Title: The University Teaching Experience: Creating Early Field Experiences Prepare Novice Teachers for the Demands of Teaching Mathematics In Schools

Date: 10/03/2016

Time: 12:00 PM - 1:00 PM

Place: 252 EH

What does it mean to be a “well-started” novice teacher of mathematics, especially in this era of accountability? Teacher preparation programs face increasing pressure to create curriculum that is responsive to the demands on new teachers in the field. Such responsiveness means that improvements to teacher preparation curriculum account for obligations on the role of teacher in schools at present, so to better develop novices’ knowledge, skills, and dispositions to manage those obligations and teach mathematics for understanding in equitable ways. In this talk, I will focus on results from a project that has designed and investigated the university teaching experience as a model for early field experience that prepares prospective teachers to learn from their teacher preparation coursework and become 'well-started novices' in their school-based placements.

Given an edge weighted directed graph embedded on a surface, the boundary measurement matrix has entries given by signed sums of paths between vertices. Postnikov (2006) first studied the boundary measurement matrix for graphs embedded on a disk, and a formula for the maximal minors (i.e. Plücker coordinates) was given by Talaska (2008). We will consider graphs embedded on any closed orientable surface with boundary and give a formula of the maximal minors of the boundary measurement matrix.

Let p = p_1 ... p_n be a permutation in the symmetric group S_n written as a sequence. The descent set of p is the set of indices i such that p_i > p_{i+1}. A classic result of MacMahon states the the number of permutations in S_n with a given descent set is a polynomial in n. But little work seems to have been done concerning the properties of these polynomials. The peak set of p is the set of indices i such that p_{i+1} < p_i > p_{i+1}. Recently Billey, Burdzy, and Sagan proved that the number of permutations in S_n with given peak set is a polynomial in n times a power of two. I will survey what is known about these two polynomials, including their degrees, roots, coefficients, and analogues for other Coxeter groups.

When A and B are two Hermitian matrices which commute, we have e^(A+B)=e^{A}e^{B}. The goal is to discuss some relationships in the non-commutative case, focusing on the Golden-Thompson inequality, and give applications such as e^A leq e^B implies A \leq B and Chernoff type inequalities for Random Matrices.

Title: Maximal Green Sequences and Quiver Mutations

Date: 10/05/2016

Time: 3:00 PM - 3:50 PM

Place: C304 Wells Hall

Quiver mutation is a combinatorial process that takes a directed graph and makes a “local change” to create a new direct graph. Fomin and Zelevinsky in 2003, utilized these combinatorics to understand a class of algebras known as cluster algebras. In this talk we will introduce quiver mutation and then explore a special sequence of mutations known as Maximal Green Sequences. The existence of these sequences plays a role in understanding the underlying algebraic structure. This topic is an interesting intersection of algebra, topology, and combinatorics with many exciting and challenging problems still open.

Title: Log-Canonical Poisson Brackets on the Algebra of Rational Functions

Date: 10/05/2016

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

On a symplectic manifold, there are always canonical coordinates around any point, where the symplectic form looks like the standard one on R^2n. In terms of Poisson geometry, this means the bracket of any two coordinate functions is constant. We ask whether such a thing is possible in the algebraic situation. That is, given a Poisson bracket, is there some change of coordinates, using only rational functions, which makes the bracket between coordinate functions constant?

Title: Braid foliations and quasipositive minimal braids

Date: 10/06/2016

Time: 2:00 PM - 2:50 PM

Place: C304 Wells Hall

Braid foliation techniques provide tools for studying knots and links and related surfaces in the three-sphere. We discuss these techniques and how they can be used to study quasipositive links, an important class of links lying at the intersection of complex, contact, and symplectic geometry. In particular, we'll use recent work of LaFountain-Menasco to show that every quasipositive link has a quasipositive minimal braid index representative.

This is the first seminar in our new series of Mathematics Career Seminars, which aims at exposing undergraduate mathematicians to various job opportunities,in industry, government, national labs, etc., which require quantitative skills and mathematical background.
The first presentation will be given by Tek Systems. Operating across the globe, TEKsystems Inc. is a leading provider of strategic IT staffing and Global Services. TEKsystems is committed to fostering your professional development and helping you to find employment on engagements you find stimulating and rewarding.
With their relationships with more than 135 successful companies in the Metro-Detroit area
alone, they are able to market your resume directly to hiring managers. The presenters will
share information about opportunities with local companies, as well as provide
understanding on how to apply your Mathematics and Actuarial Science backgrounds to
career opportunities in the field of Big Data and Analytics.

