Mon, May 4, 2026

Algebra Seminar

Xingting Wang, Louisiana State University

Quantum-symmetric equivalence and deformations of connected graded algebras

4:00 PM, 250 Math building

We introduce the notion of quantum-symmetric equivalence of two connected graded algebras, based on Morita–Takeuchi equivalences of their universal quantum groups in the sense of Manin. We give away of deforming a connected graded algebra which preserves its quantum-symmetric equivalence class. We study ring-theoretic and homological properties of connected graded algebras that are invariant under quantum-symmetric equivalence. In particular, by combining our results with the work of Raedschelders and Van den Bergh, we prove that Koszul Artin–Schelter regular algebras of fixed global dimension form a single quantum-symmetric equivalence class.

Fri, May 1, 2026

Applied Mathematics Seminar

Applied Math Seminar (Sayantan Sarkar)

From Video to PDE: Data-Driven Discovery and Latent Linear Structure in Plume Dynamics

4:00 PM, MATH 250

Extracting governing equations from raw video data poses a significant challenge in scientific machine learning. This talk presents a sparse identification framework for learning partial differential equations from observations of dye plumes. By combining Weak-form Sparse Identification of Nonlinear Dynamics  (Weak-SINDy)  with forward-rollout error minimization, a predictive transport model is developed. The implementation addresses noise mitigation, drift decoupling, and robust parameter estimation using inverse Physics-Informed Neural Networks ( iPINNs ) to aid in the identification of unknown coefficients using the  Bootstrap method . Notably, the nonlinear transport law can be transformed into a linear advection-diffusion equation, revealing that the apparent nonlinearity arises from the image-derived observable rather than from intrinsic physics, thereby highlighting latent linear structures in visual data.

Thu, Apr 30, 2026

Colloquium

Daniel Litt, University of Toronto

Braid groups and 2x2 matrices

4:00 PM, 250 Mathematics Building

Let \(X_n\) be the set of conjugacy classes of \(n\)-tuples of 2\(\times\)2 matrices, whose product is the identity matrix. There is a natural braid group action on \(X_n\), whose study goes back to work of Markov in the late 19th century. The most basic question one can ask about this action, which dates to work of Painlevé, Fuchs, Schlesinger, and Garnier in the beginning of the 20th century, is: what are its finite orbits? I’ll explain the history of this question, as well as some recent work (combining results of Bronstein-Maret with results obtained jointly with Josh Lam and Aaron Landesman), which answers it completely. Time permitting, I’ll discuss "higher genus" variants of this question, whose answer relies on non-abelian Hodge theory and the Langlands program, and resolves conjectures of Esnault-Kerz, Budur-Wang, Kisin, and Whang.

Mon, Apr 27, 2026

Algebra Seminar

Michael Brannan, University of Waterloo

Quantum symmetries of Hadamard matrices

4:00 PM, Mathematics Building

A Hadamard matrix is a square matrix of complexnumbers whose entries have modulus 1, and whose columns are pairwiseorthogonal. In this talk I will discuss various notions of quantum symmetriesand equivalences of Hadamard matrices using ideas from quantum group theory.These symmetries, in the case of finite order (i.e., Butson-type) Hadamardmatrices, turn out to have interesting applications in the theory of non-localgames from quantum information theory. This is joint work with Daniel Gromada,Roberto Hernandez-Palomares, and Nicky Priebe.

Fri, Apr 24, 2026

Topology and Geometry Seminar

Brandis Whitfield (University of Wisconsin-Madison/SLMath)

Constructing reducibly geometrically finite subgroups of the mapping class group

4:00 PM, 122 Mathematics Building

A longstanding goal within low-dimensional geometry and topology is to establish parallels between the theory of Kleinian groups and subgroups of the mapping class group. Convex-cocompact subgroups of the mapping class group have been well-studied in the past few decades; current research aims to extend this analogy to the more general class of geometric finiteness. Dowdall–Durham–Leininger–Sisto introduce parabolically geometrically finite (PGF) subgroups of the mapping class group whose coned-off Cayley graphs, with respect to a collection of twist subgroups, quasi-isometrically embed into the curve complex. In joint work with Aougab, Bray, Dowdall, Hoganson and Maloni, we extend this definition to allow the coned-off subgroups to be more generally reducible, and produce plentiful examples of this behavior. Our examples include free products of reducible subgroups and the beloved right-angled Artin constructions of Clay–Leininger–Mangahas.

