Thursday September 06, 2018 4:00 PM  5:00 PM Math Tower P131
 Mark Pollicott, University of Warwick
Dimensions of Cantor sets and Diophantine approximationThere are two recent cases where a specific knowledge of the dimension of certain Cantor set in the unit interval has had applications to problems in the Diophantine approximation. The first is on estimates by Matheus and Moriera on the size of the difference between the Lagrange and Markoff spectra, and the second is on the work of BourgainKontorovich and Huang on the Zaremba conjecture. We will talk about these problems and, in particular, about the progress that has been made in estimating the dimension of such dynamically defined Cantor sets.

Thursday September 13, 2018 4:00 PM  5:00 PM Math Tower P131

No Colloquium

Thursday September 20, 2018 4:00 PM  5:00 PM Math Tower P131
 Pace Nielsen, Brigham Young University
Why Euclidean domains are both easier and harder than you thinkWe give a gentle introduction to the proof that there is a Euclidean domain with no multiplicative norm.

Thursday September 27, 2018 4:00 PM  5:00 PM Math Tower P131
 Boris Bukh, Carnegie Mellon
Bipartite Turan problem
What is the largest graph not containing a copy of a fixed graph H? This question of Turan from 1941 remains open. The most challenging case is when H is bipartite. This case leads to interesting graph embedding problems on one hand, and mysterious algebrogeometric constructions on the other. In the talk, I will discuss the key problems and recent progress in the area.

Friday September 28, 2018 4:00 PM  5:00 PM SCGP 102
 Federico Rodriguez Hertz, Penn State University
On the geometry of invariant measuresIn this talk I will try to explain that, in dynamical systems with some hyperbolicity, some invariant measures present a very nice geometric structure. I also will explain how this geometric structures can be used to recover certain rigidity of the system in study. The goal is to show this approach through examples, so I plan to spend some time describing these examples, their relevant measures and geometry.

Thursday October 04, 2018 4:00 PM  5:00 PM Math Tower P131
 Kevin Ford, Univ. of Illinois at UrbanaChampaign
Probabilistic aspects of number theoryWe describe how probabilistic tools and reasoning are used to study problems about gaps between prime numbers in various sequences, for example the sequence of all primes, and the sequence of prime values of a polynomial.

Thursday October 11, 2018 4:00 PM  5:00 PM Math Tower P131
 Lenny Ng, Duke
Studying knots through the cotangent bundle and symplectic geometryAn old idea of Arnol'd is to examine the topology of a smooth manifold through the symplectic geometry of its cotangent bundle, building on the familiar concept of phase space from classical mechanics. One particular application of this approach yields strong invariants of knots, with some unexpected relations to topological string theory. I'll discuss the current state of affairs regarding this approach to knot invariants through symplectic geometry.

Thursday October 18, 2018 4:00 PM  5:00 PM Math Tower P131
 Ailana Fraser, UBC
Geometries that optimize eigenvaluesWhen we choose a metric on a manifold we determine the spectrum of the Laplace operator. Thus an eigenvalue may be considered as a functional on the space of metrics. For example the first eigenvalue would be the fundamental vibrational frequency. In some cases the normalized eigenvalues are bounded independent of the metric. In such cases it makes sense to attempt to find critical points in the space of metrics. For surfaces, the critical metrics turn out to be the induced metrics on certain special classes of minimal (mean curvature zero) surfaces in spheres and Euclidean balls. The eigenvalue extremal problem is thus related to other questions arising in the theory of minimal surfaces. In this talk we will give an overview of progress that has been made for surfaces with boundary, and contrast this with some recent results in higher dimensions. This is joint work with R. Schoen.

Thursday October 25, 2018 4:00 PM  5:00 PM SCGP 102
 Nigel Hitchin, University of Oxford
Semiflat hyperkaehler manifolds and their submanifoldsThis is a joint lecture with the SCGP Workshop Geometrical Aspects of Supersymmetry.

Thursday November 01, 2018 4:00 PM  5:00 PM Math Tower P131
 Leon Takhtajan, Stony Brook University
Scattering theory for mathematiciansMathematical scattering theory has its origin in quantum mechanics. It is a beautiful part of functional analysis, and was a very active area of research in the second half of the previous century. In this talk I will explain the basic notions of scattering theory and will present its applications, old and recent, to the spectral theory, nonlinear waves and number theory. No prior knowledge of the scattering theory is assumed.