Title: Level-set based structured solution for geophysical imaging

Date: 10/07/2016

Time: 4:10 PM - 5:00 PM

Place: 1502 Engineering Building

We study the inverse problems arising from geophysical imaging. A
level-set-based parametric approach is proposed to find structured solutions
with piecewise continuous structure and interface structure. We have studied
traveltime tomography, inversion of potential field data, and joint
inversion of heterogeneous data; a systematic methodology has been developed
to handle these problems in an Eulerian framework. The structural
parameterization improves resolutions of interfaces in the imaging, and our
method alleviates ambiguities in the interpretations of geophysical data.

Title: Tangent measures (D. Preiss ``Geometry of measure in R^n') - III

Date: 10/10/2016

Time: 1:40 PM - 3:10 PM

Place: C304 Wells Hall

We continue to explore Preiss's paper. We will finish Section 2 by showing that compact tangent cones are connected (in particular, a tangent cone cannot be generated by two different measures) and pass to the study of uniform measures in Section 3.

Title: Two phase free boundary problem of Chris Bishop: how singular integrals and blow up methods solve it.

Date: 10/10/2016

Time: 4:02 PM - 5:00 PM

Place: C517 Wells Hall

This is a report of a joint work with Azzam, Mourgoglou and Tolsa. Harmonic measure was in the focus of attention of many analysts (Bishop, Bourgain, Carleson, Jones, Makarov, Wolff to name a few) in 1985-1995. Beautiful metric properties of harmonic measure were revealed during this period, this concerns especially a 2D case. In higher dimensions the behavior was more enigmatic, and many problems were left for the future. Recently it turned out that the technique of non-homogeneous harmonic analysis developed by (among others) Nazarov-Treil-and the speaker (NTV) is the right tool to clarify several
outstanding questions about harmonic measure in 3 and higher dimensions. We will show how to solve yet another problem formulated by Chris Bishop in 1992.

Title: Positivity and combinatorics of some bases of cluster algebras

Date: 10/11/2016

Time: 1:00 PM - 1:50 PM

Place: C304 Wells Hall

Lots of research of cluster algebras focuses on construction of their natural bases. A desirable property of a good basis is that all elements are universally positive and that all structure constants are positive. In this talk, I will survey some bases that have this property. In particular, I will talk about the theta function bases constructed by Gross, Hacking, Keel and Kontsevich using scattering diagrams and broken lines, and give an explicit combinatorial description of these broken lines in special cases.

Title: What We Still Don’t Know About Addition and Multiplication

Date: 10/11/2016

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

How could there be something we don’t know about arithmetic? It would seem that subject was sewn up in third grade. But here’s a problem we don’t know the solution to: What is the most efficient method for multiplication? And here is another: How many different numbers appear in a large multiplication table? There are many more such problems, largely unsolved, and for which we could use some help!

Speaker: Gregory Larnell, University of Illinois - Chicago

Title: We Real Cool: Reconsidering the Cooling Out Phenomenon in a Remedial Mathematics Education Context

Date: 10/12/2016

Time: 3:30 PM - 4:20 PM

Place: 252 EH

Recognizing the gatekeeping function of mathematics as curricular discipline has been a longstanding subject of mathematics education research and scholarly debate. Difficult and complex questions about how this gatekeeping occurs, however, have been explored less frequently. Furthermore, everyday social occurrences by which mathematics gatekeeping and racialized inequity intersect have received even less attention. In this talk, I will re-introduce and discuss the cooling out phenomenon in education (COPE), a concept that originates in sociology and has re-emerged across several academic disciplines, including higher education. This time, the purpose is to understand how COPE functions amid Black undergraduates’ experiences within remedial mathematics courses. Drawing on two recent studies in which I analyzed series of interviews, I discuss the ways that COPE unfolds amid study participants’ experiences, who and what processes may be involved in COPE, and implications regarding COPE for mathematics education research more broadly.

A 2-knot is defined to be an embedding of S^2 in S^4. Unlike for knots
in S^3, the theory of concordance of 2-knots is trivial. This talk
will be framed around the related concept of 0-concordance of 2-knots.
It has been conjectured that this is also a trivial theory, that every
2-knot is 0-concordant to every other 2-knot. We will show that this conjecture is false, and in fact there are infinitely many 0-concordance classes.
We'll in particular point out how the concept of 0-concordance is
related to understanding smooth structures on S^4. The proof will
involve invariants coming from Heegaard-Floer homology, and we will
furthermore see how these invariants can be used shed light on other
properties of 2-knots such as amphichirality and invertibility.

