Liouville field theory is the model for two-dimensional quantum gravity. It was constructed rigorously using probabilistic methods by David-Kupiainen-Rhodes-Vargas in 2016. According to the conformal bootstrap conjecture n-point correlation functions can be expressed in terms of 3-point correlation functions and so-called conformal blocks.
We restrict ourselves to the case of one point correlation function of the Liouville field theory on the torus. We want to study conformal blocks. They are described using complicated asymptotic series. The probabilistic model for them was suggested by Ghosal-Remy-Sun-Sun in 2021. It allowed showing that the asymptotic series is actually converging in a small disc.
Liouville field theory has central charge c associated to it. Zamolodchikov in 1984 conjectured that conformal blocks have a limit as c goes to infinity. The limit was called classical conformal blocks. We use the probabilistic formula for conformal blocks to prove Zamolodchikov conjecture and show that the asymptotic series for them is converging in a small disc.
This is joint work with Harini Desiraju and Promit Ghosal.

The chromatic symmetric function, introduced by Stanley, counts graph colourings, recording the number of vertices of each colour. I will talk about a K-theoretic analogue of the chromatic symmetric function, in which we are colouring each vertex with a set of colours (rather than a single colour), as well as some results and open questions for this new function. Joint work with Logan Crew and Oliver Pechenik.

In general relativity, one of the most interesting ways to construct notions of energy is the method of Hamiltonian analysis. For asymptotically flat spacetimes, this approach yields the well-known ADM mass. In order to define quasi-local energy/mass for compact initial data sets, one would like to apply the Hamiltonian analysis of GR on compact spacetimes with time-like boundary. Traditionally, this has been done based on fixing the Dirichlet boundary condition of the spacetimes — one of the most well-known work along this thread is the Brown-York quasi-local mass. In this talk we will discuss in detail the relation between the study of initial boundary value problem for vacuum Einstein equations and the Hamiltonian analysis on compact spacetimes. Then we will construct a notion of quasi-local Hamiltonian (energy) based on a well-posed initial boundary value problem.

Reinforcement learning (RL) is garnering significant interest in recent years due to its success in a wide variety of modern applications. However, theoretical understandings on the non-asymptotic sample and computational efficiencies of RL algorithms remain elusive, and are in imminent need to cope with the ever-increasing problem dimensions. In this talk, we discuss our recent progress that sheds light on understanding the efficacy of popular RL algorithms in finding the optimal policy in tabular Markov decision processes.

The motivation of the work is to study topological properties of
partially hyperbolic systems which are similar to those of uniformly hyperbolic systems. We try to obtain some properties similar to these of uniformly hyperbolic systems by ``ignoring'' the motions along the center direction.
We show that any partially hyperbolic systems are quasi-stable in the sense that for any homeomorphism $g$ $C^0$-close to $f$, there exist a continuous map $\pi$ from $M$ to itself and a family of locally defined continuous maps $\{\tau_x\}$, which send points along the center direction, such that
$$\pi\circ g=\tau_{fx}\circ f\circ\pi.
$$
In particular, if $f$ has $C^1$ center foliation, then we can make the motion $\tau$ along the center foliation.
As application we obtain some continuity properties for topological entropy.

Over the past ten years, optimal transport has become a fundamental tool in statistics and machine learning: the 2-Wasserstein metric provides a new notion of distance for classifying distributions and a rich geometry for interpolating between them. In parallel, optimal transport has gained mathematical significance by providing new tools for studying stability and limiting behavior of partial differential equations, through the theory of 2-Wasserstein gradient flows.
In fact, the success optimal transport in each of these contexts ultimately relies on the same fundamental property of the 2-Wasserstein metric: as originally discovered by Otto, the 2-Wasserstein metric is unique among classical optimal transport metrics in that it has a formal Riemannian structure. In my talk, I will introduce the theory of optimal transport, explain the special geometric structure of the 2-Wasserstein metric, and illustrate the essential role it plays in how optimal transport is used in both machine learning and partial differential equations.

