The colloquium typically meets Mondays at 4:00 PM in Room 617 on the sixth floor of Wachman Hall.
The colloquium is preceded by tea starting at 3:30 in the Faculty Lounge, adjacent to Room 617. Click on title for abstract.
Kavita Ramanan, Brown University
The hyperplane conjecture in convex geometry is a statement about the volume of a convex body and that of its hyperplane sections. Taking a measure-theoretic perspective to this problem, Bourgain highlighted the importance of the notion of a $\psi_2$-convex body, which captures integrability properties of linear images of the volume measure on the body. Despite this notion being introduced more than a quarter century ago, there are not many examples of such bodies. We describe several results on the $\psi_2$ (or more generally, $\psi_\alpha$) behavior of Schatten balls and their marginals, and their relation to the hyperplane conjecture. Along the way, we also establish some properties of the Haar measure on the orthogonal group that may be of independent interest. This is joint work with Grigoris Paouris.
Ralph Kaufmann, Purdue University
Graphs are a universal tool in mathematics. Their study is interesting in its own right. They also naturally appear in the geometry of surfaces. When coupled with additional structures such as local orders or other local information in the form of so-called ribbon graphs. We will present a unified approach to these structures and explore their relationship with algebra, topology and geometry.
Samit Dasgupta, Duke University
In this talk we will discuss two central problems in algebraic number theory and their interconnections: explicit class field theory and the special values of L-functions. The goal of explicit class field theory is to describe the abelian extensions of a ground number field via analytic means intrinsic to the ground field; this question lies at the core of Hilbert's 12th Problem. Meanwhile, there is an abundance of conjectures on the special values of L-functions at certain integer points. Of these, Stark's Conjecture has special relevance toward explicit class field theory. I will describe two recent joint results with Mahesh Kakde on these topics. The first is a proof of the Brumer-Stark conjecture. This conjecture states the existence of certain canonical elements in CM abelian extensions of totally real fields. The second is a proof of an exact formula for Brumer-Stark units that has been developed over the last 15 years. We show that these units together with other easily written explicit elements generate the maximal abelian extension of a totally real field, thereby giving a p-adic solution to the question of explicit class field theory for these fields.
Frauke Bleher, University of Iowa
This is joint work with A. Lubotzky and T. Chinburg. It is a longstanding problem in Galois theory to give an explicit description of the absolute Galois group G_Q of the rationals. A famous theorem by Belyi from 1979 says that G_Q can be embedded into the automorphism group A of a free profinite group on two generators. This theorem led Grothendieck, Drinfel'd and others to try to identify G_Q inside A as a subgroup satisfying various extra relations. Each set of such relations defines a so-called Grothendieck-Teichmueller group GT inside A that is a candidate for being G_Q. It has been an open problem since the 1990's to identify natural non-abelian representations of G_Q that can be shown to lift to such GT groups. We will show that after passing to subgroups of finite index, this can be done for various natural families of representations, e.g. for the actions of subgroups of finite index in G_Q on all of the finite subgroups of the points of an elliptic curve defined over a number field. The main tool is a construction of linear representations of the automorphism group of a free profinite group on finitely many generators that generalizes work of Grunewald and Lubotzky on linear representations of the automorphism group of a finitely generated free discrete group.
Michael Tait, Villanova University
How many edges may a graph with no triangle have? Given a graph F, the Turan problem asks to maximize the number of edges in a graph on n vertices subject to the constraint that it does not contain F as a subgraph. In this talk, we will discuss constructions for this problem coming from finite geometry (eg using projective planes), additive combinatorics, and "random polynomials".
Kazim Buyukboduk, University College Dublin
Negative answer to Hilbert's 10th problem tells us that determining whether or not an algebraic variety should carry any rational points is impossibly hard (literally!). The same problem even for curves is very difficult: For elliptic curves, this is the subject of the celebrated Birch and Swinnerton-Dyer conjecture. I will survey recent results on this problem, and explain briefly an explicit p-adic analytic construction of rational points of infinite order on elliptic curves of rank one (settling a conjecture of Perrin-Riou). These final bits of the talk will be a report on joint works with Rob Pollack & Shu Sasaki, and with Denis Benois.
Natesh Pillai, Harvard University
Determining the mixing time of Kac's random walk on the n-sphere was a long-standing open problem. In this talk, I will discuss my joint work with Aaron Smith on obtaining the optimal mixing time bounds for this walk and its variants. In addition to discussing the key coupling construction underlying our proof, I will discuss its connections and applications to random matrix theory, dimension reduction methods, and other statistical applications. In particular, we will exhibit a Johnson-Lindenstrauss (JL) transform using Kac's walk that is memory-optimal and outperforms existing algorithms in certain regimes, confirming a conjecture of Ailon and Chazelle.
This is joint work with Aaron Smith, Vishesh Jain, Ashwin Sah and Mehtaab Sawhney.
Ricardo Nochetto, University of Maryland
We discuss modeling, numerical analysis and computation of liquid crystal networks (LCNs). These materials couple a nematic liquid crystal with a rubbery material. When actuated with heat or light, the interaction of the liquid crystal with the rubber creates complex shapes. Thin bodies of LCNs are natural candidates for soft robotics applications. We start from the classical 3D trace energy formula and derive a reduced 2D membrane energy as the formal asymptotic limit of vanishing thickness and characterize the zero energy deformations. We design a sound numerical method and prove its Gamma convergence despite the strong nonlinearity and lack of convexity properties of the membrane energy. This work is joint with L. Bouck and S. Yang.
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.
Kiran Kedlaya, UCSD/IAS
We classify all possible configurations of vectors in three-dimensional space with the property that any two of the vectors form an angle whose measure is a rational multiple of pi. As a corollary, we find all tetrahedra whose six dihedral angles are all rational multiples of pi. While these questions (and their answers) are of an elementary nature, their resolution will take us on a tour through cyclotomic number fields, computational algebraic geometry, and an amazing fact about the geometry of tetrahedra discovered by two physicists in the 1960s. Joint work with Sasha Kolpakov, Bjorn Poonen, and Michael Rubinstein.
Ian Tobasco, University of Illinois Chicago
Mechanical metamaterials are many-body elastic systems that deform in unusual ways, due to the interactions of essentially rigid building blocks. Examples include origami patterns with many folds, or kirigami patterns made by cutting material from a thin elastic sheet. In either case, the local deformations of the pattern involve internal degrees of freedom which must be matched with the usual global Euclidean invariances--- e.g., groups of four origami panels move by coordinated rotations and translations, but it is still possible to bend the whole origami pattern into an overall curved shape. This talk will introduce the homogenization problem for kirigami and origami metamaterials to a broad audience and describe our recent results. Our goal is to explain the link between the design of the individual cuts/folds and the bulk deformations and geometries they can produce. This is joint work with Paul Plucinsky (U. Southern California, Aerospace and Mechanical Engineering) and Paolo Celli (Stony Brook U., Civil Engineering).