Title: The ranges of some familiar arithmetic functions

Date: 10/13/2016

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

We consider 4 functions from elementary number theory:
sigma (the sum-of-divisors function), phi (Euler's function),
lambda (Carmichael's universal exponent function), and s (the
sum-of-proper-divisors function). In particular we discuss the
distribution of the values of these functions, and coincidences
of values. Most of the problems considered have a fairly long
history, some over 80 years. We report on recent progress.
(Various parts of this work are joint with Kevin Ford,
Florian Luca, Paul Pollack, and others.)

Speaker: Pierre Gremaud, North Carolina State University

Title: Data analysis for mathematicians: the example of syncope

Date: 10/14/2016

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

We will discuss four important concepts in the analysis of data:
variable importance, surrogate models, classification and clustering.
Each of these concepts will be introduced from scratch with a general
mathematical audience in mind. In particular, we will introduce methods
for nonparametric statistics as well as new results in global
sensitivity analysis.
The material of the talk will be illustrated by the analysis of syncope
data. Syncope refers to a spontaneous and transient loss of
consciousness. It is a prevalent disorder which accounts for over 1
million visits to emergency departments per year in the US alone. There
is currently no agreement on the root causes of that disorder.

Title: Harmonic measure and Geometric Measure Theory: no regularity two phase fee boundary problem

Date: 10/17/2016

Time: 4:02 PM - 4:52 PM

Place: C517 Wells Hall

Harmonic measure and Geometric Measure Theory: no regularity two phase fee boundary problem
We will show how from absolute continuity of outer and inner harmonic measures one can deduce the rectifiability of the common boundary.
This solves the problem of Bishop from the early 90's.
This requires the combination of the technique from the GMT and the new technique of non-homogeneous harmonic analysis
elaborated by Nazarov-Treil-Volberg and Tolsa and his school.
This is a joint work with J. Azzam, M. Mourgoglou and X. Tolsa.

In this talk we will discuss construction of 5-dimensional open books with 4-dimensional exotic pages, and discuss how their contact structures relate to the smooth structures of their pages.

Originally, cluster algebras were introduced to axiomatize part of the `dual canonical basis' for the ring of functions on a Lie group, and similar patterns have been found in many other contexts. This naturally begs the question: do cluster algebras have a natural basis, extending the set of cluster monomials?
Recent work by Gross, Hacking, Keel, and Kontsevich has proposed a basis of `theta functions'. These theta functions are defined by counting certain tropical curves in a `scattering diagram', a task which appears to be prohibitively difficult in practice. In this talk, I will demonstrate how to use this counting problem to restrict the behavior of the theta functions; in particular, to constrain their monomial support. In simple cases (that is, rank 2), such constraints completely characterize the theta functions, and may be used to give alternate characterizations which are far more computable in practice. Parts of this talk will cover joint work with Man Wai Cheung, Mark Gross, Gregg Musiker, Dylan Rupel, Salvatore Stella, and Harold Williams

Title: Introduction to representation theory of semisimple (complex) Lie algebras

Date: 10/18/2016

Time: 3:00 PM - 3:50 PM

Place: C304 Wells Hall

In this talk, we will define semisimple Lie algebras and their representations. We will see all the finite dimensional irreducible representations of sl(2,C) and a few ways they help us understand the structure of a semisimple Lie algebra.

Title: The Restriction-Contraction Matroid Hopf Algebra

Date: 10/18/2016

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

A graded connected Hopf algebra is often referred to as a combinatorial Hopf algebra. This is largely because many Hopf algebras of this nature arise from combinatorial settings. In general much of the information about a Hopf algebra of this type is given to you by a function called the antipode of the algebra. In this talk we will look at one such Hopf algebra which arises by looking at matroids. We will briefly go over the main concepts from matroid theory that we will need, and then construct the combinatorial Hopf algebra. After that I will present some results for computing the antipode of this Hopf algebra. This talk is designed to be fairly introductory, and shouldn't require any background either in Hopf algebras or in matroid theory.

Title: Regularity for Minimal Surfaces of Codimension 1

Date: 10/18/2016

Time: 4:10 PM - 5:00 PM

Place: C517 Wells Hall

Assuming the existence of minimizers for the area functional, we will show how to set up the problem in order to invoke De Giorgi-Nash-Moser. Time permitting, we will show why regularity
results are a much harder problem for minimal surfaces of codimension strictly bigger than 1.

