We introduce a new partial order called the maximal chain descent order on the maximal chains of any finite, bounded poset with an EL-labeling. We prove that the maximal chain descent order encodes via its linear extensions all shellings of the order complex induced by the EL-labeling strictly including the well-known lexicographic shellings. We show that the standard EL-labeling of the Boolean lattice has maximal chain descent order isomorphic to the type A weak order. We also prove that natural EL-labelings of intervals in Young's lattice give maximal chain descent orders isomorphic to partial orders on the standard Young tableaux or standard skew tableaux of a fixed shape given by swapping certain entries. We additionally show that the cover relations of maximal chain descent orders are generally more subtle than one might first expect, but we characterize the EL-labelings with the expected cover relations including many well-known families of EL-labelings.

I will discuss basics of potentially orthonormalizable modules and some related concepts, which are preliminaries for the theory of Fredholm's determinant of compact operators in non-archimedean setting.

Title: ZOOM TALK (password the smallest prime > 100) - Towards Intrinsically Low-Dimensional Models in Wasserstein Space: Geometry, Statistics, and Learning

Date: 02/02/2023

Time: 2:30 PM - 3:30 PM

Place: C304 Wells Hall

Contact: Mark A Iwen ()

We consider the problems of efficient modeling and representation learning for probability distributions in Wasserstein space. We consider a general barycentric coding model in which data are represented as Wasserstein-2 (W2) barycenters of a set of fixed reference measures. Leveraging the Riemannian structure of W2-space, we develop a tractable optimization program to learn the barycentric coordinates when given access to the densities of the underlying measures. We provide a consistent statistical procedure for learning these coordinates when the measures are accessed only by i.i.d. samples. Our consistency results and algorithms exploit entropic regularization of the optimal transport problem, thereby allowing our barycentric modeling approach to scale efficiently. We also consider the problem of learning reference measures given observed data. Our regularized approach to dictionary learning in Wasserstein space addresses core problems of ill-posedness and in practice learns interpretable dictionary elements and coefficients useful for downstream tasks. Applications to image and natural language processing will be shown throughout the talk.

In this talk, we will discuss a free probabilistic quantity called free Stein dimension and compute it for a crossed product by a finite group. The free Stein dimension is the Murray-von Neumann dimension of a particular subspace of derivations. Charlesworth and Nelson defined this quantity in the hope of finding a von Neumann algebra invariant. While it is still not known to be a von Neumann algebra invariant, it is an invariant for finitely generated unital tracial *-algebras and algebraic methods have been more successful than analytic ones in studying it. Our result continues this trend, and reveals a formula for the free Stein dimension of a crossed product by a finite group that is reminiscint of the Schreier formula for a finite index subgroups of free groups.

How good of an invariant is the Jones polynomial? The question is closely tied to studying braid group representations since the Jones polynomial can be defined as a (normalized) trace of a braid group representation.
In this talk, I will present my work developing a new theory to precisely characterize the entries of classical braid group representations, which leads to a generic faithfulness result for the Burau representation of B_4 (the faithfulness is a longstanding question since the 1930s and is equivalent to whether B_4 is a group of 3 x 3 matrices). In forthcoming work, I use this theory to furthermore explicitly characterize the Jones polynomial of all 3-braid closures and generic 4-braid closures. I will also describe my work which uses the class numbers of quadratic number fields to show that the Jones polynomial detects the unknot for 3-braid links - this work also answers (in a strong form) a question of Vaughan Jones.
I will discuss all of the relevant background from scratch and illustrate my techniques through simple examples.

In this talk, we will study the Freidlin–Wentzell LDP for the KPZ equation using the variational principle. Such an approach goes under the name of the weak noise theory in physics. We will explain how to extract various limits of the most probable shape of the KPZ equation in the setting of the Freidlin–Wentzell LDP. Some future directions will also be discussed at the end. The talk is based on several joint works with Pierre Yves Gaudreau Lamarre and Li-Cheng Tsai.

