Department of Mathematics

Co-organizers:

Angela Gibney (angela.gibney {at} gmail [dot] com)

Danny Krashen (daniel.krashen {at} gmail [dot] com)

Vladimir Retakh (vretakh {at} math [dot] rutgers [dot] edu)

Chris Woodward (ctw {at} math [dot] rutgers [dot] edu)

April 21: Nikolay Bogachev (Skoltech & MIPT)

TBA: Alessio Corti (Imperial College)

TBA: Bietmar Salamon (ETH)

This talk is devoted to the memory of my teacher, Professor Ernest Borisovich Vinberg: 1937 -- 2020. All of us are familiar with the kaleidoscope, in which multicolored glasses form an attractive and amazing reflection pattern. Such pictures are obtained by a system of reflections with respect to certain mirrors. Generalizing this concept, we may consider discrete reflection groups of finite covolume in Euclidean spaces E^n and on the spheres S^n. Such were studied by many mathematicians and finally classified completely by Coxeter in 1933 via Coxeter diagrams. They exist in all dimensions n for both E^n and S^n.

The story of reflection groups in hyperbolic Lobachevsky spaces H^n (and to the phenomenal impact of Vinberg in this theory) goes back to the 19th century, to the works of Poincare and Dyck about the classification of Fuchsian groups. However only low-dimensional examples of such groups were known. In 1967, Vinberg initiated his fundamental theory of hyperbolic reflection groups. In 1972, he suggested an algorithm for constructing the fundamental Coxeter polytope of an arbitrary hyperbolic reflection group. The Vinberg algorithm is now widely used by many people in different branches of mathematics. In 1981, Vinberg obtained the following celebrated and surprising result: there are no compact hyperbolic Coxeter polytopes and no arithmetic finite volume Coxeter polytopes in H^n with n>29. With these results, Vinberg was selected as an Invited Speaker at the ICM 1983. In 2014, he constructed the first examples of higher-dimensional non-arithmetic non-compact hyperbolic Coxeter polytopes

We will discuss the interplay between three branches of Conformal Dynamics: iteration of (anti-)rational maps, actions of Kleinian groups, and dynamics generated by Schwarz reflections in quadrature domains. We will show examples of Schwarz reflections obtained by matings between anti-quadratic maps and the triangle modular group, and examples of Julia realizations for Apollonian-like gaskets. Some of these examples can be turned into others by means of a David surgery.

The celebrated Ratner's orbit closure theorem proved around 1990 says that in a homogeneous space of finite volume, the closure of an orbit of any subgroup generated by unipotent flows is homogeneous. A special case of Ratner's theorem (also proved by Shah independently) implies that the closure of a geodesic plane in a hyperbolic manifold of finite volume is always a properly immersed submanifold. Searching for analogs of Ratner's theorem in the infinite volume setting is a major challenge. We present a continuous family of hyperbolic 3-manifolds, and a countable family of higher dimensional hyperbolic manifolds of infinite volume, for which we have an analogue of Ratner's theorem. (Based on joint work with McMullen, Mohammadi, Benoist and Lee in different parts.)

Mirror symmetry is a field of mathematics that originated in string theory, where particles are replaced by strings and six extra small dimensions are needed to describe the universe. These extra dimensions are wrapped into Calabi-Yau varieties. Mirror symmetry stems from the observation that Calabi-Yau varieties occur in dual mirror pairs which result in equivalent physical theories. Mirror pairs are of very different geometric nature, but the symplectic geometry of one variety is reflected in the algebraic geometry of its mirror. One of the main challenges in this field is to understand how to construct the mirror to a given Calabi Yau. A first step to understand a symmetry is to find and study its fixed points. In this seminar I will focus on a special class of such fixed points and describe their properties.

The Erdos-Hajnal conjecture states that for every graph H there is a constant \epsilon(H) such that every n-vertex graph with no induced subgraph isomorphic to H has a clique or a stable set of size n^{\epsilon}. This conjecture has only been verified for a few graphs H, and in particular until recently it remained open for the case when H is a cycle of length 5. This special case received a considerable amount of attention. In this talk we will survey some known results, and discuss the recent proof for the case of a cycle of length 5.

