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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.
Theresa Anderson, University of Wisconsin
Many problems at the interface of analysis and number theory involve showing that the primes, though deterministic, exhibit random behavior. The Green-Tao theorem stating that the primes contain infinitely long arithmetic progressions is one such example. In this talk, we show that prime vectors equidistribute on the sphere in the same manner as a random set of integer vectors would be expected to. We further quantify this with explicit bounds for naturally occurring maximal functions, which connects classical tools from harmonic analysis with analytic number theory. This is joint work with Cook, Hughes, and Kumchev.
Wai-Tong (Louis) Fan, University of Wisconsin-Madison
Interacting particle models are often employed to gain understanding of the emergence of macroscopic phenomena from microscopic laws in complex systems. These individual-based models capture fine details, including randomness and discreteness of individuals, that are not considered in continuum models such as partial differential equations (PDE) and integral-differential equations. The challenge is how to simultaneously retain key information in microscopic models as well as efficiency and robustness of macroscopic models.
I will discuss how this challenge can be overcome by elucidating the probabilistic connections between particle models and PDE. These connections also explain how stochastic partial differential equations (SPDE) arise naturally under a suitable choice of level of detail in mathematical modeling. I will also present novel scaling limits including SPDE on graphs and coupled SPDE. These SPDE quantify the source and the order of magnitude of stochasticity. Scaling limit theorems and new duality formulas are obtained for these SPDE, which connect phenomena across scales and offer insights about the genealogies and the time-asymptotic properties of certain population dynamics.
Mariusz Mirek, Institute for Advanced Study
In the first part of the talk we will be concerned with the problem of existence of infinitely many arithmetic progressions of length at least three in subsets which have vanishing density in the set of prime numbers $\mathbb P$. Our principal example will be the set of Piatetski--Shapiro prime numbers
$ \mathbf P_{\gamma} = \mathbb P \cap \{\lfloor n^{1 / \gamma} \rfloor: n \in \mathbb N \}, $
with $\gamma \in (71/72 , 1)$. In the second part we will explain connections of the problem raised above with some questions in the pointwise ergodic theory. Specifically, we will see the usefulness of r-variational estimates in pointwise convergence problems.
Finally, I would like to mention about some problem in pointwise ergodic theory which led us to study dimension-free bounds for maximal functions and $r$-variations corresponding to the discrete Hardy--Littlewood averaging operators defined over the cubes in $\mathbb Z^d$.
The last part is joint project with J. Bourgain, E.M. Stein and B. Wr\'obel.
Alex Townsend, Cornell University
Matrices that appear in computational mathematics are so often of low rank. Since random ("average") matrices are almost surely of full rank, mathematics needs to explain the abundance of low rank structures. We will give a characterization of certain low rank matrices using Sylvester matrix equations and show that the decay of singular values can be understood via an extremal rational problem. We will give another characterization involving the Johnson-Lindenstrauss Lemma that partially explains the abundance of low rank structures in big data.
Christopher Leininger, University of Illinois at Urbana-Champaign
In the late 70's and early 80's, Thurston's approach to studying 3-manifolds revolutionized the theory, showing that hyperbolic geometry provided a framework to more systematically study these manifolds. Specifically, he conjectured (and proved in many cases) that 3-manifolds could be canonically decomposed into geometric pieces, with hyperbolic geometry being the richest and most interesting geometric structure arising. Based on earlier work by Dehn, the key features of hyperbolic geometry were abstracted by Gromov to study more general spaces (most famously, finitely generated groups), and he has asked whether the analogue of the "hyperbolic parts" of Thurston's geometrization hold in a more general setting. In this talk, I will describe a particular instance of Gromov's "hyperbolization question", motivated by Thurston's approach, and explain some partial results in this direction. This is joint work with Bestvina, Bromberg, and Kent.
Florian Pop, University of Pennsylvania
Grothendieck's anabelian geometry originates from his famous "Esquisse d'un programme" and "Letter to Faltings". Among the topics of this program, Grothendieck proposed to give a non-tautological description of absolute Galois groups, especially of the absolute Galois group G_Q of the rational numbers. After intensive work by many -- starting with Deligne, Ihara, Drinfel'd -- this development led to the so called Ihara/Oda-Matsumoto conjecture, for short I/OM, which gave (conjecturally) a topological combinatorial description of G_Q. In the talk I will review/explain the question and present the state of the art, in particular recent refinements of I/OM, based on the so called Bogomolov (birational anabelian) Program.
Frank Farris, Santa Clara University
One possible model for the shape of the universe is the Poincaré dodecahedral space, which is a quotient of the 3-sphere by the action of the icosahedral group. To help cosmologists, Jeff Weeks adopted a method originally proposed by Klein to find all the spherical harmonics invariant under the icosahedral and other polyhedral groups. In trying to connect the method to polyhedrally-invariant functions on the 2-sphere, we discovered an interesting connection to self-mappings of the 2-sphere, opening the door to a new technique for mathematical art. (Joint work with Jeff Weeks.)
Jason Manning, Cornell University
Hyperbolic and relatively hyperbolic groups are characterized by having particularly nice kinds of proper actions on coarsely negatively curved (Gromov hyperbolic) spaces. Such a Gromov hyperbolic space has a natural compactification by its Gromov boundary -- this boundary and the group action on it can tell us various things about the group. A central question in this area is the Cannon Conjecture, which would characterize those hyperbolic groups whose boundary is a 2-sphere. I will survey some results connecting group-theoretic structure to the topology of the boundary, and, time permitting, describe some relevant recent work of mine with my collaborators Daniel Groves, Alessandro Sisto, and Oliver Wang.