Thu, Apr 23, 2026

Colloquium

Brandis Whitfield (University of Wisconsin-Madison/SLMath)

The geometries of subgroups of the mapping class group

4:00 PM, 250 Mathematics Building

Mapping class groups of surfaces play a prominent role in understanding the geometry and topology of three-manifolds. These connections have inspired decades-long research programs toward a notion of geometric finiteness for mapping class subgroups. In this talk, we’ll introduce the mapping class group, discuss methods of embedding different groups into it and how topological properties of a mapping class determine geometric data of its associated 3-manifold.

Mon, Apr 20, 2026

Algebra Seminar

Douglas Rizzolo, University of Delaware

Ordered Chinese Restaurant Process up-down chains

4:00 PM, 250 Mathematics Building

Up-Down chains on branching graphs provide an interesting link between the algebraic structure of branching graphs and stochastic processes on the boundaries of these graphs. Up-down chains on branching graphs whose vertices are given by partitions of an integer are well understood, but up-down chains on branching graphs whose vertices are compositions of an integer have only begun to be studied recently. In this talk we will discuss up-down chains on graphs of compositions whose up-steps are based on the Ordered Chinese Restaurant Process. We will show how these can be used to construct diffusions on the boundary of this graph whose generators have simple expressions in terms of quasi-symmetric functions.  We will give examples showing what sorts of new statistics can be understood based on the order structure, with an emphasis on a connection to phylogenetics where the order structure can be interpreted as the relative ages of alleles in a population.

Fri, Apr 17, 2026

Applied Mathematics Seminar

Lili Ju (U of South Carolina)

Transferable Neural Networks for Partial Differential Equations

4:00 PM, MATH250

Transfer learning for partial differential equations (PDEs) aims to develop a pre-trained neural network that can be used to solve a wide class of PDEs. Existing transfer learning approaches require much information about the target PDEs such as its formulation and/or data of its solution for pre-training. In this work, we propose to design transferable neural feature spaces for the shallow neural networks from purely function approximation perspectives without using PDE information. The construction of the feature space involves the re-parameterization of the hidden neurons and uses auxiliary functions to tune the resulting feature space. Theoretical analysis shows the high quality of the produced feature space, i.e., uniformly distributed neurons. We use the proposed feature space as the predetermined feature space of a random feature model and use existing least squares solvers to obtain the weights of the output layer. Extensive numerical experiments verify the outstanding performance of our method, including significantly improved transferability, e.g., using the same feature space for various PDEs with different domains and boundary conditions, and the superior accuracy, e.g., several orders of magnitude smaller mean squared error than the state-of-the-art methods. Finally, we discuss ongoing and future research topics along this direction.

Wed, Apr 15, 2026

Analysis Seminar

Rizwanur Khan, University of Texas at Dallas

Gaussian Behavior of Eisenstein Series

3:30 PM, 122 Mathematics building

Understanding the behavior of Laplace eigenfunctions inthe high-energy limit is a central question in analysis, with specialsignificance for number theorists when the underlying manifold is arithmetic,such as the modular surface. I will discuss recent work (joint with GoranDjanković) on the distribution of the continuous spectrum—namely, theEisenstein series—on the modular surface. We provide some rigorous evidence(through calculation of the fourth moment) that these functions exhibitGaussian random behavior, consistent with Berry’s Random Wave Conjecture andthe numerical observations of Hejhal and Rackner.

Fri, Apr 10, 2026

Applied Mathematics Seminar

Yulong Lu (U Minnesota)

In-Context Learning in Scientific Computing

4:00 PM, MATH 250

Transformer-based foundation models, pre-trained on large datasets spanning a wide range of tasks, have shown remarkable adaptability to diverse downstream applications—even in low-data regimes. A particularly striking capability is  in-context learning (ICL) : when given a prompt containing a few examples from a new task alongside a query, these models can produce accurate predictions without any parameter updates. This emergent behavior is often viewed as a paradigm shift for transformers, yet its theoretical foundations remain only partially understood. In this talk, I will present recent theoretical progress toward understanding ICL in  scientific computing . I will focus on understanding how transformer architectures can implicitly perform task adaptation in three representative problem classes: learning solution operators of PDEs, dynamical system prediction and generative modeling.

