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Jackie Lang, Temple University
I will discuss work in progress with Robert Pollack and Preston Wake about counting congruences between "vexing" modular forms.
Patrick Phelps, Temple University
Abstract: We present recent results on spatial decay and properties of non-uniqueness for the 3D Navier-Stokes equations. We show asymptotics for the ‘non-linear’ part of scaling invariant flows with data in subcritical classes. Motivated by recent work on non-uniqueness, we investigate how non-uniqueness of the velocity field would evolve in time in the local energy class. Specifically, by extending our subcritical asymptotics to approximations by Picard iterates, we may bound the rate at which two solutions, evolving from the same data, may separate pointwise. We conclude by extending this separation rate to solutions with no scaling assumption. Joint work with Zachary Bradshaw.
Valerie C. Dudley
Director of Multicultural Education and Training
Institutional Diversity, Equity, Advocacy, and Leadership, Temple University
Inclusive language puts our humanity at the center; it allows everyone to feel recognized, valued, invited, and motivated to contribute at their highest level. To become an anti-racist institution, we must take concrete action to change our culture and the experiences of members of our community. In this workshop, we will consider the impact of using respectful, identity-affirming language in creating an antiracist welcoming environment that embraces diversity as a whole.
Daniel Slonim, University of Virginia
We introduce the model of random walks in random environments (RWRE), which are random Markov chains on the integer lattice. These random walks are well understood in the nearest-neighbor, one-dimensional case due to reversibility of almost every Markov chain. For example, directional transience and limiting speed can be characterized in terms of simple expectations involving the transition probabilities at a single site. The reversibility is lost, however, if we go up to higher dimensions or relax the nearest-neighbor assumption by allowing jumps, and therefore much less is known in these models. Despite this non-reversibility, certain special cases have proven to be more tractable. Random walks in Dirichlet environments (RWDE), where the transition probability vectors are drawn according to a Dirichlet distribution, have been fruitfully studied in the nearest-neighbor, higher dimensional setting. We look at RWDE in one dimension with jumps and characterize when the walk is ballistic: that is, when it has non-zero limiting velocity. It turns out that in this model, there are two factors which can cause a directionally transient walk to have zero limiting speed: finite trapping and large-scale backtracking. Finite trapping involves finite subsets of the graph where the walk is liable to get trapped for a long time. It is a highly local phenomenon that depends heavily on the structure of the underlying graph. Large-scale backtracking is a more global and one-dimensional phenomenon. The two operate "independently" in the sense that either can occur with or without the other. Moreover, if neither factor on its own is enough to cause zero speed, then the walk is ballistic, so the two factors cannot conspire together to slow a walk down to zero speed if neither is sufficient to do so on its own. This appearance of two independent factors affecting ballisticity is a new feature not seen in any previously studied RWRE models.
Rebecca Bellovin, IAS
The Fontaine-Mazur conjecture (proved by Kisin and Emerton) says that (under certain technical hypotheses) a Galois representation \rho:Gal_Q\rightarrow GL_2(\overline{Q_p)$ is modular if it is unramified outside finitely many places and de Rham at p. I will talk about what this means, and I will discuss an analogous modularity result for Galois representations \rho:Gal_Q\rightarrow GL_2(L) when L is instead a non-archimedean local field of characteristic p.
Noa Kraitzman, Macquarie University
Sea ice is a crucial component of the Earth’s climate system, affecting the ocean circulation, the atmospheric temperature, and the marine ecosystems. However, sea ice is not a simple solid material; it is a complex mixture of ice crystals, brine pockets, and air bubbles, that changes its structure and properties depending on the environmental conditions. In this talk, I will explore how we can model and understand the behaviour of sea ice at different scales, from the microscopic interactions of ice and salt to the macroscopic effects of heat transfer and fluid flow. I will present two mathematical models: a thermodynamically consistent model for the liquid-solid phase change in sea ice that incorporates the effects of salt, using multiscale analysis to derive a quasi-equilibrium Stefan-type problem. And a new rigorous derivation of bounds on the sea ice effective thermal conductivity obtained through Padé approximates and using Stieltjes integrals.
This week’s meeting of the Temple Math Club will feature employees from the Advanced Concepts Lab about their work and the relevant mathematical topics (cryptography, statistics, discrete math, and more). This will be a good meeting to attend for any students interested to see the uses of math in industry. And, of course, there will be free pizza!
There are no conferences this week.