We study parabolic aligned elements associated with the type-B Coxeter group and the so-called linear Coxeter element. These elements were introduced algebraically in (Mühle and Williams, 2019) for parabolic quotients of finite Coxeter groups and were characterized by a certain forcing condition on inversions. We focus on the type-B case and give a combinatorial model for these elements in terms of pattern avoidance. Moreover, we describe an equivalence relation on parabolic quotients of the type-B Coxeter group whose equivalence classes are indexed by the aligned elements. We prove that this equivalence relation extends to a congruence relation for the weak order. The resulting quotient lattice is the type-B analogue of the parabolic Tamari lattice introduced for type A in (Mühle and Williams, 2019). These lattices have not appeared in the literature before. As work in progress, we will also talk about various combinatorial models and bijections between them. Joint work with Henri Mühle and Jean-Christophe Novelli.

Bohnenblust--Hille inequalities for Boolean cubes have been proven with dimension-free constants that grow sub-exponentially in the degree (Defant—Mastylo—Peres). Such inequalities have found great applications in learning low degree Boolean functions (Eskenazis—Ivanisvili). Motivated by learning quantum observables, a quantum counterpart of Bohnenblust--Hille inequality for Boolean cubes was recently conjectured in Cambyse Rouz\’e, Melchior Wirth, and Haonan Zhang: ``Quantum Talagrand, KKL and Friedgut’s theorems and the learnability of quantum Boolean functions.” arXiv preprint, arXiv:2209.07279, 2022.
Haonan Zhang and myself prove such noncommutative Bohnenblust--Hille inequalities with constants that are dimension-free and of exponential growth in the degree. As applications, we study learning problems of quantum observables.
(Speaker will present remotely)

Persistent homology, the flagship method of topological data analysis, can be used to provide a quantitative summary of the shape of data. One way to pass data to this method is to start with a finite, discrete metric space (whether or not it arises from a Euclidean embedding) and to study the resulting filtration of the Rips complex. In this talk, we will discuss several available methods for turning a time series into a discrete metric space, including the Takens embedding, $k$-nearest neighbor networks, and ordinal partition networks. Combined with persistent homology and machine learning methods, we show how this can be used to classify behavior in time series in both synthetic and experimental data.

In this talk, we will discuss foliations and their transverse invariant measures (i.e., measures on cross-sections that are invariant under the holonomy maps) from a dynamical systems point of view. We will show that for a large family of diffeomorphisms, the unstable foliations admit families of transverse measures that are naturally related to certain probability measures invariant under the dynamics. Given an unstable leaf, we will consider a dynamically defined average that captures its intersection with cross-sections and prove that this averaging will converge exponentially fast to the transverse invariant measures. This is a joint work with Ures, Viana and J. Yang.

Abstract: Any closed compact 3 manifold admits a Heegaard splitting, which splits the 3 manifold into two handlebodies. In this talk, we will use bagels to illustrate the idea of Heegaard splitting. More specifically, we will use 2 bagels to construct 3 sphere and finite many bagels to construct any 3 manifold. Besides, bagels will be provided during the talk.

I’ll explain how methods from derived algebraic geometry can be applied to give a uniform definition of special cycle classes on integral models of Shimura varieties of Hodge type, verifying some consequences of Kudla’s conjectures on the modularity of generating series of cycles on Shimura varieties of Hermitian type.

Abstract: Witten-Reshetikhin-Turaev SO(3) quantum representations are a family of representations of mapping class groups of surfaces. The family is asymptotically faithful, but each representation has kernel: indeed, r-th powers of Dehn twists are in the kernel of the level r quantum representation.
An open question is whether the kernel is generated by r-th powers of Dehn twists; we will present partial results on this question, by relating the so-called "h-adic expansion" of quantum representations to Johnson homomorphisms.

