I will show you how to construct a full factor $M$ such that $M$ and $L(F_2)$ do not have any isomorphic ultrapowers.  The construction uses a combination of techniques from deformation/rigidity and free entropy theory.  We also provide the first example of a $\mathrm{II}_1$ factor that is full such that its ultrapower is strongly $1$-bounded.  This is joint work with Adrian Ioana and Ionut Chifan.

We develop hybrid projection methods for computing solutions to large-scale inverse problems, where the solution represents a sum of different stochastic components. Such scenarios arise in many imaging applications (e.g., anomaly detection in atmospheric emissions tomography) where the reconstructed solution can be represented as a combination of two or more components and each component contains different smoothness or stochastic properties. In a deterministic inversion or inverse modeling framework, these assumptions correspond to different regularization terms for each solution in the sum. Although various prior assumptions can be included in our framework, we focus on the scenario where the solution is a sum of a sparse solution and a smooth solution. For computing solution estimates, we develop hybrid projection methods for solution decomposition that are based on a combined flexible and generalized Golub-Kahan processes. This approach integrates techniques from the generalized Golub-Kahan bidiagonalization and the flexible Krylov methods. The benefits of the proposed methods are that the decomposition of the solution can be done iteratively, and the regularization terms and regularization parameters are adaptively chosen at each iteration. Numerical results from photoacoustic tomography and atmospheric inverse modeling demonstrate the potential for these methods to be used for anomaly detection.

We discuss Bregman distance extensions of the primal-dual three-operator (PD3O) and Condat-Vu proximal algorithms. When used with standard proximal operators these algorithms include several important methods as special cases. Extensions to generalized Bregman distances are attractive if the complexity per iteration can be reduced by matching the Bregman distance to the structure in the problem. As an example, we apply the proposed method to the centering problem in sparse semidefinite programming. The logarithmic barrier function for the cone of positive semidefinite completable sparse matrices is used as a distance-generating kernel. For this distance, the complexity of evaluating the Bregman proximal operator is shown to be roughly proportional to the cost of a sparse Cholesky factorization. This is much cheaper than the standard proximal operator with Euclidean distances, which requires an eigenvalue decomposition.

LIFE HACK: Attend Food For Thought (FFT) on Wednesday at 4:00 PM. Studies show that attending FFT improves mood by 43%, attending FFT boosts cognition by 15%, attending FFT decreases stress by 28%, and that 120% of statistics that people quote are 150% true! If you attend FFT this week, we'll talk about a few other graduate student life hacks that hopefully can improve your life by just a little bit. See you there!

An invariant random order is a shift-invariant measure on the space of all orders on a group. It is easy to show that on an amenable group, any invariant random order could be invariantly extended to an invariant random total order. Recently, Glaner, Lin and Meyerovitch showed that this is no longer true for $\mathrm{SL}_3(\mathbb{Z})$. I will explain, how starting from their construction, one can show that this order extension property does not hold for non-amenable groups, and discuss an analog of this result for measure preserving equivalence relations.

We introduce a "mesoscale flatness criterion" for hypersurfaces with bounded mean curvature, discussing its relation to and differences with classical blow-up and blow-down theorems, and then we exploit this tool for a complete resolution of relative isoperimetric sets with large volume in the exterior of a compact obstacle. This is joint work with Francesco Maggi (UT Austin).

(Joint with Stefan Patrikis.) In this talk, we discuss a strong form of independence of $\ell$ for canonical $\ell$-adic local systems on Shimura varieties, and sketch a proof of this for Shimura varieties arising from adjoint groups whose simple factors have real rank $\geq 2$. Notably, this includes all adjoint Shimura varieties which are not of abelian type. The key tools used are the existence of companions for $\ell$-adic local systems and the superrigidity theorem of Margulis for lattices in Lie groups of real rank $\geq 2$.

The independence of $\ell$ is motivated by a conjectural description of Shimura varieties as moduli spaces of motives. For certain Shimura varieties that arise as a moduli space of abelian varieties, the strong independence of $\ell$ is proven (at the level of Galois representations) by recent work of Kisin and Zhou, refining the independence of $\ell$ on the Tate module given by Deligne's work on the Weil conjectures.

We will discuss about resolutions of coherent sheaves by line bundles from strong full exceptional sequences over rational surfaces. We call them Gaeta resolutions. We then apply the results towards the study of the moduli space of sheaves, in particular Le Potier's strange duality conjecture. We will show that the strange morphism is injective in some new cases. One of the key steps is to show that certain Quot schemes are finite and reduced. The next key step is to enumerate the length of the finite Quot scheme, by identifying the Quot scheme as the moduli space of limit stable pairs, where we are able to calculate the (virtual) fundamental class. This is based on joint work with Thomas Goller.

Pre-talk for graduate students: 3:30pm - 4:00pm