Colloquium
The Colloquium is held on Mondays at 4pm in Deady 208.
Spring Quarter, 2019
- April 8, No colloquium – Moursund Lecture (Amie Wilkinson)
- April 15, Mai Gehrke (Nice)
Logic in computer science: meaning versus powerAbstract: Mathematical logic was developed to provide foundations for mathematics, but it is also a subfield of mathematics, with connections to other areas such as algebra, topology and category theory. As such, it has applications and many of these are in computer science. In fact, in the same way that calculus and various parts of analysis provide the language of modern physics, so logic has been called the language of computer science [1].A program is very syntactic in nature, having variables, operations of various kinds and rules about how it can be put together. This is closely related to algebra and, in logic, to formal deductive systems. On the other hand, when one runs a program, something happens: there is a set of computational states and a dynamic unfolding of the computation among these states. This is the `meaning’ of the program. Theoretical computer scientists would like to model this meaning in order to be able to prove that their programs do what they intend them to do. This is the area of program semantics and it is closely related to semantics in logic. The main theorems here are `completeness theorems’ which connect syntactic deductive systems (or certain algebras) with `models’ (or certain spaces). Mathematically, this is closely related to Stone duality, which is due to Marshall H. Stone, an American functional analyst. See [2] for a general public introduction to the subject.
A quite distinct part of theoretical computer science concerns the `power’ or resources (such as time and space) needed to solve a problem. Here, the most famous is the P versus NP problem, which asks whether the existence of a polynomial time algorithm for checking whether a potential solution is indeed a solution implies the existence of a polynomial time algorithm for finding a solution from scratch. Not only is this problem open, but one has to go down very low in the hierarchy of complexity classes, towards the bottom of what is known as Boolean circuit classes, to find a class which has been proved separated from NP. From a mathematician’s point of view, an exciting fact is that many interesting complexity classes, such as P, NP, and the Boolean circuit classes correspond to logic fragments (within second order logic). This means we can talk about them in a mathematical setting devoid of notions of machine, computation, or time [3,4].
The translation into logic allows one to see the complexity classes as Boolean algebras of subsets of free finitely generated monoids, A*, A a finite set (A is usually called an alphabet, the elements of A* are called words, and the subsets of A* are called (formal) languages). Thus we want to separate sub-Boolean algebras of the powerset of A*. This is also the case in another area of computer science, namely automata theory, and there also interesting classes are given by logic fragments. In automata theory, researchers have developed sophisticated tools based on profinite topological algebra for separating and deciding membership for the language classes [5]. Trying to exploit this in the search for lower bounds in complexity theory dates back to the 1990’s. More recently, it was realized that these profinite algebra tools are part of Stone duality and this allows one to generalize them to the setting of the language classes of complexity theory. This is the subject of the ERC research project DuaLL, which I am responsible for [6,7].
The talk will stay at a fairly non-technical level in order to cover all this ground. It will consist roughly of three equal parts: one on logic, semantics, and Stone duality; one on logic, computational complexity classes, and language theory; one on tools from Stone duality and profinite algebra for the separation of language classes.
[1] J.Y. Halpern, R. Harper, N. Immerman, P.G. Kolaitis, M.Y. Vardi, V. Vianu, On the unusual effectiveness of logic in computer science, Bull. Symb. Log. 7 (2001), 213-236.
[2] http://www.liafa.univ-paris-diderot.fr/~mgehrke/oratie.pdf
[3] R. Fagin, Finite-model theory – a personal perspective, Theoretical Computer Science 116 (1993), 3-31.
[4] https://people.cs.umass.edu/~immerman/descriptive_complexity.html
[5] J.-É. Pin, Profinite methods in automata theory, STACS 2009, IBFI Schloss Dagstuhl, 31-50.
[6] M. Gehrke, A. Krebs, Stone duality for languages and complexity, SIGLOG News 4 (2017), 29-53.
