Title: Bipartite dimer model and minimal surfaces in the Minkowski space

Date: 09/01/2020

Time: 4:00 PM - 5:00 PM

Place: Online (virtual meeting)

We discuss a new approach to the convergence of height fluctuations in the bipartite dimer model considered on big planar graphs. This viewpoint is based upon special embeddings of weighted planar graphs into the complex plane known under the name Coulomb gauges or, equivalently, t-embeddings. The long-term motivation comes from trying to understand fluctuations on irregular graphs, notably on random planar maps equipped with the dimer (or, similarly, the critical Ising) model.
When the dimer model is considered on subgraphs of refining lattices, a classical conjecture due to Kenyon--Okounkov predicts the convergence of fluctuations to the Gaussian Free Field in a certain conformal structure. However, the latter is defined via a lattice-dependent entropy functional, which makes the analysis of irregular graphs highly problematic. To overcome this difficulty, we introduce a notion of 'perfect t-embeddings' of abstract weighted bipartite graphs and develop new discrete complex analysis techniques to handle correlation functions of the dimer model on t-embeddings. Though in full generality the existence of perfect embeddings remains an open question, we prove that - at least in some concrete cases - they reveal the relevant conformal structure in a lattice-independent way: as that of a related Lorentz-minimal surface in the Minkowski space.
Based upon joint works with Benoît Laslier, Sanjay Ramassamy and Marianna Russkikh.

Title: High dimensional approximation with trigonometric polynomials; zoom link @ https://sites.google.com/view/minds-seminar/home

Date: 09/03/2020

Time: 2:30 PM - 3:30 PM

Place:

(Part of One World MINDS seminar: https://sites.google.com/view/minds-seminar/home)
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In this talk, we present fast Fourier based methods for the approximation of multivariate functions. Our aim is to learn the support of the Fourier coefficients in the frequency domain of high-dimensional functions. We are interested in two different approximation scenarios. The first case is black-box approximation where the user is allowed to sample the unknown function at any point and in the second case we are working with fixed scattered data. For black-box approximation we employ quasi Monte-Carlo methods on rank-1 lattice points. The fast algorithms are then based on one-dimensional fast Fourier transforms (FFT). In the second case, which is much more difficult, we will couple truncated ANOVA (analysis of variance) decompositions with the fast Fourier transform on nonequispaced data (NFFT). In both cases, we present error estimates and numerical results. The presented methods can be understood as sparse high dimensional FFT’s.
This talk based on joint work with Lutz Kämmerer, Michael Schmischke, Manfred Tasche, and Toni Volkmer.

Let K be a null-homologous knot in a closed 3-manifold Y, and F be a Seifert surface. One can cap off the boundary of F with a disk in the zero surgery on K to get a closed surface F_0. If we know that F is Thurston norm minimizing, we can ask whether F_0 is also Thurston norm minimizing. A classical theorem of Gabai says that the answer is Yes when Y is the 3-sphere. Gabai's theorem can be generalized to many other 3-manifolds using Heegaard Floer homology. In this talk, we will discuss a sufficient condition for F_0 to be Thurston norm minimizing which relates this property to the 4-genus of the knot.
Zoom: https://msu.zoom.us/j/92550015779?pwd=azRER2p1Sm5CWWRML3lHbVQyWDU1QT09

A sheaf quantization is a sheaf associated to a Lagrangian brane. This sheaf conjecturally has information as much as Floer theory of the Lagrangian. On the other hand, exact WKB analysis is an analysis of differential equations containing the Planck constant hbar.
In this talk, I will explain how to construct a sheaf quantization over the Novikov ring of the spectral curve of an hbar-differential equation, by using the ideas of exact WKB analysis and spectral network. In the construction, one can see how (conjecturally) the convergence in WKB analysis are related to the convergence of Fukaya category. In degree 2, the sheaf quantization associates a cluster coordinate which is the same as Fock—Goncharov coordinate. I will also mention about some relationships to Riemann—Hilbert correspondence of D’Agnolo—Kashiwara and Kontsevich—Soibelman.
https://msu.zoom.us/j/95159415920?pwd=bUlETkdpazdiWGNjZnNkUWNIaXRFQT09

Title: On the homology of subword order. Zoom https://msu.zoom.us/j/5476724571

Date: 09/09/2020

Time: 4:10 PM - 5:00 PM

Place:

