Departmental Colloquium 2018-2019

Thursdays, 4:00 PM, JWB 335

Fall 2018

September 27:
Speaker: Raphaël Rouquier, UCLA
Title: Higher representation theory
Abstract: Representation theory is the study of vector spaces with prescribed symmetries. Replacing vector spaces by more complicated structures (of categorical, topological or geometrical origin) leads to new directions in algebra. This sheds a new light on geometric representation theory. More general moduli spaces constructions and enumerative invariants should fit in that framework. An additional motivation is the construction of invariants of low-dimensional manifolds from quantum field theories.

November 8:
Speaker: Mark Walker, University of Nebraska
Title:The Total and Toral Rank Conjectures
Abstract: Assume X is a nice space (a compact CW complex) that admits a fixed-point free action by a d-dimensional torus T. For example, X could be T acting on itself in the canonical way. The Toral Rank Conjecture, due to Halperin, predicts that the sum of the (topological) Betti numbers of X must be at least 2^d. Put more crudely, this conjecture predicts that it takes at least 2^d cells to build X by gluing. Now suppose M is a module over the polynomial ring k[x_1, \dots, x_d] that is finite dimensional as a k-vector space. The Total Rank Conjecture, due to Avramov, predicts that the sum of algebraic Betti numbers of M must be at least 2^d. Here, the algebraic Betti numbers refer to the ranks of the free modules occuring in the minimal free resolution of M. The Total Rank Conjecture is a weak form of the well-known Buchsbaum-Eisenbud-Horrocks Conjecture. In this talk I will discuss the relationship between these conjectures, some related conjectures, and recent progress toward settling them.

Special Colloquium: Tuesday November 27, 4-5pm, JWB 335:
Speaker: Eric Cator, Radboud University
Title: From WW II bombs to robust statistics.
Abstract: In this talk I will highlight some of the statistical problems that I have been working on over the years.
The first project is about unexploded bombs from World War II still in the ground in Amsterdam. There is a lot of historical information on bombardment the Allies carried out on German industry in Amsterdam, and the city would like to use this information to determine the risk of starting new building projects in these areas. We developed a new model for this, and were able to implement a Bayesian statistics approach through the Metropolis-Hastings algorithm. Quantative statements about the risk of specific areas can now be given, and this method is about to be implemented across the Netherlands.
The second project is on the Minimum Covariance Determinant (MCD) estimator that is frequently used in robust statistics. Consider a sample from some multidimensional distribution. In robust statistics, we try to estimate properties of the underlying distribution using only a fraction of the data, so that if part of the data is corrupted, this would not influence our estimators too much. The MCD tries to find that fixed fraction of the dat that has a covariance matrix with minimal determinant. It was known, for some special multivariate distributions, that this estimator has better convergence properties than the more intuitive Minimum Volume Estimator, but the asymptotic normality and even consistency for general distributions remained out of reach for more than 10 years. By extending the definition of MCD from a sample to a continuous distribution in a proper way, we were able to prove consistency and asymptotic normality, with explicit covariance structure, under mild conditions on the underlying distribution.

Special Colloquium: Monday December 3, 3-4pm, JWB 335: (Note special time)
Speaker: Nivedita Bhaskhar, UCLA
Title: On rational points, zero cycles and norm principles
Abstract: From the epitaphs of Diophantus to the margins of Fermat, the study of solutions to systems of polynomial equations has left its mark everywhere in history. In broad terms, given nice systems of polynomial equations over a field k which define varieties with tractable structure, one would like to ascertain whether the set of solutions over k is empty or not, i.e. whether the variety in question has a k-rational point.
A zero cycle on a k-variety X is any element of the free abelian group of closed points of X and its degree is the sum of its coefficients, weighted by the degrees of the residue fields. Any k-rational point of X is a zero cycle of degree one. In this talk, we discuss Serre’s injectivity question which asks whether the converse is true for torsors X under connected linear algebraic groups, i.e. whether such an X admitting a zero cycle of degree one in fact has a rational point. This naturally brings into the picture the so-called norm principles, which examine the behaviour of the images of group morphisms over field extensions from a linear algebraic group into a commutative one with respect to the norm map.

Special Colloquium: Thursday December 6, 4-5pm, JWB 335:
Speaker: Elina Robeva, MIT
Title: Orthogonal Tensor Decomposition
Abstract: Tensor decomposition has many applications. However, it is often a hard problem. In this talk I will discuss a family of tensors, called orthogonally decomposable tensors, which retain some of the properties of matrices that general tensors don't. A symmetric tensor is orthogonally decomposable if it can be written as a linear combination of tensor powers of n orthonormal vectors. Such tensors are interesting because their decomposition can be found efficiently. We study their spectral properties and give a formula for all of their eigenvectors. We also give equations defining all real symmetric orthogonally decomposable tensors. Analogously, we study nonsymmetric orthogonally decomposable tensors, describing their singular vector tuples and giving polynomial equations that define them. In an attempt to extend the definition to a larger set of tensors, we define tight-frame decomposable tensors and study their properties. Finally, I will conclude with some open questions and future research directions.

