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PDEs & Applications Seminar

Friday, September 27th, 2024

Time: 9:30 a.m.  Place: Jeffery Hall, Room 422

Speaker: Zachary Selk (¾ÅÐãÖ±²¥)

Title: Fictitious Densities of the \Phi^4 measure

Abstract: The \Phi^4 stochastic PDE \partial_t u=\Delta u-u^3+\infty u +\xi where \xi is space-time white noise is a prototypical example of a singular stochastic PDE if the space dimension is at least 2. The singularity is due to the nonlinear cubic term, the fact that SPDEs in space dimension at least 2 have distribution valued solutions, and the product of distributions is in general ill-defined - leading to the need of renormalization. The \infty u term represents a renormalization term needed to handle the ill-definedness of the term u^3, and this renormalization procedure is formalized by Hairer's theory of regularity structures (A theory of regularity structures, Inventiones Mathematicae, 2014) or Gubinelli, Imkeller and Perkowski's theory of paracontrolled calculus (Paracontrolled distributions and singular PDEs, Forum of Math. PI, 2015). 
   
   The \Phi^4 SPDE comes from stochastic quantization. In analogy with Langevin dynamics and SODEs, stochastic quantization is a way of constructing measures on infinite dimensional spaces as the invariant measure of a SPDE. The invariant measure of the \Phi^4 SPDE is an important object in mathematical physics. One potential criticism of this stochastic quantization procedure is that everything is asymptotic and the limiting object is hard to characterize. Barashkov and Gubinelli computed its Laplace transform explicitly (A variational method for \Phi_3^4, Duke Math Journal, 2020) in terms of a stochastic control problem. 
   
   In an ongoing joint work with Ioannis Gasteratos (TU Berlin) we are studying the fictitious density of this measure. In particular we have shown that the Onsager-Machlup function (which plays the role of a density function in infinite dimensions) of the invariant measure is what is expected in dimensions 1 and 2, however we suspect that in dimension 3 the Onsager-Machlup function is infinite everywhere, necessitating further generalizations of fictitious densities. We suspect this due to the measure's singularity with respect to every nonzero shift.

Math & Stats Department Colloquium
Friday, September 27, 2024

Time: 2:30 p.m.  Place: Jeffery Hall, Room 126

Speaker: Various

Title: Undergraduate Summer Projects

Abstract: This week colloquium will consists of six ten-minute presentations, starting at 2:30pm and in the following order, by:

• Jeremy Hare-Chang, "Product Formation in Sequential Batch Reactors".
• Evelyn Hubbard, "Mathematics of Reinforcement Learning under Partial Information".
• Denis Khatnyuk and Simon Nair, "Markov Genealogy Processes".
• Don Kim, "Numerical approximation of moving sets".
• Tony Luo and Bernd Zwanziger, "Investigating Secant Varieties". 
• Jinghan Yu, "Preferential Attachment Networks via Polya Contagion".
   

Algebra & Geometry Seminar
Tuesday, September 17, 2024

Time: 3:30 p.m.  Place: Jeffery Hall, Room 422

Speaker: Charles Paquette (RMC & Queen’s University)

Title: Inflations for quotient root systems, and applications to decomposing inversion sets - PART 2

Abstract: This is a report on joint work with I. Dimitrov, C. Gigliotti, E. Ossip and D. Wehlau. In the work of Dewji - Dimitrov - McCabe - Roth - Wehlau - Wilson, the notion of inflation of a permutation of the symmetric group Sn+1 was used to better understand inversion sets and how they yield decompositions of the set of all positive roots of the corresponding root system An. This led to nice geometric applications. A key observation is that inflations for the symmetric group can be defined by combining quotient root systems (QRSs) and subsystems, which are again of type A. In the first part of this talk, after reviewing the case of the symmetric group and type A root systems, we will define inversion sets and inflations for any QRS. Using induction and some new combinatorial objects that we can assign to an inversion set, we explain how to decompose the set of positive roots of a QRS into a disjoint union of inversion sets.