This is joint work with Alex Scott, Paul Seymour and Sophie Spirkl.

Fano manifolds are basic building blocks in algebraic geometry, and their classification is a long-standing open problem. There is exactly 1 one-dimensional Fano manifold, the line; there are 10 two-dimensional Fano manifolds, the del Pezzo surfaces; and there are 105 three-dimensional Fano manifolds. Almost nothing is known about the classification in higher dimensions. I will describe a program to find and classify Fano manifolds using mirror symmetry, which should work in higher dimensions as well. This is joint work with Corti, Galkin, Golyshev, Gross, Kalashnikov, Kasprzyk, Prince, and others.

Over the last forty years, most progress in four-dimensional topology came from gauge theory and related invariants. Khovanov homology is an invariant of knots of a different kind: its construction is combinatorial, and connected to ideas from representation theory. There is hope that it can tell us more about smooth 4-manifolds; for example, Freedman, Gompf, Morrison and Walker suggested a strategy to disprove the smooth 4D Poincare conjecture using Rasmussen's invariant from Khovanov homology. It is yet unclear whether their strategy can work, and I will explain some of its challenges. I will also review other topological applications of Khovanov homology, with regard to smoothly embedded surfaces in 4-manifolds.

The Hasse principle is a useful guiding philosophy in arithmetic geometry that relates "global" questions to analogous "local" questions, which are often easier to understand. A simple incarnation of the Hasse principle says that a given polynomial equation has a solution in the rational numbers (i.e., is "globally soluble") if and only if it has a solution in the real numbers and in the p-adic numbers for all primes p (i.e., is "everywhere locally soluble"). While this principle holds for many "simple" such polynomials, it is a very difficult question to classify the polynomials (or more generally, algebraic varieties) for which the principle holds or fails.

In this talk, we will discuss problems related to the Hasse principle for some classes of varieties, especially certain genus one curves. We will describe how to compute the proportion of these curves that are everywhere locally soluble (joint work with Tom Fisher and Jennifer Park), and we will explain why the Hasse principle fails for a positive proportion of these curves (joint work with Manjul Bhargava).

Recall that an algebraic variety is said to be __rational__ if it contains a Zariski-open subset isomorphic to a Zariski-open subset of projective space. Subtle questions of rationality have seen a great deal of recent interest and progress, but most varieties aren't rational. I will survey a developing body of work in a complementary direction, centered around measures of irrationality for varieties whose non-rationality is known.

There is a long history of networked dynamical systems that models the spread of opinions over social networks, with the graph Laplacian playing a lead role. One of the difficulties in modelling opinion dynamics is the presence of polarization: not everyone comes to consensus. This talk will describe work joint with Jakob Hansen [OSU] introducing a new model for opinion dynamics using sheaves of vector spaces over social networks. The graph Laplacian is enriched to a Hodge Laplacian, and the resulting dynamics on *discourse sheaves* can lead to some very interesting and perhaps more realistic outcomes.

In this talk, I will discuss "quantum symmetry" from an algebraic viewpoint, especially for symmetries of algebras. The term "quantum" is used as algebras here are usually noncommutative. I will mention some interesting results on when symmetries of algebras must factor or do not factor through symmetries of classical gadgets (such as groups or Lie algebras), that is, when we must enter the realm of quantum groups (or Hopf algebras) to understand symmetries of a given algebra. This all fits neatly into the framework of studying algebras in monoidal categories, and if time permits, I will give some recent results in this direction. I aim to keep the level of the talk down-to-earth by including many basic definitions and examples.

