1;95;0c EveryTopic Seminar

### EveryTopic Seminar, Fall 2019

Thursdays 11:00-12:00pm, Goldsmith 300.
Organizers: Corey Bregman, Rahul Krishna.

The EveryTopic Seminar is the Brandeis math department graduate student colloquium. Talks are 50 minutes long and aimed at a broad audience of mathematicians.

 Date Speaker (Affiliation) Title and Abstract Sep 26 Rahul Krishna (Brandeis) An introduction to the Gan-Gross-Prasad conjectures Recently there has been a flurry of activity around the global Gan-Gross-Prasad conjecture and its relatives. In this talk I will try to explain some of the motivation behind these conjectures, which relate central values of certain $L$-functions to period integrals of automorphic forms. I will also survey some of the recent progress on these conjectures. This talk will involve an overly-speedy review of some of the basic objects involved in these statements, and should be largely accessible to graduate students. Oct 10 TBD TBA Oct 17 Satyan Devadoss (UCSD) Unfolding Mathematics at Burning Man A 2-ton interactive sculpture came to life at Burning Man 2018, the world’s most influential large-scale sculpture showcase. Rising 12 feet tall with an 18-foot wingspan in the Nevada desert, the unfolding dodecahedron was illuminated by 16,000 LEDs, requiring 6500 person hours and $50,000 in funds. Its interior, large enough to hold 15 people, was fully lined with massive mirrors, alluding to a possible shape of our universe. The heart of this artwork can be traced back 500 years to Albrecht Durer, and the tantalizing open problem of discovering a geometric unfolding for every convex polyhedron. We develop a combinatorial algorithm and show that *every* unfolding of an n-cube is valid, with elegant relationships to integer partitions and chord diagrams. Numerous unsolved questions remain, with connections to group theory, topology, and computational geometry. Oct 24 TBD TBA Oct 31 (Joint with Number Theory and Dynamics) Marius Lemm (Harvard) Global eigenvalue distribution of matrices defined by the skew-shift We consider large Hermitian matrices whose entries are defined by evaluating the exponential function along orbits of the skew-shift$\binom{j}{2}\omega+j\cdot y+x \mod 1$for irrational frequency$\omega\$. We prove that the global eigenvalue distribution of these matrices converges to the corresponding distributions from random matrix theory, namely, the Wigner semicircle law for square matrices and the Marchenko-Pastur law for rectangular matrices. The result evidences the quasi-random nature of the skew-shift dynamics. This is joint work with Arka Adhikari and Horng-Tzer Yau. Nov 7 Matt Stoffregen (MIT) Heegaard Floer homology, 4-Manifolds, and Triangulations In 2013, Manolescu showed that for all n greater than or equal to 5, there exist topological manifolds of dimension n that cannot be triangulated. We'll talk about a constellation of topics related to this result, including smooth 4-manifolds, the homology cobordism group, and Floer homology, as well as giving a brief rendition of Manolescu's proof. Then we'll give a sketch of how Heegaard Floer homology works, and finally, a sketch of how Heegaard Floer homology does not work. Nov 14 Claire Burrin (Rutgers) Discrete lattice orbits in the plane Take a lattice in SL(2,R) and let it act linearly on the plane. Its orbits will be either discrete or dense. If we take a discrete lattice orbit; how are its points distributed in the plane? We will see examples, illustrate what makes this question challenging, and what can be said using some ideas from number theory. The latter bit is based on recent work with Amos Nevo, Rene Rühr, and Barak Weiss. Nov 21 Lisa Piccirillo (MIT and Brandeis) Knot concordance and 4-manifolds There is a rich interplay between the fields of knot theory and 3- and 4-manifold topology. In this talk, I will describe knot concordance (a weak notion of equivalence for knots) and highlight some historical and recent connections between knot concordance and the study of 4-manifolds, with a particular emphasis on applications of knot concordance to the construction and detection of small 4-manifolds which admit multiple smooth structures. Dec 5 Ryan Goh (BU) Growth and patterns The interplay between growth processes and spatial patterns has intrigued researchers in many areas, such as directional quenching in alloy melts, growing organisms in biological systems, moving masks in ion milling, eutectic lamellar crystal growth, and traveling reaction fronts, where such processes have been shown to select and control spatially periodic patterns, while mediating defects. Mathematically, they can be encoded in a step-like parameter dependence that allows patterns in a subset of the spatial domain, and suppresses them in the complement, while the boundary of the pattern-forming region propagates with fixed normal velocity. In this talk, I will show how techniques from dynamical systems, functional analysis, and numerical continuation, can be used to study the effect of these traveling heterogeneities on patterns left in the wake. I will also show how periodic wrinkles can form on top of pure stripes, with frequency behavior similar to that of a saddle-node on a limit cycle. I will explain this approach in the context of the Swift-Hohenberg PDE, a prototypical model for many pattern forming systems, posed in one and two spatial dimensions. I will also discuss recent work which uses techniques from matrix Ricatti equations and modulational theory to study the stability and dynamics of these structures.

Directions to the Brandeis math department.