## UC Riverside Department of Mathematics Fridays 1–2pm in Surge 284

### Organizers

 Joe Moeller (email hidden, enable JavaScript) Dylan Noack (email hidden, enable JavaScript) Mike Pierce (email hidden, enable JavaScript)

### Scheduled Talks, Fall 2017

 1 December 2017 TBA Lawrence Mouillé TBA
 17 November 2017 Analysis on Manifolds via Li-Yau Gradient Estimates Xavier Ramos Olivé We are always told that the motivation for defining a smooth structure on a manifold is to be able to do calculus and analysis on manifolds. But how exactly is this done, and why? Will analysis give us information about our manifold? In this talk we will see how to define some natural differential equations on Riemannian manifolds, and how studying their solutions we can get topological information of the underlying manifold. We will do this via an example: by studying the so called Li-Yau gradient estimates of the heat kernel, with a particular focus to their relationship to the Ricci curvature. These estimates can be used to derive some bounds on the Betti numbers of the manifold. If time permits, we will explore some different strategies to derive the gradient estimate under different curvature assumptions, although to protect our sanity, we will skip all the messy computations. No previous knowledge about the concept of curvature will be required for the talk.
 3 November 2017 Covariance Computations for the Active Subspace Method Applied to a Wind Model Jolene Britton The method of active subspaces is an effective means for reducing the dimensions of a multivariate function $f$. This method enables experiments and simulations that would otherwise be too computationally expensive due to the high-dimensionality of $f$. By using a covariance matrix composed of the gradients of $f,$ one can find the directions in which $f$ varies most strongly, i.e. the active subspace. The current standard for estimating these covariance matrices is the Monte Carlo estimator. Due to the slow convergence of Monte Carlo methods, we propose alternative algorithmic approaches. The first utilizes a separated representation of $f,$ while the second uses polynomial chaos expansions. Such representations have well-defined sampling strategies and allow for the analytic computation of entries of the covariance matrix. Experimental results demonstrate how the Monte Carlo methods compare to our proposed alternative approaches as applied to a function representing power output of a wind turbine.
 27 October 2017 Better Faster Strongly Jacobson Modules Tim McEldowney Inventing new math is hard. However, there is a nice work around. Take old math and add an adjective. In this talk, I will build up to my most recent result which looks suspiciously like another theorem. I will start by talking about the base structures I study called ‘$G$-domains’ which are integral domains which are close to being their field of fractions. Next, I will define ‘$G$-ideals’ and ‘Hilbert rings’ which are made from these $G$-domains with some clear examples of these structures from common rings. Afterwards, I talk about the ‘strongly’ adjective and what that does to these objects. Lastly, I close with a game I like to call pin the adjective on your adviser’s theorem.
 20 October 2017 Network Models Joe Moeller A network is a complex of interacting systems which can often be represented as a graph equipped with extra structure. Networks can be combined in many ways, including by overlaying one on top of the other or sitting one next to another. We introduce network models — which are formally a simple kind of lax symmetric monoidal functor — to encode these ways of combining networks. By applying a general construction to network models, we obtain operads for the design of complex networked systems.
 13 October 2017 Can a nice variety of variety exist? Ethan Kowalenko Algebraic geometry is notorious for being difficult, as it is broadly the study of the zero sets of polynomials through some assigned rings. I will attempt to describe a very computable class of such zero sets, called toric varieties, by showing literal computations. Like, explicitly.
 6 October 2017 An Introduction to Hopf Algebras Dane Lawhorne What happens when you take the commutative diagrams that define an algebra and reverse all the arrows? The result is called a coalgebra, and with a few more axioms, you get a Hopf Algebra. In this talk, we will examine the role of Hopf algebras in representation theory. In particular, we will see that the category of left modules over a Hopf algebra has both tensor products and dual modules.

### Talks from Previous Years

 2016–2017 2015–2016 2014–2015 2013–2014 2012–2013