*THURSDAY* 17 October 2019

Federico Bocci (Rice University)

*Phenotypic heterogeneity driven by cell-cell signaling: principles of pattern formation and implications for cancer metastasis*

Cells communicate with their neighbors to establish cell fate in physiological and pathological conditions. In this talk, I will discuss the operating principles of cellular patterning mediated by Notch signaling and its implications in cancer metastasis. Notch is an evolutionary conserved signaling pathway that regulates cell fate and multicellular patterning through binding of ligands and receptors of neighboring cells. Binding of the receptor with different ligands leads to different principles of pattern formation. Neighboring cells either differentiate between opposite phenotypes or maintain similar phenotypes. Our mathematical model of the Notch pathway predicts a sharp transition between these opposite regimes that is confirmed experimentally in the context of blood vessel formation. Moreover, Notch couples with multiple signaling ‘axes’ of cancer progression to stabilize highly aggressive phenotypes. We devise a dynamical model to investigate the coupling between Notch and the gene regulatory networks that regulate cancer cell migration and proliferation. Notch signaling facilitates the acquisition of migratory and proliferative traits that maximize the cells’ metastatic potential. This framework can be generalized to identify molecular perturbations that stabilize/destabilize aggressive cancer phenotypes and predict the spatial patterning of cancer cells.

*WEDNESDAY* 23 October 2019

Kamil Szpojankowski (Warsaw University of Technology)

*Characterization problems in classical and non-commutative probability*

The most prominent characterization by independence property is Bernstein’s Theorem, and says that for independent random variables X, Y, if X+Y and X-Y are independent, then X and Y are necessarily Gaussian.

Since Bernstein’s result many similar characterisations were proved.

In my talk I shall present most important examples, their generalizations and applications in statistics.

In the second part of the talk I shall discuss a surprising parallel results in the framework of free probability, which is a non commutative counterpart of the classical probability theory.

**Friday 25 October 2019**

David Sondak (Harvard University)

*An Autoencoder for Learning Reduced Dynamics of the Kuramoto-Sivashinsky Equation*Fluids play a central role in science and engineering. Understanding their behavior opens avenues to predict quantities (such as heat transport) that are important for designing physical systems. Fluid behavior is complicated by the phenomenon of turbulence. Although most fluid flows are turbulent, traditional techniques to probe turbulence have been challenged, in large part due to its high dimensionality. In the present work, a machine learning approach is proposed for learning the low dimensional dynamics of a chaotic system. The system in question is the Kuramoto-Sivashinsky equation, which exhibits spatio-temporal chaos, but whose dynamics are known to fall on an inertial manifold. This talk will begin with a review of the key ideas of fluid mechanics and turbulence and then provide some background on machine learning and neural networks in particular. Following this, an autoencoder architecture with latent space penalization will be introduced and results on the Kuramoto-Sivashinksy equation will be presented. It is shown that the autoencoder is able to learn a finite-dimensional reduced space while reproducing the essential dynamics. The talk will close with a discussion of future work and thoughts on applications to real turbulent flows.

**Friday 1 November 2019**

Gabriel Nagy (Kansas State University)

*The Spectral Theorem from Scratch*

We provide an elementary approach to the development of the continuous functional calculus both for a single bounded normal operator, as well as for commuting tuples of bounded self-adjoint operators on a Hilbert space. The only tools needed are College Algebra, Elementary Topology, and the definition of the adjoint. As such, the talk will be accessible to undergraduate Math majors.

**Friday 15 November 2019**

Stefania Patrizi (UT Austin)

*In some segregation models*

Segregation phenomena occurs in many areas of mathematics and science: from equipartition problems in geometry, to social and biological processes (cells, bacteria, ants, mammals) to finance (sellers and buyers). Segregation problems model a situation of high competition for resources and involve a combination of diffusion and annihilation between populations. We present different models: segregation at distance, segregation involving fully nonlinear diffusion, segregations where competing species follow local vs nonlocal propagation diffusion equations. These works are in collaborations with Luis Caffarelli, Veronica Quitalo, Monica Torres.

**Friday 22 November 2019**

Matthew Thomas (Ithaca College)

*TBA*

TBA