Fri, Apr 10, 2026

Topology and Geometry Seminar

Roberta Shapiro (University of Michigan)

Geometry, topology, and combinatorics of fine curve graphs

4:00 PM, 122 Mathematics Building

The fine curve graph of a surface is a graph that encodes information about curves on a surface and their interaction. This is similar to the more classical curve graph, which encodes information about the isotopy classes of curves on a surface. In this talk, we construct both graphs and compare and contrast some properties, such as the groups that act on them, their geometry, their topology, and their combinatorics. Some shared results will be work joint with Ryan Dickmann, Zachary Himes, and Alex Nolte.

Wed, Apr 8, 2026

Analysis Seminar

Jakob Streipel, SUNY at Buffalo

Second moments and zero-density estimates

4:00 PM, 250 Mathematics building

The Riemann Hypothesis for an L-function says that its nontrivial zeros lie on a certain line. If this were true, it simplifies lots of applications of these L-functions simply because a certain part of the calculation now becomes fixed instead of potentially variable. We don’t knowhow to prove the Riemann Hypothesis for any of the L-functions we study, but sometimes for certain applications or computations the second best thing suffices: knowing that, in some quantifiable way, at most very few of the zeros aren’t on the line where they ought to be. That way, even though there might be outliers, they are rare enough to not ruin the overall picture too much. Such a result is known as a zero-density estimate. This talk will be about zero-density estimates in general, how one might find them, and how second moments of L-functions are a natural way to do so. In doing so we will discuss recent work, joint with Sheng-Chi Liu, in which we compute a zero-density estimate for the L-functions associated with GL(2)Hecke–Maass cusp forms.

Mon, Mar 30, 2026

Algebra Seminar

Dmytro Matvieievskyi, UMass Amherst

Symplectic duality, nilpotent cones, and fixed points

4:00 PM, 250 Mathematics Building

Conical symplectic singularities provide a large class of Poisson varieties that give a geometric framework to study representation theory of many interesting non-commutative algebras. An important topic, extensively studied in the last 15 years, is the phenomenon of symplectic duality, establishing a connection between geometric properties of two different symplectic singularities. Lately, a systematic study of a more specific class of symplectic singularities was initiated. Namely, assume that a conical symplectic singularity \(X\) has a Poisson involution, or more generally, an action of a finite group of Poisson automorphisms. Then the fixed points with respect to this involution constitute a symplectic singularity, playing an important role in studying representation theory of algebras, such as twisted current Lie algebras, or \(i\)-quantum groups. The relation of taking fixed points to symplectic duality is, however, unclear. In my talk I will show how symplectic duality and the fixed-points construction naturally appear inside the nilpotent cones of semisimple complex groups, and what that suggests about the symplectic dual to the fixed-points construction.

Fri, Mar 13, 2026

Topology and Geometry Seminar

Melissa Zhang (UC Davis)

Skein lasagna modules with 1-dimensional inputs

4:00 PM, 122 Mathematics Building

In this talk I will describe joint work with Qiuyu Ren, Ian Sullivan, Paul Wedrich, and Michael Willis, where we define a new version of Morrison–Walker–Wedrich’s skein lasagna modules by replacing the input balls with neighborhoods of 1-complexes.   This version is more amenable to computations for 4-manifolds with 1-handles. The strategy is to use an isomorphism discovered in previous joint work with Ian Sullivan, where we related the skein lasagna module of \(S^2 \times D^2\) to Rozansky–Willis homology, a version of Khovanov homology for links in connected sums of \(S^2 \times S^1\).

Mon, Mar 9, 2026

Algebra Seminar

Michaela Vancliff, University of Texas at Arlington

Generalizing classical Clifford algebras, graded Clifford algebras and their associated geometry

4:00 PM, 250 Mathematics Building

Graded Clifford algebras are non-commutative graded algebras related to classical Clifford algebras, and certain properties of such an algebra can be deduced from certain commutative geometric data associated to it. In particular, a standard result is that a graded Clifford algebra \(C\) is quadratic and Artin-Schelter regular with Hilbert series equal to that of a polynomial ring if and only if a certain quadric system associated to \(C\) is base-point free. About two decades ago, T. Cassidy and the speaker introduced a generalization of such an algebra, called a graded skew Clifford algebra, and they found that many results concerning graded Clifford algebra shave analogues in the case of graded skew Clifford algebras, provided the appropriate non-commutative geometric data is defined. More recently, T.Cassidy and the speaker defined a "skew" version of classical Clifford algebras, and related such algebras to graded skew Clifford algebras. Indeed,just as (classical) Clifford algebras are the Poincaré-Birkhoff-Witt (PBW) deformations of exterior algebras, skew Clifford algebras may be viewed as \(\mathbb{Z}_2\)-graded PBW deformations of quantum exterior algebras.