This talk will be a combination of a meta discussion about how I developed my research program and the main contributions of some of my projects. Specifically, I will share my journey to find a research program that simultaneously engages both equity and cognitive research. I will discuss connections between my work and Funds of Knowledge, as well as other anthropology-informed work, like Ethnomathematics and studies of the mathematics of Indigenous communities. I will share how one of my current projects with Marta Civil, Project AdeLanTe, implements the principles of anti-deficit learning
and teaching, while also building on principles from Funds of Knowledge and Culturally Relevant Pedagogy. In Room 252 Erickson and on Zoom: https://msu.zoom.us/j/98177166186
Password: PRIME

This talk is aimed more at the general audience.
A fundamental question in the representation theory of semisimple Lie groups is to classify their irreducible unitary representations. A guiding principle here is the
Orbit method, first discovered by Kirillov in the 60's for nilpotent Lie groups. It states that the irreducible unitary representations should be related to coadjoint orbits, i.e., the orbits of the Lie group action in the dual of its Lie algebra.
Passing from orbits to representations could be thought of as a quantization problem and it is known that in this setting this is very difficult. For semisimple Lie groups it makes sense to speak about nilpotent orbits, and one could try to study representations that should correspond to these orbits via the yet undefined Orbit method. These representations are called unipotent: they are expected to be nicer than general ones, while one hopes to reduce the study of general representations to that of unipotent ones. I will concentrate on the case of complex Lie groups. I will explain how recent advances in the study of deformation quantizations of singular symplectic varieties allow to define unipotent representations and obtain some results about them. The talk is based on the joint work with Lucas Mason-Brown and Dmytro Matvieievskyi.

In analysis we tend to focus on the "small scale" structure of a space. For example, both derivatives and continuity only depend on a very small neighborhood around a point. Coarse geometry on the other hand focuses on the "large scale" structure of a space. Coarse spaces generalize metric spaces in a way that provides an appropriate framework to study large-scale geometry. Coarse geometry is used to study: higher index theory, elliptical operators, the coarse Baum-Connes conjecture and as a consiquence the Novikov conjecture.
In this talk we will discuss what a coarse structure is, both in terms of metric spaces and in full generality. Then we will look at a few examples. Next, We will introduce uniform Roe algebras and examine their relationship to coarse structures along with recent advances in solving the rigidity problem. Then, time permitting, we will look at uniform Roe modules.

Abstract: Preparation of entangled states via engineered open quantum systems is proven to be successful. In our work, we initiate a study of engineered open quantum systems which drive the states to a subspace. In other word, our system will be non-ergodic. We prove some stability results and large deviation phenomenon in this setting, under some symmetry condition on the Liouvillian. This is joint work with Marius Junge and Nicholas Laracuente.

Given a Legendrian Knot L in a contact 3 manifold, one can associate a so-called LOSS invariant to L which lives in the knot Floer homology group. We prove that the LOSS invariant is natural under the positive contact surgery. In this talk I will review some background and definition, get the idea of the proof and try to focus on the application which is about new examples of non-simple knots.

In their celebrated proof of the Property P Conjecture and its sequel, Kronheimer and Mrowka proved that the fundamental group of r-surgery on a nontrivial knot in the 3-sphere admits an irreducible SU(2)-representation whenever r is at most 2 in absolute value (which implies in particular that surgery on a nontrivial knot is never a homotopy 3-sphere). They asked whether the same is true for other small values of r -- in particular, for r = 3 and 4 -- noting that it's false for r = 5 since 5-surgery on the right-handed trefoil is a lens space. I'll describe recent work which answers their question in the affirmative. Our proof involves Floer homology and also the dynamics of surface homeomorphisms. All of this work is joint with Steven Sivek, and significant parts are also joint with Zhenkun Li and Fan Ye.

Over the last thirty years, there has been various work on simplicial complexes defined from graphs, much from a topological viewpoint. In this talk I will present recent work (with many collaborators) on the topology of two families of simplicial complexes. One is the matching complex, the complex whose faces are sets of edges that form a matching in a graph, with new results on planar graphs coming from certain tilings. The other is the cut complex, where the facets are sets of vertices whose complements induce disconnected graphs.

