Title: Understanding sieve via additive combinatorics

Date: 01/11/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Xuancheng Fernando Shao, Oxford

Many of the most interesting problems in number theory can be phrased under the general framework of sieve problems. For example, the ancient sieve of Eratosthenes is an algorithm to produce primes up to a given threshold. Sieve problems are in general very difficult, and a class of clever techniques have been discovered in the last 100 years to yield stronger and stronger results. In this talk I will discuss the significance of understanding general sieve problems, and present a novel approach to study them via additive combinatorics. This is joint work with Kaisa Matomaki.

Title: Projective coordinates for the analysis of data

Date: 01/12/2017

Time: 2:00 PM - 2:50 PM

Place: C304 Wells Hall

Speaker: Jose Perea, MSU

Barcodes - the persistent homology of data - have been shown to be effective quantifiers of multi-scale structure in finite metric spaces. Moreover, the universal coefficient theorem implies that (for a fixed field of coefficients) the barcodes obtained with persistent homology are identical to those obtained with persistent cohomology. Persistent cohomology, on the other hand, is better behaved computationally and allows one to use convenient interpretations such as the Brown representability theorem. We will show in this talk how one can use persistent cohomology to produce maps from data to (real and complex) projective space, and conversely, how to use these projective coordinates to interpret persistent cohomology computations.

Title: Random discrete structures: Phase transitions, scaling limits, and universality

Date: 01/13/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Sanchayan Sen, McGill University

The aim of this talk is to give an overview of some recent results in two interconnected areas:
a) Random graphs and complex networks: The last decade of the 20th century saw significant growth in the availability of empirical data on networks, and their relevance in our daily lives. This stimulated activity in a multitude of fields to formulate and study models of network formation and dynamic processes on networks to understand real-world systems.
One major conjecture in probabilistic combinatorics, formulated by statistical physicists using non-rigorous arguments and enormous simulations in the early 2000s, is as follows: for a wide array of random graph models on n vertices and degree exponent \tau>3, typical distance both within maximal components in the critical regime as well as on the minimal spanning tree on the giant component in the supercritical regime scale like n^{\frac{\tau\wedge 4 -3}{\tau\wedge 4 -1}}. In other words, the degree exponent determines the universality class the random graph belongs to. The mathematical machinery available at the time was insufficient for providing a rigorous justification of this conjecture.
More generally, recent research has provided strong evidence to believe that several objects, including
(i) components under critical percolation,
(ii) the vacant set left by a random walk, and
(iii) the minimal spanning tree,
constructed on a wide class of random discrete structures converge, when viewed as metric measure spaces, to some random fractals in the Gromov-Hausdorff-Prokhorov sense, and these limiting objects are universal under some general assumptions. We report on recent progress in proving these conjectures.
b) Stochastic geometry: In contrast, less precise results are known in the case of spatial systems. We discuss a recent result concerning the length of spatial minimal spanning trees that answers a question raised by Kesten and Lee in the 90's, the proof of which relies on a variation of Stein's method and a quantification of the classical Burton-Keane argument in percolation theory.
Based on joint work with Louigi Addario-Berry, Shankar Bhamidi, Nicolas Broutin, Sourav Chatterjee, Remco van der Hofstad, and Xuan Wang.

Gauge theory is a subject that has emerged from theoretical physics. It has deep links with many areas of mathematics (including partial differential equations, representation theory, algebraic geometry, differential geometry and topology). Most of the mathematical work on gauge theory has focused on low dimensions, where one can exploit the anti-self-dual Yang-Mills equation and the analytic difficulties are still quite tractable.
In this talk I will discuss three concrete questions that arise in gauge theory in higher dimensions. First, I will discuss gauge theory on Kähler manifolds, with a particular focus on singular Hermitian Yang-Mills connections. Afterwards, I will move on to the more exotic topic of gauge theory on G2-manifolds. I will discuss a method to construct solutions of the Yang-Mills equation on a class of G2-manifolds called twisted connected sums. Finally, I will talk about the prospects of defining enumerate gauge theoretical invariants for G2-manifolds and the difficulties arising from codimension four bubbling.

Title: Free dynamics of a tracer particle in a Fermi sea

Date: 01/19/2017

Time: 11:00 AM - 11:50 AM

Place: C304 Wells Hall

Speaker: Soeren Petrat, IAS and Princeton

The talk is about the dynamics of a tracer particle coupled strongly to a dense non-interacting electron gas in one or two dimensions. I will present a recent result that shows that for high densities the tracer particle moves freely for very long times, i.e., the electron gas becomes transparent. However, the correct phase factor is non-trivial. To leading order, it is given by mean-field theory, but one also has to include a correction coming from immediate recollision diagrams.

Title: Turaev-Viro invariants of links and the colored Jones polynomial

Date: 01/19/2017

Time: 2:00 PM - 2:50 PM

Place: C304 Wells Hall

Speaker: Renaud Detcherry, MSU

In a recent work by Tian Yang and Qingtao Chen, it has been observed that one can recover the hyperbolic volume from the asymptotic of Turaev-Viro invariants of 3-manifolds at a specific root of unity. This is reminiscent of the volume conjecture for the colored Jones polynomial.
In the case of link complements, we derive a formula to express Turaev-Viro invariants as a sum of values of colored Jones polynomial, and get a proof of Yang and Chen's conjecture for a few link complements. We also discuss the link between this conjecture and the volume conjecture. This is joint work with Effie Kalfagianni and Tian Yang.

Rectifiable spaces are a class of metric measure spaces that are Lipschitz analogues of differentiable manifolds (for example, they admit a parameterization by Lipschitz charts) and arise naturally in many areas of analysis and geometry. Due to the important works of Federer, Mattila, Preiss, and many others, we now have a good understanding of the geometric properties of rectifiability in Euclidean spaces. In this talk, we examine some generalizations of rectifiability to the setting of non-Euclidean spaces and discuss the similarities and differences between rectifiability in the Euclidean setting and these generalizations.

Title: What matters for learners in linguistically diverse classrooms?

Date: 01/25/2017

Time: 3:30 PM - 5:00 PM

Place: 252 EH

Speaker: M. Alejandra Sorto, Texas State University

In this talk, I will discuss which factors contribute to mathematics gains of learners in Texas middle schools including teacher education, mathematical knowledge for teaching (MKT), mathematical quality of instruction (MQI), and quality of instruction in linguistically diverse classrooms. Teachers’ time spent in professional development activities and general quality of instruction had positive and significant effects for all learners, with much larger effect on learners that are English proficient. For students that are learning English as a second language, teachers’ practices affording their linguistic diversity had a positive and significant effect.

Let S be a hyperbolic surface. We will give a history of counting results for geodesics on S. In particular, we will give estimates that fill the gap between the classical results of Margulis and the more recent results of Mirzakhani. We will then give some applications of these results to the geometry of curves. In the process we highlight how combinatorial properties of curves, such as self-intersection number, influence their geometry.

Title: Orders from $\widetilde{PSL_2(\mathbb{R})}$ Representations and Non-examples

Date: 01/26/2017

Time: 2:00 PM - 2:50 PM

Place: C304 Wells Hall

Speaker: Xinghua Gao, UIUC

It is still unknown for the case of rational homology 3-sphere whether the left-orderability of its fundamental group and it not be a Heegaard-Floer L-space are equivalent. Let $M$ be an integer homology 3-sphere. One way to study left-orderability of $\pi_1(M)$ is to construct a non-trivial representation from $\pi_1(M)$ to $\widetilde{PSL_2(\mathbb{R})}$. However this method does not always work. In this talk, I will give examples of non L-space irreducible integer homology 3-spheres whose fundamental groups do not have nontrivial $\widetilde{PSL_2(\mathbb{R})}$ representations.

Title: Representations, Combinatorics, and Configurations.