Sat, Feb 28, 2026

Algebra Seminar

Title: TBD

Time: TBD, Room: TBD

Fri, Feb 13, 2026

Applied Mathematics Seminar

Di Qi (Purdue University)

Reduced-order data assimilation models for predicting probability distributions of multiscale turbulent systems

4:00 PM, MATH 250

A new strategy is presented for the statistical forecasts of multiscale nonlinear systems involving non-Gaussian probability distributions. The capability of using reduced-order models to capture key statistical features is investigated. A closed stochastic-statistical modeling framework is proposed using a high-order statistical closure enabling accurate prediction of leading-order statistical moments and probability density functions in multiscale complex turbulent systems. A new efficient ensemble forecast algorithm is developed dealing with the nonlinear multiscale coupling mechanism as a characteristic feature in high-dimensional turbulent systems. To address challenges associated with closely coupled spatio-temporal scales in turbulent states and expensive large ensemble simulation for high-dimensional complex systems, we introduce efficient computational strategies using the random batch method. Effective nonlinear ensemble filters are developed based on the nonlinear coupling structures of the explicit stochastic and statistical equations, which satisfy an infinite-dimensional Kalman-Bucy filter with conditional Gaussian dynamics. It is demonstrated that crucial principal statistical quantities in the most important large scales can be captured efficiently with accuracy using the new reduced-order model in various dynamical regimes of the flow field with distinct statistical structures.

Wed, Feb 11, 2026

Analysis Seminar

Didier Lesesvre, Université de Lille

A connection between zeros and central values of \(L\)-functions

4:00 PM, Mathematics Building

\(L\) -functions appear as generating functions encapsulating information about various objects such as Galois representations, elliptic curves, arithmetic functions, modular forms, Mass forms, etc. Studying  \(L\) -functions is therefore of utmost importance in number theory at large. Two of their attached data carry critical information: their zeros, which govern the distributional behavior of underlying objects; and their central values, which are related to invariants such as the class number of a field extension. We will discuss the important conjectures, one concerning the distribution of the zeros and one concerning the distribution of the central values, and explain a general principle that any restricted result towards the first conjecture can be refined to show that most corresponding central value shave the typical distribution predicted by the second conjecture. We will instantiate this general principle for a wide class of  \(L\)- functions, and provide a more detailed discussion in the case of  \(L\) -functions attached to modular forms.

Mon, Dec 8, 2025

Algebra Seminar

Mihai Fulger, University of Connecticut

Infinitesimal successive minima and convex geometry

4:00 PM, 250 Mathematics Building

We introduce infinitesimal successive minima of a line bundle at a point. We define them in terms of base loci and show that they are also the lengths of the largest simplex contained in the generic infinitesimal Newton-Okounkov body (iNObody) of the line bundle at the point.  We characterize when the generic iNObody is simplicial. When the point is sufficiently general, we prove that the body is Borel-shaped, a property inspired by generic initial ideals. In particular, it satisfies simplicial lower bounds and polytopal upper bounds determined by its widths, which are again the infinitesimal successive minima. This is joint work with Victor Lozovanu

Mon, Dec 1, 2025

Algebra Seminar

Mahdi Asgari, Oklahoma State University and Cornell

Some dyadic Hecke algebras

4:00 PM, Mathematics Building

The Iwahori-Hecke algebras (in their various incarnations) are ubiquitous in the representation theory of both finite groups of Lie type, thanks to the works of Lusztig and others, and \(p\)-adic groups,following work of Howe-Moy and Bushnell-Kutzko among many others. In this talk I will focus on these objects in the often overlooked case of \(p=2\) and report on current joint work with Dan Barbasch. I hope to also explain our motivation for this study, which stems from studying certain interesting representations of non-linear cover groups.