[7] https://math.unice.fr/~mgehrke/DuaLL.htm - April 22, Melissa Liu (Columbia)
Mirror Symmetry for Calabi-Yau threefoldsAbstract: A Calabi-Yau manifold is a Kahler manifold with a nowhere vanishing holomorphic volume form. Mirror symmetry relates the A-model on a Calabi-Yau manifold (defined in terms of the symlectic structure) to the B-model on a mirror Calabi-Yau manifold (defined in terms of the complex structure). In this talk I will survey some recent progress and open problems on mirror symmetry for 3-dimensional Calabi-Yau manifolds. - May 6, No colloquium – Niven Lecture (Alison Etheridge)
- May 14, Beverly K Berger (Stanford)
Special Time: 4pm in 208 Deady
The Mystery of Expanding Galileo SpacetimesAbstract: For the past 15 years or so, Jim Isenberg and I have been trying to understand the attractor-like behavior we observed in certain mathematical cosmologies (solutions in general relativity with cosmological boundary conditions) in the expanding direction. While others had found similar behavior, we were unable to make any progress, especially with regard to mathematical statements. Finally, Jim’s student Adam Layne (now a postdoc in Sweden) found the key ingredient we had missed. I will introduce the Galileo spacetimes, describe our recent results, and say a few words about Jim. - May 28, Cengiz Pehlevan (Harvard)
Winter Quarter, 2019
- January 8, Otis Chodish (Princeton University and the Institute for Advanced Study in Mathematics)
The multiplicity one conjecture on 3-manifoldsAbstract: Minimal surfaces are critical points of the area functional on the space of surfaces. Thus, it is natural to try to construct them via Morse theory. However, there is a serious issue when carrying this out, namely the occurrence of “multiplicity.” I will explain this issue and recent joint work with C. Mantoulidis ruling this out for generic metrics. As a consequence of this, for generic metrics, we are able to give a new proof of Yau’s conjecture on infinitely many minimal surfaces, obtaining new geometric information as predicted by Morse theoretic considerations. - January 9, Laura Fredrickson (Stanford)
The asymptotic geometry of the Hitchin moduli spaceAbstract: Hitchin’s equations are a system of gauge theoretic equations on a Riemann surface that are of interest in many areas including representation theory, Teichmuller theory, and the geometric Langlands correspondence. The Hitchin moduli space carries a natural hyperkahler metric. A conjectural description of its asymptotic structure appears in the work of physicists Gaiotto-Moore-Neitzke and there has been a lot of progress on this recently. I will discuss some recent results. - January 11, Konstantin Matveev (Brandeis University)
ART and the Kerov’s conjectureAbstract: Asymptotic Representation Theory is concerned with taking limits within the context of Representation Theory. It was a key insight of Vershik and Kerov from the 1970s that such limiting questions should be viewed and analyzed as the study of statistical mechanical systems. This idea in turn gave rise to Integrable Probability: the study of probabilistic models with underlying algebraic structure. I will talk about one of the recent developments in this field: the solution of the Kerov’s conjecture from 1992 classifying Gibbs measures on the Young graph with Macdonald multiplicities; its connections to total positivity and vertex models. - January 14, Matthew Junge (Duke University)
Coexistence in chase-escapeAbstract: Imagine barnacles and mussels spreading across the surface of a rock. Barnacles propagate to adjacent unfilled spots. Mussels too, but they can only attach to barnacles. Barnacles with a mussel on top no longer spread. What conditions on the rock geometry (i.e. graph) and spreading rates ensure that barnacles can survive? Coexistence is closely related to the classical but still mysterious Richardson growth model. We will explain this connection and the extent of our rigorous understanding. - January 16, Sylvie Corteel (Université Paris Diderot, UC Berkeley)
Combinatorics of multivariate orthogonal polynomialsAbstract: The classical q-hypergeometric orthogonal polynomials are assembled into a hierarchy called the q-Askey scheme. It is now a classical subject to study the combinatorics of their coefficients and their moments. The polynomials admit a generalization leading to remarkable orthogonal polynomials in several variables. The most general family is the Macdonald-Koornwinder polynomials and Macdonald polynomials associated to any classical root system can be expressed as limits or special cases of Macdonald-Koornwinder polynomials. Understanding the combinatorics of these polynomials is an important open problem. In this talk we will show some recent progress related to special cases of these polynomials. We will highlight combinatorial formulas for
1. Certain Macdonald-Koornwinder polynomials using exclusion processes with open boundaries and
2. Macdonald polynomials of type A using exclusion processes and multiline queues (arXiv:1811.01024)
3. Multivariate q-Little Jacobi polynomials thanks to Lecture Hall Tableaux (arXiv:1804.02489)
This talk will be about enumerative, algebraic and asymptotics combinatorics.
