A hybrid system is a dynamical system that exhibits both continuous and discrete dynamic behavior. The state of a hybrid system changes either continuously by the flow described by a differential equation or discretely following some jump conditions. A canonical example of a hybrid system is the bouncing ball, imagined as a point-mass, under the influence of gravity. In this talk, we explore the solutions and algorithms to the extensions of this example in 3-dimension where the body of interest is rigid and convex in general. In particular, the solutions utilize the theory of nonsmooth Lagrangian mechanics to derive the differential equations and jump conditions, which heavily depend on the collision detection function. The proposed algorithm called Lie group variational collision integrator is developed using the combination of techniques and knowledge from variational collision integrators and Lie group variational integrators. If time permits, we can discuss how some of these techniques can be used in optimal control and robotics.

A hyperoperation on a set M is an operation that associates to each pair of elements a subset of M. Hypergroups and hyperrings are two examples of structures defined in terms of hyperoperations. While they were respectively defined in the 1930s and 1950s, they have recently gained prominence through various appearances in number theory, combinatorics, and absolute algebraic geometry. However, to date there has been relatively little attention given to categories of hyperstructures. I will discuss several categories of hyperstructures that generalize hypergroups. A common theme is that in order for these categories to enjoy good properties like (co)completeness, we must allow for the product or sum of two elements to be the empty subset, which cannot occur in a hypergroup. In particular, I will introduce a category of hyperstructures called mosaics whose subcategory of commutative objects possess a closed monoidal structure reminiscent of the tensor product of abelian groups. This is joint work with So Nakamura.

 We present the Partition of Unity Method (PUM) and its implementation in Fraunhofer SCAI’s PUMA software framework. The fundamental idea and benefit of the PUM is to reduce the necessary number of degrees of freedom of a simulation while attaining the required accuracy of the application by using application-dependent enrichment functions which can resolve highly localized behavior of the solution instead of using mesh-refinement and standard piecewise polynomial basis functions. We discuss how such application-dependent enrichments can be constructed a-priori and on-the-fly during a simulation for e.g. laminated composites, shells and additive manufacturing problems.

Schubert polynomials are distinguished representatives of Schubert cells in the cohomology of the flag variety. Pipedreams (PD) and bumpless pipedreams (BPD) are two combinatorial models of Schubert polynomials. There are many classical perspectives to view PDs: Fomin and Stanley represented each PD as an element in the NilCoexter algebra; Lenart and Sottile converted each PD into a labeled chain in the Bruhat order. In this talk, we unravel the BPD analogues of both viewpoints. One application of our results is a simple bijection between PDs and BPDs via Lenart's growth diagram.

In the first half, I will introduce the subject of random conformal geometry. Schramm-Loewner evolution (SLE) is a random planar curve describing the scaling limits of interfaces in statistical physics models (e.g. percolation, Ising model). Liouville quantum gravity (LQG) is a random 2D surface arising as the scaling limit of random planar maps. These fractal geometries have deep connections to bosonic string theory and conformal field theories. LQG and SLE exhibit a rich interplay: cutting LQG by independent SLE gives two independent LQG surfaces [Sheffield '10, Duplantier-Miller-Sheffield '14]. In the second half, I will present extensions of these LQG/SLE theorems and give several applications.

To any group G is associated the space Ch(G) of all characters on G. After defining this space and discussing its interesting properties, I'll turn to discuss dynamics on such spaces. Our main result is that the action of any arithmetic group on the character space of its amenable/solvable radical is stiff, i.e, any probability measure which is stationary under random walks must be invariant. This generalizes a classical theorem of Furstenberg for dynamics on tori. Relying on works of Bader, Boutonnet, Houdayer, and Peterson, this stiffness result is used to deduce dichotomy statements (and 'charmenability') for higher rank arithmetic groups pertaining to their normal subgroups, dynamical systems, representation theory and more. The talk is based on a joint work with Uri Bader.

The class number formula describes the behavior of the Dedekind zeta function at s = 0 and s = 1. The Stark and Gross conjectures extend the class number formula, describing the behavior of Artin L-functions and p-adic L-functions at s = 0 and s = 1 in terms of units. The Harris–Venkatesh conjecture describes the residue of Stark units modulo p, giving a modular analogue to the Stark and Gross conjectures while also serving as the first verifiable part of the broader conjectures of Venkatesh, Prasanna, and Galatius.
In this talk, I will draw an introductory picture, formulate a unified conjecture combining Harris–Venkatesh and Stark for weight one modular forms, and describe the proof of this in the imaginary dihedral case.

Over the last few years there have been significant developments in the techniques used to understand singularities within minimal submanifolds. I will discuss this circle of ideas and explain how they enable us to reconnect the study of these geometric singularities with more classical PDE techniques, such as those used in unique continuation.

 In this talk, we consider a conjecture by Gross and Kudla that relates the derivatives of triple-product L-functions  for three modular forms and the height pairing of the Gross—Schoen cycles on Shimura curves.

Then, we sketch a proof of a generalization of this conjecture for Hilbert modular forms in the spherical case. This is a report of work in progress with Xinyi Yuan and Wei Zhang, with help from Yifeng Liu.

 In this talk, we first consider a formula proved in the 2010s that relates the height pairing of the Gross—Schoen cycles on the product of a curve over a number field and the self-product of the relative dialyzing sheaves. Then, we describe a recent application given by Xinyi Yuan on the uniform Bogomolov conjecture and the uniform Mordell—Lang conjecture.