The famous knot complement theorem of Gordon and Luecke states that two knots in the three-sphere are equivalent if and only if the complements are homeomorphic. This was proved more than 30 years ago using combinatorial methods. In this talk, we will prove some extended results using techniques from gauge theory. If there is time…
In this talk, I will discuss the evolution of rigid bodies in a perfect incompressible fluid, and the limit systems that can be obtained as the bodies shrink to points. The model is as follows: the fluid is driven by the incompressible Euler equation, while the solids evolve according Newton’s equations under the pressure force on…
Spatio-temporal dynamical systems, such as fluid flows or vibrating structures, are prevalent across various applications, from enhancing user comfort and reducing noise in HVAC systems to improving cooling efficiency in electronic devices. However, these systems are notoriously hard to optimize and control due to the infinite dimensionality and nonlinearity of their governing partial differential equations…
The Fisher-KPP equation was introduced in 1937 to model the spread of an advantageous gene through a spatially distributed population. Remarkably precise information on the traveling front has been obtained via a connection with branching Brownian motion, beginning with works of McKean and Bramson in the 70s. I will discuss an extension of this probabilistic…
Determining the sensitivity of model outputs to input parameters is an important precursor to developing informative parameter studies, building surrogate models, and performing rigorous uncertainty quantification. A prominent class of models in many applications is dynamical systems whose trajectories lie on or near some attracting set after a sufficiently long time, and many quantities of…
Minimal surfaces are fundamental geometric objects which have been studied intensively since the 1700's. Classes of minimal surfaces of particular interest are the complete embedded ones in Euclidean space, closed (compact boundaryless) embedded in the round three-sphere, free boundary compact embedded ones in the unit Euclidean three-ball, and self-shrinkers of the mean curvature flow. Since…
Determinantal systems are systems of polynomial equations which encode a rank deficiency of a given matrix with polynomial entries over the solution set to other polynomial equations. Such systems arise in a number of areas of computational mathematics such as polynomial optimization, real algebraic and enumerative geometry and engineering sciences such as robotics and biology.…
Invariants withstand transformations and, therefore, represent the essence of objects or phenomena. In mathematics, transformations often constitute a group action. Since the 19th century, studying the structure of various types of invariants and designing methods and algorithms to compute them remains an active area of ongoing research with an abundance of applications, in particular, to…
A "frieze" is an infinite strip of numbers satisfying certain determinantal identities, or any one of several generalizations of this idea. In this talk, I will give an introduction to friezes whose shape is determined by a Dynkin diagram (motivated by their exceptional properties as well as connections to representation theory and cluster algebras). One…
It is with great enthusiasm that I have the opportunity to announce to the College this year's nominees for the Staff Awards for Excellence. Your colleagues, listed below, have been nominated by supervisors and peers for the most prestigious honor bestowed upon non-faculty employees. This award recognizes the outstanding accomplishments and contributions of individual employees, above…
End of year awards and tea.
A convex continuous function can be determined, up to a constant, by its remoteness (distance of the subdifferential to zero). Based on this result, we discuss possible extensions in three directions: robustness (sensitivity analysis), slope determination (in the Lipschitz framework) and general determination theory. Zoom meeting: Link
Transition paths connecting metastable states are significant in science and engineering, such as in biochemical reactions. In this talk, I will present a stochastic optimal control formulation for transition path problems over an infinite time horizon, modeled by Markov jump processes on Polish spaces. An unbounded terminal cost at a stopping time and a running…
1. Kelsey Hanser Title : Greedy Kohnert Posets Abstract : K-Kohnert polynomials form a large family of polynomials which generalize Lascoux polynomials. Each K-Kohnert polynomial encodes a certain collection of diagrams which is formed from an initial seed diagram by applying what are called “Kohnert" and “ghost moves." In particular, Kohnert polynomials are the restrictions…
Invariants withstand transformations and, therefore, represent the essence of objects or phenomena. In mathematics, transformations often constitute a group action. Since the 19th century, studying the structure of various types of invariants and designing methods and algorithms to compute them remains an active area of ongoing research with an abundance of applications, in particular, to…
Please save the date for the Department of Mathematics Spring 2024 Graduation Ceremony. Commencement starts at 5:30PM, with guest seating starting at 5:00PM.
Discover the beauty of mathematics and its wide applicability and power in everyday life! Grade Level: High School Students Schedule of Events: 11-11:15 am Drop Off 11:15-11:30 am Introductions 11:30-12:30 pm Applied Math and Math Modeling 12:30-1:15 pm Lunch 1:15-2:15 pm Control and Optimization in Biomedicine 2:15-2:45 pm Closing Remarks and Snacks 2:45-3 pm…
Cauchy's determinantal identity (1840s) expands via Schur polynomials the determinant of the matrix f, where f(t) = 1/(1-t) is applied entrywise to the rank-one matrix u v^T = (u_i v_j). This theme has resurfaced in the 2010s in analysis (following a 1960s computation by Loewner), in the quest to find polynomials p(t) with a negative coefficient that entrywise preserve…