The Murnaghan-Nakayama formula expresses the product of a
Schur function with a Newton power sum in the basis of Schur
functions. An important generalization of Schur functions are
Schubert polynomials (both classical and quantum). For these, a
Murnaghan-Nakayama formula is geometrically meaningful. In
previous work with Morrison, we established a Murnaghan-Nakayama
formula for Schubert polynomials and conjectured a quantum
version. In this talk, I will discuss some background and then
some recent work proving this quantum conjecture. This is joint
work with Benedetti, Bergeron, Colmenarejo, and Saliola.

Given a spin rational homology sphere equipped with a cyclic group action, I will introduce equivariant refinements of Manolescu's kappa invariant, derived from the equivariant K-theory of the Seiberg--Witten Floer spectrum. These invariants give rise to equivariant relative 10/8-ths type inequalities for equivariant spin cobordisms between rational homology spheres. I will explain how these inequalities provide applications to knot concordance, obstruct cyclic group actions on spin fillings, and give genus bounds for knots in punctured 4-manifolds. If time permits I will explain how these invariants are related to equivariant eta-invariants of the Dirac operator, and describe work-in-progress which provides explicit formulas for the $S^1$-equivariant eta-invariants on Seifert-fibered spaces.

The quantum trace homomorphism connects two competing quantizations for the $SL_n$-character variety of a surface, consisting of $SL_n$-local systems over the surface. The first quantization is through the $SL_n$-skein algebra, which is intrinsic but difficult to work with. The second quantization is based on a quantization of Thurston-Fock-Goncharov local coordinates, and is algebraically easier to handle but depends on choices. I will focus on the construction of this quantum trace in the case where $n=2$.

While driver telematics has gained attention for risk classification in auto
insurance, scarcity of observations with telematics features has been problematic, which
could be owing to either privacy concern or adverse selection compared to the data points
with traditional features. To handle this issue, we propose a data integration technique based
on calibration weights. It is shown that the proposed technique can efficiently integrate the
so-called traditional data and telematics data and also cope with possible adverse selection
issues on the availability of telematics data. Our findings are supported by a simulation study
and empirical analysis on a synthetic telematics dataset.

DP-coloring (also called correspondence coloring) of graphs is a generalization of list coloring of graphs that has been widely studied in recent years after its introduction by Dvorak and Postle in 2015. Intuitively, DP-coloring is a variation on list coloring where each vertex in the graph still gets a list of colors, but identification of which colors are different can change from edge to edge. DP-coloring has been investigated from both the extremal (DP-chromatic number) and the enumerative (DP-color function) perspectives.
In this talk, we will give an overview of questions arising with regard to when the DP-color function equals the chromatic polynomial (or any polynomial), and how the polynomial method, through the Combinatorial Nullstellensatz and the Alon-Furedi theorem for the number of non-zeros of a polynomial, can be applied to both extremal and enumerative problems in DP-coloring. Many open problems and conjectures will be presented.

We consider the Schramm-Loewner evolution (SLE_kappa) with kappa=4, the critical value of kappa>0 at or below which SLE_kappa is a simple curve and above which it is self-intersecting. We show that the range of an SLE_4 curve is a.s. conformally removable, answering a question posed by Sheffield. In order to establish this result, we give a new sufficient condition for a set X in the complex plane to be conformally removable which applies in the case that X is not necessarily the boundary of a simply connected domain. This is based on a recent joint work with Jason Miller and Lukas Schoug.

Learning an optimal Individualized Treatment Rule (ITR) is a very important problem in precision medicine. In this talk, we consider the challenge when the number of treatment arms is large, and some groups of treatments in the large treatment space may work similarly for the patients. Motivated by the recent development of supervised clustering, we propose a novel adaptive fusion-based method to cluster the treatments with similar treatment effects together and estimate the optimal ITR simultaneously through a single convex optimization. We establish the theoretical guarantee of recovering the underlying true clustering structure of the treatments for our method. Finally, the superior performance of our method will be demonstrated via both simulations and a real data application on cancer treatment.
This is joint work with Haixu Ma and Donglin Zeng at UNC-Chapel Hill.

