# Math Calendar

## Double Threshold Digraphs

*Jerry Spinrad, Vanderbilt University*

Location: Stevenson 1432

In a semiorder, elements are associated with weights, and there is a threshold t such that x > y iff w(x) > w(y)+t. We justify a new model in which there are 2 thresholds t1 and t2; if the difference in weights is smaller than t1, the elements are unrelated; if the difference w(x)-w(y) > t2 then x >y; if t2 > w(x)-w(y) > t1 then x may or may not be greater than y. In general, the goal will be to represent a directed acyclic graph using the minimum possible ratio t2/t1. We will discuss various properties of the class of graphs defined using different ratios t2/t1.

## On Conformally Kähler Surfaces

*Caner Koca, Vanderbilt University*

Location: Stevenson 1308

The famous Frankel Conjecture in complex geometry, which was proved by Siu and Yau in the 1981, asserts that the only compact complex n-manifold that admits a Kahler metric of positive (bi)sectional curvature is the complex projective n-space. In this talk, we prove this conjecture in complex dimension 2 under weaker hypotheses: Namely, a compact complex "surface", which admits a "conformally Kahler" metric g of "positive orthogonal holomorphic bisectional curvature" is biholomorphic to the complex projective plane. Our theorem also has a nice corollary: if, In addition, g is an Einstein metric, then the biholomorphism can be chosen to be an isometry, via which g becomes a multiple of the Fubini-Study metric. (Joint work with M. Kalafat.)

## How to Use Calculus in Your Everyday Lives

*Josh Sparks, Vanderbilt University*

Location: Stevenson 1206

For many undergraduate students, calculus is the first course one takes, and (sadly) often their last. While not the easiest of subjects, the biggest question that boggles course takers is, "How are we going to use this in real life?" This presentation will not only apply calculus to the real world, but to things that actually mean something, like falling in love and eating lots of food. By delving upon the three main aspects of introductory calculus -- the limit, the derivative, and the integral -- we'll start off the year with a fun presentation that will actually help you use calculus in your everyday lives.

## Graduate Student Tea

Location: Stevenson 1425

## Phase Retrieval: Approaching the Theoretical Limits in Practice

*Dustin Mixon, Air Force Institute of Technology*

Location: Stevenson 1432

In many areas of imaging science, it is difficult to measure the phase of linear measurements. As such, one often wishes to reconstruct a signal from intensity measurements, that is, perform phase retrieval. Very little is known about how to design injective intensity measurements, let alone stable measurements with efficient reconstruction algorithms. This talk helps to fill the void - I will discuss a wide variety of recent results in phase retrieval, including various conditions for injectivity and stability (joint work with Afonso S. Bandeira (Princeton), Jameson Cahill (Duke) and Aaron A. Nelson (AFIT)) as well as measurement designs based on spectral graph theory which allow for efficient reconstruction (joint work with Boris Alexeev (Princeton), Afonso S. Bandeira (Princeton) and Matthew Fickus (AFIT)). In particular, I will show how Fourier-type tricks can be leveraged in concert with this graph-theoretic design to produce pseudorandom aperatures for X-ray crystallography and related disciplines (joint work with Afonso S. Bandeira (Princeton) and Yutong Chen (Princeton)).

## Some Unitary Representations of the Thompson Groups

F and T

*Vaughan Jones, Vanderbilt University*

Location: Stevenson 1310

In a "naive" attempt to create algebraic quantum field theories on the circle, we obtain a family of unitary representations of Thompson's groups T and F for any subfactor. In the simplest case the coefficients of the representations are polynomial invariants of links and the question arises of just what links the Thompson group produces.

