We survey some recent observations and ongoing work motivated by a hope to better understand p-adic K-theory. More specifically, we discuss arithmetic problems—and potential approaches—related to syntomic cohomology in positive and mixed characteristics. At the level of the structure sheaf, syntomic cohomology is an 'intelligent version' of p-adic étale Tate twists at the characteristic and (among other things) provides a motivic filtration on p-adic étale K-theory via the theory of trace invariants.

Let R be a polynomial ring with m x n many indeterminate over the complex numbers. We can think of the indeterminates as a matrix X of size m x n.  

Consider the group G = Gl(m) x Gl(n). Then G acts on R via the group action (A,B)X =AXB^{-1}. In 1980, DeConcini, Eisenbud, and Procesi introduced the ideals that are invariant under this group action.

In the same paper, they described various properties of those ideals, e.g., associated primes, primary decomposition, and integral closures.  In recent work with Sudipta Das, Tài Huy Hà, and Jonathan Montaño, we described their rational powers and proved that they satisfy the binomial summation formula. In an ongoing work, Alexandra Seceleanu and I are formulating symbolic properties of these ideals. In this talk, I will describe these ideals and the properties we are interested in. I will also showcase some results from my collaborations.

We introduce BiLO (Bilevel Local Operator Learning), a novel neural network-based approach for solving inverse problems in partial differential equations (PDEs). BiLO formulates the PDE inverse problem as a bilevel optimization problem: at the upper level, we optimize PDE parameters by minimizing data loss, while at the lower level, we train a neural network to locally approximate the PDE solution operator near given PDE parameters. This localized approximation enables accurate descent direction estimation for the upper-level optimization. We apply gradient descent simultaneously on both the upper and lower level optimization problems, leading to an effective and fast algorithm. Additionally, BiLO can infer unknown functions within PDEs by introducing an auxiliary variable. Extensive experiments across various PDE systems demonstrate that BiLO enforces strong PDE constraints, is robust to sparse and noisy data, and eliminates the need for manually balancing residual and data loss, a common challenge in soft PDE constraints. We also discuss how to apply the BILO for uncertainty quantification in a Bayesian framework.

In his seminal paper, Ozawa demonstrated the solidity property for ${\rm II}_1$ factor arising from Biexact groups. In this talk, I will discuss a relative version of the solidity property for biexact groups in the setting of measure equivalence and its applications to measure equivalence rigidity. This is a joint work with Daniel Drimbe.

In 1995, Kuperberg introduced a remarkable collection of trivalent web bases which encode tensor invariants of $U_q(\mathfrak{sl}_3)$. Extending these bases to general $\mathfrak{sl}_r$ has been an open problem ever since. We present a solution to the $r=4$ case by introducing hourglass plabic graphs - a new generalization of Postnikov's plabic graphs. Joint work with Christian Gaetz, Oliver Pechenik, Jessica Striker and Joshua Swanson.

For q a prime power, let F_q be the finite field of order q. There are a finite number of isomorphism classes of (smooth, projective, geometrically irreducible) curves of genus g over F_q. Can one give a closed form expression for this number? We discuss how to correctly interpret this question; how to generalize it by also counting marked points; what is known for small g; and what information can be gained by making complete tables of curves of a given genus.

Solving linear systems is a crucial subroutine and challenge in the large-scale data setting. In this presentation, we introduce an iterative method for approximating the solution of large-scale multi-linear systems, represented in the form A*X=B under the tensor t-product. Unlike previously proposed randomized iterative strategies, such as the tensor randomized Kaczmarz method (row slice sketching) or the tensor Gauss-Seidel method (column slice sketching), which are natural extensions of their matrix counterparts, our approach delves into a distinct scenario utilizing frontal slice sketching. In particular, we explore a context where frontal slices, such as video frames, arrive sequentially over time, and access to only one frontal slice at any given moment is available. This talk will present our novel approach, shedding light on its applicability and potential benefits in approximating solutions to large-scale multi-linear systems.
 