In this talk, we will share some of our past, present, and future efforts to support students’ defining and conjecturing activity. We will engage in some of the tasks that we are currently implementing with two calculus students. We will also discuss two future directions of our work: optimizing our task design for the whole class setting to promote equitable participation and developing science-based motivational tasks that elicit informal ideas about calculus concepts.

Big data has revolutionized the landscape of computational mathematics and has increased the demand for new numerical linear algebra tools to handle the vast amount of data. One crucial task is to efficiently capture inherent structure in data using dimensionality reduction and feature extraction. Tensor-based approaches have gained significant traction in this setting by leveraging multilinear relationships in high-dimensional data. In this talk, we will describe a matrix-mimetic tensor algebra that offers provably optimal compressed representations of multiway data via a family of tensor singular value decompositions (SVDs). Moreover, using the inherited linear algebra properties of this framework, we will prove that these tensor SVDs outperform the equivalent matrix SVD and two closely related tensor decompositions, the Higher-Order SVD and Tensor-Train SVD, in terms of approximation accuracy. Throughout the talk, we will provide numerical examples to support the theory and demonstrate practical efficacy of constructing optimal tensor representations.
This presentation will serve as an overview of our PNAS paper "Tensor-tensor algebra for optimal representation and compression of multiway data" (https://www.pnas.org/doi/10.1073/pnas.2015851118).

In this talk we consider the entropy theory for singular vector fields with all singularities hyperbolic and non-degenerate. We will construct a countable partition with the property that the metric entropy for any ergodic invariant measure is finite. For singular star flows, we will show that this partition is generating. This is a joint work with Yi Shi and Jiagang Yang.

A fundamental question for any knot invariant asks which knots it detects (if any). For example, it is famously open whether the Jones polynomial detects the unknot. I'll focus in this talk on the detection question for knot invariants coming from Floer theory and the Khovanov--Rozansky link homology theories. I'll survey the progress made on this question over the past twenty years, and will gesture at some of the topological ideas that go into my recent work with Sivek. I'll end with applications of our results to problems in Dehn surgery, explaining in particular how we use them to extend some of Gabai's work from the eighties.

The application of Artificial Intelligence/Machine Learning (AI/ML) in drug development is expanding rapidly. AI/ML have the potential to improve the efficiency of drug development and advance precision medicine. However, there are unique challenges. The presentation will mainly focus on the topic of AI/ML applications in clinical trials, including the following parts:
1. The increasing numbers of submissions over years.
2. Hot therapeutic areas of AI/ML submissions.
3. Types of analysis and objectives in AI/ML in submissions.
4. Case examples.
5. Challenges and outlooks.
In the end of the presentation, opportunities of FDA-ORISE fellowship will be introduced to senior PhD students.

Speaker: Terry Haut, Lawerence Livermore National Lab

Title: An Overview of High-Order Finite Elements for Thermal Radiative Transfer

Date: 03/17/2023

Time: 4:00 PM - 5:00 PM

Place: C304 Wells Hall

Contact: Mark A Iwen ()

In this talk, I will give an overview of numerical methods for thermal radiative transfer (TRT), with an emphasis on the use of high-order finite elements for their solution. The TRT equations constitute a (6+1)-dimensional set of nonlinear PDEs that describe the interaction of a background material and a radiation field, and their solution is critical for modeling Inertial Confinement Fusion and astrophysics applications. Due to their stiff nature, they are typically discretized implicitly in time, and their solution often accounts for up to 90% of the runtime of multi-physics simulations. I will discuss some recently developed linear solvers, physics-informed preconditioners, and methods for preserving positivity that are used to make the solution to the TRT equations efficient and robust.

Fibered knots show up all over low-dimensional topology, as they provide a robust way to investigate interactions between phenomena of different dimensions. In this talk, I'll survey what they are, why you should care, and how to identify them. Then, as time permits, I'll also sketch a proof that positive braid knots are fibered. I will assume very little background for this talk -- all are welcome!