Date: 01/27/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Linhui Shen, Northwestern

We briefly recall Knutson-Tao’s hive model that calculates the Littlewood-Richardson coefficients. We consider the configuration spaces of decorated flags introduced by Fock and Goncharov. The configuration spaces admit natural functions called potentials introduced by Goncharov and myself. We prove that the tropicalization of configuration spaces with potentials recovers Knutson-Tao’s hives. As an application, Hong and I solve the Saturation problem for the Lie algebra so(2n+1). If time permits, I will further explain their deep connections with geometric Satake correspondence, homological mirror symmetry, and Donaldson-Thomas theory.

Title: New bounds for equiangular lines and spherical two-distance sets

Date: 01/31/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Wei-Hsuan Yu, MSU

The set of points in a metric space is called an s-distance set if pairwise distances between these points admit only s distinct values. Two-distance spherical sets with the set of scalar products {alpha, -alpha}, alpha in [0,1), are called equiangular. The problem of determining the maximal size of s-distance sets in various spaces has a long history in mathematics. We determine a new method of bounding the size of an s-distance set in two-point homogeneous spaces via zonal spherical functions. This method allows us to prove that the maximum size of a spherical two-distance set in R^n is n(n+1)/2 with possible exceptions for some n = (2k+1)^2-3, k a positive integer. We also prove the universal upper bound ~ 2 n a^2/3 for equiangular sets with alpha = 1/a and, employing this bound, prove a new upper bound on the size of equiangular sets in an arbitrary dimension. Finally, we classify all equiangular sets reaching this new bound.

Speaker: Keerthi Madapusi Pera, University of Chicago

Periods are a special class of complex numbers, arising as integrals of differential forms on algebraic varieties. L-functions are analytic objects that generalize the Riemann zeta function. Both are objects admitting deceptively simple definitions, but carry deep arithmetic information.
In this talk, I'll explain a relationship between periods of abelian varieties with complex multiplication, and certain Artin L-functions, originally conjectured by P. Colmez, and sketch a proof of it that arose out of joint work with Andreatta, Goren and Ben Howard. Among other applications, this relationship has led to a proof by J. Tsimerman of the Andre-Oort conjecture for Siegel modular varieties.

Speaker: Leonid Chekhov, Steklov Mathematical Institute

We identify the Teichmuller space $T_{g,s,n}$ of (decorated) Riemann
surfaces $\Sigma_{g,s,n}$ of genus $g$, with $s>0$ holes and $n>0$
bordered cusps located on boundaries of holes uniformized by Poincare with
the character variety of $SL(2,R)$-monodromy problem. The effective
combinatorial description uses the fat graph technique; observables are
geodesic functions of closed curves and $\lambda$-lengths of paths
starting and terminating at bordered cusps decorated by horocycles. Such
geometry stems from special 'chewing gum' moves corresponding to colliding
holes (or sides of the same hole) in a Riemann surface with holes. We
derive Poisson and quantum structures on sets of observables relating them
to quantum cluster algebras of Berenstein and Zelevinsky. A seed of the
corresponding quantum cluster algebra corresponds to the partition of
$\Sigma_{g,s,n}$ into ideal triangles, $\lambda$-lengths of their sides
are cluster variables constituting a seed of the algebra; their number
$6g-6+3s+2n$ (and, correspondingly, the seed dimension) coincides with the
dimension of $SL(2,R)$-character variety given by
$[SL(2,R)]^{2g+s+n-2}/\prod_{i=1}^n B_i$,
where $B_i$ are Borel subgroups associated with bordered cusps. I also discuss the
very recent results enabling constructing monodromy matrices of SL(2)-connections out of
the corresponding cluster variables.
The talk is based on the joint papers with with M.Mazzocco and V.Roubtsov

Title: Cutting plane theorems for Integer Optimization and computer-assisted proofs

Date: 02/03/2017

Time: 10:00 AM - 11:00 AM

Place: 1502 Engineering Building

Speaker: Yuan Zhou, UC Davis

Optimization problems with integer variables form a class of mathematical
models that are widely used in Operations Research and Mathematical Analytics.
They provide a great modeling power, but it comes at a high price: Integer
optimization problems are typically very hard to solve, both in theory and practice.
The state-of-the-art solvers for integer optimization problems use cutting-plane
algorithms. Inspired by the breakthroughs of the polyhedral method for
combinatorial optimization in the 1980s, generations of researchers have studied the
facet structure of convex hulls to develop strong cutting planes. However, the
proofs of cutting planes theorems were hand-written, and were dominated by
tedious and error-prone case analysis.
We ask how much of this process can be automated: In particular, can we use
algorithms to discover and prove theorems about cutting planes? I will present our
recent work towards this objective. We hope that the success of this project would
lead to a tool for developing the next-generation cutting planes that answers the
needs prompted by ever-larger applications and models.

Title: Linearly Preconditioned Nonlinear Solvers for Phase Field Equations

Date: 02/06/2017

Time: 10:00 AM - 11:00 AM

Place: C100 Wells Hall

Speaker: Wenqiang Feng, The University of Tennessee, Knoxville

Many unconditionally energy stable schemes for the physical models will lead to a highly nonlinear elliptic PDE systems which arise from time discretization of parabolic equations. I will discuss two efficient and practical preconditioned solvers- Preconditioned Steepest Descent (PSD) solver and Preconditioned Nonlinear Conjugate Gradient (PNCG) solver - for the nonlinear elliptic PDE systems. The main idea of the preconditioned solvers is to use a linearized version of the nonlinear operator as a pre-conditioner, or in other words, as a metric for choosing the search direction. Based on certain reasonable assumptions of the linear pre-conditioner, a geometric convergence rate is shown for the nonlinear PSD iteration. Numerical simulations for some important physical application problems - including Cahn-Hillilard equation, epitaxial thin film equation with slope selection, square phase field crystal equation and Functionalized Cahn-Hilliard equation- are carried out to verify the efficiency of the solvers.

This is a continuation of the previous talks. We will be working through the paper of Karp and Williams, to understand the Amplituhedron as the image of a matrix map from the totally positive Grassmannian. This talk will work through the basic definitions of the amplituhedron, and then walk through some examples when k=1. It will follow closely to section 1 and section 3 of the Karp and Williams paper.

This is an expository talk and no background will be assumed. Given two integral vectors R = (r_1,...,r_m) and S = (s_1,...,s_n) we wish to know whether there exists an m x n matrix A whose ith row has sum r_i and whose jth column has sum s_j for all i, j. Such matrices have applications via the Transportation Problem. We will discuss the fundamental results in this area, including the Gale-Ryser Theorem.

Title: Spectral dimension and quantum dynamical bound for quasiperiodic Schrodinger operators.

Date: 02/09/2017

Time: 11:00 AM - 11:50 AM

Place: C304 Wells Hall

Speaker: Shiwen Zhang, MSU

We introduce a notion of beta-almost periodicity and prove quantitative lower spectral/quantum dynamical bounds for general bounded beta-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.

Speaker: Christine Lee, University of Texas, Austin

A Jones surface for a knot in the three-sphere is an essential surface whose boundary slopes, Euler characteristic, and number of sheets correspond to quantities defined from the asymptotics of the degrees of colored Jones polynomial. The Strong Slope Conjecture by Garoufalidis and Kalfagianni-Tran predicts that there are Jones surfaces for every knot.
A link diagram D is said to be a Murasugi sum of two links D' and D' if a state graph of D has a cut vertex, which separates the graph into two state graphs of D' and D', respectively. We may obtain a state surface in the complement of the link K represented by D by gluing the state surface for D and the state surface for D' along the disk filling the circle represented by the cut vertex in the state graph. The resulting surface is called the Murasugi sum of the two state surfaces.
We consider near-adequate links which are Murasugi sums of certain non-adequate link diagrams with an adequate link diagram along their all-A state graphs with an additional graphical constraint. For a near-adequate knot, the Murasugi sum of the corresponding state surface is a Jones surface by the work of Ozawa. We discuss how this proves the Strong Slope Conjecture for this class of knots and raises interesting questions about constraints on the possible Murasugi sum-decompositions of a link diagram.