Cameron Gordon, University of Texas at Austin
The fundamental group is a more or less complete invariant of a 3-dimensional manifold. We will discuss how the purely algebraic question of whether or not this group has a left-invariant total order appears to be related to two other, seemingly quite different, properties of the manifold, one geometric and the other essentially analytic.
Nick Crawford, Technion
I'll discuss some probabilistic models of random walks which interact with their environments. These models are interesting for various reasons. They are relatively natural, yet extremely challenging to control mathematically. Indeed, the simplest variant, Linearly Reinforced Random Walk (LRRW), was introduced by P. Diaconis in the mid 1980’s but general recurrence and transience results only appeared in the last 5 years. Notably, variants (LRRW) have popped up in such diverse subjects as the modeling of path optimization by ant colonies and (the supersymmetric approach to) Anderson localization, first uncovered empirically through the work of M. Disertori, T. Spencer, and M. Zirnbauer and made precise by C. Sabot and P. Tarres.
In this talk I will provide an introduction to LRRW, and a few other walks with reinforcement for the purposes of comparison. Then I will explain the probabilistic concept of exchangeability, which provides LRRW with features that can be exploited. Finally I will survey the recent results on recurrence and transience, obtained by two groups of authors — one group being the 5 authors mentioned above and the other group being O. Angel, myself and G. Kozma.
Gilbert Strang, MIT
The "fundamental theorem of linear algebra" tells us about orthogonal bases for the row space and column space of any matrix. More than that, it identifies the most important part of the matrix -- which is a central goal for a matrix of data. Since data matrices are normally rectangular, singular values must replace eigenvalues. This talk will be partly about the underlying theory and partly about some of its applications to understanding what the matrix tells us. For several one-zero matrices we have open questions about the rank. Alex Townsend has identified an important class of large matrices that have rapidly decaying singular values --- allowing superfast algorithms.
Joan Birman, Columbia University
In the early 1980’s William Harvey, a mathematician working on Teichmuller spaces, introduced a finite dimensional simplicial complex C(S) on which the mapping class group M(S) of a surface S acts. His hope was that C(S) would play a role for M(S) analogous to the role of buildings in the work of Tits on linear groups. The group M(S) is of broad interest in mathematics because of its role in topology, analysis, geometric group theory, algebraic geometry.... The simplicial complex C(S) is connected, has infinite diameter, and an index 2 extension M*(S) turns out to be its automorphism group. (In M(S) maps preserve orientation, in M*(S) one allows orientation-reversing maps too.) We will discuss recent efforts, including a computer program, to use elementary tools to understand the local topology of C(S), focussing on its 1-skeleton, a metric graph in which every edge has length 1.
Timo Seppalainen, University of Wisconsin Madison
This talk begins with a reminder of the law of large numbers and the central limit theorem for classic random walk and then proceeds to models of random paths currently studied in probability and statistical mechanics. In particular, we discuss directed percolation and directed polymer models. Subadditive ergodic theory gives deterministic large scale limiting shapes for these models, but properties of these limits have remained a challenge for decades. We describe some new variational formulas that characterize these limits and connections with other features of the models such as fluctuation exponents. Percolation and polymer models are expected to obey Kardar-Parisi-Zhang (KPZ) universality, as opposed to the Gaussian universality of random walk.
Philip Maini, Oxford University
Collective cell movement is a phenomenon that occurs in normal development, wound healing and disease (such as cancer). In many cases, the ability of cell populations to move large distances coherently arises due to a structure of "leaders" and "followers" within the population. I will present two such examples: (i) angiogenesis -- this the process by which new blood vessels form in response to injury, or in response to a cancerous tumour's demand for more nutrient. We systematically derive a discrete cell-based model for the "snail-trail" phenomenon of blood vessel growth and show that this leads to a novel partial differential equation model. We compare and constrast this model with those in the literature. (ii) neural crest cell invasion - this is the process by which cells move to target locations within the embryo to begin construction of body parts. Through an interdisciplinary research project we show how a hybrid discrete-cell-based mathematical model, and an experimental model, combine to allow us to gain new insights into this phenomenon.
This special event celebrates the professional accomplishments of two distinguished colleagues, Cristian Gutierrez and Martin Lorenz.
Julia Bergner, University of Virginia
The notion of a 2-Segal space was recently defined by Dyckerhoff and Kapranov, and independently by Galvez-Carrillo, Kock, and Tonks under the name of decomposition space. Unlike Segal spaces, which encode the structure of a category up to homotopy, 2-Segal spaces encode a more general structure in which composition need not exist or be unique, but is still associative. Both sets of authors above proved that the output of the Waldhausen $S_\bullet$-construction is a 2-Segal space. In joint work with Osorno, Ozornova, Rovelli, and Scheimbauer, we look at a discrete version of this construction whose output is a 2-Segal set. We show that, via this construction, the category of 2-Segal sets is equivalent to the category of augmented stable double categories. In this talk, I'll introduce 2-Segal sets and spaces, discuss this result and a conjectured homotopical generalization, and, time permitting, look at some other interesting features of 2-Segal spaces.
Tom Church, Stanford University and IAS
Representation theory over Z is famously intractable, but "representation stability" provides a way to get around these difficulties, at least asymptotically, by enlarging our groups until they behave more like commutative rings. Moreover, it turns out that important questions in topology / number theory / representation theory / ... correspond to asking whether familiar algebraic properties hold for these "rings". I'll explain how these connections work; describe what we know and don't know; and give a wide sampling of concrete applications in different fields. No knowledge of representation theory will be required -- indeed, that's sort of the whole point!
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