Dimensionality reduction is the transformation of data from a high-dimensional space into a low-dimensional space so that the low-dimensional representation retains some meaningful properties of the original data, ideally close to its intrinsic dimension.
A classical embedding result is the well-know “Johnson–Lindenstrauss”. The JL lemma shows how a $n$-set of points in $\mathbb{R}^N$ can be embedded into a smaller dimensional space. In this talk we present a result similar to the JL-embedding in the interesting case where instead of a discrete set we embed a compact $d$-dimensional submanifold $\mathcal{M}$ of $\mathbb{R}^N$ into $\mathbb{R}^m $ where $m$ depends on the volume, reach and dimension of $\mathcal{M}.$
$\\$
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 .

A vector bundle on projective space is called "Steiner" if it can be recognized simply as the cokernel of a map given by a matrix of linear forms. Such maps arise from various geometric setups and one can ask: from the Steiner bundle, can we recover the geometric data used to construct it? In this talk, we will mention an interesting Torelli-type result of Dolgachev and Kapranov from 1993 that serves as an origin of this story, as well as other work that this inspired. We'll then indicate our contribution which amounts to analogous Torelli-type statements for certain tautological bundles on the very ample linear series of a polarized smooth projective variety. This is joint work with R. Lazarsfeld.

In classical probability theory, Fisher information is one of the important concepts. Voiculescu introduced the free probability analogue of this quantity, called free fisher information. In this talk, we will discuss how Free Fisher information helps us to understand a von Neumann algebra.

Cloning systems are a method for generalizing Thompson's groups, for example $V_d$, that result in a family of groups, $\mathcal{T}_d(G_*)$, whose group von Neumann algebras have been intensely studied by Bashwinger and Zarmesky in recent years. We consider the group actions of a large class of $\mathcal{T}_d(G_*)$ and show they are stable, that is, $G \sim_{OE} G \times \mathbb{Z}.$ As a corollary, we answer Bashwinger and Zaremsky question about when $\mathcal{T}_d(G_*)$ is a McDuff Group in the sense of Deprez and Vaes. As a contrasting result, we show $L(V_d)$ is a prime II$_1$ factor. This is joint work with Rolando de Santiago and Krishnendu Khan.

In this talk, I’ll describe a braid word theoretic property, called “twist positivity”, which often puts strong restrictions on quantitative and geometric properties of a braid. I’ll describe some old and new results about twist positivity, as well as some new applications towards knot concordance. In particular, I’ll describe how using a suite of numerical knot invariants (including the braid index) in tandem allows one to prove that there is an infinite family of L-space knots (containing all positive torus knots and also an infinite family of hyperbolic knots) where every knot represents a distinct smooth concordance class. This confirms a prediction of the slice-ribbon conjecture. Everything I’ll discuss is joint work with Hugh Morton. I will assume little background about knot invariants for this talk – all are welcome!

In this talk, we will explore and make comparisons between various models that exist for spherical tensor categories associated to the category of representations of the quantum group $U_q(sl_n).$ In particular, we will discuss the combinatorial model of Murakami-Ohtsuki-Yamada (MOY), the n-valent ribbon model of Sikora and the trivalent spider category of Cautis-Kamnitzer-Morrison (CKM). We conclude by showing that the full subcategory of the spider category from CKM, whose objects are monoidally generated by the standard representation and its dual, is equivalent as a spherical braided category to Sikora's quotient category. This proves a conjecture of Le and Sikora and also answers a question from Morrison's Ph.D. thesis.

Given a finite poset that is not completely ordered, is it always possible to find two elements x and y, such that the probability that x is less than y in the random linear extension of the poset, is bounded away from 0 and 1? Kahn-Saks gave an affirmative answer and showed that this probability falls between 3/11 (0.273) and 8/11 (0.727). The currently best known bound is 0.276 and 0.724 by Brightwell-Felsner-Trotter, and it is believed that the optimal bound should be 1/3 and 2/3, also known as the 1/3-2/3 Conjecture. Most notably, log-concave and cross product inequalities played the central role in deriving both bounds. In this talk we will discuss various generalizations of these results together with related open problems. This talk is joint work with Igor Pak and Greta Panova, and is intended for the general audience.

