• Speaker: Betty Phillips, MSU; AJ Edson, MSU
• Title: Embedding the CMP Curriculum into a Digital Collaborative Platform
• Date: 11/11/2019
• Time: 12:00 PM - 1:00 PM
• Place: 115 Erickson Hall

The context for the work is transitioning the Connected Mathematics, an established problem-based curriculum, to a digital environment. In CMP, mathematical understandings are embedded in tasks which are carefully sequenced to build deep understanding of important mathematical ideas. In this session, we report on curriculum research and development efforts to leverage digital technologies to support student collaboration and enhance students’ productive disciplinary engagement in mathematics. Bring a laptop to partake in the collaborative environment.

• Speaker: Wenchuan Tian, MSU
• Title: Properties of Busemann function on manifolds with nonnegative sectional curvature outside of a compact set
• Date: 11/11/2019
• Time: 3:00 PM - 3:50 PM
• Place: C304 Wells Hall

Busemann functions are useful. Cheeger and Gromoll used them to prove the splitting theorem for manifolds with nonnegative ricci curvature that contains a line. Yau used them to prove that complete noncompact manifolds with nonnegative Ricci curvature have at least linear volume growth. In a paper called "Positive Harmonic Functions on Complete Manifolds with Non-Negative Curvature Outside a Compact Set" Peter Li and Luen-Fai Tam also used Busemann function to show the existence of positive harmonic functions. I will talk about Li and Tam's proof of properties of Busemann function. The proof only uses Toponogov theorem and cosine law. The results of the proof is useful for the subsequent analysis part of the paper.

• Speaker: Joshua Ruiter, Michigan State
• Title: Field norm for algebraic groups, with a view towards non-split tori
• Date: 11/11/2019
• Time: 4:30 PM - 5:30 PM
• Place: C304 Wells Hall

Field norm maps are useful in many areas of algebra, such as Galois theory. Using the language of (affine) algebraic groups, I will place the field norm in a larger context, as a particular instance of a certain natural transformation. This will set us up for my talk the following week, on special subgroups of algebraic groups called tori, and what it means for such tori to be split or non-split. In particular, the generalized norm will provide a (somewhat) concrete example of a non-split torus.

• Speaker: Lev Tovstopyat, MSU
• Title: On the surgery exact triangle in Heegaard Floer homology
• Date: 11/12/2019
• Time: 12:00 PM - 1:30 PM
• Place: C117 Wells Hall

No abstract available.

• Speaker: Robert Bell, MSU
• Title: Cop number and edge deletion, addition, or subdivision
• Date: 11/13/2019
• Time: 3:00 PM - 3:50 PM
• Place: C304 Wells Hall

We present new and old results about the effect of edge operations on the cop number of a finite graph. This project was part of the SURIEM summer REU program in 2019 at MSU.

• Speaker: Farhan Abedin, Michigan State University
• Title: Regularity results for a class of Kolmogorov-Fokker-Planck equations in non-divergence form
• Date: 11/13/2019
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

The Kolmogorov-Fokker-Planck equation is a degenerate parabolic equation arising in models of gas dynamics from kinetic theory. The operator is of the form $$\mathcal{L}_Au := \mathrm{tr}(A(v,y,t) D^2_v u) + v \cdot \nabla_yu - \partial_tu,$$ where $$u(v,y,t): \mathbb{R}^{2d+1} \to \mathbb{R} \text{ and } 0 < \lambda \mathbb{I}_d \leq A \leq \Lambda \mathbb{I}_d.$$ It is an open problem if non-negative solutions of $\mathcal{L}_A u = 0$ in $\mathbb{R}^{2d+1}$ satisfy a scale-invariant Harnack inequality, assuming the matrix coefficient $A$ is merely bounded and measurable. I will discuss recent joint work with Giulio Tralli in which progress is made on partially solving this problem.

• Speaker: Dapeng Zhan, MSU
• Title: Time-reversal of multiple-force-point SLE$_\kappa(\underline\rho)$ with all force points lying on the same side
• Date: 11/14/2019
• Time: 3:00 PM - 3:50 PM
• Place: C506 Wells Hall

We define intermediate SLE$_\kappa(\underline \rho)$ and reversed intermediate SLE$_\kappa(\underline\rho)$ processes using Appell-Lauricella multiple hypergeometric functions, and use them to describe the time-reversal of multiple-force-point chordal SLE$_\kappa(\underline \rho)$ curves in the case that all force points are on the boundary and lie on the same side of the initial point, and $\kappa $ and $\underline \rho=(\rho_1,\dots,\rho_m)$ satisfy that either $\kappa\in(0,4]$ and $\sum_{j=1}^k \rho_j>-2$ for all $1\le k\le m$, or $\kappa\in(4,8)$ and $\sum_{j=1}^k \rho_j\ge \frac{\kappa}{2}-2$ for all $1\le k\le m$.

• Speaker: Jihye Hwang, Michigan State University
• Title: What do you mean by "show"? Is it the same as "prove"?
• Date: 11/14/2019
• Time: 4:00 PM - 5:00 PM
• Place: C304 Wells Hall

Students face a mathematical task whenever they involve in doing mathematics. We wondered whether the word used in posing question impact students’ response and we hypothesize that task provider’s intention and students’ response can be different. This talk focus on students interpretation of the different prompts, such as “prove,” “show,” “explain,” and “convince a classmate,” for proving tasks. This talk focuses on the two prompts, “prove” and “show” as the prompts are regarded as synonymous by many people, especially mathematicians. Although there exist similarities between the prompts, the result mainly demonstrates students possibly have different meaning for “prove” and “show” depending on individual and one possible relationship between the two prompts.

• Speaker: Marius Lemm, Harvard
• Title: Spectral gaps without frustration
• Date: 11/15/2019
• Time: 10:30 AM - 11:30 AM
• Place: C304 Wells Hall

In quantum spin systems, the existence of a spectral gap above the ground state has strong implications for the low-energy physics. We survey recent results establishing spectral gaps in various frustration-free spin systems by verifying finite-size criteria. The talk is based on collaborations with Abdul-Rahman, Lucia, Mozgunov, Nachtergaele, Sandvik, Yang, Young, and Wang.

• Speaker: Yao Yao, Georgia Tech
• Title: Uniqueness and non-uniqueness of steady states of aggregation-diffusion equation
• Date: 11/15/2019
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

In this talk, I will discuss a nonlocal aggregation equation with degenerate diffusion, which describes the mean-field limit of interacting particles driven by nonlocal interactions and localized repulsion. When the interaction potential is attractive, it is previously known that all steady states must be radially decreasing up to a translation, but uniqueness (for a given mass) within this class was open, except for some special interaction potentials. For general attractive potentials, we show that the uniqueness/non-uniqueness criteria are determined by the power of the degenerate diffusion, with the critical power being m=2. Namely, for m>=2, we show the steady state for any given mass is unique for any attractive potential, by tracking the associated energy functional along a novel interpolation curve. And for 1<m<2, we construct examples of smooth attractive potentials, such that there are infinitely many radially decreasing steady states of the same mass. This is a joint work with Matias Delgadino and Xukai Yan.