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.