The holonomy group of a Riemannian manifold exhibits various geometric structures compatible with the metric. In 1955, M.Berger classified all possible Riemannian holonomy groups. Studying all these are more than one semester subject. So, in this talk after a brief introduction we overview very basics of these holonomy groups.

Title: Arithmetic intersection theory and Arakelov's Hodge Index Theorem

Date: 11/04/2019

Time: 4:30 PM - 5:30 PM

Place: C304 Wells Hall

The famous Mordell-Weil conjecture was first proved by Faltings in a classical way, then Vojta gave an alternative proof using arithmetic Arakelov geometry, which is one big motivation for developing Arakelov theory into a mature tool. In this talk I will introduce Neron functions and divisors, which is an arithmetic approach to define divisors rather than classical algebraic geometry. We shall also cover arithmetic chow groups and the arithmetic intersection number. In the end I will present Neron symbols and use it to give a sketch proof of Arakelov’s Hodge Index Theorem.

Speaker: Carl Wang-Erickson, University of Pittsburg

Title: Bi-ordinary modular forms

Date: 11/05/2019

Time: 3:00 PM - 4:00 PM

Place: C304 Wells Hall

It is known that p-ordinary cuspidal Hecke eigenforms give rise to 2-dimensional global Galois representations which become reducible after restriction to a decomposition group at p. For which such forms is this restriction not only reducible but also splittable? Complex multiplication (CM) forms satisfy this p-local property, but is such a restrictive global property as CM necessary? In classical weights at least 2, it is expected that this is the case. We present a construction of "bi-ordinary" p-adic modular forms, which can measure exceptions to this expectation. We also give evidence that there are non-CM but p-locally splittable forms in p-adic weights. This is joint work with Francesc Castella.

Normal rulings are combinatorial structures associated to the front diagrams of 1-dimensional Legendrian knots in R^3. They were introduced independently by Fuchs and Chekanov-Pushkar in the context of augmentations of the Legendrian DG-algebra and generating families. In this talk I will present joint work with B. Henry in which we construct a decomposition of the augmentation variety into disjoint pieces indexed by normal rulings. The pieces of the decomposition are products of algebraic tori and affine spaces with dimensions determined by the combinatorics of the ruling. As a consequence, the ruling polynomial invariants of Chekanov-Pushkar are seen to be equivalent to augmentation number invariants defined by counting augmentations to finite fields. The construction of the decomposition is based on considering Morse complex sequences which are combinatorial analogs of generating families.

Title: Variations of cops and robbers on infinite graphs

Date: 11/06/2019

Time: 3:00 PM - 3:50 PM

Place: C304 Wells Hall

The game of cops and robbers is a two player pursuit and evasion game played on a discrete graph G. We study a variation of the classical rules which leads to a different invariant when G is an infinite graph. In this variation, called "weak cops and robbers," the cops win by preventing the robber from visiting any vertex infinitely often. In the classical game, if G is connected and planar, then the cops can always win if there are at least three cops. We prove that this is true in the weak game if G is a locally finite plane graph with no vertex accumulation points.

Speaker: Ioakeim Ampatzoglou, University of Texas, Austin

Title: Derivation of a ternary Boltzmann system

Date: 11/06/2019

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

In this talk work we present a rigorous derivation of a new kinetic equation describing the limiting behavior of a classical system of particles with three particle instantaneous interactions, which are modeled using
a non-symmetric version of a ternary distance. The equation, which we call ternary Boltzmann equation, can be understood as a step towards modeling a dense gas in non-equilibrium. This is a joint work with Natasa Pavlović.

Speaker: Marios Velivasakis, University of Western Ontario

Title: Schubert Varieties in Partial Flag Manifolds and Generalized Severi-Brauer Varieties

Date: 11/07/2019

Time: 11:00 AM - 12:00 PM

Place: C329 Wells Hall

Schubert varieties form one of the most important classes of singular algebraic varieties. They are also a kind of moduli spaces. One problem is that these varieties are not easy to understand and manipulate using only their geometric nature. In this talk, we will discuss about Schubert varieties and present a way to characterize them combinatorially. In addition, we will discuss how they relate to Severi-Brauer varieties SB(d,A) and how we can use their combinatorial description to answer questions about subvarieties of SB(d,A)

Title: Localization for the Anderson--Bernoulli model on the integer lattice

Date: 11/07/2019

Time: 11:30 AM - 12:30 PM

Place: C304 Wells Hall

Abstract: I will give a brief mathematical introduction to Anderson localization followed by a discussion of my recent work with Jian Ding. In our work we establish localization near the edge for the Anderson Bernoulli model on the two dimensional lattice. Our proof follows the program of Bourgain--Kenig and uses a new unique continuation result inspired by Buhovsky--Logunov--Malinnikova--Sodin. I will also discuss recent work of by Li and Zhang on the three dimensional case.

