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PDE and Differential Geometry Seminar

Title: Regularity theory for Type I Ricci flows

Speaker: Pannagiotis Gianniotis (University of Waterloo)

Time: Monday, March 27, 2017 at 2:30 pm

Place: MONT 214Abstract: A Ricci flow exhibits a Type I singularity if the curvature blows up at a certain rate near the singular time. Type I singularities are abundant and in fact it is conjectured that they are the generic singular behaviour for the Ricci flow on closed manifolds. In this talk, I will describe some new integral curvature estimates for Type I flows, valid up to the singular time. These estimates partially extend to higher dimensions an estimate that was recently shown to hold in dimension three by Kleinert-Lott, using Ricci flow with surgery. In our case, we rely on a monotonicity formula that is available which allows us to adapt the technique of quantitative stratification of Cheeger-Naber to Type I Ricci flows.

Title: Crystal structure of certain PBW bases

Speaker: Ben Salisbury (Central Michigan University)

Time: Wednesday, March 29, 2017 at 11:15 am

Place: MONT 214Abstract: Lusztig's theory of PBW bases gives a way to realize the crystal $B(\infty)$ for any complex-simple Lie algebra where the underlying set consists of Kostant partitions. In fact, there are many different such realizations: one for each reduced expression of the longest element of the Weyl group. There is an explicit algorithm to calculate the actions of the crystal operators, but it can be quite complicated. In this talk, we will explain how, for certain reduced expressions, the crystal operators can also be described by a much simpler bracketing rule. Conditions describing these reduced expressions will be given in every type except $E_8$, $F_4$ and $G_2$ and several examples will be provided. This is joint work with Peter Tingley and Adam Schultze.

Title: Symmetric powers of the Legendre and Kloosterman family

Speaker: Douglas Haessig (University of Rochester)

Time: Wednesday, March 29, 2017 at 2:25 pm

Place: MONT 214Abstract: This talk will discuss two families, the Legendre family of elliptic curves and the Kloosterman family of exponential sums. While the two families are quite different they share many of the same arithmetic properties. For instance, the L-function of symmetric powers of both families are polynomials of known degree and satisfy functional equations once trivial factors are accounted for. This suggests that results on one family may shed light on analogous results for the other. The aim of this talk is to explore this line of reasoning.

Title: Optimal Investment in Incomplete Markets

Speaker: Oleksii Mostovyi (University of Connecticut)

Time: Wednesday, March 29, 2017 at 2:30 pm

Place: MONT 111Abstract: We will discuss some topics related to optimal investment

Title: Infinite mutations on marked surfaces

Speaker: Sira Gratz (University of Oxford, UK)

Time: Wednesday, March 29, 2017 at 3:30 pm

Place: MONT 313

Title: Error-Correcting Codes

Speaker: Angelynn Alvarez (University of Connecticut)

Time: Wednesday, March 29, 2017 at 5:45 pm

Place: MONT 321Abstract: Whenever data is transmitted across a channel, errors are likely to occur and the received message may not be identical to the original message. An *error-correcting code* is an algorithm for expressing a sequence of elements so that any errors formed during data transmission can be detected and possibly corrected. These codes have many real-world applications, such as in cell phones, credit cards, CDs, high-speed modems, and ISBN numbers. In this talk, I will discuss the basic construction of different types of error-correcting codes.Comments: Free pizza and drinks!

Title: Prime Numbers: What is Known and Unknown

Speaker: Keith Conrad (University of Connecticut)

Time: Thursday, March 30, 2017 at 2:30 pm

Place: MONT 214Abstract: Prime numbers are among the most elementary and mysterious objects in mathematics. They appear random (when is the next one?) while in the long run obeying statistical laws (all primes but 2 and 5 have units digit 1, 3, 7, or 9 and each choice occurs with 25% probability). In this talk we will discuss how to find primes, how to use primes, what we know about primes, and what we don't know about primes.

Title: Fermat's Last Theorem (Number Theory Day)

Speaker: Ken Ribet (University of California, Berkeley)

Time: Thursday, March 30, 2017 at 4:00 pm

Place: MONT 104Abstract: This talk will explain the history of Fermat Last Theorem, including its resolution in the mid-1990s by Andrew Wiles. The talk will describe some of the ideas used in the proof and the continuing influence of these ideas in current research.

Title: The Fermat Equation for Polynomials ...it's as easy as ABC

Speaker: Bobby McDonald (University of Connecticut)

Time: Friday, March 31, 2017 at 12:20 pm

Place: MONT 111Abstract: Many problems in number theory are easy to state but are very hard to prove. A famous example, taking over 350 years to settle, is Fermat's last theorem: there are no solutions $x, y, z$ in positive integers to the equation $$ x^n+y^n=z^n $$ when $n\geq 3$. In this talk, we will discuss analogies between integers and polynomials, and see that Fermat's Last Theorem can be proved very simply for polynomials using just a little calculus! We will also see some other results for integers that are either extremely hard to prove or still unknown, but which have been proved in a simple way for polynomials.

Title: Conformal equivalence of visual metrics in pseudoconvex domains

Speaker: Luca Capogna (WPI)

Time: Friday, March 31, 2017 at 1:30 pm

Place: MONT 226Abstract: We will present recent joint work with Enrico Le Donne (Jyvaskyla) in which we refine estimates introduced by Balogh and Bonk, to show that the boundary extensions of isometries between smooth strongly pseudoconvex domains in $\mathbb{C}^n$ are conformal with respect to the sub-Riemannian metric induced by the Levi form. As a corollary we obtain an alternative proof of a result of Fefferman on smooth extensions of biholomorphic mappings between pseudoconvex domains. The proofs are inspired by Mostow's proof of his rigidity theorem and are based both on the asymptotic hyperbolic character of the Kobayashi or Bergman metrics and on the Bonk-Schramm hyperbolic fillings.

Title: Linearity of Stability Conditions -- Northeastern-UConn Joint Seminar in Representation Theory

Speaker: Kiyoshi Igusa (Brandeis University)

Time: Friday, March 31, 2017 at 3:00 pm

Place: MONT 214Abstract: In 2002 Markus Reineke conjectured that there is a linear stability condition (given by a central charge Z) making all roots of a Dynkin quiver stable. This is reported to be solved by Hille and Juteau in an unpublished work. In 2014 Yu Qiu posted a solution of this conjecture along with many other results in the Dynkin case. The point is that there is an easy nonlinear stability condition, given by a maximal green sequence (MGS), which makes all roots stable. This becomes obvious given the equivalence between MGSs and Harder-Narasimhan (HN) stratification of mod H for any hereditary H, a result obtained by Yu Qiu in the Dynkin case. This talk is an updated version of talks with the same title given in Sherbrooke and in Hong Kong. New since Hong Kong: we extended results (same proofs!) to any finite dimensional algebra over any field. Namely, there is a notion of maximal green sequence for any algebra. Key properties of these generalized maximal green sequences are presented with an application: In joint work with PJ Apruzzese, an undergraduate at Brandeis, we found the upper bound on the length of a MGS with any cyclic quiver including the oriented cycle and give an explicit central charge attaining the upper bound showing that it is given by a linear MGS for any orientation of a circular quiver.

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.