Department Seminar

Vertex operator algebras are the chiral algebras of two-dimensional conformal field theory. Their representations are closely related to algebraic curves. Recently, Damiolini–Gibney–Tarasca proved that representations of rational and C₂-cofinite VOAs give rise to vector bundles over the moduli space of stable genus-g algebraic curves with n marked points, generalizing the Verlinde bundles defined by representations of affine Lie algebras (affine VOAs) at positive integral levels. Moreover, these bundles are globally generated if the VOA is generated in degree one. In this talk, we will discuss a generalization of this construction to nonrational and C₂-cofinite VOAs and give a criterion for global generation beyond VOAs generated in degree one. This talk is based on joint work with Viktor Alekseev and Avik Chakravarty.

Symmetries are fundamental for our understanding of geometry. In complex geometry, the symmetries are the holomorphic automorphisms. In this talk I would like to focus on complex manifolds with very large (i.e. infinite-dimensional) groups of holomorphic automorphisms, and present some of my own research in this area. The talk is aimed at a general audience of mathematical researchers.

We introduce a class of good endofunctors of C*-algebras, endow it with a structure of a bimonoidal category, and define homotopies of natural transformations between such endofunctors. For every pair of C*-algebras and a good endofunctor, we construct a commutative monoid of generalized morphisms, and endow it with a bilinear composition. This construction generalizes the homotopy category of asymptotic homomorphisms used in the definition of the Connes–Higson E-theory. We also introduce the notion of asymptotically adjoint good endofunctors, which has interesting applications to E-theory and K-homology.

The Markoff equation x² + y² + z² = 3xyz is a classical Diophantine equation whose integer solutions encode optimal approximation properties of irrational numbers. In this talk, we review the structure of Markoff triples, their generation via Vieta involutions, and their relation to Diophantine approximation.

We then describe connections with hyperbolic geometry via trace identities in SL₂(R), and outline more recent developments linking Markoff-type structures to the Painlevé VI equation, double affine Hecke algebras, and cluster algebras.

This is a talk aimed at second year undergraduate students in applied mathematics. I will introduce finite groups and their representations and their character tables. Then I will explain how to classify all representations of the symmetric group. Finally I will discuss what a field with one element is and use that to show why the symmetric group has some analogies with the matrix group GL_n.

I aim to give an accessible and entertaining introduction to Morse theory. Morse theory allows mathematicians to analyze the topology of higher-dimensional objects using ideas from analysis and forms the basis for some of the most active areas of modern pure math research.

This talk will provide an expository overview of Kleinian (ADE) surface singularities, their commutative and noncommutative deformations, and the role of quiver varieties in their geometric representation theory. If time permits, I will also describe our recent results on the Morita equivalence of noncommutative deformations of Kleinian singularities.

This seminar introduces my two research areas briefly as follows:

1. A statistical model for information retrieval that competed at TREC (Text Retrieval Conference) is introduced. It is a Bayes classification model based on the multinomial distribution, approximated by a maximum-entropy distribution under partial information constraints. An iterative proportional fitting algorithm utilizing the cross-entropy measure is proposed to estimate the maximum entropy distribution.

2. The web-based free software system at www.estat.me, eStat, supports teaching Statistics and Data Science. The eStat not only has web data processing capability but also numerous simulation modules to support teaching. It contains data visualization, statistical distributions, confidence intervals, hypothesis testing, analysis of variance, nonparametric tests, regression analysis, forecasting, supervised learning, and unsupervised learning models. The eStat is linked to a web-book system with embedded modules, which include many examples, presentation files, and lecture videos. Data Science modules are also supported by R and Python commands for practice.

In the past two decades 3d N=4 mirror symmetry has been widely studied by physicists and mathematicians. It is also closely related to symplectic duality proposed by Braden et al. One of the conjectures in this field, initiated by Hikita, says that the cohomology of one symplectic manifold is isomorphic to the coordinate ring of the torus-fixed scheme of its symplectic dual. In this talk we show a quantum version of Hikita’s conjecture between Kleinian singularities and the corresponding Slodowy slices. This talk is based on a joint work with He and Yu from Sun Yat-sen University.

An ordered regular semigroup S is said to be principally ordered if, for every element x in S, there exists x* = max{ y in S : xyx ≤ x }.

We begin by reviewing basic concepts from semigroup theory and discussing the origins of principally ordered regular semigroups. Several examples and fundamental properties will illustrate this class of ordered semigroups.

In particular, we show that in a principally ordered regular semigroup S, the map x ↦ x* is antitone (that is, for all x, y in S, if x ≤ y then y* ≤ x*) if and only if S is naturally ordered. The latter means that for every pair of idempotents e and f in S satisfying ef = e = fe, we have e ≤ f.

We also explore the existence and significance of a greatest idempotent in a principally ordered regular semigroup.

Finally, we present recent results for principally ordered regular semigroups S satisfying the additional condition that, for all idempotents e in S, we have e = e*.

For an associative algebra A, the Loday–Quillen–Tsygan isomorphism relates the cyclic homology of A and the primitive part of the Lie algebra homology of the infinite matrices gl(A). The main goal here is to study a deformation and quantization of this isomorphism.