In this talk we examine the homology representation of the symmetric group $S_n$ on rank-selected subposets of subword order. We show that the action on the rank-selected chains is a nonnegative integer combination of tensor powers of the reflection representation $S_{(n-1,1)}$ indexed by the partition $(n-1,1)$, and that its Frobenius characteristic is $h$-positive and supported on the set $T_{1}(n)=\{h_\lambda: \lambda=(n-r, 1^r), r\ge 1\}.$
We give an explicit formula for the homology module for words of bounded length, as a sum of tensor powers of $S_{(n-1,1)}$. This recovers, as a special case, a theorem of Bj\"orner and Stanley for words of length at most $k.$ We exhibit a curious duality in homology in the case when one rank is deleted. We also show that in many cases, the rank-selected homology modules, modulo one copy of the reflection representation, are $h$-positive and supported on the set $T_{2}(n)=\{h_\lambda: \lambda=(n-r, 1^r), r\ge 2\}.$
Our analysis of the homology also uncovers curious enumerative formulas that may be interesting to investigate combinatorially.

We'll introduce and motivate techniques of Galois descent for classifying twisted forms of algebras, using non-abelian group cohomology. We'll also describe a connection to relative Brauer groups.
Zoom link: https://msu.zoom.us/j/96111069403

Title: The phase factor: recovery from magnitudes of signal representations; zoom link @ https://sites.google.com/view/minds-seminar/home

Date: 09/10/2020

Time: 2:30 PM - 3:30 PM

Place:

We study the problem of phase retrieval from a deterministic viewpoint, in which the magnitudes of a time-frequency or time-scale representation of a signal are known. From an inverse problems perspective, the questions of uniqueness and stability are crucial to theoretically guarantee meaningful reconstruction. In this talk, we present results on these two questions for Gabor frames and wavelet frames and conclude by discussing some open problems.

Title: Relative Reshetikhin-Turaev invariants and hyperbolic cone metrics on 3-manifolds

Date: 09/15/2020

Time: 3:00 PM - 4:00 PM

Place: Online (virtual meeting)

We propose the Volume Conjecture for the relative Reshetikhin-Turaev invariants of a closed oriented 3-manifold with a colored framed link inside it whose asymptotic behavior is related to the volume and the Chern-Simons invariant of the hyperbolic cone metric on the manifold with singular locus the link and cone angles determined by the coloring, and prove the conjecture for a number of families of examples. This provides a possible approach of solving the Volume Conjecture for the Reshetikhin-Turaev invariants of closed oriented hyperbolic 3-manifolds. A large part of this work is joint with Ka Ho Wong. https://msu.zoom.us/j/92550015779?pwd=azRER2p1Sm5CWWRML3lHbVQyWDU1QT09

Some physical and mathematical theories have the unfortunate feature that if one takes them at face value, many quantities of interest appear to be infinite! What's worse, this doesn't just happen for some exotic theories, but in the standard theories describing some of the most fundamental aspects of nature. Various techniques, usually going under the common name of “renormalisation” have been developed over the years to address this, allowing mathematicians and physicists to tame these infinities. We will dip our toes into some of the conceptual and mathematical aspects of these techniques and we will see how they have recently been used in probability theory to study equations whose meaning was not even clear until now.

Speaker: Samantha Dahlberg, Arizona State University

Title: Diameters of Graphs of Reduced Words of Permutations, Zoom https://msu.zoom.us/j/5476724571

Date: 09/16/2020

Time: 3:00 PM - 3:50 PM

Place: Online (virtual meeting)

It is a classical result that any two reduced words of a permutation in the symmetric group can be transformed into one another by a sequence of long braid moves and commutation moves. In this talk we will discuss the diameters of these connected graphs formed from the reduced words connected by single moves. Recently, the diameter has been calculated for the longest permutation $n\ldots 21$ by Reiner and Roichman as well as Assaf. In this talk we present our results on diameters for certain classes or permutation . We also make progress on conjectured bounds of the diameter by Reiner and Roichman, which are based on the underlying hyperplane arrangement.

Classical (contravariant) Dieudonne theory establishes an antiequivalence of categories between finite commutative group schemes over a perfect field $k$ and Dieudonne modules. In this talk we will talk about this antiequivalence and some simple applications of it.