Title: Quantum groups from quantum character varieties

Date: 10/19/2016

Time: 2:30 PM - 3:50 PM

Place: C304 Wells Hall

It was shown by Fock and Goncharov that moduli spaces of local systems on decorated surfaces provide examples of cluster varieties and thus admit canonical quantizations. I will describe a joint work with Gus Schrader where we embed the quantum group U_q(sl_n) into the quantized moduli space of $SL(n)$-local systems on a punctured disk with two marked points. This embedding endows the quantum group with a system of quantum cluster coordinates. It also allows us to realize the adjoint action of the $R$-matrix as a half Dehn twist of a twice punctured disk and factor it into a sequence of cluster mutations. If time permits, I will discuss applications of our construction to the representation theory.
The talk will be preceded by a 20-25 minutes refresher on quantum groups. I will recall the definition and discuss how one might think of a quantum group as a quantization of a Poisson-Lie group on one hand, and a deformation of a universal enveloping algebra on the other. The main part of the talk will be independent of the first one, all necessary definitions will be reintroduced.

Speaker: Luis Caffarelli, University of Texas at Austin

Title: The geometric structure of mathematical models involving minimal surfaces and phase transition problems.

Date: 10/19/2016

Time: 5:30 PM - 6:30 PM

Place:

In the first lecture I will discuss some of the main ideas in the
theory of regularity for minimal surfaces and for free boundary problems
and how they influence each other.
In the next two lectures I will dwell in more detail on the main ideas of
the mathematical theory and several problems where phase transitions and
minimal surfaces interact, mainly when the phase transition energy
involves bulk energy in each phase and transition energy along the
interphase.

The splitting number of a link is the minimal number of crossing changes between different components required to convert it into a split union of its components. Since 2012, this invariant has been studied using various tools such as Khovanov homology, covering link calculus, the Alexander polynomial and Heegaard-Floer homology.
After briefly reviewing some of these methods, we will show how (multivariable) signatures give strong lower bounds on the splitting number. This is a joint work with David Cimasoni and Kleopatra Zacharova.

Speaker: Luis Caffarelli, University of Texas at Austin

Title: The geometric structure of mathematical models involving minimal surfaces and phase transition problems.

Date: 10/20/2016

Time: 4:00 PM - 5:00 PM

Place: 115 International Center

In the first lecture I will discuss some of the main ideas in the
theory of regularity for minimal surfaces and for free boundary problems
and how they influence each other.
In the next two lectures I will dwell in more detail on the main ideas of
the mathematical theory and several problems where phase transitions and
minimal surfaces interact, mainly when the phase transition energy
involves bulk energy in each phase and transition energy along the
interphase.

Speaker: Luis Caffarelli, University of Texas at Austin

Title: The geometric structure of mathematical models involving minimal surfaces and phase transition problems.

Date: 10/21/2016

Time: 4:00 PM - 5:00 PM

Place: C304 Wells Hall

In the first lecture I will discuss some of the main ideas in the
theory of regularity for minimal surfaces and for free boundary problems
and how they influence each other.
In the next two lectures I will dwell in more detail on the main ideas of
the mathematical theory and several problems where phase transitions and
minimal surfaces interact, mainly when the phase transition energy
involves bulk energy in each phase and transition energy along the
interphase.

Title: Stochastic Three-Dimensional Rotating Navier-Stokes Equations: Averaging, Convergence and Regularity

Date: 10/24/2016

Time: 4:02 PM - 5:00 PM

Place: C517 Wells Hall

We consider stochastic three-dimensional rotating Navier-Stokes equations and prove averaging theorems for stochastic problems in the case of strong rotation. Regularity results are established by bootstrapping from global regularity of the limit stochastic equations and convergence theorems. The energy injected in the system by the noise is large, the initial condition has large energy, and the regularization time horizon is long. Regularization is the consequence of a precise mechanism of relevant three-dimensional nonlinear interactions. We establish multiscale averaging and convergence theorems for the stochastic dynamics. References [1] Flandoli F. , Mahalov A. , “Stochastic 3D Rotating Navier-Stokes Equations: Averaging, Convergence and Regularity,” Archive for Rational Mechanics and Analysis, 205, No. 1, 195–237 (2012). [2] Cheng B. , Mahalov A. , “Euler Equations on a Fast Rotating Sphere – Time- Averages and Zonal Flows,” European Journal of Mechanics B/Fluids, 37, 48-58 (2013). [3] Mahalov A. Multiscale modeling and nested simulations of three-dimensional ionospheric plasmas: Rayleigh-Taylor turbulence and nonequilibrium layer dynamics at fine scales, Physica Scripta, Phys. Scr. 89 (2014) 098001 (22pp), Royal Swedish Academy of Sciences.

