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Title: Mathematical Relativity and the Geometry of Initial Data Sets for the Einstein Equations I

Speaker: Armando Cabrera (University of Connecticut)

Time: Friday, February 17, 2017 at 3:30 pm

Place: MONT 214Abstract: The Einstein Equations, which describe the interaction between the matter and the geometry of the universe, can be interpreted as a complicated system of hyperbolic differential equations. One way to study its solutions is by considering an initial value problem, which naturally leads to the understanding of the geometry of special Riemannian manifolds with non-negative scalar curvature, called initial data sets. In this series of lectures, we will briefly describe some important concepts in mathematical relativity, and then we will focus on the geometry of initial data sets by going over some classical results and moving towards recent developments.

Title: Mathematical Relativity and the Geometry of Initial Data Sets for the Einstein Equations II

Speaker: Armando Cabrera (University of Connecticut)

Time: Friday, February 24, 2017 at 3:30 pm

Place: MONT 111Abstract: The Einstein Equations, which describe the interaction between the matter and the geometry of the universe, can be interpreted as a complicated system of hyperbolic differential equations. One way to study its solutions is by considering an initial value problem, which naturally leads to the understanding of the geometry of special Riemannian manifolds with non-negative scalar curvature, called initial data sets. In this series of lectures, we will briefly describe some important concepts in mathematical relativity, and then we will focus on the geometry of initial data sets by going over some classical results and moving towards recent developments.

Title: Mathematical Relativity and the Geometry of Initial Data Sets for the Einstein Equations III

Speaker: Armando Cabrera (University of Connecticut)

Time: Friday, March 3, 2017 at 3:30 pm

Place: MONT 214Abstract: The Einstein Equations, which describe the interaction between the matter and the geometry of the universe, can be interpreted as a complicated system of hyperbolic differential equations. One way to study its solutions is by considering an initial value problem, which naturally leads to the understanding of the geometry of special Riemannian manifolds with non-negative scalar curvature, called initial data sets. In this series of lectures, we will briefly describe some important concepts in mathematical relativity, and then we will focus on the geometry of initial data sets by going over some classical results and moving towards recent developments.

Title: Algebraic Probability and Stochastic Processes I

Speaker: Arthur Parzygnat (University of Connecticut)

Time: Friday, March 24, 2017 at 3:30 pm

Place: MONT 313Abstract: Certain algebraic structures reproduce familiar notions from probability theory. These are states on C∗-algebras and completely positive maps between them. In the case of commutative algebras, these reproduce spaces with probability density functions and stochastic processes between them. The non-commutative analogues can be interpreted as non-commutative probability theory. To set the stage, the first lecture is about finite probability spaces, stochastic matrices, and their algebraic analogues. In the second lecture, we discuss the Gelfand-Naimark Theorem, which provides an equivalence between commutative C∗-algebras and compact Hausdorff topological spaces. In the third lecture, we come back to studying states on arbitrary (not necessarily finite-dimensional) commutative C∗-algebras. Occasional references will be made to quantum mechanics.

Title: Algebraic Probability and Stochastic Processes II

Speaker: Arthur Parzygnat (University of Connecticut)

Time: Friday, March 31, 2017 at 3:30 pm

Place: MONT 313Abstract: Certain algebraic structures reproduce familiar notions from probability theory. These are states on C∗-algebras and completely positive maps between them. In the case of commutative algebras, these reproduce spaces with probability density functions and stochastic processes between them. The non-commutative analogues can be interpreted as non-commutative probability theory. To set the stage, the first lecture is about finite probability spaces, stochastic matrices, and their algebraic analogues. In the second lecture, we discuss the Gelfand-Naimark Theorem, which provides an equivalence between commutative C∗-algebras and compact Hausdorff topological spaces. In the third lecture, we come back to studying states on arbitrary (not necessarily finite-dimensional) commutative C∗-algebras. Occasional references will be made to quantum mechanics.

Title: Algebraic Probability and Stochastic Processes III

Speaker: Arthur Parzygnat (University of Connecticut)

Time: Friday, April 14, 2017 at 3:30 pm

Place: MONT 313Abstract: Certain algebraic structures reproduce familiar notions from probability theory. These are states on C∗-algebras and completely positive maps between them. In the case of commutative algebras, these reproduce spaces with probability density functions and stochastic processes between them. The non-commutative analogues can be interpreted as non-commutative probability theory. To set the stage, the first lecture is about finite probability spaces, stochastic matrices, and their algebraic analogues. In the second lecture, we discuss the Gelfand-Naimark Theorem, which provides an equivalence between commutative C∗-algebras and compact Hausdorff topological spaces. In the third lecture, we come back to studying states on arbitrary (not necessarily finite-dimensional) commutative C∗-algebras. Occasional references will be made to quantum mechanics.

Title: Groups and Probability I

Speaker: Behrang Forghani (University of Connecticut)

Time: Friday, April 21, 2017 at 3:30 pm

Place: MONT 313Abstract: Traditionally, the areas of Algebra, Analysis, and Probability have attracted the attention of both mathematicians and scientists from other fields. The theory of random walks on groups ties these three fundamental classical mathematical fields together. Recent work on random walks expanded mathematical field significantly (e.g. the first example of expander graphs) and the results can be applied not only to mathematical fields but also to other branches of science (e.g. biology) and industry (e.g. gambling machines at casinos). In this series of talks, I will start with the definition of a random walk on a discrete group. The construction of the Poisson boundary and its relation with the entropy and harmonic functions associated with a random walk will be discussed. I will also show how algebraic properties (e.g. amenability) of a discrete group is related to the behaviours of a random walk at infinity.

Title: Groups and Probability II

Speaker: Behrang Forghani (University of Connecticut)

Time: Friday, April 28, 2017 at 3:30 pm

Place: MONT 313Abstract: Traditionally, the areas of Algebra, Analysis, and Probability have attracted the attention of both mathematicians and scientists from other fields. The theory of random walks on groups ties these three fundamental classical mathematical fields together. Recent work on random walks expanded mathematical field significantly (e.g. the first example of expander graphs) and the results can be applied not only to mathematical fields but also to other branches of science (e.g. biology) and industry (e.g. gambling machines at casinos). In this series of talks, I will start with the definition of a random walk on a discrete group. The construction of the Poisson boundary and its relation with the entropy and harmonic functions associated with a random walk will be discussed. I will also show how algebraic properties (e.g. amenability) of a discrete group is related to the behaviours of a random walk at infinity.

Title: Groups and Probability III

Speaker: Behrang Forghani (University of Connecticut)

Time: Tuesday, May 2, 2017 at 1:30 pm

Place: MONT 313Abstract: Traditionally, the areas of Algebra, Analysis, and Probability have attracted the attention of both mathematicians and scientists from other fields. The theory of random walks on groups ties these three fundamental classical mathematical fields together. Recent work on random walks expanded mathematical field significantly (e.g. the first example of expander graphs) and the results can be applied not only to mathematical fields but also to other branches of science (e.g. biology) and industry (e.g. gambling machines at casinos). In this series of talks, I will start with the definition of a random walk on a discrete group. The construction of the Poisson boundary and its relation with the entropy and harmonic functions associated with a random walk will be discussed. I will also show how algebraic properties (e.g. amenability) of a discrete group is related to the behaviours of a random walk at infinity.

Organizer: Matthew Badger