Title: Spatial and stochastic dynamics in development and regeneration

Date: 02/09/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Qing Nie, UC Irvine

In developing and renewing tissues, the properties of cells are controlled by secreted molecules from cells, their actions to the downstream regulators and genes, and transitions among different types of cells. The multiscale and stochastic nature of such spatial and dynamic systems presents tremendous challenges in synthesizing experimental observations and their understanding. In this talk, I will present several mathematical modeling frameworks with different complexity for systems ranging from single cells to multistage cell lineages. Questions of our interests include roles of feedbacks in regeneration speed, stem cell niche for tissue spatial organization, and crosstalk between epigenetic and genetic regulations. In addition to comparing our modeling outputs with experimental data, we will emphasize development of various mathematical and computational tools critical to success of using models in analyzing complex biological systems.

Title: Polytopes and the Problem with Pick's Theorem

Date: 02/10/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Robert Davis, MSU

Polytopes are a generalization of convex polygons into higher dimensions. In two dimensions, we have Pick's Theorem: a simple way of relating the area of a polygon with the number of ordered pairs it contains where each coordinate is an integer. So, why not try to do the same thing with polytopes in general? Problems quickly arise, even in three dimensions. In this talk, I will describe what these problems are and how we can overcome them. The answers aren't obvious, but we will see just how much it pays off in the end.

Speaker: Joshua Mike, The University of Tennessee, Knoxville

In this talk, we will explore two applications: The first considers simplicial cohomology as a tool to investigate and eliminate inequity in kidney paired donation (KPD). The KPD pool is modeled as a graph wherein cocycles represent fair organ exchanges. Helmholtz decomposition is used to split donation utilities into gradient and harmonic portions. The gradient portion yields a preference score for cocycle allocation. The harmonic portion is isomorphic to the 1-cohomology and is used to guide a new algorithmic search for exchange cocycles. We examine correlation between a patient’s chance to obtain a kidney and their score under various allocation methods and conclude by showing that traditional methods are biased, while our new algorithm is not.
The second considers the persistent homology of a smoothed noisy dynamic. The machinery of persistent homology yields topological structure for discrete data within a metric space. Homology in dynamical systems can capture important features such as periodicity, multistability, and chaos. We consider a hidden Markov dynamic and compare particle filter to optimal smoothing a posteriori. We conclude with a stability theorem for the convergence of the persistent homology of the particle filtered path to that of the optimal smoothed path.

We will continue reading Preiss's paper [Geometry of measures in R^n: Distribution, rectifiability, and densities]. We will first review some necessary notations and properties of uniformly distributed measure discussed last semester and then study whether a measure is flat or curved at infinity.

Title: Matrices with given row and column sums, II

Date: 02/14/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Bruce Sagan, MSU

In this continuation of the first talk, I will discuss new results by Brualdi and myself where we impose various symmetry conditions on the desired matrix A via the action of the dihedral group of the square.

Title: Topological recursion for matrix models and abstract topological recursion (a course of 3 lectures)

Date: 02/15/2017

Time: 3:00 PM - 3:50 PM

Place: C304 Wells Hall

Speaker: Leonid Chekov, Steklov Institute and MSU

I will begin with the description of a generating function for numbers of
Grothendieck's dessins d'enfant, or Belyi pairs. This generating function
is given by a random matrix model integral, I describe what is a
topological expansion (AKA genus expansion) of such models. The method
allowing finding corrections in all orders of the genus expansion is
Topological Recursion formulated in its present form by Eynard, Orantin
and the speaker in 2005-2006. This method had already found numerous
applications in mathematics and mathematical physics, so I describe the
general construction underlying the topological recursion and present an
(incomplete) list of its applications. Very recently, this method was
developed into an abstract topological recursion by Kontsevich and
Soibelman. In my last lecture I explain their construction and our
interpretation of it (forthcoming paper by J.Andersen, G.Borot, L.Ch., and
N.Orantin).

Title: How does racial identity matter in the mathematics classroom?

Date: 02/15/2017

Time: 3:30 PM - 5:00 PM

Place: 252 EH

Speaker: Maria del Rosario Zavala, San Francisco State University

In this talk I will use contemporary research in mathematics learning as a racialized form of experience to take stock of where the field is in relation to the question of “How does racial identity matter in the mathematics classroom?” and examine where we are headed in research and classroom practice. One aspect of my talk is to complicate the master narrative of achievement motivation, which ascribes high achievement solely to effort, and analyze the racial dimensions of such narratives using examples from research with Latinx high school students.

First, I will give the definition of CAT(0) spaces. Then I will describe the boundary of these spaces and how the geodesic flow behave on these spaces.

Title: Non-equilibrium transitions between metastable patterns in populations of motile bacteria

Date: 02/16/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Eric Vanden-Eijnden, NYU

Active materials can self-organize in many more ways than their
equilibrium counterparts. For example, self-propelled particles whose
velocity decreases with their density can display motility-induced
phase separation (MIPS), a phenomenon building on a positive feedback
loop in which patterns emerge in locations where the particles slow
down. Here, we investigate the effects of intrinsic fluctuations in
the system's dynamics on MIPS, using a field theoretic description
building on results by Cates and collaborators. We show that these
fluctuations can lead to transitions between metastable patterns. The
pathway and rate of these transitions is analyzed within the realm of
large deviation theory, and they are shown to proceed in a very
different way than one would predict from arguments based on
detailed-balance and microscopic reversibility. Specifically, we show
that these transitions involve fluctuations in diffiusivity of the
bacteria followed by fluctuations in their population, in a specific
sequence. The methods of analysis proposed here, including their
numerical components, can be used to study noise-induced
non-equilibrium transitions in a variety of other non-equilibrium
set-ups, and lead to predictions that are verifiable experimentally.

We will continue reading Theorem 3.14 of Preiss's paper [Geometry of measures in R^n: Distribution, rectifiability, and densities]. We will study whether a uniformly distributed measure is flat or curved at infinity. The proof is based on the previous Lemma 3.13 and some basic properties of a symmetric bi-linear form.

Title: On Distance Preserving and Sequentially Distance Preserving Graphs

Date: 02/21/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Emad Zahedi, MSU

A graph H is an isometric subgraph of G if d_H(u,v) = d_G(u,v), for every pair u,v in V(H), where d denotes distance. A graph is distance preserving (dp) if it has an isometric subgraph of every possible order. We consider how to add a vertex to a dp graph so that the result is a dp graph. This condition implies that chordal graphs are dp. A graph is sequentially distance preserving (sdp) if its vertices can be ordered such that deleting the first i vertices results in an isometric subgraph, for all i at least 1. We give an equivalent condition to sequentially distance preserving based upon simplicial orderings. Using this condition, we prove that if a graph does not contain any induced cycles of length 5 or greater, then it is sdp. In closing, we discuss our results, other work and open problems concerning dp graphs.

