All colloquia are on Fridays at 4:00 pm in Van Vleck B239, unless otherwise indicated.
|Jan 23, 4pm||Saverio Spagnolie (Brown)||Hydrodynamics of Self-Propulsion Near a Boundary: Construction of a Numerical and Asymptotic Toolbox||Jean-Luc|
|Jan 27||Ari Stern (UCSD)||Numerical analysis beyond Flatland: semilinear PDEs and problems on manifolds||Jean-Luc / Julie|
|Feb 3||Akos Magyar (UBC)||On prime solutions to linear and quadratic equations||Street|
|Feb 8||Lan-Hsuan Huang (Columbia U)||Positive mass theorems and scalar curvature problems||Sean|
|Feb 10||Melanie Wood (UW Madison)||Counting polynomials and motivic stabilization||local|
|Feb 17||Milena Hering (University of Connecticut)||The moduli space of points on the projective line and quadratic Groebner bases||Andrei|
|Feb 24||Malabika Pramanik (University of British Columbia)||Analysis on Sparse Sets||Benguria|
|March 9||Eftychios Sifakis (UW-Madison, CS Dept.)||TBA||Nigel|
|March 19||Colin Adams and Thomas Garrity (Williams College)||Which is better, the derivative or the integral?||Maxim|
|March 23||Martin Lorenz (Temple University)||Prime ideals and group actions in noncommutative algebra||Don Passman|
|March 30||Wilhelm Schlag (University of Chicago)||TBA||Street|
|April 6||Spring recess|
|April 13||Ricardo Cortez (Tulane)||TBA||Mitchell|
|April 18||Benedict H. Gross (Harvard)||TBA||distinguished lecturer|
|April 19||Benedict H. Gross (Harvard)||TBA||distinguished lecturer|
|April 20||Robert Guralnick (University of Southern California)||Maps from the Generic Riemann surface to the Riemann sphere||Shamgar|
|April 27||Gui-Qiang Chen (Oxford)||TBA||Feldman|
|May 4||Mark Andrea de Cataldo (Stony Brook)||TBA||Maxim|
|May 11||Tentatively Scheduled||Shamgar|
Mon, Jan 23: Saverio Spagnolie (Brown)
"Hydrodynamics of Self-Propulsion Near a Boundary: Construction of a Numerical and Asymptotic Toolbox"
The swimming kinematics and trajectories of many microorganisms are altered by the presence of nearby boundaries, be they solid or deformable, and often in perplexing fashion. When an organism's swimming dynamics vary near such boundaries a question arises naturally: is the change in behavior fluid mechanical, biological, or perhaps due to other physical laws? We isolate the first possibility by exploring a far-field description of swimming organisms, providing a general framework for studying the fluid-mediated modifications to swimming trajectories. Using the simplified model we consider trapped/escape trajectories and equilibria for model organisms of varying shape and propulsive activity. This framework may help to explain surprising behaviors observed in the swimming of many microorganisms and synthetic micro-swimmers. Along the way, we will discuss the numerical tools constructed to analyze the problem of current interest, but which have considerable potential for more general applicability.
Fri, Feb 3: Akos Magyar (UBC)
On prime solutions to linear and quadratic equations
The classical results of Vinogradov and Hua establishes prime solutions of linear and diagonal quadratic equations in suﬃciently many variables. In the linear case there has been a remarkable progress over the past few years by introducing ideas from additive combinatorics. We will discuss some of the key ideas, as well as their use to obtain multidimensional extensions of the theorem of Green and Tao on arithmetic progressions in the primes. We will also discuss some new results on prime solutions to non-diagonal quadratic equations of suﬃciently large rank. Most of this is joint work with B. Cook.
Wed, Feb 8: Lan-Hsuan Huang (Columbia U)
Positive mass theorems and scalar curvature problems
More than 30 years ago, Schoen-Yau and later Witten made major breakthroughs in proving the positive mass theorem. It has become one of the most important theorems in general relativity and differential geometry. In the first part of the talk, I will introduce the positive mass theorem and present our recent work that extends the classical three-dimensional results to higher dimensions. In the second part, I will discuss how the observation from general relativity enables us to solve classical geometric problems related to the scalar curvature.
Fri, Feb 10: Melanie Wood (local)
Counting polynomials and motivic stabilization
We will begin with the problem of counting polynomials modulo a prime p with a given pattern of root multiplicity. Here we will discover phenomena that point to vastly more general patterns in configuration spaces of points. To see these patterns, one has to work in the ring of motives--so we will describe this place where a space is equivalent to the sum of its pieces. We will then be able to describe how these patterns in the ring of motives are related to theorems in topology on the homological stability of configuration spaces. This talk is based on joint work with Ravi Vakil.
Fri, Feb. 17: Milena Hering (UConn)
The moduli space of points on the projective line and quadratic Groebner bases
The ring of invariants for the action of the automorphism group of the projective line on the n-fold product of the projective line is a classical object of study. The generators of this ring were determined by Kempe in the 19th century. However, the ideal of relations has been only understood recently in work of Howard, Millson, Snowden and Vakil. They prove that the ideal of relations is generated by quadratic equations using a degeneration to a toric variety. I will report on joint work with Benjamin Howard where we further study the toric varieties arising in this degeneration. As an application we show that the second Veronese subring of the ring of invariants admits a presentation whose ideal admits a quadratic Groebner basis.
Fri, Feb. 24: Malabika Pramanik (UBC)
Analysis on Sparse Sets
Fri, March 23: Martin Lorenz (Temple University)
Prime ideals and group actions in noncommutative algebra
Having originated from number theory, the notion of a prime ideal has become central in many different branches of algebra. This talk will focus on the role of prime ideals in the representation theory of noncommutative algebras and the use of group actions as an efficient tool in organizing the spectrum of all prime ideals of a given algebra. In many cases, the group in question is an affine algebraic group and geometric methods are essential.
Fri, April 20: Bob Guralnick (USC)
Maps from the Generic Riemann surface to the Riemann sphere
Zariski, in his thesis, proved that any map from the generic Riemann surface of genus g > 6 could not be solvable. Using more sophisticated permutation group theory, we can prove a much stronger result: if f is an indecomposable map of degree n from the generic Riemann surface of genus g > 3 to the Riemann sphere, then the monodromy group of f is either the symmetric group of degree n with n > (g+2)/2 or the alternating group of degree n with n > 2g. We will discuss the main ideas used in solving this problem and some related problems. We will also discuss the analog of this problem in positive characteristic.