## Basic Reproduction Number in Population Models with Periodic Forcing

*Cameron Browne, Vanderbilt University*

Location: Stevenson 1307

Seasonality and periodic control measures may be important features of certain systems arising in epidemiology and ecology. I will give an overview of the basic reproduction number for structured population models in periodic environments. The definition of the basic reproduction number involves the spectral radius of an integral operator derived from the linearized partial differential equation system which describes the population dynamics. Then I will consider two examples in which calculating the reproduction number can provide insight on optimal timing of periodic interventions. Specifically, I study a within-host HIV model with combination antiviral drug treatment and a multi-patch epidemic model with periodic pulse vaccinations.

## Compact Complex Surfaces and Locally Conformally Flat Metrics

*Caner Koca, Vanderbilt University*

Location: Stevenson 1308

The question of existence of Einstein metrics on compact smooth 4-manifolds is a classical problem in Differential Geometry. Significant progress has been achieved in recent decades if one looks for Einstein metrics on complex surfaces. Inspired by this, we are interested in the question of existence of another important family of canonical metrics, called locally conformally flat metrics (LCF, for short), on compact complex surfaces. Our first result in this direction reduces the question down to a list of well-known cases: If a compact complex surface admits a LCF metric, then it cannot contain a smooth rational curve of odd self intersection. In particular, the surface has to be minimal. We will also give a list of possibilities. Whether or not each possibility in our list is realized by an example is an interesting open problem. (Joint work with M. Kalafat.)

## Talk Title TBA

*Charley Conley, Vanderbilt University*

Location: Stevenson 1206

## Talk Title TBA

*Andrei Martinez-Finkelshtein, Vanderbilt University*

Location: Stevenson 1432

## Commuting Degree for Infinite Groups

*Yago Antolin Pichel, Vanderbilt University*

Location: Stevenson 1310

There is a classical result saying that, in a finite group, the probability that two elements commute is never between 5/8 and 1 (i.e. if it is greater than 5/8 the group is abelian). In this talk we present a generalization of this result for infinite finitely generated groups. The main result is the following The main one is the following: "A polynomially growing group G has positive commuting degree if and only if it is virtually abelian". This is a Joint work with Enric Ventura and Armando Martino.

## Talk Title TBA

*Kun Wang, Vanderbilt University*

Location: Stevenson 1310

## Shock Formation in Solutions to 3D Wave Equations

*Jared Speck, Massachusetts Institute of Technology*

Location: Stevenson 5211

I will provide an overview of the formation of shock waves, developing from small, smooth initial conditions, in solutions to quasilinear wave equations in 3 spatial dimensions. I will first describe prior contributions from many researchers including F. John, S. Alinhac, and especially D. Christodoulou. I will then describe some results from my recent book, in which I show that for two important classes of wave equations, a necessary and sufficient, condition for the phenomenon of small-data shock-formation is the failure of S. Klainerman's classic null condition. I will highlight some of the main ideas behind the analysis including the critical role played by geometric decompositions based on true characteristic hypersurfaces. Some aspects of this work are joint with G. Holzegel, S. Klainerman, and W. Wong. Tea at 3:30 pm in SC 1425.

## Talk Title TBA

*Brent Nelson, UCLA
*

Location: Stevenson 1432

## Talk Title TBA

*Efim Zelmanov, University of California, San Diego*

Location: Stevenson 5211

## Universal Single Qubit and Qutrit Gates in the Kauffman-Jones Version of SU(2) Chern-Simons Theory at Level 4

*Claire Levaillant, UCSB
*

Location: Stevenson 1432

This is a recent development regarding universal topological quantum computation in a specific anyonic system, as appearing in the title, and joint work with Michael Freedman and Station Q. The anyonic system we use is hoped to become physically realizable. Our starting point are two Jones unitary representations of the braid group on four strands. One representation arises from braiding four anyons of respective topological charges 1,2,2,1 and the second representation occurs when braiding four anyons of identical topological charge 2. Both representations have a finite image and this image yields a finite subgroup of SU(2) and SU(3) respectively whose elements are called quantum gates. By protocols involving both braids and measurements, we show how to make in each case an additional quantum gate. In the qubit case, this new gate generates an infinite subgroup of SU(2) and in the qutrit case, the new gate enlarges the size of the finite SU(3) group issued from braiding only. Our method uses ancilla preparation with adequate norms and interesting relative phases and fusion of the ancilla into the input in order to form the gate.