Special Colloquium: Tuesday December 11, 4-5pm, JWB 335:
Speaker: Natalie Sheils, University of Minnesota
Title: Interface Problems Using the Unified Transform Method
Abstract: Interface problems for partial differential equations are initial boundary value problems for which the solution of an equation in one domain prescribes boundary conditions for the equations in adjacent domains. These types of problems occur widely in applications including heat transfer, quantum mechanics, and mathematical biology. These problems, though linear, are often not solvable analytically using classical approaches. In this talk I present an introduction to the Unified Transform Method as well as an extension appropriate for solving interface problems.

Special Colloquium: Thursday December 13, 4-5pm, JWB 335:
Speaker: Priyam Patel, UC Santa Barbara
Title: Quantitative methods in hyperbolic geometry
Abstract: Peter Scott’s famous result states that the fundamental groups of hyperbolic surfaces are subgroup separable, which has many powerful consequences. For example, given any closed curve on such a surface, potentially with many self-intersections, there is always a finite cover to which the curve lifts to an embedding. It was shown recently that hyperbolic 3-manifold groups share this separability property, and this was a key tool in Ian Agol's resolution to the Virtual Haken and Virtual Fibering conjectures for hyperbolic 3-manifolds.
I will begin this talk by giving some background on separability properties of groups, hyperbolic manifolds, and these two conjectures. There are also a number of interesting quantitative questions that naturally arise in the context of these topics. These questions fit into a recent trend in low-dimensional topology aimed at providing concrete topological and geometric information about hyperbolic manifolds that often cannot be gathered from existence results alone. I will highlight a few of them before focusing on a quantitative question regarding the process of lifting curves on surfaces to embeddings in finite covers.

Special Colloquium: Thursday January 10, 4-5pm, JWB 335:
Speaker: Will Feldman, University of Chicago
Title: Interfaces in inhomogeneous media: pinning, hysteresis, and facets
Abstract: I will discuss some models for the shape of liquid droplets on rough solid surfaces. The framework of homogenization theory allows to study the large scale effects of small scale surface roughness, including interesting physical phenomena such as contact line pinning, hysteresis, and formation of facets.

Special Colloquium: Tuesday January 15, 4-5pm, JWB 335:
Speaker: Sean Howe, Stanford University
Title: Probabilistic structures in the topology and arithmetic of moduli spaces
Abstract: The average smooth surface in P^3 is a plane, the universal curve of genus g approaches a Poisson random variable, and the probability that a random hypersurface is smooth is given by a special value of a zeta function! In this talk, we explain how to make sense of these claims using the language of motivic random variables to adapt basic concepts from probability, like moment generating functions and independence, to the study of the topology and arithmetic of moduli spaces. This framework leads to new results on the topology of moduli spaces of hypersurface sections, a new perspective on representation stability for configuration spaces of algebraic varieties, and the computation of higher order terms in Katz-Sarnak style statistics for the zero-spacings of zeta functions; we will explain some of these applications, and speculate about future directions.

Special Colloquium: Thursday January 17, 4-5pm, JWB 335:
Speaker: Ian Tobasco, University of Michigan
Title: The cost of crushing: curvature-driven wrinkling of thin elastic shells
Abstract: How much energy does it take to stamp a thin elastic shell flat? Motivated by recent experiments on wrinkling patterns formed by thin floating shells, we develop a rigorous method (via Gamma-convergence) for evaluating the cost of crushing to leading order in the shell’s thickness and other small parameters. The observed patterns involve regions of well-defined wrinkling alongside totally disordered regions where no single direction of wrinkling is preferred. Our goal is to explain the appearance of such “wrinkling domains”. Our analysis proves that energetically optimal patterns maximize their projected planar area subject to a shortness constraint. This purely geometric variational problem turns out to be explicitly solvable in many cases of interest, and a straightforward scheme for predicting wrinkle patterns results. We demonstrate our methods with concrete examples and offer comparisons with simulation and experiment.
This talk will be mathematically self-contained, not assuming prior background in elasticity or calculus of variations.

Special Colloquium: Thursday February 7, 4-5pm, JWB 335:
Speaker: Pedro Maia, UC San Francisco
Title: Mathematical models and methods in computational neurology
Abstract: The emerging field of computational neurology provides an important window of opportunity for modeling of complex biophysical phenomena, for scientific computing, for understanding functionality disruption in neural networks, and for applying machine-learning methods for diagnosis and personalized medicine. In this talk, I will illustrate some of our latest results across different spatial scales spanning a broad array of mathematical techniques such as: (i) numerical methods for nonlinear PDEs for solving inhomogeneous active cable equations, (ii) spike-train metrics for quantifying information loss on compromised neural signals, (iii) applied dynamical systems for modeling biological neural networks, (iv) decision-making models, (v) applied inverse-problem techniques for finding the origins of neurodegeneration, and (vi) data methods in medical imaging. (The target audience for this talk is the broader applied mathematics community and no previous knowledge in neuroscience is required. The clinical/biological significance of our findings will be explored in more details on the follow-up seminar "Computational neurology and translational modeling of brain disorders.")