A rotating cosmic string spacetime has a singularity along a timelike curve corresponding to a one-dimensional source of angular momentum. Such spacetimes are not globally hyperbolic: they admit closed timelike curves near the so-called "string". This presents challenges to studying the existence of solutions to the wave equation via conventional energy methods. In this work, we show that forward solutions to the wave equation (in an appropriate microlocal sense) do exist. Our techniques involve proving a statement on propagation of singularities and using the resulting estimates to show existence of solutions. This is joint work with Jared Wunsch.

The two-scale relation in wavelet analysis dictates that a square-integrable function can be written as a linear combination of scaled and shifted copies of itself. This fact is equivalent to the existence of square-integrable functions whose time-scale shifts are linearly dependent. By contrast, by replacing the scaling operator with a modulation operator one would think that the linear dependency of the resulting time-frequency shifts of a square-integrable function might be easily inferred. However, more than two decades after C.~Heil, J.~Ramanatha, and P.~Topiwala conjectured that any such finite collection of time-frequency shifts of a non-zero square-integrable function on the real line is linearly independent, this problem (the HRT Conjecture) remains unresolved.
The talk will give an overview of the HRT conjecture and introduce an inductive approach to investigate it. I will highlight a few methods that have been effective in solving the conjecture in certain special cases. However, despite the origin of the HRT conjecture in Applied and Computational Harmonic Analysis, there is a lack of experimental or numerical methods to resolve it. I will present an attempt to investigate the conjecture numerically.

We define a natural, purely geometrical bijection between the set solutions of Bethe ansatz equations for the Gaudin magnet chain and the set of arc diagrams of Frenkel-Kirillov-Varchenko. The former set is in natural bijection with monodromy-free sl_2-opers (aka projective structures) on the projective line with the prescribed type of regular singularities at prescribed real marked points (according to Feigin and Frenkel), while the latter indexes the canonical base in a tensor product of U_q(sl_2)-modules (via the Schechtman-Varchenko isomorphism). Both sets carry a natural action of the cactus group, i.e., the fundamental group of the real Deligne-Mumford space of stable rational curves with marked points (by monodromy of solutions to Bethe ansatz equations on the former and by crystal commuters on the latter). We prove that our bijection is compatible with this cactus group action. This is joint work with Nikita Markarian.

Deep learning has had transformative impacts in many fields including computer vision, computational biology, and dynamics by allowing us to learn functions directly from data. However, there remain many domains in which learning is difficult due to poor model generalization or limited training data. We'll explore two applications of representation theory to neural networks which help address these issues. Firstly, consider the case in which the data represent an $G$-equivariant function. In this case, we can consider spaces of equivariant neural networks which may more easily be fit to the data using gradient descent. Secondly, we can consider symmetries of the parameter space as well. Exploiting these symmetries can lead to models with fewer free parameters, faster convergence, and more stable optimization.

High-dimensional function approximation suffers from the "curse of dimensionality": an exponential dependence on the dimension of the function's domain in an algorithm's runtime. For problems with large dimensions (e.g., >10), approximation based on standard tensor-product-like approaches of one-dimensional theory can be intractable.
$\\$
In this talk, we present an alternative algorithm that by passes the computational curse of dimensionality, so long as the function of interest has a sparse Fourier series (or at least can be well approximated by one). Additionally, we show a new way to use this high-dimensional sparse Fourier transform in a classic application of Fourier transforms: spectral methods for solving PDE. Our method allows us to solve extremely high-dimensional (>1000) and multiscale elliptic PDE.
$\\$
This is the practice for my dissertation defense! Please come, give feedback, and help make my presentation better!
$\\$
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 .

"Nice spaces", say smooth Riemannian manifolds, contain a great deal of geometric structure. We can examine their tangent and normal bundles. We can look at their commutative algebra of smooth functions, which are dense in the commutative algebra of continuous functions. In fact, from the continuous functions (which are a commutative C*-algebra) we can recover the topology of the manifold. The primary motivation of noncommutative geometry is to extend this notion of commutative duality to the noncommutative setting. To find an analog of De Rham cohomology. To find suitable definitions of differential forms and multivector fields on a noncommutative space. A natural class of noncommutative spaces to investigate are coarse spaces, which are dual to uniform Roe algebras.
In this talk we will define uniform Roe algebras and uniform Roe modules and look at a few examples. We will also define Hochschild (co)homology which will be our analog for differential forms and multivector fields. Lastly, we will discuss current advancements in our understanding of the Hochschild cohomology of uniform Roe algebras.