## Talk Title TBA

*Andrew Sale, Vanderbilt University*

Location: Stevenson 1310

## Nonlocal Phenomena in Partial Differential Equations

*Glenn Webb, Vanderbilt University*

Location: Stevenson 1307

Four examples of nonlocal phenomena in partial differential equations will be presented: (1) partial differential equations with time delay (non-local in time – the future depends not only on the present time, but on a history before the present time; (2) age structured population models (nonlocal in the boundary condition – offspring are born at age 0 from a mother with age in a specified age range); (3) cell-cell adhesion models (nonlocal in the transport term – cells have a spatial sensing radius, on the order of several cell diameters, that modulates their adhesion to other cells within their sensing radius); (4) interference phenomena in quantum mechanics (nonlocal in the probability density of spatial position – the detection of a quantum particle is determined only probabilistically).

## Talk Title TBA

*Brian Simanek, Vanderbilt University*

Location: Stevenson 1206

## Talk Title TBA

*Johanna Stromberg, Vanderbilt University*

Location: Stevenson 1206

## Banded Matrices and Fast Inverses

*Gilbert Strang, Massachusetts Institute of Technology*

Location: Stevenson 5211

The inverse of a banded matrix A has a special form which we can find directly from the "Nullity Theorem." Then the inverse of that matrix A^-1 is the original A -- which can be found by a remarkable "local" inverse formula. This formula uses only the banded part of A^-1 and it offers a very fast algorithm to produce A. That fast algorithm has a potentially valuable application. Start now with a banded matrix B (possibly the positive definite beginning of a covariance matrix C -- but covariances outside the band are unknown or too expensive to compute). It is a poor idea to assume that those covariances are zero. Much better to complete B to C by a rule of maximum entropy which for Gaussians means maximum determinant. As statisticians and also linear algebraists discovered, the optimally completed matrix C is the inverse of a banded matrix. Best of all, the matrix actually needed in computations is that banded C^-1 (which is not B !).And C^-1 comes quickly and efficiently from the local inverse formula. A very special subset of banded matrices contains those whose inverses are also banded. These arise in studying orthogonal polynomials and also in wavelet theory -- the wavelet transform and its inverse are both banded ( = FIR filters). We describe a factorization for all banded matrices that have banded inverses. Tea at 3:30 pm in SC 1425.

## Talk Title TBA

*James Benn, Notre Dame University*

Location: Stevenson 1307

## Talk Title TBA

*Zach Gaslowitz, Vanderbilt University*

Location: Stevenson 1206

## Talk Title TBA

*Mikhail Ershov, University of Virginia*

Location: Stevenson 5211

## Talk Title TBA

*Alexandr Kazda, Vanderbilt University*

Location: Stevenson 1206

## Talk Title TBA

*Russell Lyons, Indiana University*

Location: Stevenson 5211

## Talk Title TBA

*Jeremy LeCrone, Kansas State University*

Location: Stevenson 1307

## Formation of Trapped Surfaces in General Relativity

*Xinliang An, Rutgers University*

Location: Stevenson 1307

The first is a simplified approach to Christodoulou’s monumental result which showed that trapped surfaces can form dynamically by the focusing of gravitational radiation from past null infinity. We extend the methods of Klainerman-Rodnianski, who gave a simplified proof of this result in a finite region. The second result extends the theorem of Christodoulou by allowing for weaker initial data but still guaranteeing that a trapped surface forms in the causal domain. In particular, we show that a trapped surface can form dynamically from initial data which is merely large in a scale-invariant way. The second result is obtained jointly with Luk.

## Talk Title TBA

*Morwen Thistlethwaite, University of Tennessee, Knoxville*

Location: Stevenson 5211

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