In a joint work with D. Hoff and A. Ioana, we have discovered the following product rigidity phenomenon: if $\Gamma$ is an icc group measure equivalent
to a product of non-elementary hyperbolic groups, then any tensor product decomposition of the II$_1$ factor $L(\Gamma)$ arises only from the canonical
direct product decomposition of $\Gamma$. Subsequently, I. Chifan, R. de Santiago and W. Sucpikarnin classified all the tensor product decompositions
for group von Neumann algebras arising from a large class of amalgamated free products. In this talk we will give an overview of these results and discuss
about a similar rigidity phenomenon that appears in the context of von Neumann algebras arising from actions. More precisely, we prove that if $\Gamma$
is a product of certain groups and $\Gamma\curvearrowright (X,\mu)$ is an arbitrary free ergodic measure preserving action, then we show that any tensor
product decomposition of the II$_1$ factor $L^\infty(X)\rtimes\Gamma$ arises only from the canonical direct product decomposition of the underlying
action $\Gamma\curvearrowright X.$

In this talk, we discuss a construction of the discretization of classical field theories, within the Lagrangian and Hamiltonian frameworks, which preserve the various underlying structures inherent to the physical theories. Preservation of structure under discretization is desirable as it ensures similar behavior between the discretized field dynamics and the actual field dynamics, and often provides computational benefits such as long-term stability and reduction of numerical artifacts. We present a Discrete Lagrangian and Discrete Hamiltonian approach to structure-preserving discretization of field theories. As a motivating example, we apply these methods, in conjunction with discretization spaces from the Finite Element Exterior Calculus, to construct a discretization of classical Yang-Mills theories arising in particle physics. As a simple numerical example, we discretize electromagnetism coupled to particle-in-cell plasma dynamics. To conclude, we briefly discuss
future directions for research, including group-equivariant interpolation applied to the discretization of gauge theories, connections with the discretization of quantum field theories, and methods for studying the (generally nonlinear) dynamics of the discretized fields.

We consider a certain class of finite-dimensional Gorenstein algebras associated to matroids. We show the Lefschetz property in the case where the matroid corresponds to a modular geometric lattice. Our result implies that the modular geometric lattice has the Sperner property. We also discuss the Grobner fan of the defining ideal of our Gorenstein algebra.

We'll discuss an extension of the work of Kudla-Millson on the modularity of special cycles on a non-compact Shimura variety associated to U(n,1) over a split CM field. The volume of their intersections with a diagonally embedded Shimura subvariety is related to Fourier coefficients of a Hilbert modular form coming from the restriction of an Eisenstein series on U(n,n). The main new idea is an application of the regularized Siegel-Weil formula of Gan-Qiu-Takeda.

In the 1970s, C. Fefferman imitated a program of constructing geometric invariants of bounded complex domains by using the canonical Einstein-Kähler metric on it. The program has been generalized to the construction of conformal invariants via complete Einstein metric with prescribed conformal structure on the boundary at infinity.

Later, in 1997, J. Maldacena applied this picture to theoretical physics; it is now known as AdS/CFT correspondence and soon become a very active area of research. Then ideas from physics were imported to Fefferman’s original program on complex domains. In this talk, I will explain some of global invariants of strictly pseudoconvex domains recently obtained in this program, including, renormalized volume and Q-prime curvature.