and commutator, to deal with the sensor fusion problem. Its application to the sleep dynamics analysis and fetal electrocardiogram analysis will be discussed.

Title: Elastic graphs and Ahlfors regular conformal dimension

Date: 04/23/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

One measure of the complexity of a Julia set is its fractal Hausdorff dimension, or more generally various notions of "conformal dimension". We show how to estimate the Ahlfors regular conformal dimension of Julia sets from above and below by using certain energies of maps between graphs. This extends an earlier characterization of which topological maps from a sphere to itself are realized by a rational map.

We will discuss templates for RCPD contract, exam correction, testing center, and other forms we are considering implementing to smooth communications with students.

With the development of multiple mathematics pathways for students at MSU, the population of College Algebra students are now primarily calculus bound. It is important that we do our best to ensure that these students leave College Algebra with the mathematical concepts and skills, as well as learning and self-assessment strategies required to be successful in subsequent STEM courses. This presentation will share the lessons learned over a 3-year journey of curriculum development and provide an overview of the content and pedagogy included in the one and two semester versions of college algebra that will be offered beginning fall semester 2018.

Title: Cluster Duality for Grassmannians and Cyclic Sieving

Date: 04/24/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

The Grassmannian Gr(k,n) parametrizes k-dimensional subspaces in C^n. Due to work of Scott, the homogenous coordinate ring C[Gr(k,n)] of Gr(k,n) is a cluster algebra of geometric type. In this talk, we introduce a periodic configuration space X(k,n) equipped with a natural potential function W. We prove that the topicalization of (X(k,n), W) canonically parametrizes a linear basis of C[Gr(k,n)], as expected by a duality conjecture of Fock-Goncharov. We identify the tropical set of (X(k,n), W) with the set of plane partitions. As an application, we show a cyclic sieving phenomenon involving the latter. This is joint work with Jiuzu Hong and Daping Weng.

Topological Hochschild Homology (THH) is a tool used to approximate algebraic K-theory. In this talk we will first discuss Hochschild Homology (HH), a classic invariant of rings, and look at some examples. Then we will examine its topological analog, THH, and talk about the tools we can use to study it

I will talk about some aspects of the spectral analysis of operators that model quasicrystals. The discussion will center around the continuum Fibonacci Hamiltonian and its generalizations to higher dimensions.

Title: Building Pathologies/Solving Inverse Problems

Date: 04/26/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

As a student, I was told to spend half my time disproving what I wanted to prove. In a broad inverse problem, you are given some data and hope to reconstruct the object from the data. In this talk, both solving inverse problems and building pathologies will be discussed. I will focus on algebraic topics though connections to geometry/topology and analysis will be discussed.