Thursday November 08, 2018 4:00 PM  5:00 PM Math Tower P131
 Nikita Nekrasov, SCGP
Lefschetz thimbles from Fermi curvesThe steepest descent method of evaluating oscillatory integrals has been recently connected to the localization computations in topological field theories. The infinitedimensional analogues of such computations lead to exact quantization conditions in quantum mechanics. In the talk I will review the recent progress in the next step of this program. We study the functional Lefschetz thimbles, i.e. the possible path integral contours, in the examples of the O(N) model of two dimensional quantum field theory. Specifically, we find a large class of complex critical points of the sigma model actions which are relevant for the theory in finite volume at finite temperature, using the theory of finitegap potentials. Based on the joint work with I. Krichever.

Tuesday November 27, 2018 4:00 PM  5:00 PM Math Tower P131
 Theodore Drivas, Princeton University
Dissipation and Unpredictability in Onsager's Theory of "Ideal" TurbulenceWe describe some recent advances in mathematical turbulence theory, with a focus on the observable phenomena of enhanced dissipation, strong mixing, and unpredictability.

Thursday November 29, 2018 4:00 PM  5:00 PM Math Tower P131
 Kristen Hendricks, MSU
New invariants of homology cobordismThis is a talk about 3manifolds and knots. We will begin by reviewing some basic constructions and motivations in lowdimensional topology, and will then introduce the homology cobordism group, the group of 3manifolds with the same homology as the 3dimensional sphere up to a reasonable notion of equivalence. We will discuss what is known about the structure of this group and its connection to higher dimensional topology. We will then discuss some existing invariants of the homology cobordism group coming from gauge theory and symplectic geometry, particularly Floer theory. Finally, we will construct a new invariant of homology cobordism coming from an equivariant version of the computationallyfriendly Floertheoretic 3manifold invariant Heegaard Floer homology, and use it to construct a new filtration on the homology cobordism group and derive some structural applications. Parts of this talk are joint work with C. Manolescu and I. Zemke; more recent parts of this talk are joint work with J. Hom and T. Lidman.

Tuesday December 04, 2018 4:00 PM  5:00 PM Math Tower P131
 Felix Janda, Michigan
Enumerative geometry: old and newFor 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 have 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 GromovWitten invariants of the quintic threefold, a CalabiYau manifold. I will explain a recent breakthrough in understanding the higher genus invariants that stems from a seemingly unrelated problem related to the study of holomorphic differentials on Riemann surfaces.

Thursday December 06, 2018 4:00 PM  5:00 PM Math Tower P131
 Max Engelstein, MIT
The role of Energy in RegularityThe calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the EulerLagrange equation, which is a partial differential equation satisfied by the critical points of the energy.
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.
We will then turn the tables, and examine PDEs which look like they should be an EulerLagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.

Wednesday December 12, 2018 2:30 PM  3:30 PM Math Tower P131
 Manuel Rivera, University of Miami
A new point in topologyOne knows in algebraic topology the homotopical properties of geometric spaces can be recast into the language of infinite dimensional topological groups determined by function spaces of closed loops in the geometric spaces. For example the zeroth homology of the function space of based loops can be naturally identified with the group algebra of the fundamental group of the geometric space. This group algebra has a compatible coproduct determined by saying the group elements are grouplike for the coproduct. Conversely the group like elements in this coalgebra form a group equivalent to the given one.
The new point in topology says this bialgebra determining the fundamental group and higher dimensional aspects can, in complete generality, be determined directly from algebraic structure on geometric chains in the geometric space itself. The algebraic construction that does this produces a free differential algebra from a differential coalgebra and was introduced sixty years ago for simply connected spaces. Remarkably it is understood only now to work fine for all geometric spaces if one adds something to it.
The new idea beyond technique is to combine the algebraic construction from the past with the infinite homotopical symmetry of chain approximations to the diagonal mapping of the geometric space which is itself perfectly symmetrical. This makes the chain coalgebra on the geometric space cocommutative in a derived sense so that the construction from the past becomes enriched to a bialgebra in a derived sense. One now sees the fundamental group in the zeroth homology of the enhanced algebraic construction and furthermore one sees higher dimensional homotopical information about the geometric space in the homotopy type of the enhanced algebraic construction.

Thursday December 13, 2018 4:00 PM  5:00 PM Math Tower P131
 Bhargav Bhatt, University of Michigan
Interpolating padic cohomology theoriesIntegration of differential forms against cycles on a complex manifold helps relate de Rham cohomology to singular cohomology, which forms the beginning of Hodge theory. The analogous story for padic manifolds, which is the subject of padic Hodge theory, is richer due to a wider variety of available cohomology theories (de Rham, etale, crystalline, and more) and torsion phenomena. In this talk, I will give a bird's eye view of this picture, guided by the recently discovered notion of prismatic cohomology that provides some cohesion to the story. (Based on joint work with Morrow and Scholze as well as work in progress with Scholze.)