No prior knowledge is required. Open problems will be presented. - January 18, Chris Schafhauser (York University)
An Embedding Theorem of C*AlgebrasAbstract: A C*-algebra consists of an algebra of bounded linear operators acting on a Hilbert space which is closed the adjoint operation (roughly, the transpose) and is complete in a certain metric. Typical examples include the ring of n x n complex matrices and the ring C(X) of representation of continuous functions from a compact space X to the complex numbers. Many more interesting examples arise from various dynamical objects (e.g. group and group actions) and from various geometric/topological constructions.The structure of finite dimensional C*-algebras is well understood: they are finite direct sums of complex matrix algebras. The class of approximately finite-dimensional (AF) C*-algebras, ones which may be written as (the closure of) an increasing union of f1inite-dimensional subalgebras, are also well understood: they are determined up to isomorphism by their module structure. However, the class of subalgebras of AF-algebras is still rather mysterious; it includes, for instance, all commutative C*-algebras and all C*-algebras generated by amenable groups. It is a long-standing problem to find an abstract characterization of subalgebras of AF-algebras.
I will discuss the AF-embedding problem for C*-algebras and a recent partial solution to this problem which gives a nearly complete characterization of C*-subalgebras of simple AF-algebras.
- January 22, Zhongyang Li (University of Connecticut)
Phase transitions and scaling limits in lattice modelsAbstract: The perfect matching is a subset of a graph where each vertex is incident to exactly one edge. It is a natural mathematical model for molecule structures, and can provide exact solutions to various other statistical mechanical models, including the celebrated Ising model and the 1-2 model. We will discuss the limit shape of the perfect matching when a rescaled graph approximates a certain simply- connected domain in the plane, as well as the frozen boundary, which is the boundary separating the frozen region and the liquid region.A closely related model is the 1-2 model, which is a probability measure on subgraphs of the hexagonal lattice where each vertex is incident to 1 or 2 edges. With the help of the dimer model, we can obtain a sharp phase transition result for the 1-2 model. We will also discuss the exact formula to compute the probability that a path occurs in a 1-2 model configuration, and almost sure non-existence of an infinite path, with the help of the mass-transport principle.