We study the regularity of the conjugacy between an irreducible Anosov automorphism $A$
on torus and its small perturbation $f$.
We say that $f$ and $A$ has the same periodic data if the
derivatives of the return maps of $f$ and $A$ at the corresponding periodic points are
conjugate. We demonstrate that if $f$ is a $C^s$ diffeomorphism with $s$ sufficiently large and has the same periodic data as $A$, then the conjugacy is $C^{s-\epsilon}$. This completes the characterization of the most elementary $C^1$-invariant for local smooth rigidity.
We also give the first example of cocycle rigidity over fibers with conjugate periodic data.

A smooth complex projective variety is rational if it can be obtained from projective space by algebraic surgeries, i.e. blowups and blowdowns. It is stably rational if it becomes rational after takinga product with some projective space.
Consider a family of such varieties over a connected base. Which members are rational? Stably rational? We focus on recent general results and also outstanding questions that remain. These are illustrated in several key examples, including hypersurfaces of low
degree.
Joint work with Kresch, Pirutka, and Tschinkel.

Tensor method can be used for compressing high-dimensional functions arising from partial differential equations (PDE). In this talk, we focus on using these methods for the simulation of transition processes between metastable states in chemistry applications, for example in molecular dynamics. To this end, we also propose a novel generative modeling procedure using tensor-network without the use of any optimization.

Everyone seems to be talking about racial equity and justice these days. Increasingly, scholars in mathematics education are recognizing the need to center the voices of those most affected—i.e., Black, Latine, Asian, and Indigenous children and families—in these discussions. Our current project explores participatory design research (PDR) as a tool for building school, university, student, and parent capacity for centering children of color and their families as researchers and designers of middle school mathematics learning, in a small but diverse Midwestern city. In this talk, we will discuss the challenges we are experiencing and what we are learning about PDR, racial justice, and ourselves, as we work to bring youth of color to the table with us to eat, learn, and act together. Join Zoom Meeting:
https://msu.zoom.us/j/94209936218 Passcode: PRIME

The problem of finding a barycenter in the Wasserstein space is a nonlinear interpolation between several probability measures. In this talk we will discuss the notion of barycenters in the Wasserstein space under a capacity constraint on the mass transported and its dual formulation.
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This will be a hybrid seminar and take place in C329 Wells Hall and via Zoom at https://msu.zoom.us/j/99426648081?pwd=ZEljM3BPUXg2MjVUMVM5TnlzK2NQZz09 .

Suppose that a group acts on a variety. When can the variety and the action be resolved so that all stabilizers are finite? Kirwan gave an answer to this question in the 1980s through an explicit blowup algorithm for smooth varieties with group actions in the context of Geometric Invariant Theory (GIT). In this talk, we will explain how to generalize Kirwan's algorithm to Artin stacks in derived algebraic geometry, which, in particular, include classical, potentially singular, quotient stacks that arise from group actions in GIT. Based on joint work with Jeroen Hekking and David Rydh.

The lattice of non-crossing partitions plays an important role in the theory of free probability. In particular, it allows one to define the so-called free cumulants, which capture the same information as a non-commutative distribution. In this talk I will provide an introduction to these ideas and show how cumulants offer a characterization of free independence as well as an easy proof of the free central limit theorem.

I will discuss the technical details of the construction of the quantum trace homomorphism, going from the SL_2-skein algebra to the quantum Teichmüller space of Chekhov-Fock.

Early in his career, John Milnor defined his seminal link invariants, now called Milnor's $\overline{\mu}$-invariants. They are topological concordance invariants of links in $S^3$, and much is known about them. However, until recently, few results have extended Milnor's work to links in other closed orientable 3-manifolds, and such extensions have done so for special classes of 3-manifolds or specific types of links. In this talk, I will discuss an extension of these invariants to concordance invariants of knots and links in any closed orientable 3-manifold, discuss some theorems that justify calling them ``Milnor's invariants", and study their properties.