The prime spectrum of a commutative ring is the underlying set of prime ideals of the ring together with the Zariski topology. A theorem proven by Reyes states that any extension of the set-valued prime spectrum functor on the category of commutative rings to the category of (not necessarily commutative) rings must assign the empty set to the n by n matrix algebra with complex entries when n is greater than 2. This suggests that sets do not serve as the underlying structure of a spectrum of a noncommutative ring. It is argued in his recent paper that coalgebras can be viewed as the underlying object of a noncommutative spectrum.

In this talk, we introduce coalgebras equipped with a ringed-space-like structure, which we call ringed coalgebras. These objects arise from fully residually finite-dimensional (RFD) algebras and schemes locally of finite type over a field k. The construction uses the Heyneman--Sweedler finite dual coalgebra and the Takeuchi underlying coalgebra. We will discuss that, if k is algebraically closed, the formation of ringed coalgebras gives a fully faithful functor out of the category of fully RFD algebras, as well as a fully faithful functor out of the category of schemes locally of finite type. The restrictions of these two functors to the category of (commutative) finitely generated algebras are isomorphic. In this way, ringed coalgebras can be thought of as a generalization of RFD algebras and schemes locally of finite type.

I show that the unitary group of any SOT-separable II_1 factor M, with the strong operator topology, is contractible. Combined with several old results, this implies that the same is true for any SOT-separable von Neumann algebra with no type I_n direct summands (n < infinity). The proof for the II_1-factor case uses regularization via free convolution and Popa's theorem on the existence of approximately free Haar unitaries in II_1 factors.  I will also explain some of the bigger picture of the free probability ingredients.

Mixed finite element methods are widely used in numerical partial differential equations. By introducing an auxiliary variable, a second-order PDE can be rewritten as a first-order linear system, enabling more robust discretizations than the original formulation. However, such discretizations typically rely on discrete inf–sup conditions for the finite element spaces, which can be difficult to verify. A remedy is the T-coercivity approach: construct a bijective linear operator T that transforms the saddle point problem into a coercive one. In this talk, I will present a T-coercive framework for designing stable mixed finite elements for nonlocal saddle-point problems, along with convergence theory and numerical experiments. We establish convergence as the nonlocal horizon tends to zero and/or as the discretization parameter vanishes.

The Horton-Strahler number (HS) is a measure of branching complexity of rooted trees, introduced in hydrology and later studied in parallel computing under the name register function. I consider this statistic for butterfly trees -- binary trees constructed from butterfly permutations, a rich class of separable permutations with origins in numerical linear algebra. I establish a central limit theorem for HS numbers of butterfly trees -- a result that has remained elusive for standard rooted planar models.

Let $C$ be a smooth, projective, geometrically integral hyperelliptic curve of genus $g \geq 2$ over a number field $k$. To study the distribution of degree $d$ points on $C$, we introduce the notion of $\mathbb{P}^1$- and AV-parameterized points, which arise from natural geometric constructions. These provide a framework for classifying density degree sets, an important invariant of a curve that records the degrees $d$ for which the set of degree $d$ points on $C$ is Zariski dense. Zariski density has two geometric sources: If $C$ is a degree $d$ cover of $\mathbb{P}^1$ or an elliptic curve $E$ of positive rank, then pulling back rational points on $\mathbb{P}^1$ or $E$ give an infinite family of degree $d$ points on $C$. Building on this perspective, we give a classification of the possible density degree sets of hyperelliptic curves.

The motion of particles is influenced by various physical effects. One of the most challenging problems in particle dynamics is forcing inference, which requires determining the unknown forcing function from measured data, such as particle trajectories or flow observations. If the forcing function can be determined accurately, it reveals the physical effects that dominate the particles' motion.

In this talk, we formulate the forcing inference problem as an optimization problem. The cost function measures the difference between the measured and simulated particle distributions. The constraints are expressed by both the particle dynamic equations and characteristic ODEs. To update the parameters representing the forcing function, we use a gradient-based method. During this process, we derive the gradient of the cost function using the adjoint method to avoid the heavy computation involved in directly calculating derivatives. This involves constructing the Lagrangian function and deriving the corresponding adjoint equations. Numerical experiments verify the effectiveness of the proposed adjoint-based method.

Randomized Kaczmarz (RK) is a well-known solver for linear least-squares problems. RK iteratively processes blocks of rows in order to update an approximation to the least-squares solution. Recent work suggests that RK converges rapidly when each block of rows is sampled from the square volume distribution defined by the target matrix. Additionally, there are reports of accelerated convergence when the RK iterates produced in the tail part of the algorithm are averaged together. I will clarify the theoretical convergence guarantees for square volume sampling Kaczmarz both with and without tail-averaging.

We study how data geometry shapes generalization in overparameterized neural networks. The analysis focuses on solutions reached under stable training dynamics and the induced, data-dependent form of regularization. We link capacity to geometric features of the input distribution. This view explains when training prefers shared representations versus memorization. We present a decomposition based on depth-type notions to separate regions where learning is data-rich from regions where activation is scarce. For the uniform distribution on the ball, the framework predicts the curse of dimensionality. For mixtures supported on low-dimensional subspaces, it predicts adaptation to the intrinsic dimension. Experiments on synthetic data and MNIST support these trends. The results provide a unified account of how stability and geometry interact to govern effective capacity of GD-trained neural networks.

Toric varieties are the quintessential connection between algebraic geometry and combinatorics. Projective toric varieties are compactifications of the algebraic torus for which the volume form has poles at every boundary divisor. In this talk, we will introduce a new class of projective varieties: cluster type varieties. These are compactifications of the algebraic torus for which the volume form has no zeros at the boundary divisor. We will explain how to understand these varieties from the perspective of birational geometry, together with some applications of this perspective. Time permitting, we will explain some connections with varieties coming from combinatorics.