Title: Density of States and Gap of Generic Quantum Hamiltonians

Date: 02/23/2017

Time: 11:00 AM - 11:50 AM

Place: C304 Wells Hall

Speaker: Ramis Movassagh, IBM TJ Watson Division of Mathematical Sciences

We propose a method, inspired by Free Probability Theory and Random Matrix Theory, that predicts the eigenvalue distribution of quantum many-body systems with generic interactions [1]. At the heart is a 'Slider', which interpolates between two extremes by matching fourth moments. The first extreme treats the non-commuting terms classically and the second treats them 'free'. By 'free' we mean that the eigenvectors are in generic positions. We prove that the interpolation is universal. We then show that free probability theory also captures the density of states of the Anderson model with an arbitrary disorder and with high accuracy [2]. Theory will be illustrated by numerical experiments.
[Joint work with Alan Edelman]
Time permitting we will prove that quantum local Hamiltonians with generic interactions are gapless [3]. In fact, we prove that there is a continuous density of states arbitrary close to the ground state. The Hamiltonian can be on a lattice in any spatial dimension or on a graph with a bounded maximum vertex degree. We calculate the scaling of the gap with the system's size in the case that the local terms are distributed according to gaussian β−orthogonal random matrix ensemble.
References:
- - - - - - - - -
[1] Phys. Rev. Lett. 107, 097205 (2011)
[2] Phys. Rev. Lett. 109, 036403 (2012)
[3] R. Movassagh 'Generic Local Hamiltonians are Gapless', (2017)
arXiv:1606.09313v2 [quant-ph]

Title: Abundant quasifuchsian surfaces in cusped hyperbolic 3-manifolds

Date: 02/23/2017

Time: 2:00 PM - 2:50 PM

Place: C304 Wells Hall

Speaker: David Futer, Temple University

I will discuss a proof that every finite volume hyperbolic 3-manifold M contains an abundant collection of immersed, $\pi_1$-injective surfaces. These surfaces are abundant in the sense that their lifts to the universal cover separate any pair of disjoint geodesic planes. The proof relies in a major way on the corresponding theorem of Kahn and Markovic for closed 3-manifolds. As a corollary, we recover Wise's theorem that the fundamental group of M is acts properly and cocompactly on a cube complex. This is joint work with Daryl Cooper.

Title: Supercritical Entanglement: counter-example to the area law for quantum matter

Date: 02/23/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Ramis Movassagh, MIT

In recent years, there has been a surge of activities in proposing exactly solvable quantum spin chains with the surprisingly high amount of entanglement entropies (super-logarithmic violations of the area law). We will introduce entanglement and discuss these models. These models have rich connections with combinatorics, random walks, and universality of Brownian excursions. Lastly, we develop techniques for proving the gap and conclude that these models do not have a relativistic conformal field theory description.

Title: AIMS: Access, Agency, and Allies in Mathematical Systems

Date: 02/27/2017

Time: 12:00 PM - 1:00 PM

Place: 133G Erickson Hall

Speaker: Bartell and Herbel-Eisenmann, MSU

In this presentation, we will share information about the AIMS project, on which faculty and doctoral students from eight universities are studying how mathematics teacher educators, mathematics teachers, and students work together to support the fair distribution of opportunities to learn. We will highlight our theoretical focus on access, agency, and allies and the ways in which this has translated into professional development design and will share the story of how two teachers’ adaptations of a PD task supported them in connecting to authentic student experiences and supporting students’ opportunities to learn rigorous mathematics.

Rachael, Andy, and Tsveta will be leading a discussion about RCPD accommodations. The goal of the discussion is to gather questions regarding common RCPD accommodations and bring those to the RCPD staff. We hope this will result in developing a guide for instructors and GTAs about how to help students with various RCPD accommodations. For example:
• What are some suggestions for providing 50% extended time on a 30-minute quiz if you have another class 20-minutes later?
• What are good ways to provide reduced distraction seating for exams?
• What are some good ways to work with students who have unexpected absences due to their condition who might miss a quiz or exam that you plan to hand back before they return to school?
We invite all of you to share your questions with us so we can bring those to the RCPD to develop a partnership with them to better serve our students.

Title: On the motion of a slightly compressible liquid

Date: 02/27/2017

Time: 4:10 PM - 5:00 PM

Place: C517 Wells Hall

Speaker: Chenyun Luo, Johns Hopkins University

I would like to go over some recent results on the compressible Euler equations with free boundary. We first provide a new a priori energy estimates which are uniform in the sound speed, which leads to the convergence to the solutions of the incompressible Euler equations. This is a joint work with Hans Lindblad. On the other hand, the energy estimates can be generalized to the compressible water wave problem, i.e., the domain that occupied by the fluid is assumed to be unbounded. We are also able to prove weighted energy estimates for a compressible water wave. Our method requires the detailed analysis of the geometry of the moving boundary.

Speaker: Hugh Thomas, University of Quebec at Montreal

Amplituhedra were introduced by Arkani-Hamed and Trnka as part of a program to provide an alternative to the classical Feynman diagram approach to scattering amplitudes, via a surprising link with Grassmannian geometry. I will attempt to provide some physics context, and then give the original definition of the amplituhedron (as a certain image of a totally non-negative Grassmannian) and a new proposed definition. This talk is based on joint work with Nima Arkani-Hamed and Jaroslav Trnka.

Title: On Distance Preserving and Sequentially Distance Preserving Graphs

Date: 02/28/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Emad Zahedi, MSU

A graph H is an isometric subgraph of G if d_H(u,v) = d_G(u,v), for every pair u,v in V(H), where d denotes distance. A graph is distance preserving (dp) if it has an isometric subgraph of every possible order. We consider how to add a vertex to a dp graph so that the result is a dp graph. This condition implies that chordal graphs are dp. A graph is sequentially distance preserving (sdp) if its vertices can be ordered such that deleting the first i vertices results in an isometric subgraph, for all i at least 1. We give an equivalent condition to sequentially distance preserving based upon simplicial orderings. Using this condition, we prove that if a graph does not contain any induced cycles of length 5 or greater, then it is sdp. In closing, we discuss our results, other work and open problems concerning dp graphs.

Title: Topological recursion for matrix models and abstract topological recursion (a course of 3 lectures)

Date: 03/01/2017

Time: 3:00 PM - 3:50 PM

Place: C304 Wells Hall

Speaker: Leonid Chekov, Steklov Institute and MSU

I will begin with the description of a generating function for numbers of
Grothendieck's dessins d'enfant, or Belyi pairs. This generating function
is given by a random matrix model integral, I describe what is a
topological expansion (AKA genus expansion) of such models. The method
allowing finding corrections in all orders of the genus expansion is
Topological Recursion formulated in its present form by Eynard, Orantin
and the speaker in 2005-2006. This method had already found numerous
applications in mathematics and mathematical physics, so I describe the
general construction underlying the topological recursion and present an
(incomplete) list of its applications. Very recently, this method was
developed into an abstract topological recursion by Kontsevich and
Soibelman. In my last lecture I explain their construction and our
interpretation of it (forthcoming paper by J.Andersen, G.Borot, L.Ch., and
N.Orantin).

Title: The genus of a special cube complex and its applications

Date: 03/02/2017

Time: 2:00 PM - 2:50 PM

Place: C304 Wells Hall

Speaker: Corey Bregman, Rice University

Recently, the geometry of non-positively curved (NPC) cube complexes has featured prominently in low-dimensional topology. We introduce an invariant of NPC special cube complexes called the genus, which generalizes the classical notion of genus for a closed orientable surface. We then show that having genus one characterizes special cube complexes with abelian fundamental group and discuss some applications.

In this talk, I will give an introduction to the so called mutation operation of a 3-manifold, which can be described as cutting the 3-manifold along a given surface and then sew it back in a different way. In particular, this induces corresponding mutation operations on knots obtained by mutating the 3-sphere along a genus 2 surface in the knot complement. I will survey a few properties and conjectures regarding mutant knots.

In a classical paper, Cafferelli, Gidas and Spruck discussed positive solutions of the Yamabe equation, corresponding to the positive scalar curvature of the conformal metrics, with a nonremovable isolated singularity. They proved that solutions are asymptotic to radial singular solutions. Korevaar, Mazzeo, Pacard, and Schoen expanded solutions to the next order. In this talk, we discuss how to expand solutions up to arbitrary order. We also discuss positive solutions of the Yamabe equation, corresponding to the negative scalar curvature of the conformal metrics, that become singular in an (n-1)-dimensional set.