Math & Stats Number Theory Seminar
Thursday, September 19th, 2024

Time: 4:30 p.m.  Place: Jeffery Hall, Room 202

Speaker: Nic Fellini (¾ÅÐãÖ±²¥)

Title: Congruence relations of Ankeny–Artin–Chowla type for real quadratic fields

Abstract: In 1951, Ankeny, Artin, and Chowla released a short note containing four congruence relations involving the arithmetic invariants of Q(sqrt(d)) for d = 1 mod 4. They proved three of these relations the following year, in a paper published in the Annals of Mathematics. In this talk, I will discuss how the work of Ankeny, Artin, and Chowla generalizes to any real quadratic field, as well as present some recent results that connect their work to Iwasawa theory.

Dynamics, Geometry and Groups Seminar
Friday, September 20th, 2024

Time: 11:30 a.m.  Place: Jeffery Hall, Room 202

Speaker: Neil MacVicar (¾ÅÐãÖ±²¥)

Title: When are intersections of Cantor sets self-similar?

Abstract: Let C be the classical middle thirds Cantor set. The set C is an example of a self-similar set. In layman's terms, it is made up of smaller versions of itself. Given a real number t, we will look at conditions on t given by Deng, We, and Hen (2008) that imply that the intersection of C and C + t is also self-similar. We will see how their techniques have been generalized for other Cantor sets.

Algebra & Geometry Seminar
Tuesday, September 17, 2024

Time: 3:30 p.m.  Place: Jeffery Hall, Room 422

Speaker: Charles Paquette (RMC & Queen’s University)

Title: Inflations for quotient root systems, and applications to decomposing inversion sets

Abstract: This is a report on joint work with I. Dimitrov, C. Gigliotti, E. Ossip and D. Wehlau. In the work of Dewji - Dimitrov - McCabe - Roth - Wehlau - Wilson, the notion of inflation of a permutation of the symmetric group Sn+1 was used to better understand inversion sets and how they yield decompositions of the set of all positive roots of the corresponding root system An. This led to nice geometric applications. A key observation is that inflations for the symmetric group can be defined by combining quotient root systems (QRSs) and subsystems, which are again of type A. In the first part of this talk, after reviewing the case of the symmetric group and type A root systems, we will define inversion sets and inflations for any QRS. Using induction and some new combinatorial objects that we can assign to an inversion set, we explain how to decompose the set of positive roots of a QRS into a disjoint union of inversion sets.

PDEs & Applications Seminar

Friday, September 20th, 2024

Time: 9:30 a.m.  Place: Jeffery Hall, Room 422

Speaker: Teresa Chiri (¾ÅÐãÖ±²¥)

Title: Mean-field Limit and Optimal Control for a Hybrid Multi-Population Traffic Flow Model

Abstract: Heterogeneous and multi-lane traffic flow modeling is fundamental to understanding the dynamics and control of complex traffic systems. In this talk, we consider three populations of vehicles: two classes of human-driven vehicles (cars and trucks) and autonomous vehicles (AV). We first develop a finite-dimensional hybrid system that relies on continuous Bando-Follow-the-Leader dynamics coupled with discrete events motivated by lane-change maneuvers. Then we rigorously prove that the mean-field limit is given by a system of Vlasov-type PDE with source terms generated by the lane-change maneuvers of the human-driven vehicles. The PDEs are coupled with ODEs for the dynamic of AVs. Using Γ-convergence, we prove the well-posedness of an optimal control problem for the mean-field limit. This is a Joint work with X. Gong (Amherst College) and B. Piccoli (Rutgers)

Math & Stats Department Colloquium
Friday, September 20, 2024

Time: 2:30 p.m.  Place: Jeffery Hall, Room 234

Speaker: Kathryn Mann (Cornell)

Title: From the plane to infinity and back again

Abstract: A "bifoliation" of a two-dimensional space is a way of covering it with local charts to the Euclidean plane $\mathbb R^2$ so that overlap maps in $\mathbb R^2$ match up the vertical and horizontal coordinate directions. Such objects arise naturally in many dynamical contexts such as Anosov diffeomorphisms on surfaces or flows on 3-manifolds.

A trick due to Mather lets one compactify a bifoliated plane with a "circle at infinity" using the data of the bifoliation. In recent work with Barthelm\'e and Bonatti, we studied the inverse question: what is the minimum amount of data from infinity that allows one to reverse this procedure and uniquely reconstruct a bifoliation of the plane?

This talk will introduce bifoliations and where they come from, and then answer the question above. While we are motivated by problems in dynamics, the talk will be mostly delightfully low-tech, low-dimensional topology, with lots of believable pictures.