This talk is centered around a symplectic approach to eigenvalue problems for systems of ordinary differential operators (e.g., Sturm-Liouville operators, canonical systems, and quantum graphs), multidimensional elliptic operators on bounded domains, and abstract self-adjoint extensions of symmetric operators in Hilbert spaces. The symplectic view naturally relates spectral counts for self-adjoint problems to the topological invariant called the Maslov index. In this talk, the notion of the Malsov index will be introduced in analytic terms and an overview of recent results on its role in spectral theory will be given.

The skein algebra of a marked surface admits the basis of bracelet elements constructed by Fock-Goncharov and Musiker-Schiffler-Williams. As a cluster algebra, it also admits the theta basis from the cluster scattering diagram by Gross-Hacking-Keel-Kontsevich. In a joint work with Travis Mandel, we show that the two bases coincide except for the once-punctured torus. Our results extend to quantum cluster algebras with coefficients arising from the surface even in punctured cases. Long-standing conjectures on strong positivity and atomicity follow as corollaries. We also connect our results to Bridgeland's stability scattering diagrams.

Regular Hessenberg varieties are subvarieties of the flag variety with important connections to representation theory, algebraic geometry, and combinatorics. Like Schubert varieties, the structure of Hessenberg varieties can be studied using the combinatorics of the symmetric group, but there are also other combinatorial parameters to incorporate. In this talk, we give a criterion for identifying singular points in a particular collection of regular Hessenberg varieties. Our results lead to a classification of all singular Hessenberg varieties in this collection. This talk is based on joint work with Erik Insko and Alex Woo.

I will present an upcoming work with J. Luk (Stanford), where we develop a general method for understanding the late time tail for solutions to wave equations on asymptotically flat spacetimes with odd spatial dimensions, which is applicable to nonlinear problems on dynamical backgrounds. In addition to its inherent interest, such information is crucial for studying problems involving the interaction of waves with a spatially localized object; indeed, our motivation for developing this method comes from the Strong Cosmic Censorship Conjecture. I will explain how our method recovers and refines Price's law for linear problems on stationary backgrounds, and also how it shows that the late time tails are in general different(!) from the linear stationary case in the presence of nonlinearity and/or a dynamical background.

In this talk, I present a rational homotopy group and and its construction using minimal models given by Sullivan.
After briefly describing Sullivan's theorem, I will consider the specific example of S^n v S^n and compute few low-dimensional rational homotopy groups.
In the second part, I introduce the Novikov integral formula for the rational homotopy group and provide analytic bound obtained from the integral formula.
The talk will end with specific examples.

The principle of ‘maximum entropy’ states that the probability distribution which best represents the current state of knowledge about a system is the one with largest entropy with respect to a given prior (data) distribution. It was first formulated in the context of statistical physics in two seminal papers by E. T. Jaynes (Physical Review, Series II. 1957), and thus constitutes an information theoretic manifestation of Occam’s razor. We bring the idea of maximum entropy to bear in the context of linear inverse problems in that we solve for the probability measure which is close to the (learned or chosen) prior and whose expectation has small residual with respect to the observation. Duality leads to tractable, finite-dimensional (dual) problems. A core tool, which we then show to be useful beyond the linear inverse problem setting, is the ‘MEMM functional’: it is an infimal projection of the Kullback- Leibler divergence and a linear equation, which coincides with Cramér’s function (ubiquitous in the theory of large deviations) in most cases, and is paired in duality with the cumulant generating function of the prior measure. Numerical examples underline the efficacy of the presented framework.
The talk encompasses joint work with Rustum Choksi (McGill), Ariel Goodwin (McGill), Carola-Bibiane Schönlieb (Cambridge), and Yakov Vaisbourd (McGill).