Title: Math Strong: Amplifying Equity and Justice in Mathematics Education Research and Practice

Date: 03/15/2017

Time: 3:30 PM - 5:00 PM

Place: 252 EH

Speaker: Julia Aguirre, University of Washington, Tacoma

In this interactive talk, Dr. Aguirre will challenge math education researchers to amplify equity in research and practice. We will discuss fundamental questions about mathematics education and its role in perpetuating systems of power, privilege and oppression. Through exploration of various empirical, instructional, and organizational tools, we will identify concrete actions as math educators to disrupt the negative effects of these systems and re-align our intent and impact to cultivate a more just and equitable mathematics education experience for our nation’s youth.

We will introduce mapping class groups of surfaces, give several specific examples, and then discuss finite generating sets for mapping class groups of surfaces that are oriented, connected, compact, with possibly finitely many punctures.

Title: Using geometry and combinatorics to move robots quickly.

Date: 03/16/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Federico Ardilla, SFSU

How do we move a robot quickly from one position to another? To answer this question, we need to understand the 'space of possibilities” containing all possible positions of the robot. Unfortunately, these spaces are tremendously large and high-dimensional, and are very difficult to visualize. Fortunately, geometers and algebraists have encountered and studied these kinds of spaces before. Thanks to the tools they’ve developed, we can build “remote controls” to navigate these complicated spaces, and move (some) robots optimally.
This talk is based on joint work with my students Arlys Asprilla, Tia Baker, Hanner Bastidas, César Ceballos, John Guo, and Rika Yatchak. It will be accessible to undergraduate students, and assume no previous knowledge of the subject.

Title: Composition algebras: from a Dublin bridge to your cellphone

Date: 03/17/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Jon Hall, MSU

In 1843 Sir William Rowan Hamilton scratched
the quaternions onto a bridge in Dublin,
and in 1998 Siavash M. Alamouti used them as
a coding scheme, now embedded in hundreds
of millions of cellphones. In discussing
this we also touch on combinatorial
designs, sums of squares, and those odd
towers in your local supermarket parking
lot.

Title: When medical challenges meet time series analysis and manifold learning

Date: 03/17/2017

Time: 4:10 PM - 5:00 PM

Place: 1502 Engineering Building

Speaker: Hau-Tieng Wu, University of Toronto

Adaptive acquisition of correct features from massive datasets
is at the core of modern data analysis. One particular interest in
medicine is the extraction of hidden dynamics from an observed time series
composed of multiple oscillatory signals. The mathematical and statistical
problems are made challenging by the structure of the signal which
consists of non-sinusoidal oscillations with time varying amplitude and
time varying frequency, and by the heteroscedastic nature of the noise. In
this talk, I will discuss recent progress in solving this kind of problem.
Based on the cepstrum-based nonlinear time-frequency analysis and manifold
learning technique, a particular solution will be given along with its
theoretical properties. I will also discuss the application of this method
to two medical problems – (1) the extraction of a fetal ECG signal from a
single lead maternal abdominal ECG signal; (2) the simultaneous extraction
of the instantaneous heart rate and instantaneous respiratory rate from a
PPG signal during exercise. If time permits, an extension to multiple-time
series will be discussed.

'Knot theory' is the study of closed, embedded curves in
three-dimensional space. Classically, knots can be studied via a
various computable polynomial invariants, such as the Alexander
polynomial. In this first talk, I will recall the basics of knot
theory and the Alexander polynomial, and then move on to a more modern
knot invariant, 'knot Floer homology', a knot invariant with more
algebraic structure associated to a knot. I will describe applications
of knot Floer homology to traditional questions in knot theory, and
sketch its definition. This knot invariant was originally defined in
2003 in joint work with Zoltan Szabo, and independently by Jake
Rasmussen. A combinatorial formulation was given in joint work with
Ciprian Manolescu and Sucharit Sarkar in 2006.

This talk is the third and final part of our working through the paper by Karp and Williams. We will discuss (finally) the poset structure that you can put on the elements of the m=1 amplituhedron and how they can be made into a hyperplane arrangement.

Title: Log-Canonical Coordinates for Poisson Brackets

Date: 03/21/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Nicholas Ovenhouse, MSU

In symplectic geometry, Darboux's theorem gives so-called 'canonical' local coordinates, in which the Poisson bracket (obtained from the symplectic structure) takes a particularly simple form: all brackets of coordinate functions are constants. We investigate whether something analogous is true for Poisson varieties (when the coordinate change is only allowed to be rational functions), and give a negative answer in the case of log-canonical coordinates.

Title: Bordered techniques in Heegaard Floer homology.

Date: 03/21/2017

Time: 5:20 PM - 6:20 PM

Place: 115 International Center

Speaker: Peter Ozsvath, Princeton

Heegaard Floer homology is a closed three-manifold invariant, defined
in joint work with Zoltan Szabo, using methods from symplectic
geometry (specifically, the theory of pseudo-holomorphic disks). The
inspiration for this invariant comes from gauge theory. In joint work
with Robert Lipshitz and Dylan Thurston from 2008, the theory was
extended to an invariant for three-manifolds with boundary,
'bordered
Floer homology'. I will describe Heegaard Floer homology, motivate
its construction, list some of its key properties and applicat

I begin with solving the loop equations in matrix models. The TR allows constructing in a very algorithmic way all correlation functions $W_s^{(g)}(x_1,\dots,x_s)$ of a model on the base of the spectral curve $\Sigma(x,y)=0$ obtained as a solution of a master loop equation in the planar approximation; the variable $y$ is identified with $W_1^{(0)}$. We are then able to construct all $W_s^{(g)}$ out of this, not very abundant, set of data supplied with the two-point correlation function $W_2^{(0)}(x_1,x_2)dx_1dx_2$, which is a universal Bergmann 2-differential on the spectral curve. We are also able to construct terms of the free energy $F^g$ using the Ch.-Eynard integration formula applied to $W_1^{(g)}$. Finally, I will describe a Feynman-like diagrammatic technique for evaluation all $W_s^{(g)}$.

I will describe a bordered construction of knot Floer homology,
defined as a computable, combinatorial knot invariant. Generators
correspond to Kauffman states, and the differentials have an algebraic
interpretation in terms of a certain derived tensor product. I will
also explain how methods from bordered Floer homology prove that this
invariant indeed computes the holomorphically defined knot Floer
homology. This is joint work with Zoltan Szabo.

In this talk, we introduce Cartan geometries which are spaces
that are locally modeled on homogeneous spaces. We express what
conformal geometries are in this language and then move on to study
signature (2,3) conformal geometry with holonomy in the split form of
G_2 . The exposition will mainly follow https://arxiv.org/abs/1002.1767
chapters 4 and 5.

Title: A new physical space approach to improved decay for linear waves with applications

Date: 03/27/2017

Time: 4:10 PM - 5:00 PM

Place: C517 Wells Hall

Speaker: Yannis Angelopoulos, UCLA

I will present a vector field method for the linear wave equation on a wide class of spacetimes through which one can obtain almost optimal energy and pointwise decay results, as well as precise asymptotics. I will also talk about applications of the above method to problems in general relativity. This is joint work with S. Aretakis and D. Gajic.

Title: Math Learning Center and Supplemental Instruction in 2017-18

Date: 03/27/2017

Time: 4:10 PM - 5:00 PM

Place: C109 Wells Hall

Speaker: MLC team and SI team, MSU

We are going to discuss the expected changes to the MLC to be implemented this coming fall. We would also welcome input from faculty and GTAs regarding alternative formats to assist students, including problem/review sessions, etc.
The team on Supplemental Instruction will also be join us to acquaint us with plans for SI in Fall '17 and Spring '18. We will discuss possible ways to coordinate these efforts with instructors of large lectures and course coordinators.

Title: What polytopes tell us about toric varieties

Date: 03/28/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Robert Davis, MSU

Polytopes are among the oldest mathematical objects that have been studied. Often, people want to find their volumes, identify triangulations, and describe their lattice points, and more. But why bother doing this? From a combinatorial perspective, the data often answer counting questions that one might have. However, there is much more depth from an algebro-geometric standpoint: this information is often useful for learning about certain toric varieties.
In the first of these talks, I will give the background needed to understand what a normal projective toric variety is and how to model them using polytopes. In the second talk, I will define several properties that an algebraic geometer may want to know about a toric variety, and explain how to detect these properties from a purely polytopal perspective.