Hilbert's third problem asks: do there exist two polyhedra with the same volume which are not scissors congruent? In other words, if $P$ and $Q$ are polyhedra with the same volume, is it alwayspossible to write $P = \bigcup_{i=1}^n P_i$ and $Q = \bigcup_{i=1}^nQ_i$ such that the $P$'s and $Q$'s intersect only on the boundaries and such that $P_i \cong Q_i$? In 1901 Dehn answered this question in the negative by constructing a second scissors congruence invariant now called the "Dehn invariant," and showing that a cube and a regular tetrahedron never have equal Dehn invariants, regardless of their volumes. We can then restate Hilbert's third problem: do the volume and Dehn invariant separate the scissors congruence classes? In 1965 Sydler showed that the answer is yes; in 1968 Jessen showed that this result extends to dimension 4, and in 1982 Dupont and Sah constructed analogs of such results in spherical and hyperbolic geometries. However, the problem remains open past dimension 4. By iterating Dehn invariants Goncharov constructed a chain complex, and conjectured that the homology of this chain complex is related to certain graded portions of the algebraic K-theory of the complex numbers, with the volume appearing as a regulator. In joint work with Jonathan Campbell, we have constructed a new analysis of this chain complex which illuminates the connection between the Dehn complex and algebraic K-theory, and which opens new routes for extending Dehn's results to higher dimensions. In this talk we will discuss this construction and its connections to both algebraic and Hermitian K-theory, and discuss the new avenues of attack that this presents for the generalized Hilbert's third problem.

Time-frequency analysis seeks to study functions or signals in both space and frequency. In this talk, I will focus on one time-frequency tool: the Weyl-Heisenberg or Gabor systems. The origin of these systems can be partially attributed to Dennis Gabor, who, in 1946, claimed that any square Lebesgue integrable function can be written as an infinite linear combination of time and frequency shifts of the standard Gaussian. Since then, decomposition methods for larger classes of functions or distributions in terms of various elementary building blocks have led to an impressive body of work in harmonic analysis. In this talk, I will give a brief introduction to the theory of Gabor frames and will survey some related easily stated yet unresolved open problems.

Hilbert's 13th Problem (H13) is a fundamental open problem about polynomials in one variable. It is part of a beautiful (but mostly forgotten) story going back 3 thousand years. In this talk I will explain how H13 (and related problems) fits into a wider framework that includes problems in enumerative algebraic geometry and the theory of modular functions. I will then report on some recent progress, joint with Mark Kisin and Jesse Wolfson. While some fancy objects will appear in this talk, much of it should (I hope) be understandable to undergraduate math majors.

This talk will offer information on K-12 math education in the US, how this connects with the job of a university math department, and roles that individual concerned mathematicians can play. The speaker, whose primary activity is math research, is also Vice Chair of the US National Commission on Math Instruction (NAS).

I will discuss geometric and dynamical properties of actions of discrete groups on Riemannian symmetric spaces. I will highlight some aspects of the interplay between geometry and dynamics, and present some recent results which generalize theorems of Sullivan, Bridgeman-Taylor, McMullen for convex cocompact subgroups acting on hyperbolic space in the framework of discrete subgroups of Lie groups of higher rank.

We will consider the inverse problem of determining the sound speed or index of refraction of a medium by measuring the travel times of waves going through the medium. This problem arises in global seismology in an attempt to determine the inner structure of the Earth by measuring travel times of earthquakes. It has also several applications in optics and medical imaging among others.

The problem can be recast as a geometric problem: Can one determine the Riemannian metric of a Riemannian manifold with boundary by measuring the distance function between boundary points? This is the boundary rigidity problem. The linearization of this problems involves the integration of a tensor along geodesics, similar to the X-ray transform.

We will also describe some recent results, joint with Plamen Stefanov and Andras Vasy, on the partial data case, where you are making measurements on a subset of the boundary.

No previous knowledge of Riemannian geometry will be assumed.

The first meeting of the department colloquium is tomorrow at 3:30; please see the Zoom link below. The first few weeks will be short talks by new postdocs, to get to know and welcome them to the department (as best we can virtually). Tomorrow's speakers are: Abid Ali, Yunbai Cao, and Chris Lutsko. We invite all to meet at "tea" (gather.town) as usual at 3, and head over to zoom at 3:30. The schedule is then: 10 min lecture, 5 min questions/discussion, 5 mins break (if people like, they can go back into the "lounge", gather.town), repeating for each of the 3 speakers.

Best,

Alex Kontorovich, on behalf of co-organizers Angela Gibney, Danny Krashen, and Chris Woodward.

This page is maintained by Chris Lutsko.