Wed, Nov 19, 2025

Analysis Seminar

Alexandru Chirvasitu (UB)

Spectrum incompressibility and continuous commutativity preservers

4:00 PM, Mathematics Building

The Kaplansky-Aupetit question of whether Jordan epimorphisms between unital semisimple Banach algebras can be characterized as linear spectrum-shrinking surjections has spawned a vast literature on adjacent problems having to do with characterizing associative/Jordan morphisms as maps preserving various properties orinvariants. The talk’s central result is to the effect that continuous, commutativity-preserving, spectrum-shrinking maps from \(X\) to the \(n\times n\)matrices are either conjugations or transpose conjugations whenever \(n\ge 3\)and \(X\) is any one of: the general linear, special linear or unitary \(n\times n\) group, the set of semisimple matrices in either of the first two, or the set of \(n\times n\) normal matrices. Such maps in particular automatically preserve spectra, hence the title’s “incompressibility”. The proof leverages among other things the Fundamental Theorem of Projective Geometry, characterizing isomorphisms between lattices of subspaces as those induced by semi-linear isomorphisms.(joint with I. Gogić and M. Tomašević)

Fri, Nov 14, 2025

Applied Mathematics Seminar

Teng Wu (UB, School of Engineering and Applied Sciences)

AI-Empowered Wind and Hurricane Engineering

4:00 PM, MATH 250

Recent advancements in performance-based wind engineering have placed new demands on wind characterization (e.g., duration consideration), aerodynamics modeling (e.g., transient feature) and structural analysis (e.g., nonlinear response). While conventional approaches in computational and experimental wind engineering provide valuable tools to overcome many of these emerging challenges, noticeable increase in use of artificial intelligence (AI) suggests its great promise in facilitating the implementation of performance-based wind design methodology. This talk will discuss state-of-the-art machine learning tools (e.g., knowledge-enhanced deep learning and deep reinforcement learning) that are successfully applied to wind climate analysis, transient aerodynamics, nonlinear structural dynamics, shape optimization and vibration control. The final part of this talk will extend the application of AI tools to enhance the coastal city resilience under hurricane hazards (wind, rain, and surge).

Mon, Nov 10, 2025

Algebra Seminar

Stephen Landsittel, Hebrew University and Harvard

Analytic spread of binomial edge ideals

4:00 PM, 250 Mathematics Building

To an ideal \(J\) in a polynomial ring \(R\) over afield \(\mathbb{K}\) we associate its analytic spread \(\ell(J)\), the dimension of the fiber cone \(F(J)\) of \(J\). When \(J\) is graded and generated in a single degree \(d\), \(F(J)\) is a finite-type \(\mathbb{K}\)-algebra. To a graph \(G\) we associate its binomial edge ideal: \(J_G := \left(x_i y_j - x_jy_i\ :\ (i,j)\text{ is an edge of $G$}\right)\). We will discuss recent work where sharp bounds are given for \(\ell(J_G)\) and we compute the exact value when \(G\) is apseudo-forest. We accomplish this by computing the transcendence degree\(\mathrm{trdeg}_{\mathbb{K}} F(J)\) of the fiber cone over \(\mathbb{K}\).

Mon, Nov 3, 2025

Algebra Seminar

James Upton, UC Santa Cruz

Geometric Iwasawa theory via counting lattice points

4:00 PM, 250 Mathematics Building

Geometric Iwasawa theory studies the variation of class groups in \(\mathbb{Z}_p\)-towers of function fields. A new program initiated by Booher and Cais suggests replacing the role of the class group by (the p-divisible group of) the Jacobian. In this talk, we discuss the p-torsion part of the Jacobian, whose structure is described concretely in terms of certain numerical invariants called a-numbers. For suitably nice towers, we give a formula for the a-number as the number of lattice points in a region of the plane. A careful analysis of this region leads to an Iwasawa-style formula for these numbers. This is joint work with Joe Kramer-Miller, Jeremy Booher, and Bryden Cais.

Wed, Oct 29, 2025

Analysis Seminar

Byung-Jay Kahng (Canisius University)

Manageability of multiplicative partial isometries and quantum groupoids

4:15 PM, 250 Mathematics Department

Since the early attempts at generalizing the Pontryagin duality to the level of non-LCA groups and more recently locally compact quantum groups, the notion of a multiplicative unitary operator has been playing a central role, both in the theory and in constructing examples. In this talk, we will extend the notion to consider multiplicative partial isometries. The conditions include the pentagon equation, but more is needed. By also generalizing Woronowicz’s notion of manageability, it can be shown that the partial isometry determines a dual pair of C*-algebraic quantum groupoids of separable type. We will discuss how this is carried out, but also discuss the limitations of the framework, and possible ways to overcome them to achieve a full generalization.