I will describe the relation between global and local models. Local models are those based on a signle branchpoint. Although a zoo of global models is huge, we have just two types of local models: the Kontsevich model (the Airy kernel) associated with ``dynamic' branchpoints (zeros $\mu_\alpha$ of $dx$ such that $y(\mu_\alpha)\ne \{0,infty\}$) and the Brezin-Gross-Witten model (the Bessel kernel) associated with a 'hard' edge (a zero $a_\alpha$ of $dx$ such that $ydx$ is regular at $a_\alpha$). I will describe the canonical action of quadratic operators establishing exact relations of Ch.-Givental form between a global model and the direct product of local models.

Title: Hedgehog Bases: A taste of the mathematics and physics of scattering amplitudes

Date: 03/30/2017

Time: 11:00 AM - 11:50 AM

Place: C304 Wells Hall

Speaker: Daniel Parker, UC Berkeley

Scattering amplitudes are the heart of particle physics, forming a bridge between theory and experiment. The last decade has seen the rise of 'amplitudeology', a program for uncovering the hidden mathematical structures of scattering amplitudes. By importing recent results in mathematics, amplitudeology has produced new insights into scattering amplitudes - such as the 'amplitudehedron' - and translated them into practical computational techniques. In this talk, I will discuss a cluster algebra structure of scattering amplitudes in N=4 Super Yang-Mills theory and deep connections with Goncharov polylogarithms. In order to develop a computational framework which exploits this connection, I show how to construct bases of Goncharov polylogarithms that can be used to describe 6-particle scattering at any order in perturbation theory.

Title: A Martingale Approach for Fractional Brownian Motions

Date: 03/30/2017

Time: 3:00 PM - 3:50 PM

Place: C405 Wells Hall

Speaker: Jianfeng Zhang, University of Southern California

Empirical studies show that the volatilities could be rough, which typically go beyond the semimartinagle framework and the fractional Brownian Motion (fBM) becomes a natural tool. Compared with BM, fBM has two features: (i) non-Markoivan; (ii) non-semimartingale (when the Hurst parameter $H< {1\over 2}$). We shall show that the recent development of path dependent PDEs provides a convenient tool to extend the standard literature of pricing/hedging derivatives to an fBM framework. This is a joint work with Frederi Viens.

In this talk we briefly go through harmonic map and Dirac-harmonic map.
Especially we see the bubble behavior of harmonic and Dirac-harmonic map and estimates needed.

Title: Mathematics courses in Lyman Briggs College and reform efforts

Date: 04/03/2017

Time: 4:10 PM - 5:00 PM

Place: C109 Wells Hall

Speaker: R. Bell and R.A. Edwards, Mathematics and Lyman Briggs, MSU

We will give an overview of the mathematics courses offered in Lyman Briggs College (LBC), the demographics of our students, and examples of curricular reform efforts. We will also discuss the alignment of LBC and MTH courses and some of the challenges that these pose.

Title: What polytopes tell us about toric varieties

Date: 04/04/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Robert Davis, MSU

Polytopes are among the oldest mathematical objects that have been studied. Often, people want to find their volumes, identify triangulations, and describe their lattice points, and more. But why bother doing this? From a combinatorial perspective, the data often answer counting questions that one might have. However, there is much more depth from an algebro-geometric standpoint: this information is often useful for learning about certain toric varieties.
In the first of these talks, I will give the background needed to understand what a normal projective toric variety is and how to model them using polytopes. In the second talk, I will define several properties that an algebraic geometer may want to know about a toric variety, and explain how to detect these properties from a purely polytopal perspective.

I will descirbe the newly developed abstract TR authored by Kontsevich and Soibelman in 2017. The main statement of the abstract TR (as presnted in the very recent paper by Andersen, Borot, L.Ch. and Orantin) is the inverse of TR for $W_s^{(g)}$: given a TR based on the set of abstract variables $\xi_k$ (which in the geometrical case can be identified with Krichever-Whitham 1-differentials based at zeros of $dx$) and imposing a single additional restriction of a total symmetricity of $W_s^{(g)}$ for all $g$ and $s$ we have a set of operators $L_k$ linear-quadratic in $\{\xi_r, \partial_{\xi_r}\}$ (one operator per one variable) all of which annihilate the partition function $Z=e^F$ that is the generating function for $W_S^{(g)}$. I present different examples of this construction including those not based on geometrical spectral curves.

Title: A solvable family of driven-dissipative many-body systems

Date: 04/06/2017

Time: 11:00 AM - 11:50 AM

Place: C304 Wells Hall

Speaker: Mohammad Maghrebi, MSU

Exactly solvable models have played an important role in establishing the sophisticated modern understanding of equilibrium many-body physics. And conversely, the relative scarcity of solutions for non-equilibrium models greatly limits our understanding of systems away from thermal equilibrium. We study a family of nonequilibrium models, described by Lindbladian dynamics, where dissipative processes drive the system toward states that do not commute with the Hamiltonian. Surprisingly, a broad subset of these models can be solved efficiently in any number of spatial dimensions. We leverage these solutions to prove a no-go theorem on steady-state phase transitions in many-body models.

Speaker: Tye Lidman, North Carolina State University

Although not every knot in the three-sphere can bound a smooth embedded disk in the three-sphere, it must bound a PL disk in the four-ball. This is not true for knots in the boundaries of arbitrary smooth contractible manifolds. We give new examples of knots in homology spheres that cannot bound PL disks in any bounding homology ball and thus not concordant to knots in the three-sphere. This is joint work with Jen Hom and Adam Levine.

We use Minkowski content (i.e., natural parametrization) of SLE to construct several types of SLE$_\kappa$ loop measures for $\kappa\in(0,8)$. First, we construct rooted SLE$_\kappa$ loop measures in the Riemann sphere $\widehat{\mathbb C}$, which satisfy M\'obius covariance, conformal Markov property, reversibility, and space-time homogeneity, when the loop is parameterized by its $(1+\frac \kappa 8)$-dimensional Minkowski content. Second, by integrating rooted SLE$_\kappa$ loop measures, we construct the unrooted SLE$_\kappa$ loop measure in $\widehat{\mathbb C}$, which satisfies M\'obius invariance and reversibility. Third, we extend the SLE$_\kappa$ loop measures from $\widehat{\mathbb C}$ to subdomains of $\widehat{\mathbb C}$ and to Riemann surfaces using Brownian loop measures, and obtain conformal invariance or covariance of these measures. Finally, using a similar approach, we construct SLE$_\kappa$ bubble measures in simply/multiply connected domains rooted at a boundary point. The SLE$_\kappa$ loop measures for $\kappa\in(0,4]$ give examples of Malliavin-Kontsevich-Suhov loop measures for all $c\le 1$. The space-time homogeneity of rooted SLE$_\kappa$ loop measures in $\widehat{\mathbb C}$ answers a question raised by Greg Lawler.

Finsler metrics are a generalization of Riemannian metrics (a norm in each tangent space) and occur naturally in various areas in physics and mathematics. Unlike for Riemannian metrics, there exists a large interesting class of Finsler metrics with constant (flag) curvature. We discuss joint work with R.Bryant, P. Foulon, S. Ivanov and V. S. Matveev on a characterization of the geodesic flow of such metrics in terms of the length of the shortest periodic orbit.

Title: Consensus and clustering in opinion formation on small-world networks

Date: 04/07/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Todd Kapitula, Calvin College

Ideas that challenge the status quo either evaporate and are forgotten, or eventually
become the new status quo. Mathematically, an ODE model was developed by Strogatz et al. for
the propagation of one idea moving through one group of a large number of interacting individuals
(a 'city'). Recently, the Strogatz model was extended to include interacting multiple cities at SUMMER@ICERM 2016
at Brown University. The one and two city models are analyzed to determine the circumstances under
which there can be consensus. The case of three or more cities is analyzed to determine when, and under
what conditions, clustering occurs. Preliminary results will be presented.