A celebrated theorem of Eskin–Mozes–Shah gives an asymptotic counting formula for the number of integral (n x n)-matrices with a prescribed irreducible (over the integers/rationals) integral characteristic polynomial. We obtain a power saving error term for the counting problem for (3 x 3)-matrices. We do this by using the connection to homogeneous dynamics and proving effective equidistribution of translates of tori in SL_3(R)/SL_3(Z). A key tool is that the limiting Lie algebra corresponding to the translates of tori is a certain nilpotent Lie algebra. This allows us to use the recent breakthrough work of Lindenstrauss–Mohammadi–Wang–Yang on effective versions of Shah's/Ratner's theorems. We actually study the phenomenon more generally for any semisimple Lie group which we may discuss if time permits.

We consider scalar field theories on the line with Ginzburg-Landau  (double-well) self-interaction potentials. Prime examples include the  $\phi^4$ model and the sine-Gordon model. These models feature simple  examples of topological solitons called kinks. The study of their = asymptotic stability leads to a rich class of problems owing to the  combination of weak dispersion in one space dimension, low power  nonlinearities, and intriguing spectral features of the linearized  operators such as threshold resonances or internal modes.

We present a perturbative proof of the full asymptotic stability of the sine-Gordon kink outside symmetry under small perturbations in weighted Sobolev norms. The strategy of our proof combines a space-time resonances approach based on the distorted Fourier transform to capture modified scattering effects with modulation techniques to take into account the invariance under Lorentz transformations and under spatial translations. A major difficulty is the slow local decay of the radiation term caused by the threshold resonances of the non-selfadjoint linearized matrix operator around the modulated kink. Our analysis hinges on two remarkable null structures that we uncover in the quadratic nonlinearities of the evolution equation for the radiation term as well as of the modulation equations.

The entire framework of our proof, including the systematic development of the distorted Fourier theory, is general and not specific to the sine-Gordon model. We conclude with a discussion of potential applications in the generic setting (no threshold resonances) and with a discussion of the outstanding challenges posed by internal modes such as in the well-known $\phi^4$ model.

This is joint work with Gong Chen (GeorgiaTech).

In this talk, I will present two recent pieces of research that are connected by the common theme of multiscale models for motile microorganisms. 

In the first part, I will discuss the orientational dynamics of microscopic organisms, such as bacteria, swimming in biofluids with properties that differ from those of isotropic Newtonian fluids, instead exhibiting characteristics of liquid crystals. These environments have a preferred direction, which forces the swimmers to align with it. However, certain types of bacteria can overcome this external torque and swim across the preferred direction. I will present a nonlinear PDE system that couples liquid crystal hydrodynamics with a model of a prototypical microswimmer. This model identifies the conditions for non-trivial reorientation dynamics and allows for deriving the homogenized limit, effectively describing the dynamics of the microswimmer colony. This is the joint work with I. Aronson (PSU), L. Berlyand (PSU), H. Chi (PSU), A. Yip (Purdue U.), and L. Zhang (SJTU). 

In the second part of the talk, I will focus on a computational model that describes how motile cancer cells interact with the extracellular matrix (ECM) during the initial invasion phase, including ECM degradation and mechanical remodeling. The model highlights the role of elastic interactions in the dynamics of cell clusters, including their shapes, sizes, and orientations. These results are joint work with O. Kim (Virginia Tech), Y. Klymenko (Indiana U.), M. Alber (UCR), and I. Aranson (PSU).

If G is a finite group then the collection of all finite dimensional complex representations of G carries two important operations: direct sum and tensor product. A tensor category is an abstraction of this situation. Finding new examples of tensor categories is a very difficult problem. In recent work with Harman, we gave a general construction of tensor categories based on oligomorphic groups, a class of infinite permutation groups best known in model theory. I will give an overview of our work.

The Chern filtration is a natural filtration that can be defined on the cohomology of moduli spaces of sheaves. Its definition was originally made for the moduli of Higgs bundles, motivated by a comparison with the perverse and weight filtrations, but it also makes sense for the very classical moduli spaces of bundles on curves. A vanishing result conjectured by Newstead and proved by Earl-Kirwan in the 90s is secretly a statement about the Chern filtration. I will explain a new approach to this vanishing which is based on parabolic bundles: it turns out that enriching the problem with a parabolic structure gives access to powerful tools, such as wall-crossing, Hecke transforms and Weyl symmetry — together, these give a new proof of the Newstead-Jefrey-Kirwan vanishing and a related "d independence" statement. Part of the talk is based on work with W. Lim and W. Pi.