Special Colloquium: Tuesday February 12, 4-5pm, JWB 335:
Speaker: Andreas Malmendier, Utah State University
Title: Mathematical physics as a catalyst for students’ engagement and learning
Abstract: In this talk, I will present on my experience and vision related to teaching and mentoring across three different institutions (USU, Colby College, and UCSB) . First, I will discuss my approach to lower division undergraduate math classes that emphasizes interdisciplinary applications of mathematics to physics and engineering. I will also describe my proposal for a data-driven, scale-able intervention to improve students' outcomes in the critical gateway course, vector calculus. Second, I will talk about my philosophy and experiences with engaging undergraduate and graduate students in pure mathematics research. I will showcase the results of my recent student advisees with respect to 1) period integrals on Kummer surfaces and their relation to special function identities and count pointing, 2) explicit relations between theta functions of genus two and 1 from geometry, and 3) new normal forms of non-principally polarized Kummer surfaces and string dualities in mathematical physics. I will also discuss our ongoing efforts to integrate basic research with vertical peer mentoring, community outreach, and global engagement programs.

February 21:
Speaker: Felix Janda, University of Michigan
Title: Enumerative geometry: old and new
Abstract: For as long as people have studied geometry, they have counted geometric objects. For example, Euclid's Elements starts with the postulate that there is exactly one line passing through two distinct points in the plane. Since then, the kinds of counting problems we are able to pose and to answer has grown. Today enumerative geometry is a rich subject with connections to many fields, including combinatorics, physics, representation theory, number theory and integrable systems. In this talk, I will show how to solve several classical counting questions. Then I will describe a more modern problem with roots in string theory which has been the subject of intense study for the last two decades, namely the study of the Gromov-Witten invariants of the quintic threefold, a Calabi-Yau manifold. I will explain a recent break-through in understanding the higher genus invariants that stems from a seemingly unrelated problem related to the study of holomorphic differentials on Riemann surfaces.

April 4: Joint CSME/Math Colloquium
Speaker: Philip Kutzko, University of Iowa
Title: Just Walk Away, René: Cultural Issues in Broadening Participation in Mathematics
Abstract: Science, as we know it today, developed in a particular time and place for reasons that have never fully been explained. The concept of a function—a concept that underlies all of modern science—first appears in Descartes’ La Géometrie in 1637; within a generation, Newton and Leibniz had developed the calculus and Newton had laid the foundation for modern physics. Similar transformative advances occurred shortly thereafter in chemistry, biology and medicine. This is the context in which we do science today; a West European, Cartesian context in an increasingly non-European nation. The Western approach to science embodies certain cultural values, among them skepticism, objectivity, secularism and a belief in progress as an unmixed virtue. These values are by no means universally accepted, either internationally or within our own country. Further, they have sometimes been used to justify aggression and sometimes worse by Europeans and their descendants in the Americas against other ethnic groups and even against certain groups of European ethnicity. This, it would seem, is reason enough for underrepresented minority groups and other Americans who have not historically been invited to the table to steer clear of European science. Any approach toward broadening participation in science that fails to take into account this cultural context can only go so far. Examples are afforded by standardized testing and affirmative action each of which is ultimately motivated by the same goal: to remove impediments to access caused by overt ethnic and class discrimination (standardized tests) and by the consequences of such discrimination (affirmative action). Both have been valuable in extending access to ethnic and national groups who have found Western science culturally appealing as well as to individuals with similar proclivities from underrepresented groups; indeed, the use of standardized testing transformed the populations doing science during the Sputnik era while affirmative action has been responsible for similar transformations in more recent times. However, these and other strategies that have focused largely on removing barriers to inclusion may be nearing the limit of their utility. One of the distinctive features of the University of Iowa math department’s initiative to broaden participation in our graduate program is the awareness we have developed of the cultural context in which this initiative takes place. I will discuss this cultural context in my talk and argue that an understanding of this context can lead to new strategies, strategies which, in our case, have transformed a traditional mathematics department in an ethnically homogeneous state into what some have called a model for what an American math department should look like in the twenty-first century.

April 11: Distinguished Lecture Series Colloquium
Speaker: Lai-Sang Young, Courant Institute
Title: Dynamics of cortical neurons with a hint of statistical mechanics
Abstract: I will start with a homogeneously connected group of neurons similar to those in local circuits in many parts of the cerebral cortex. Such neuronal systems have the flavor of interacting particle systems, except that there are two kinds of neurons, excitatory and inhibitory, that provide opposite-signed inputs to their postsynaptic neurons. The competition between these two groups leads to new complexities some of which (e.g. the emergence of a rhythm) I will discuss. In the real brain, both external stimuli and internal connectivities are highly inhomogeneous, but the ideas above continue to be relevant. In analogy with nonequilibrium statistical mechanics, steady state cortical responses can be seen as comprised of responses from local populations varying continuously across 2D cortical surfaces, each in dynamical equilibrium with synaptic currents from surrounding and external regions. For illustration I will present a computational model of the monkey visual cortex, and show fMRI-like images (or movies) of its neuronal activity in response to drifting gratings.