Title: The Formation of Shock Singularities in Solutions to Wave Equation Systems with Multiple Speeds

Date: 04/10/2017

Time: 4:10 PM - 5:00 PM

Place: C517 Wells Hall

Speaker: Jared Speck, Massachusetts Institute of Technology

In this talk, I will describe my recent work on the formation of shock singularities in solutions to quasilinear wave equation systems in 2D with more than one speed, that is, systems with at least two distinct wave operators. In the systems under study, the fast wave forms a shock singularity while the slow wave remains regular, even though the two waves interact all the way up to the singularity. This work represents an extension of the remarkable proofs of shock formation for scalar quasilinear wave equations provided by S. Alinhac, as well as the breakthrough sharpening of Alinhac's results by D. Christodoulou for the scalar wave equations of irrotational fluid mechanics. Both results crucially relied on the construction of geometric vectorfields that are adapted to the wave characteristics, whose intersection drives the singularity formation. THe key new difficulty for systems with multiple speeds is that the geometric vectorfields are, by necessity, precisely adapted to the characteristics of the shock-forming (fast) wave. Thus, there is no freedom left to adapt the vectorfields to the characteristics of the slow wave, and for this reason, they exhibit very poor commutation properties with the slow wave operator. To overcome this difficulty, we rely in part on some ideas from our recent joint work with J. Luk, in which we proved a shock formation result for the compressible Euler equations with vorticity, which we formulated as a wave-transport system featuring precisely one wave operator.

Large course meetings have been inefficient in the past, especially when meetings focus on delivering updates about course administration and policies that are effectively communicated via email. Instead, TAs need an opportunity to discuss their teaching in a smaller group where they can get feedback and guidance about their specific issues. We are working to design a structure that will require less time from course supervisors and lecturers than weekly course meetings by having our Lead TAs run smaller meetings that are more focused on preparing lessons and developing teaching strategies.

As part of or program on noncommutative laurent phenomenon, we
introduce and study noncommutative Catalan 'numbers' as Laurent
polynomials in infinitely many free variables and related theory of
noncommutative binomial coefficients. We also study their (commutative
and noncommutative) specializations, relate them with Garsia-Haiman
(q,t)-versions, and establish total positivity of the corresponding
Hankel matrices. Joint work with Arkady Berenstein (Univ. of Oregon).

In a partially ordered set P, let a pair of elements (x,y) be called alpha-balanced if the proportion of linear extensions that has x before y is between alpha and 1-alpha. The 1/3-2/3 Conjecture states that every finite poset which is not a chain has some 1/3-balanced pair. While the conjecture remains unsolved, we extend the list of posets that satisfy the conjecture by adding certain lattices, including products of two chains, as well as posets that correspond to Young diagrams.

Khovanov homology is a combinatorially-defined knot invariant which refines the Jones polynomial. After recalling the definition of Khovanov homology we will sketch a construction of a stable homotopy refinement of Khovanov homology. We will conclude with some modest applications and some work in progress. This is joint work with Tyler Lawson and Sucharit Sarkar. Another construction of the Khovanov stable homotopy type was given by Hu-Kriz-Kriz.

Persistent homology is a method for computing topological features of a space at different spatial resolutions. In 2004, Zomorodian and Carlsson figured out an algorithm to compute persistent homology when the coefficient ring is a field F. I will mostly be focusing on this. It is supposed to be a very basic talk.

Title: Uniformly and Strongly Consistent Estimation for the Hurst Function of a Linear Multifrational Stable Motion

Date: 04/13/2017

Time: 3:00 PM - 3:50 PM

Place: C405 Wells Hall

Speaker: Antoine Ayache, University of Lille 1, France

Multifractional processes have been introduced in the 90's in order to overcome some limitations of the well-known Fractional Brownian Motion (FBM) due to the constancy in time of its Hurst parameter; in their context, this parameter becomes a Hölder continuous function. Global and local path roughness of a multifractional process are determined by values of this function; therefore, several authors have been interested in their statistical estimation, starting from discrete variations of the process. Because of the complex dependence structure of variations, showing consistency of estimators is a tricky problem which often requires hard computations.
The main goal of our talk, is to introduce in the setting of the non-anticipative moving average Linear Multifractional Stable Motion (LMSM) with a stability parameter 'alpha' strictly larger than 1, a new strategy for dealing with the latter problem. In contrast with the previous strategies, this new one, does not require to look for sharp estimates of covariances related to variations; roughly speaking, it consists in expressing them in such a way that they become independent up to negligible remainders.
This is a joint work with Julien Hamonier at University of Lille 2.

Title: The algebra of box splines, hyperplane arrangements, and zonotopes

Date: 04/13/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Olga Holtz, UC Berkeley

Zonotopal algebra is a framework for studying various algebraic, combinatorial, and analytic objects associated to a linear map $\phi: \R^N \rightarrow \R^n$, where $n\le N$. This unified perspective gives formulas for volumes and lattice point enumerators of certain zonotopes, among other things.
This framework was inspired by the theory of box splines, which are piecewise-polynomial functions supported on zonotopes, whose chambers are determined by the matrix $X$ of the map $\phi$. Box splines can be thought of as fiber volume functions, as they measure the volume of the $(N-n)$-dimensional preimage of their $n$-dimensional argument, where the preimage is restricted
to the ``box' $[0,1]^N$.
I'll explain how the theory of zonotopal algebra connects these analytic phenomena to the algebraic properties of the linear map $X$. In particular, how the matroidal structure of $X$ is related to:
1. a family of polynomial ideals associated to $X$,
2. the kernels of those ideals, i.e., the spaces of polynomials annihilated by those ideals,
3. the discrete geometry of the associated hyperplane arrangement, and
4. the tilings of the associated zonotope.
This new line of research allows to study combinatorial and algebraic objects using techniques of analysis. Examples include recent results of de Concini, Procesi, Vergne, Moci, Lenz, and others.

Speaker: Olga V. Holtz, University of California, Berkeley

Ever since the invention of the first algorithms, mathematicians wondered how 'complex' such computational procedures are. This talk will offer an excursion into the world of complexity. How fast can we determine if a given number is prime, find the greatest common divisor of two
polynomials, or multiply two matrices? What problems are solvable in polynomial time? What are randomized algorithms and how complex are they? What is communication complexity? And why should we care whether or not P equals NP?

Title: Extended Somos and Gale-Robinson sequences, dual numbers, and cluster superalgebras

Date: 04/18/2017

Time: 2:00 PM - 2:50 PM

Place: C304 Wells Hall

Speaker: Valentin Ovsienko, Reims

In 1980, Michael Somos invented integer sequences that have later been popularized and generalized (among others) by David Gale. A certain mystery around this class of sequences is probably due to their relation with a wealth of different topics, such as: elliptic curves, continued fractions, and more recently with cluster algebras and integrable systems.
I will describe a way to extend Somos-4 and Somos-5 and more general Gale-Robinson sequences, and construct a great number of new integer sequences that also look quite mysterious. The construction is based on the notion of 'cluster superalgebra' (which can be used as a machine to produce integer sequences).
Most of the talk will be accessible to non-experts in any of the above mentioned subjects.

I will begin by defining and giving examples of combinatorial species. I will then explain how they are related to generating functions and how to view some common operations on generating functions in this context. Time permitting I will talk about how combining combinatorial species with the idea of a monoidal category leads to a generalization of Hopf algebras.

Title: Constructing Sard-Smale Fundamental Classes

Date: 04/20/2017

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

Speaker: Thomas H. Parker, MSU

The moduli spaces in gauge theory usually arise as generic fibers of a universal moduli space, and invariants are constructed using cobordisms between generic fibers.
I will describe a topological setting that, in important cases, produces a fundamental class on all fibers, and gives an alternative perspective on the resulting invariants.
This is joint work with E. Ionel.