Title: Learning Interaction laws in particle- and agent-based systems; zoom link @ https://sites.google.com/view/minds-seminar/home

Date: 09/17/2020

Time: 2:30 PM - 3:30 PM

Place:

Interacting agent-based systems are ubiquitous in science, from modeling of particles in Physics to prey-predator and colony models in Biology, to opinion dynamics in economics and social sciences. Oftentimes the laws of interactions between the agents are quite simple, for example they depend only on pairwise interactions, and only on pairwise distance in each interaction. We consider the following inference problem for a system of interacting particles or agents: given only observed trajectories of the agents in the system, can we learn what the laws of interactions are? We would like to do this without assuming any particular form for the interaction laws, i.e. they might be “any” function of pairwise distances. We consider this problem both the mean-field limit (i.e. the number of particles going to infinity) and in the case of a finite number of agents, with an increasing number of observations, albeit in this talk we will mostly focus on the latter case. We cast this as an inverse problem, and study it in the case where the interaction is governed by an (unknown) function of pairwise distances. We discuss when this problem is well-posed, and we construct estimators for the interaction kernels with provably good statistically and computational properties. We measure their performance on various examples, that include extensions to agent systems with different types of agents, second-order systems, and families of systems with parametric interaction kernels. We also conduct numerical experiments to test the large time behavior of these systems, especially in the cases where they exhibit emergent behavior. This is joint work with F. Lu, J.Miller, S. Tang and M. Zhong.

Title: Operator Spaces and Operator Systems: An Exposition

Date: 09/21/2020

Time: 2:00 PM - 2:50 PM

Place: Online (virtual meeting)

During this lecture I will give an overview of the history and theory of operator spaces and operator systems. These "matrix normed spaces'' and "matrix ordered $*$-vector spaces'' arose in a somewhat natural fashion and the study of these objects is motivated by problems that arise when studying the "classical'' theory such as C*-algebras. After going over necessary background for both objects I will discuss how operator space and operator system theory were applied to approaching and solving problems in operator algebras.
Join via Zoom: https://msu.zoom.us/j/95716797501

Title: An Abstract Characterization for Projections in Operator Systems

Date: 09/21/2020

Time: 3:00 PM - 3:50 PM

Place: Online (virtual meeting)

Given an abstract operator system V it is not clear how one would go about defining the notion of a projection. During this talk I will present an answer and some recent results on this question. This is done by first considering abstract compression operator systems associated with a positive contraction in V and then determining when we have a realization of V in such an abstract compression operator system. It then follows that there is a one-to-one correspondence between abstract and concrete projections, and in particular, that every abstract projection is a concrete projection in the C*-envelope of V. I will then conclude with some applications to quantum information theory. In particular, the study of certain correlation sets. This is joint work with Travis Russell (West Point).
Join via Zoom: https://msu.zoom.us/j/95716797501

Title: A geometric approach to fractional powers of the Laplacian and sharp Sobolev trace inequalities

Date: 09/22/2020

Time: 3:00 PM - 4:00 PM

Place: Online (virtual meeting)

A seminal paper of Caffarelli and Silvestre identifies fractional powers of the Laplacian on Euclidean space as Dirichlet-to-Neumann operators. In this talk, I will use conformal geometry to generalize their approach to Riemannian manifolds. More specifically, I will present multiple (equivalent) definitions of (conformally covariant) operators with principal symbol that of a fractional power of the Laplacian. I will also discuss how these operators lead to a simple derivation of a broad family of sharp Sobolev trace inequalities.
https://msu.zoom.us/j/92550015779?pwd=azRER2p1Sm5CWWRML3lHbVQyWDU1QT09

Title: Polynomial interpolation is harder than it sounds

Date: 09/22/2020

Time: 4:00 PM - 5:00 PM

Place: Online (virtual meeting)

Suppose that $(x_1,y_1),\ldots,(x_r,y_r)$ is a set of points in the plane. Given a degree $d$ and multiplicities $m_i$, does there a nonzero polynomial in two variables of degree at most $d$ which vanishes to order at least $m_i$ at $(x_i,y_i)$? What is the dimension of the space of such polynomials, and how does it vary with the parameters? I will explain some of the basic results and conjectures and show how this problem is connected to some questions of current interest in algebraic geometry.