Title: Sketching and Clustering Metric Measure Spaces

Date: 11/08/2019

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Two important optimization problems in the analysis of geometric data sets are clustering and simplification (sketching) of data. Clustering refers to partitioning a dataset, according to some rule, into sets of smaller size with the aim of extracting important information from the data. Sketching, or simplification of data, refers to approximating the input data with another dataset of much smaller size in such a way that properties of the input dataset are retained by the smaller dataset. In this sense, sketching facilitates understanding of the data.
There are many clustering methods for metric spaces (mm spaces) already present in literature, such as k-center clustering, k-median clustering, k-means clustering, etc. A natural method for obtaining a k-sketch of a metric space (mm space) is by viewing the space of all metric spaces (mm space) as a metric under Gromov-Hausdorff (Gromov-Wasserstein) distance, and then determining, under this distance, the k point metric space (mm space) closest to the input metric space (mm space).
These two problems of sketching and clustering, a priori, look completely unrelated. However, we establish a duality i.e. an equivalence between these notions of sketching and clustering. For metric spaces, we consider the case where the clustering objective is minimizing the maximum cluster diameter. We show that the ratio between the sketching and clustering objectives is constant over compact metric spaces.
We extend these results to the setting of metric measure spaces where we prove that the ratio of sketching to clustering objectives is bounded both above and below by some universal constants. In this setting, the clustering objective involves minimizing various notions of the $\ell_p$-diameters of the clusters.
We also identify procedures/maps that transform a solution of the sketching problem to a solution of the clustering problem, and vice-versa. These maps give rise to algorithms for performing these transformations and, by virtue of these algorithms, we are able to obtain an approximation to the k-sketch of a metric measure space (metric space) using known approximation algorithms for the respective clustering objectives. This is joint work with Facundo Memoli and Anastasios Sidiropoulos, and is available online at https://arxiv.org/abs/1801.00551.

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.

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.

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.

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.

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.

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 $1\le k\le m$.

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.

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.

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.

Title: Computations in Topological CoHochschild Homology

Date: 11/18/2019

Time: 3:00 PM - 3:50 PM

Place: C304 Wells Hall

Hochschild homology (HH) is a classical algebraic invariant of rings that can be extended topologically to be an invariant of ring spectra, called topological Hochschild homology (THH). There exists a dual theory for coalgebras called coHochschild homology (coHH), and in recent work Hess and Shipley defined an invariant of coalgebra spectra called topological coHochschild homology (coTHH). In this talk we will discuss coTHH calculations and the tools needed to do them.

Tori are an important structural aspect of algebraic groups, and "split" vs "non-split" tori are especially important. Unfortunately, "non-split" phenomena only occur over non-algebraically closed fields, so not all the traditional tools of classical algebraic geometry apply. Using the generalized field norm map from my talk last week, I'll describe a concrete example of a non-split torus. Then we'll try to use that example to try and understand the importance of non-split tori.

In this learning seminar style talk, I will define the notion of a derivation on a von Neumann algebra. I will also discuss their history and how they factor into modern research in operator algebras.

Title: Applications of augmentations in contact topology

Date: 11/19/2019

Time: 3:00 PM - 4:00 PM

Place: C204A Wells Hall

Chekanov introduced a differential graded algebra as an invariant for Legendrian knots in standard contact manifold R^3. An augmentation is a rank 1 representation of the dga. Augmentations are accessible invariants, and the moduli space of augmentations carries important properties from both algebraic and geometric perspectives. In this talk, I will review some problems in contact topology and discuss the applications of augmentations.

Title: Rigidity for higher dimensional expanding maps

Date: 11/19/2019

Time: 3:50 PM - 4:50 PM

Place: C304 Wells Hall

Expanding maps are self covers of smooth compact manifolds which expand the lengths of all non-zero tangent vectors. Classification of such maps up to topological conjugacy is known due to work of Shub, Franks and Gromov. Classification up to smooth conjugacy should be quite different because periodic points of expanding maps carry invariants of $C^1$ conjugacy. Shub and Sullivan classified expanding maps on the circle up to smooth conjugacy on the circle. I will explain smooth classification of expanding maps in higher dimensions on and open dense set in the space of expanding maps. Joint work with F. Rodriguez Hertz.

Title: On the modularity of elliptic curves over imaginary quadratic fields

Date: 11/20/2019

Time: 3:00 PM - 4:00 PM

Place: C304 Wells Hall

The Langlands-Tunnell theorem is an important starting point in Wiles's proof of the modularity of semistable elliptic curves over the rationals. Over imaginary quadratic fields it is unclear how to similarly use the Langlands-Tunnell theorem and Wiles's strategy runs into problems right from the start. I will motivate and explain the subtle but fundamental issues that arise, and indicate how they can be circumvented in many cases. As an application, we deduce that a positive proportion of elliptic curves over imaginary quadratic fields are modular. This is joint work with Chandrashekhar Khare and Jack Thorne.