Title: New developments in the theory of smooth actions.

Date: 04/20/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Federico Rodriguez Hertz, PSU

In resent years several new advances in the theory of lattice actions have been made. In this talk I will present some of the key ingredients to these advances. I plan to keep the talk at an elementary level so only some basic notions of measure theory and differentiation on manifolds should be needed.

Title: Numerical methods for energy-based models and its applicability to mixtures of isotropic and nematic flows with anchoring and stretching effects

Date: 04/21/2017

Time: 4:10 PM - 5:00 PM

Place: 1502 Engineering Building

Speaker: Giordano Tierra, Temple University

The study of interfacial dynamics between two different components has become the key role to understand the behavior
of many interesting systems. Indeed, two-phase flows composed of fluids exhibiting different microscopic structures
are an important class of engineering materials. The dynamics of these flows are determined by the coupling among
three different length scales: microscopic inside each component, mesoscopic interfacial morphology and macroscopic
hydrodynamics. Moreover, in the case of complex fluids composed by the mixture between isotropic (newtonian fluid)
and nematic (liquid crystal) flows, its interfaces exhibit novel dynamics due anchoring effects of the liquid crystal
molecules on the interface.
In this talk I will introduce a PDE system to model mixtures composed by isotropic fluids and nematic liquid
crystals, taking into account viscous, mixing, nematic, stretching and anchoring effects and reformulating the corre-
sponding stress tensors in order to derive a dissipative energy law. Then, I will present new linear unconditionally
energy-stable splitting schemes that allows us to split the computation of the three pairs of unknowns (velocity- pres-
sure, phase field-chemical potential and director vector-equilibrium) in three different steps. The fact of being able
to decouple the computations in different linear sub-steps maintaining the discrete energy law is crucial to carry out
relevant numerical experiments under a feasible computational cost and assuring the accuracy of the computed results.
Finally, I will present several numerical simulations in order to show the efficiency of the proposed numerical
schemes, the influence of the shape of the nematic molecules (stretching effects) in the dynamics and the importance
of the interfacial interactions (anchoring effects) in the equilibrium configurations achieved by the system.
This contribution is based on joint work with Francisco Guill´ en-Gonzal´ ez (Universidad de Sevilla, Spain) and Mar´ıa
´
Angeles Rodr´ıguez-Bellido (Universidad de Sevilla, Spain)

Speaker: Gabriel Nagy, Mathematics, MSU; Tsveta Sendova, Mathematics, MSU

We will lead a conversation regarding the current MTH 235 curriculum, reform efforts, and our vision for the revised course, to be piloted in the fall of 2017.

Machine learning, which draws from a diversity of fields including computer science, mathematics, and physics, has been taking the world by storm due to its flourishing industrial applications. We present a live demo of machine learning in action, where we train a neural network to classify handwritten digits to an appreciable degree in real time. We then proceed to give an introductory overview of the ingredients that go into this task: logistic regression, stochastic gradient descent, and deep learning. (Part II, which connects machine learning to approximation theory and quantum physics, will take place the following week.)

Title: Braid Group Symmetries of Grassmannian Cluster Algebras

Date: 04/25/2017

Time: 2:00 PM - 2:50 PM

Place: C304 Wells Hall

Speaker: Chris Fraser, IUPUI

We define an action of the k-strand braid group on the set of cluster variables for the Grassmannian Gr(k, n), whenever k divides n. The action sends clusters to clusters, preserving the underlying quivers, defining a homomorphism from the braid group to the cluster modular group for Gr(k, n). Then we apply our results to the Grassmannians of 'finite mutation type'. We prove the n = 9 case of a conjecture of Fomin-Pylyavskyy describing the cluster combinatorics for Gr(3, n), in terms of Kuperberg’s basis of non-elliptic webs, and prove a similar result for the Grassmannian Gr(4,8).

Title: Bipolar filtration of topologically slice knots

Date: 04/25/2017

Time: 3:00 PM - 4:00 PM

Place: C304 Wells Hall

Speaker: Min Hoon Kim, KIAS

We show that the bipolar filtration of the smooth concordance group of topologically slice knots introduced by Cochran, Harvey and Horn has nontrivial graded quotients at every stage. To detect a nontrivial element in the quotient, the proof uses Cheeger-Gromov $L^2$ $\rho$-invariants and infinitely many Heegaard Floer correction term invariants simultaneously.

Title: Feynman integrals and multiple polylogarithms

Date: 04/27/2017

Time: 11:00 AM - 12:00 PM

Place: C304 Wells Hall

Speaker: Andreas v. Manteuffel, MSU Physics

Precision predictions for high energy experiments at the Large Hadron Collider
require the evaluation of Feynman integrals. In this talk I will discuss how
Feynman integrals can be evaluated in terms of multiple polylogarithms using
differential equations and the coproduct.

Speaker: Anna Marie Bohmann, Vanderbilt University

Equivariant cohomology theories are cohomology theories incorporate a group action on spaces. These types of cohomology theories are increasingly important in algebraic topology but can be difficult to understand or construct. In recent work, Angelica Osorno and I have developed a construction for building them out of purely algebraic data by controlling pieces with different isotropy types under the group action. Our method is philosophically similar to classical work of Segal on building nonequivariant cohomology theories.

Title: Multipoint estimates for radial and whole plane SLE

Date: 04/27/2017

Time: 3:00 PM - 3:50 PM

Place: C405 Wells Hall

Speaker: Ben Mackey, MSU

We prove upper bounds for the probability that a radial SLE$_{\kappa}$ curve comes within specified radii of $n$ different points in the unit disc. Using this estimate, we then prove a similar upper bound for the probability that a whole plane SLE$_{\kappa}$ passes near any $n$ points in the complex plane. We can then use these estimates to show that the lower Minkowski content of both the radial and whole plane SLE$_{\kappa}$ curves have finite moments of any order.

There is a long history of counting permutations according to statistics such as descents and peaks, with connections to geometry and algebra. A descent in a permutation w is a position i such that w(i)>w(i+1), while a peak is a position i such that w(i-1)<w(i)>w(i+1). The number of permutations with a fixed descent set is well-known, and not too long ago Billey, Burdzy, and Sagan explored the analogous question for peak sets. In recent work with Rob Davis, Sarah Nelson, and Bridget Tenner, we study what happens when we record not "i" but rather "w(i)" for each peak. (A similar variation on descents can be found in work of Ehrenborg and Steingrimsson.) We call these values "pinnacles" and ask the basic question: How many permutations have a given set of pinnacles?

Title: Fedor Nazarov: Fine approximation of convex bodies by polytopes.

Date: 04/27/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker:

We will show that for every convex body $K\subset R^d$
($d\ge 2$) with the center of mass at the origin and every
$a\in (0,1/2)$, one can find a convex polytope $P$ with at most
$(C/a)^{(d-1)/2}$ vertices such that $(1-a)K\subset P\subset K$.
This is a joint work with Marton Naszodi and Dmitry Ryabogin.

Title: Random Walk Models and Applied Chemical Ecology

Date: 04/28/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Jeffrey Schenker, Michigan State University

Farmers and pest managers make decisions about when to apply chemical insecticide sprays based on numbers of pests captured in monitoring traps. However until very recently there has not been a sound scientific understanding of the quantitative relationship between capture numbers and actual population densities. Using a combination of computer random walk models and marked release recapture field experiments we have begun to develop a mechanistic theory of insect trapping. The analysis of the random walk models has led to some mathematical surprises. In this talk I will describe the scientific background briefly and then turn to the mathematical story of the associated random walk models. I will present strong numerical evidence for a conjectured "renormalization" of trap radii that is needed for the central limit/invariance principle analysis for these random walks. (Joint work with Jim Miller and Chris Adams in MSU Entomology)