# Math Calendar

## Rich Subgraphs of a Graph

*Victor Falgas-Ravry, Vanderbilt University*

Location: Stevenson 1432

Let G be a graph on n vertices with edge density p. We say that an m-vertex subgraph H of G is /rich/ if its minimum degree is at least p(m-1). In this talk we shall answer the following question: what is the largest m=m(n,p) for which we can guarantee the existence of a rich subgraph on m vertices? This is joint work with Klas Markström (Umeå) and Jacques Verstraete (UCSD).

## Yamabe Invariant of Symplectic 4-Manifolds of General Type

*Ioana Suvaina, Vanderbilt University*

Location: Stevenson 1308

We compute the Yamabe invariant for a class of symplectic 4-manifolds obtained by taking the rational blow-down of Kahler surfaces. In particular, for any point on the half-Noether line we show that there is a minimal symplectic manifold with known Yamabe invariant.

## Graduate Student Tea

Location: Stevenson 1425

## The Fundamental Group at Infinity for a Finitely Presented Group

*Mike Mihalik, Vanderbilt University*

Location: Stevenson 1310

If a finitely presented group satisfies a certain asymptotic condition called semistability at infinity, then the fundamental group at an end of that group is independent of base ray converging to that end (in analogy with a space being path connected so that fundamental group is independent of base point). The following are long standing open (and associated) questions: Question 1: Are all finitely presented groups semistable at infinity? Question 2: Is H2(G, ZG) free abelian for all finitely presented groups G? We begin this talk with motivation, history, examples and classical results associated with these questions. We end the talk with a proof of the following: Theorem. If a finitely presented group G contains an infinite, finitely generated sub-commensurated subgroup of infinite index, then G is semistable at infinity and H2(G,ZG) is free abelian. This result generalizes many of the classical results on semistability. If H is a subgroup of a group G then H is commensurated in G if for all g in G, the intersection of gHg-1 and H has finite index in both. So commensurated is weaker than normal.

## Approximation Properties for Groups and C*-Algebras

*Uffe Haagerup, University of Copenhagen*

Location: Stevenson 5211

It is a classical result in Fourier analysis, that the Fourier series of a continuous function my fail to converge uniformly or even pointwise to the given function. However if one use a summation method as e.g. convergence in Cesaro mean, one actually gets uniform convergence of the Fourier series. This result can easily be generalized to amenable locally compact groups, where in the non-abelian case, the continuous functions on dual group G^ must be replaced by the reduced group C*-algebra of G. In 1994 Jon Kraus and I introduced a new approximation property (AP) for locally compact groups. The groups having (AP) is the largest class of locally compact groups for which a generalized Cesaro mean convergence theorem can hold. Amenable groups as well as the group SL(2,R) has property (AP), but it was proved by Vincent Lafforgue and Mikael de la Salle in 2011, that SL(n,R) fails to have (AP) for n = 3,4,... In two recent joint works with Tim de Laat we have extend their result by proving that Sp(2,R) and more generally all simple connected Lie groups of real rank >=2 does not have the (AP). In the talk I will give an introduction to amenability and to the property (AP) for locally compact groups, and their relation to other group properties (e. g. weak amenability and Property T). The corresponding properties for C*-algebras will also be discussed. Tea at 3:30 pm in SC 1425.

## Weighted L^2-Extension of Holomorphic Functions from Singular Hypersurfaces

*Vamsi P. Pingali, Johns Hopkins University*

Location: Stevenson 1307

I shall speak about my results on the following problem: “Given a holomorphic function (which is square-integrable with respect to some weighted measure) on a complex hypersurface of C^n, extend it in a square-integrable manner to C^n.” The techniques used involve the so-called “d-bar” PDE. This is joint work with Dror Varolin.

## More Polish Full Groups

*François Le Maître, École Normale Supérieure de Lyon
*

Location: Stevenson 1432

Full groups were introduced in Dye's visionary paper of 1959 as subgroups of Aut(X,m) stable under cutting and gluing their elements along a countable partition of the probability space (X,m). However, the focus has since then been rather on full groups generated by countable groups, which are Polish for the uniform topology. This situation is justified by the very nice interplay between these full groups and von Neumann algebras of countable measure preserving equivalence relations. On the other hand, note that the group Aut(X,m) itself is a full group, Polish for the weak topology, and its topological properties are still an important subject of investigation. In a work in progress with A. Carderi, we investigate Polish full groups of a new kind, whose topology is intermediate between the uniform and the weak topology. They arise as full groups of equivalence relations generated by the Borel action of a Polish group on (X,m), and we will discuss some of their topological properties such as topological rank or amenability.

## How to Use Calculus in Your Everyday Lives

*Josh Sparks, Vanderbilt University*

Location: Stevenson 1206

## 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)).

## 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.

## Talk Title TBA

*Charley Conley, Vanderbilt University*

Location: Stevenson 1206

## Talk Title TBA

*Andrei Martinez-Finkelshtein, Vanderbilt University*

Location: Stevenson 1432

## 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

*Alexandr Kazda, Vanderbilt University*

Location: Stevenson 1206

## Talk Title TBA

*Mikhail Ershov, University of Virginia*

Location: Stevenson 5211

## 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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