Speaker: Samin Aref, Max Planck Institute for Demographic Research

Title: Structural analysis of signed graphs: a talk on methods and applications, Zoom https://msu.zoom.us/j/5476724571

Date: 09/23/2020

Time: 3:00 PM - 3:50 PM

Place: Online (virtual meeting)

This talk focuses on positive and negative ties in networks (signed graphs) resulting in a common structural configuration. We analyze signed networks from the perspective of balance theory which predicts structural balance as a stable configuration. A signed network is balanced iff its set of vertices can be partitioned into two groups such that positive edges are within the groups and negative edges are between the groups.
The scarcity of balanced configurations in networks inferred from empirical data (real networks) requires us to define the notion of partial balance in order to quantify the extent to which a network is balanced. After evaluating several numerical measures of partial balance, we recommend using the frustration index, which equals the minimum number of edges whose removal results in a balanced network [arxiv.org/abs/1509.04037].
We use the definition of balance to optimally partition nodes of signed networks into two internally solidary but mutually hostile groups. An optimal partitioning leads to an exact value for the frustration index. We tackle the intensive computations of finding an optimal partition by developing efficient mathematical models and algorithms [arxiv.org/abs/1710.09876] [arxiv.org/abs/1611.09030]. We then extend the concepts of balance and frustration in signed networks to applications beyond the classic friend-enemy interpretation of balance theory in the social context. Using a high-performance computer, we analyze large networks to investigate a range of applications from biology, chemistry and physics to finance, international relations, and political science [arxiv.org/abs/1712.04628].
In another project manly focused on a political science application, we focus on the challenge of quantifying political polarization in the US Congress, and analyzing its relationship to the fraction of introduced bills that are passed into law (bill passage rate). We use signed graph models of political collaboration among legislators to show that changes in bill passage rates are better explained by the partisanship of a chamber's largest coalition, which we identify by partitioning signed networks of legislators according to balance theory [arxiv.org/abs/1906.01696].
In another project, we expand the evaluation of balance to incorporate directionality of the edges and consider three levels of analysis: triads, subgroups, and the whole network. Through extensive computational analysis, we explore common structural patterns across a range of social settings from college students and Wikipedia editors to philosophers and Bitcoin traders. We then apply our multilevel framework of analysis to examine balance in temporal and multilayer networks which leads to new observations on balance with respect to time and layer dimensions [arxiv.org/abs/2005.09925].

Title: Algebraic groups with good reduction. Zoom https://msu.zoom.us/j/97573873209

Date: 09/23/2020

Time: 4:00 PM - 5:00 PM

Place: Online (virtual meeting)

Techniques involving reduction are very common in number theory and arithmetic geometry. In particular, elliptic curves and general abelian varieties having good reduction have been the subject of very intensive investigations over the years. The purpose of this talk is to report on recent work that focuses on good reduction in the context of reductive linear algebraic groups over finitely generated fields. In addition, we will highlight some applications to the study of local-global principles and the analysis of algebraic groups having the same maximal tori. (Parts of this work are joint with V. Chernousov and A. Rapinchuk.)

A path connected topological space is simply connected if the space of based paths is path connected. Equivalently, the fundamental group is zero or any connected covering space is trivial. However, these notions do not capture the correct notion in the world of algebraic geometry. For example, if $X$ is a Riemann surface then the Zariski topology (the usual topology in algebraic geometry) on $X$ is equivalent to the cofinite topology, so $X$ is simply connected.
In this talk, we will introduce a few definitions of simply connectedness in algebraic geometry - each corresponding to one of the equivalent definitions above. We will then compare these definitions and discuss how their consequences differ from their topological counterparts.