Mon, Oct 27, 2025

Algebra Seminar

Nathan H. Fox, Canisius University

Mathematical models of genome rearrangement

4:00 PM, 250 Mathematics Building

In biology, a major driver of evolution is mutations in DNA structure. Given two organisms with similar but not identical genetic structure, biologists are interested in quantifying how “related” these organisms are. They are also interested in exploring what sequences of mutations might have led to these organisms. In this talk, I will give an overview of three mathematical models used to study genome rearrangement: the Single Cut or Join, Single Cut-and-Join, and Double Cut-and-Join models. Then,I will frame some of the biological questions in the language of these models and discuss some results. Then, time permitting, I will go into depth on some of the mathematical tools used to analyze these models.

Fri, Oct 24, 2025

Applied Mathematics Seminar

Julia Bernatska (U Connecticut)

Methods of algebraic geometry in application to completely integrable hierarchies

4:00 PM, Room 250

For many soliton-type equations, like the Korteweg-de-Vries (KdV), sine-Gordon, modified KdV, Boussinesq equations, etc. hierarchies of integrable hamiltonian systems were constructed. Each hamiltonian system possesses a spectral curve, and obtaining solutions is closely related to the uniformization problem for the curve. Solutions in terms of multiply periodic functions will be presented, along with a brief introduction to algebraic geometry and methods of constructing these solutions.

Wed, Oct 15, 2025

Analysis Seminar

Kehe Zhu (SUNY Albany)

The Bargmann transform and its applications

4:00 PM, Mathematics Building room 250

The Bargmann transform is an integral operatorthat maps \(L^2\) of the real line unitarily onto the Fock space \(F^2\) of thecomplex plane. Thus it establishes a correspondence between operators on \(L^2\)and those on \(F^2\), and serves as a bridge between real analysis, complexanalysis, harmonic analysis, and functional analysis. I will talk about theaction of the Bargmann transform on several classical operators on \(L^2\),including the Fourier transform, the Hilbert transform, and linear canonicaltransforms. These examples lead to several natural classes of operators on theFock space that were studied by various authors in the past few years,including my recent joint work with Xingtang Dong of Tianjin University inChina.

Fri, Sep 12, 2025

Algebra Seminar

Kay Rülling

Tame cohomology of the structure sheaf in mixed characteristic

4:00 PM, 250 Mathematics Department

This is joint work in progress with Alberto Merici and Shuji Saito. Let  R be a complete discrete valuation ring of mixed characteristic with fraction field K. We show that the tame cohomology of Hübner-Schmidt on smooth K-schemes relative to R of(a twist of) the structure sheaf is ^1-invariant and is a finite R-module up to bounded torsion. This induces a canonical R-lattice in the cohomology of the structure sheaf of smooth proper K-schemes. Hence for example the eigenvalues of an automorphism of X acting on O-cohomolgy are integral over R. If X has a regular model over R and a mild version of resolutions of singularities in mixed characteristic holds, then this lattice would be the cohomology of this regular model. The interesting point is that weget the existence of such a lattice also in case X has no regular model and without using resolutions. To avoid resolutions in mixed characteristic we use classical results by Bartenwerfer and van der Put in rigid geometry.If time allows I can explain how we try to extend the above result to Hodge and de Rham cohomology.

Sun, Sep 7, 2025

Algebra Seminar

Mengwei Hu, Yale

On certain Lagrangian subvarieties in minimal resolutions of Kleinian singularities

Time: TBD, Mathematics Building

Kleinian singularities are quotients of \(\mathbb{C}^2\) by finite subgroups of \(SL_2(\mathbb{C})\). They are in bijection with the ADE Dynkin diagrams via the McKay correspondence. In this talk, I will introduce certain singular Lagrangian subvarieties in the minimal resolutions of Kleinian singularities that are related to the geometric classification of certain unipotent Harish-Chandra \((\mathfrak{g},K)\)-modules. The irreducible components of these singular Lagrangian subvarieties are \(\mathbb{P}^1\)s and \(\mathbb{A}^1\)s. I will describe how they intersect with each other through the realization of Kleinian singularities as Nakajima quiver varieties. Time permitting, we will also discuss the deformations of these singular Lagrangian subvarieties.