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

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

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

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

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

Title: Concordance in homology spheres

Date: 04/06/2017

Time: 2:00 PM - 2:50 PM

Place: C304 Wells Hall

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

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.

Speaker: Jared Speck, Massachusetts Institute of Technology

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

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.

Speaker: Antoine Ayache, University of Lille 1, France

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

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.

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

Date: 04/13/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

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

Title: The blessings and curses of complexity

Date: 04/15/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

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

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

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

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

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

Title: MTH 235 Reform

Date: 04/24/2017

Time: 4:10 PM - 5:00 PM

Place: C109 Wells Hall

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

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

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

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.

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.

Moduli spaces of two different J-holomorphic curves representing a fixed homology class are cobordant to each other. But until we establish some compactness it does not mean very much (since any manifold is cobordant to empty manifold via a noncompact cobordism). We will try to discuss cases where the moduli space becomes compact under certain assumptions.

Title: Multipoint estimates for radial and whole plane SLE

Date: 04/27/2017

Time: 3:00 PM - 3:50 PM

Place: C405 Wells Hall

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

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.