Title: Regularization in Infinite-Width ReLU Networks; zoom link @ https://sites.google.com/view/minds-seminar/home

Date: 09/24/2020

Time: 2:30 PM - 3:30 PM

Place: Online (virtual meeting)

A growing body of research illustrates that neural network generalization performance is less dependent on the network size (i.e. number of weights or parameters) and more dependent on the magnitude of the weights. That is, generalization is not achieved by limiting the size of the network, but rather by explicitly or implicitly controlling the magnitude of the weights. To better understand this phenomenon, we will explore how neural networks represent functions as the number of weights in the network approaches infinity. Specifically, we characterize the norm required to realize a function f as a single hidden-layer ReLU network with an unbounded number of units (infinite width), but where the Euclidean norm of the weights is bounded, including precisely characterizing which functions can be realized with finite norm. This was settled for univariate functions in Savarese et al. (2019), where it was shown that the required norm is determined by the L1-norm of the second derivative of the function. We extend the characterization to multivariate functions (i.e., networks with d input units), relating the required norm to the L1-norm of the Radon transform of a (d+1)/2-power Laplacian of the function. This characterization allows us to show that all functions in certain Sobolev spaces can be represented with bounded norm and to obtain a depth separation result. These results have important implications for understanding generalization performance and the distinction between neural networks and more traditional kernel learning.

Title: Fundamental groups of certain property (T) factors

Date: 09/28/2020

Time: 2:00 PM - 2:50 PM

Place: Online (virtual meeting)

Calculation of fundamental groups of type $\rm II_1$ factor is, in general, an extremely hard and central problem in the field of von Neumann algebras. In this direction, a conjecture due to A. Connes states that the fundamental group of the group von Neumann algebra $L(\Gamma)$ associated to any icc property (T) group $\Gamma$ is trivial. Up to now, there was no single example of property (T) group factor satisfying the conjecture. In this talk, I shall provide the first examples of property (T) group factors with trivial fundamental group. This talk is based on a joint work with Ionut Chifan, Sayan Das and Cyril Houdayer.
Join via Zoom: https://msu.zoom.us/j/98441498789

Title: Skein theory in the geometric Langlands TFT

Date: 09/29/2020

Time: 3:00 PM - 4:00 PM

Place: Online (virtual meeting)

I will overview several appearances (some recently established, some conjectural) of skein theory in the so-called quantum geometric Langlands fully extended TFT. The talk will be mostly elementary, and I'll highlight an application to a conjecture of Witten concerning the finite-dimensionality of skein modules of 3-manifolds at generic values of the quantum parameter. https://msu.zoom.us/j/92550015779?pwd=azRER2p1Sm5CWWRML3lHbVQyWDU1QT09

Informally, Kakeya type problems ask whether tubes with different positions and directions can overlap a lot. One usually expects the answer to be no in an appropriate sense. Thanks to the uncertainty principle, such a quantified non-overlapping theorem would often see powerful applications in analysis problems that have Fourier aspects. Perhaps the most well-known Kakeya type problem is the Kakeya conjecture. It remains widely open in $\Bbb{R}^n (n>2)$ as of today. Nevertheless, in the recent few decades people have been able to prove new Kakeya type theorems that led to improvements or complete solutions to analysis problems that appeared out of reach before. I will give an introduction to Kakeya type problems/theorems and analysis problems that see their applications. Potentially reporting some recent progress joint with Du, Guo, Guth, Hickman, Iosevich, Ou, Rogers, Wang and Wilson.

Title: Whitney Duals of Graded Posets, Zoom https://msu.zoom.us/j/5476724571

Date: 09/30/2020

Time: 3:00 PM - 3:50 PM

Place: Online (virtual meeting)

To each graded poset one can associate two sequences of numbers; the Whitney numbers of the first kind and the Whitney numbers of the second kind. One sequence keeps track of the Möbius function at each rank level and the other keeps track of the number of elements at each rank level. The Whitney numbers appear in many contexts in combinatorics. For example, they appear as the coefficients of the chromatic polynomial of a graph and can be used to compute the number of regions in a real hyperplane arrangement.
We say that posets P and Q are Whitney duals if the Whitney numbers of the first kind of P are the Whitney numbers of the second kind of Q and vice-versa. In this talk, we will discuss a method to construct Whitney duals using edge labelings and quotient posets. We will also discuss some applications of Whitney duals.
This is joint work with Rafael S. González D'León.

Schramm-Loewner evolutions (SLE) are probabilistic models of simple planar curves. They first arise as interfaces in scaling limits of 2D statistical mechanics lattice models which exhibit conformal invariance. In this talk, I will explain how asymptotic behaviors of SLE give rise to an interesting quantity (multichordal Loewner potential), which connects to rational function, zeta-regularized determinants of Laplacian, and Belavin-Polyakov-Zamolodchikov equations in conformal field theory. This is a joint work with Eveliina Peltola (Bonn).