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Colloquium
 Xuancheng Fernando Shao, Oxford
 Understanding sieve via additive combinatorics
 01/11/2017
 4:10 PM  5:00 PM
 C304 Wells Hall
Many of the most interesting problems in number theory can be phrased under the general framework of sieve problems. For example, the ancient sieve of Eratosthenes is an algorithm to produce primes up to a given threshold. Sieve problems are in general very difficult, and a class of clever techniques have been discovered in the last 100 years to yield stronger and stronger results. In this talk I will discuss the significance of understanding general sieve problems, and present a novel approach to study them via additive combinatorics. This is joint work with Kaisa Matomaki.

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Geometry and Topology
 Jose Perea, MSU
 Projective coordinates for the analysis of data
 01/12/2017
 2:00 PM  2:50 PM
 C304 Wells Hall
Barcodes  the persistent homology of data  have been shown to be effective quantifiers of multiscale structure in finite metric spaces. Moreover, the universal coefficient theorem implies that (for a fixed field of coefficients) the barcodes obtained with persistent homology are identical to those obtained with persistent cohomology. Persistent cohomology, on the other hand, is better behaved computationally and allows one to use convenient interpretations such as the Brown representability theorem. We will show in this talk how one can use persistent cohomology to produce maps from data to (real and complex) projective space, and conversely, how to use these projective coordinates to interpret persistent cohomology computations.

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Colloquium
 Sanchayan Sen, McGill University
 Random discrete structures: Phase transitions, scaling limits, and universality
 01/13/2017
 4:10 PM  5:00 PM
 C304 Wells Hall
The aim of this talk is to give an overview of some recent results in two interconnected areas:
a) Random graphs and complex networks: The last decade of the 20th century saw significant growth in the availability of empirical data on networks, and their relevance in our daily lives. This stimulated activity in a multitude of fields to formulate and study models of network formation and dynamic processes on networks to understand realworld systems.
One major conjecture in probabilistic combinatorics, formulated by statistical physicists using nonrigorous arguments and enormous simulations in the early 2000s, is as follows: for a wide array of random graph models on n vertices and degree exponent \tau>3, typical distance both within maximal components in the critical regime as well as on the minimal spanning tree on the giant component in the supercritical regime scale like n^{\frac{\tau\wedge 4 3}{\tau\wedge 4 1}}. In other words, the degree exponent determines the universality class the random graph belongs to. The mathematical machinery available at the time was insufficient for providing a rigorous justification of this conjecture.
More generally, recent research has provided strong evidence to believe that several objects, including
(i) components under critical percolation,
(ii) the vacant set left by a random walk, and
(iii) the minimal spanning tree,
constructed on a wide class of random discrete structures converge, when viewed as metric measure spaces, to some random fractals in the GromovHausdorffProkhorov sense, and these limiting objects are universal under some general assumptions. We report on recent progress in proving these conjectures.
b) Stochastic geometry: In contrast, less precise results are known in the case of spatial systems. We discuss a recent result concerning the length of spatial minimal spanning trees that answers a question raised by Kesten and Lee in the 90's, the proof of which relies on a variation of Stein's method and a quantification of the classical BurtonKeane argument in percolation theory.
Based on joint work with Louigi AddarioBerry, Shankar Bhamidi, Nicolas Broutin, Sourav Chatterjee, Remco van der Hofstad, and Xuan Wang.

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Seminar in Cluster algebras
 John Machacek, MSU
 Introducing the Combinatorics Behind the Amplituhedron
 01/17/2017
 2:00 PM  3:00 PM
 C304 Wells Hall
No abstract available.
Colloquium
 Thomas Walpuski, MIT
 Gauge Theory in Higher Dimensions
 01/17/2017
 4:10 PM  5:00 PM
 C304 Wells Hall
Gauge theory is a subject that has emerged from theoretical physics. It has deep links with many areas of mathematics (including partial differential equations, representation theory, algebraic geometry, differential geometry and topology). Most of the mathematical work on gauge theory has focused on low dimensions, where one can exploit the antiselfdual YangMills equation and the analytic difficulties are still quite tractable.
In this talk I will discuss three concrete questions that arise in gauge theory in higher dimensions. First, I will discuss gauge theory on Kähler manifolds, with a particular focus on singular Hermitian YangMills connections. Afterwards, I will move on to the more exotic topic of gauge theory on G2manifolds. I will discuss a method to construct solutions of the YangMills equation on a class of G2manifolds called twisted connected sums. Finally, I will talk about the prospects of defining enumerate gauge theoretical invariants for G2manifolds and the difficulties arising from codimension four bubbling.

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Mathematical Physics and Gauge Theory
 Soeren Petrat, IAS and Princeton
 Free dynamics of a tracer particle in a Fermi sea
 01/19/2017
 11:00 AM  11:50 AM
 C304 Wells Hall
The talk is about the dynamics of a tracer particle coupled strongly to a dense noninteracting electron gas in one or two dimensions. I will present a recent result that shows that for high densities the tracer particle moves freely for very long times, i.e., the electron gas becomes transparent. However, the correct phase factor is nontrivial. To leading order, it is given by meanfield theory, but one also has to include a correction coming from immediate recollision diagrams.
Geometry and Topology
 Renaud Detcherry, MSU
 TuraevViro invariants of links and the colored Jones polynomial
 01/19/2017
 2:00 PM  2:50 PM
 C304 Wells Hall
In a recent work by Tian Yang and Qingtao Chen, it has been observed that one can recover the hyperbolic volume from the asymptotic of TuraevViro invariants of 3manifolds at a specific root of unity. This is reminiscent of the volume conjecture for the colored Jones polynomial.
In the case of link complements, we derive a formula to express TuraevViro invariants as a sum of values of colored Jones polynomial, and get a proof of Yang and Chen's conjecture for a few link complements. We also discuss the link between this conjecture and the volume conjecture. This is joint work with Effie Kalfagianni and Tian Yang.