Speaker: Michelle Cirillo, University of Delaware; Jennifer Reed, Odyssey Charter Middle School, Delaware

Title: Learning Together Through Collaborative Research: The Case of Proof in Secondary Mathematics

Date: 11/20/2019

Time: 3:30 PM - 5:00 PM

Place: 252 EH

The Proof in Secondary Classrooms (PISC) project is a design and development research study focused on secondary students’ success with mathematical proof. The goal of this project was to develop a new and improved intervention to support the teaching and learning of proof. The central research objective of this project was to develop a pedagogical framework and a corresponding set of lesson plans and support materials to guide teachers toward improving students’ success with proof. The primary educational objective of this project was to support mathematics educators in understanding particular sub-goals of proof and developing strategies for teaching them. We present data and findings from our three-year collaboration on this project (2016-2019), which made use of ideas from design research and lesson study, and we discuss lessons learned through our collaboration.

Title: The graph isomorphism game for quantum graphs

Date: 11/21/2019

Time: 11:30 AM - 12:30 PM

Place: C304 Wells Hall

Non-local games give us a way of observing quantum behaviour through the observation of only classical data. The graph isomorphism game is one such non-local game played by Alice and Bob which involves two finite, undirected graphs. A winning strategy for the game is called quantum if it utilizes some shared resource of quantum entanglement between the players. We say two graphs are quantum isomorphic if there is a winning quantum strategy for the graph isomorphism game. We show that the *-algebraic, C*-algebraic, and quantum commuting (qc) notions of a quantum isomorphism between classical graphs X and Y are all equivalent. This is based on joint work with M. Brannan, A. Chirvasitu, S. Harris, V. Paulsen, X. Su, and M. Wasilewski.

Title: Stability of traveling planewave solutions to Lorentzian vanishing mean curvature flow

Date: 11/21/2019

Time: 1:00 PM - 1:50 PM

Place: C517 Wells Hall

Lorentzian minimal submanifolds of Minkowski space are the dynamic analogue of minimal surfaces in the elliptic regime. They are defined by the vanishing of mean curvature, which can be expressed as a system of geometric PDEs. With the requirement that the submanifold be Lorentzian, that is, that the induced metric is Lorentzian, the equations have a hyperbolic nature. Consequently, the natural approach to study them is via the Cauchy initial value problem. In this talk we discuss stability properties of traveling planewave solutions to these equations, and highlight the difficulties introduced by the "infinite energy" planewave background. This is joint work with Willie Wong.

Title: Braid invariant relating knot Floer homology and Khovanov homology

Date: 11/21/2019

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

Khovanov homology and knot Floer homology are two knot invariants that were defined around the same time, and despite their different constructions, share many formal similarities. After reviewing the construction of Khovanov homology and some of these similarities, we will discuss an algebraic braid invariant which is closely related to both Khovanov homology and the refinement of knot Floer homology into tangle invariants. This is a joint work with Nathan Dowlin.

The AMS Grad Student Chapter is putting on a Git, GitHub, and Zotero workshop. If you want to know more about version control / staying organized / making .bib files / best collaboration practices, this workshop is for you! Please bring your laptop. Snacks will be provided!

Speaker: Exchange Program and MSU Student Research Teams

Title: Undergraduate MTH and STT Research Project Presentations

Date: 11/22/2019

Time: 3:30 PM - 7:00 PM

Place: C304 Wells Hall

Special event: Undergraduate MTH and STT Research Project Presentations
Exchange Program and MSU Student Research Teams will present the results of their research projects

Speaker: Yeonhyang Kim, Central Michigan University

Title: Q-ball imaging using predicted diffusion gradient directions

Date: 11/22/2019

Time: 4:10 PM - 5:00 PM

Place: A203 Wells Hall

We review the theory of q-ball imaging and describe a simple linear matrix formulation for the q-ball reconstruction based on spherical harmonic basis function interpolation. In this talk, we present a novel method to improve the q-ball reconstruction by determining appropriate positions at which a synthesized signal is obtained by combining surrounding signals in given diffusion gradient directions.

Title: Local rigidity for partially hyperbolic toral automorphisms

Date: 11/26/2019

Time: 3:50 PM - 4:50 PM

Place: C304 Wells Hall

We study perturbations of an irreducible ergodic toral automorphism $L$ with non-trivial center. For a small perturbation $f$ of $L$ with smooth center foliation we obtain results on regularity of the leaf conjugacy to $L$ and of a Holder topological conjugacy to $L$, when it exists. As a corollary, we show that for any symplectic perturbation a Holder conjugacy to $L$ must be smooth. For a totally irreducible $L$ with two-dimensional center, we establish various equivalent conditions that ensure smooth conjugacy between $L$ and $f$. This is joint work with Andrey Gogolev and Victoria Sadovskaya