Title: How to define the Torelli group of a surface with boundary?

Date: 10/24/2016

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

The Torelli group of a closed surface S is defined as the group of diffeomorphisms of S acting trivially on homology and considered up to isotopy. These groups naturally arise
in topology of 3-manifolds and in algebraic geometry. At the same time they are quite interesting groups by themselves. In order to study them, it is highly desirable to have also Torelli groups of surfaces with non-empty boundary. It turns out that the naive definition is not a good one.

This talk will review the perspective of strings on quivers and mutations.
Starting from a brief mention of the closed string which was well studied in the 80s and early 90s we will move to a discussion on the open string and the natural way it gives rise to the notion of quivers - the so called quiver gauge theories. This perspective allows the study of various dynamical questions in brane physics, but also brings contact with fields in mathematics in representation theory and algebraic geometry. We will discuss quivers that arise on branes at singularities, brane tilings, and the phenomenon of Seiberg duality - the physics counterpart of a quiver mutation.

Speaker: Maitreyee Kulkarni, Louisiana State University

Title: Categorification of cluster structure on partial flag varieties

Date: 10/25/2016

Time: 1:00 PM - 1:50 PM

Place: C304 Wells Hall

Let G be a Lie group of type ADE and P be a parabolic subgroup. It is known that there exists a cluster structure on the coordinate ring of the partial flag variety G/P (see the work of Geiss, Leclerc, and Schroer). Since then there has been a great deal of activity towards categorifying these cluster algebras. Jensen, King, and Su gave a direct categorification of the cluster structure on the homogeneous coordinate ring for Grassmannians (that is, when G is of type A and P is a maximal parabolic subgroup). In this setting, Baur, King, and Marsh gave an interpretation of this categorification in terms of dimer models. In this talk, I will give an analog of dimer models for groups in other types by introducing a technique called “constructing sheets over Dynkin diagrams”, which can (conjecturally) be used to generalize the result of Baur, King, and Marsh.

We will give an introduction to Poisson algebras and Poisson manifolds, and discuss the relationship with symplectic geometry and the Hamiltonian formalism in classical mechanics. We will also state some well-known results in the subject, and, time-permitting, discuss some related algebraic questions pertaining to Poisson varieties, which are defined analogously in the algebraic category.

Title: Curvature free rigidity for higher rank three manifolds.

Date: 10/27/2016

Time: 2:00 PM - 2:50 PM

Place: C304 Wells Hall

Fixing K=-1,0, or 1, a complete Riemannian manifold is said to have higher rank if each geodesic admits a parallel vector field making curvature K with the geodesic.
Locally symmetric spaces provide examples. Rank rigidity theorems aim to show that these are the only examples of manifolds of higher rank, usually with additional curvature assumptions.
After discussing historical results, I'll discuss how rank rigidity results hold in dimension three without additional curvature assumptions.

Speaker: Tom Chou, University of California at Los Angeles

Title: Path integral-based inference of PDEs and bond energies and mobility in Dynamic Force Spectroscopy

Date: 10/28/2016

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

A Bayesian interpretation is given for regularization terms for
parameter functions in inverse problems. Fluctuations about the
extremal solution depend on the regularization terms - which encode
prior knowledge - provide quantification of uncertainty. After
reviewing a general path-integral framework, we discuss an application
that arises in molecular biophysics: The inference of bond energies
and bond coordinate mobilities from dynamic force spectroscopy
experiments.

Title: Arithmetic criteria of spectral dimension for quasiperiodic Schrodinger operators.

Date: 10/31/2016

Time: 4:02 PM - 4:52 PM

Place: C517 Wells Hall

Shiwen Zhang (msu), joint work with Svetlana Jitomirskaya (uci)
Abstract: We introduce a notion of β-almost periodicity and prove quantitative lower spectral/
quantum dynamical bounds for general bounded β-almost periodic potentials.
Applications include a sharp arithmetic criterion of full spectral dimensionality for analytic
quasiperiodic Schrodinger operators in the positive Lyapunov exponent regime
and arithmetic criteria for families with zero Lyapunov exponents, with applications
to Sturmian potentials and the critical almost Mathieu operator.

Title: How to define the Torelli group of a surface with boundary-2

Date: 10/31/2016

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

The first talk was devoted mostly to the motivation. The second talk will be devoted to the extension problem in Torelli topology and some speculations about the possible answer to the question posed in the talk title.