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Colloquium
 Sean Li, University of Chicago
 Beyond Euclidean rectifiability
 01/23/2017
 4:10 PM  5:00 PM
 C304 Wells Hall
Rectifiable spaces are a class of metric measure spaces that are Lipschitz analogues of differentiable manifolds (for example, they admit a parameterization by Lipschitz charts) and arise naturally in many areas of analysis and geometry. Due to the important works of Federer, Mattila, Preiss, and many others, we now have a good understanding of the geometric properties of rectifiability in Euclidean spaces. In this talk, we examine some generalizations of rectifiability to the setting of nonEuclidean spaces and discuss the similarities and differences between rectifiability in the Euclidean setting and these generalizations.

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Seminar in Cluster algebras
 Rob Davis, MSU
 Plabic Graphs Part 1
 01/24/2017
 2:00 PM  2:50 PM
 C304 Wells Hall
No abstract available.

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Mathematics Education Colloquium Series
 M. Alejandra Sorto, Texas State University
 What matters for learners in linguistically diverse classrooms?
 01/25/2017
 3:30 PM  5:00 PM
 252 EH
In this talk, I will discuss which factors contribute to mathematics gains of learners in Texas middle schools including teacher education, mathematical knowledge for teaching (MKT), mathematical quality of instruction (MQI), and quality of instruction in linguistically diverse classrooms. Teachers’ time spent in professional development activities and general quality of instruction had positive and significant effects for all learners, with much larger effect on learners that are English proficient. For students that are learning English as a second language, teachers’ practices affording their linguistic diversity had a positive and significant effect.
Colloquium
 Jenya Sapir, UIUC
 Geodesics on surfaces
 01/25/2017
 4:10 PM  5:00 PM
 C304 Wells Hall
Let S be a hyperbolic surface. We will give a history of counting results for geodesics on S. In particular, we will give estimates that fill the gap between the classical results of Margulis and the more recent results of Mirzakhani. We will then give some applications of these results to the geometry of curves. In the process we highlight how combinatorial properties of curves, such as selfintersection number, influence their geometry.

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Student Algebra
 Sebastian Troncoso, MSU
 An Introduction to Elliptic Curves
 01/26/2017
 11:30 AM  12:20 PM
 C117 Wells Hall
No abstract available.
Geometry and Topology
 Xinghua Gao, UIUC
 Orders from $\widetilde{PSL_2(\mathbb{R})}$ Representations and Nonexamples
 01/26/2017
 2:00 PM  2:50 PM
 C304 Wells Hall
It is still unknown for the case of rational homology 3sphere whether the leftorderability of its fundamental group and it not be a HeegaardFloer Lspace are equivalent. Let $M$ be an integer homology 3sphere. One way to study leftorderability of $\pi_1(M)$ is to construct a nontrivial representation from $\pi_1(M)$ to $\widetilde{PSL_2(\mathbb{R})}$. However this method does not always work. In this talk, I will give examples of non Lspace irreducible integer homology 3spheres whose fundamental groups do not have nontrivial $\widetilde{PSL_2(\mathbb{R})}$ representations.

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Colloquium
 Linhui Shen, Northwestern
 Representations, Combinatorics, and Configurations.
 01/27/2017
 4:10 PM  5:00 PM
 C304 Wells Hall
We briefly recall KnutsonTao’s hive model that calculates the LittlewoodRichardson coefficients. We consider the configuration spaces of decorated flags introduced by Fock and Goncharov. The configuration spaces admit natural functions called potentials introduced by Goncharov and myself. We prove that the tropicalization of configuration spaces with potentials recovers KnutsonTao’s hives. As an application, Hong and I solve the Saturation problem for the Lie algebra so(2n+1). If time permits, I will further explain their deep connections with geometric Satake correspondence, homological mirror symmetry, and DonaldsonThomas theory.

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Seminar in Cluster algebras
 Rob Davis, MSU
 Plabic Graphs Part 2
 01/31/2017
 2:00 PM  2:50 PM
 C304 Wells Hall
No abstract available.
Combinatorics and Graph Theory
 WeiHsuan Yu, MSU
 New bounds for equiangular lines and spherical twodistance sets
 01/31/2017
 4:10 PM  5:00 PM
 C304 Wells Hall
The set of points in a metric space is called an sdistance set if pairwise distances between these points admit only s distinct values. Twodistance spherical sets with the set of scalar products {alpha, alpha}, alpha in [0,1), are called equiangular. The problem of determining the maximal size of sdistance sets in various spaces has a long history in mathematics. We determine a new method of bounding the size of an sdistance set in twopoint homogeneous spaces via zonal spherical functions. This method allows us to prove that the maximum size of a spherical twodistance set in R^n is n(n+1)/2 with possible exceptions for some n = (2k+1)^23, k a positive integer. We also prove the universal upper bound ~ 2 n a^2/3 for equiangular sets with alpha = 1/a and, employing this bound, prove a new upper bound on the size of equiangular sets in an arbitrary dimension. Finally, we classify all equiangular sets reaching this new bound.