We show that if A is a Koszul Calabi–Yau algebra, then the primitive part of this Lie algebra homology has a Lie bialgebra structure, which is induced from the Poincaré duality of A, and quantizes this Lie algebra homology to a Hopf algebra. The Loday–Quillen–Tsygan isomorphism lifts to the quantum level, which can be interpreted as a quantization of the tangent map from the tangent complex of BGL to the tangent complex of K-theory.

In this talk, we first explain various models of reaction-diffusion systems characterized by high competition rates. We investigate the existence and uniqueness of solutions for each model and the numerical approximation of their singular limits. Building on insights from these systems, we explore graph-based semi-supervised learning that utilizes properties of these Partial Differential Equations (PDEs) to classify data when only limited labels are available. Furthermore, we enhance the accuracy of several well-known algorithms, adapting them to handle large datasets with sparse labeled data more effectively.

This will be a report on my joint project with Mehrzad Ajoodanian, who gave an introduction on it on 17 October. We will give a fresh look at why one should think of curvature as a measure of non-commutativity using a Lie algebra and not emphasising on a metric. I will try to make it understandable to everyone. Students are especially welcome.

The finite element method (FEM) is one of the most powerful tools for solving partial differential equations; however, its implementation can become slow and cumbersome, especially in high-level languages like MATLAB or Python. This talk demonstrates how vectorization—replacing explicit loops with compact array operations—enables faster and cleaner implementations of FEM assemblies. We will examine how this approach accelerates the computation of stiffness and mass matrices, and it extends naturally to nonlinear models, including the two-phase obstacle problem with a posteriori error control. I will also touch on Poincaré-type inequalities for operators involving grad, curl, and div, illustrating how computational efficiency and mathematical structure can go hand in hand.

This is a talk for the general audience. It is another application of non abelian gauge theory to study vector bundles on Riemann surfaces (algebraic curves). This is a report of a joint work with Amir Jafari.

In this talk we visit Frobenius theorem, its complex counterpart, and visit some applications to Arts.

Since their inception by Berenstein, Fomin, and Zelevinsky in the early 2000s, cluster algebras have emerged as a unifying structure connecting diverse areas of mathematics and physics. These include mathematical physics, combinatorics, representation theory, algebraic geometry, Teichmüller theory, differential equations, number theory, knot theory, and symplectic geometry. Remarkably, cluster algebras are also computationally accessible due to their rich combinatorial framework. This interplay between theory and explicit computations makes cluster algebras a rewarding subject for both mathematicians and physicists. In this talk, we will introduce the basic ideas of the theory.

I will describe recent progress in understanding instanton equations on hyperkähler manifolds. This project is motivated by the observation that the deformed Sp(n)-instanton equation arising from the real Fourier–Mukai transform of a complex Lagrangian graph coincides with the classical Sp(n)-instanton equation. Results include new relationships between gauge-theoretic equations on hyperkähler cones, their 3-Sasakian links, and various dimensional reductions, as well as a compact and non-compact Lewis-type theorem. Lastly, I will present explicit examples of Sp(2)-instantons, Hermitian Yang–Mills connections, and Spin(7)-instantons in the cohomogeneity-one setting of the Calabi manifold T*CP².

This talk is based on joint work with Jesse Madnick and Izar Alonso:

The classification of tuples of matrices up to simultaneous conjugation is a classical problem in linear algebra and representation theory. While the case of a single matrix leads to the well-known Jordan normal form, the situation becomes far more intricate for tuples, where the problem is considered “wild” and therefore not classifiable in any reasonable sense. Geometric Invariant Theory (GIT) provides a powerful framework for studying such spaces through quotient varieties, which parametrize semisimple representations of free associative algebras. In this talk, after briefly reviewing recent results on invariants of 4×4 matrix pairs—with applications to the invariant commuting variety and the fourth Calogero–Moser space—the focus will shift to the GIT quotient of the space of anti-commuting matrix pairs. This variety exhibits a rich and highly reducible geometric structure and differs in significant ways from the commuting variety. The algebra of invariants on the anti-commuting variety will be studied, along with a general generating set, proven to be minimal for matrix sizes up to five. An algorithmic method for deriving the defining relations among these generators will be presented, and using computational tools such as Macaulay2, these relations are verified case by case, thereby providing a complete description of the invariant ring for matrices of size up to five.

Swarms of unmanned aerial vehicles (UAVs) often operate in dynamic environments in which given information changes over time and new information only becomes available locally. This necessitates the development and use of mathematical models and corresponding distributed computational frameworks that take uncertainty into account. In addition, most UAVs do not operate fully autonomously: they have to communicate with each other or with a group of remote pilots. This increased complexity is reflected in additional decisions that have to be taken with respect to the usage of the electromagnetic spectrum (EM) for communication by the UAVs. In this talk we discuss the corresponding challenges and opportunities and provide examples for such mathematical models in the realm of defense. In defense, judicious use of EM resources is particularly important, as adversaries and own forces will attempt to intercept, spoof, and jam electronic communications in a highly dynamic and rapidly evolving environment.