- January 25, Sean Howe (Stanford University)
Probabilistic structures in the topology and arithmetic of moduli spacesAbstract: The average smooth surface in P^3 is a plane, the universal curve of genus g approaches a Poisson random variable, and the probability that a random hypersurface is smooth is given by a special value of a zeta function! In this talk, we explain how to make sense of these claims by adapting basic concepts from probability, like moment generating functions and independence, to describe rich structures in the topology and arithmetic of moduli spaces. This framework leads to new cohomological stabilization results for moduli of hypersurface sections, a new perspective on representation stability for configuration spaces of algebraic varieties, and the computation of higher order terms in Katz-Sarnak style statistics for the zero-spacings of zeta functions; we will touch on some of these applications, and speculate about future directions. - January 28, Patricia Hersh (North Carolina State University)
Fibers of maps to totally nonnegative spaces and the Fomin-Shapiro ConjectureAbstract: Anders Björner and Michelle Wachs (and independently Matthew Dyer) proved that each interval in Bruhat order is the partially ordered set (poset) of closure relations of a regular CW complex. This led to the question of finding regular CW complexes naturally arising from representation theory having the intervals in Bruhat order as their posets of closure relations. Sergey Fomin and Michael Shapiro conjectured a solution. I will briefly discuss the proof of the Fomin-Shapiro Conjecture, a proof that utilized an interpretation of these stratified spaces as images of an intriguing family of maps — maps also arising in work of Lusztig related to canonical bases. I will also discuss a bigger picture this appears to fit within involving several other stratified spaces of interest as well as very recent joint work with Jim Davis and Ezra Miller regarding the structure of the fibers of these same maps. Background will be provided along the way. - January 30, Ben Krause (Cal Tech)
Discrete Analogues in Harmonic Analysis: Oscillation and SymmetryAbstract: Although discrete harmonic analysis — harmonic analysis on the integers — was initially motivated by problems in pointwise ergodic theory, the field has developed into its own right: the current program of linking the behavior of even the most classical operators in harmonic analysis — those designed to detect oscillation and symmetry — to their discrete counterparts is still in its early stages. This talk will emphasize the similarities and differences between the discrete and continuous theories, and will present some new results in discrete harmonic analysis with this perspective in mind. - February 8, Merav Stern (University of Washington)
From Connectivity Matrices to Rate DynamicsAbstract: Mean-field theory is commonly used to analyze the dynamics of large neural network models. In this approach, the interactions of the original network are replaced by appropriately structured noise driving uncoupled units in a self-consistent manner. This allows properties of the network dynamics to be predicted and the behavior of the network to be understood as a whole. Results in random matrix theory have been used to relate the structure of the connectivity of neural networks to their mean-field dynamics. In my talk I will explain the mean-field approach, discuss its relation to random matrix theory, and analyze how the dynamics of neural network models are related to their connectivity structure. I will provide examples of networks that the mean-field theory describes accurately as well as examples, analyzed with the use of matrix theory, in which small modifications in the connectivity matrix can result in large deviations from mean-field predictions. - February 25, Dan Margalit (Georgia Tech)
Algebraic, Geometric, and Dynamical Aspects of SurfacesAbstract: To each homeomorphism of a surface we can associate a real number, called the entropy, which encodes the amount of mixing being effected. This number can be studied from topological, geometrical, dynamical, analytical, and algebraic viewpoints. We will start by explaining Thurston’s beautiful insight for how to compute the entropy and explain a new, fast algorithm based on his ideas. We will also discuss some classical results and recent work concerning homeomorphisms with small entropy. One theme is that algebraic complexity and geometric complexity both imply dynamical complexity. - March 4, Craig Westerland (Minnesota)
Topology and arithmetic statisticsAbstract: There are many questions in number theory and arithmetic geometry of the sort “Does the following situation ever occur?” For instance, the inverse Galois problem asks whether every finite group occurs as the Galois group of an extension of the rationals. Similarly, one might ask whether one expects the rank of elliptic curves to be unbounded. Arithmetic statistics, broadly speaking, pursues the more quantitative question of how often such situations occur. The extension of the inverse Galois problem to this setting is a conjecture of Malle’s, which predicts an asymptotic formula for the number of occurrences of a given finite group G as the Galois group of a number field, as a function of the discriminant (there are analogous statistical conjectures regarding the rank of elliptic curves ordered by height).In this talk, we will give an introduction to these sort of questions, focusing on Malle’s conjecture. Additionally, we will explain how to formulate function field analogues of this conjecture and transform this conjecture into a problem in algebraic topology (about the homology of certain moduli spaces of branched covers of P^1). In joint work with Ellenberg and Tran, we partially solved this problem, giving the upper bound in Malle’s conjecture. Details on this homological calculation will be given in the topology seminar on 5 March.
Fall Quarter, 2018
- September 24, No Colloquium
- October 1, Justin Sawon (University of North Carolina)
Moduli of vector bundlesAbstract: A vector bundle on an algebraic variety associates a vector space to each point of the variety. A moduli space of vector bundles is a space parametrizing all vector bundles of a particular kind. A simple example is the Jacobian of a curve: the Jacobian is the moduli space of all line bundles of degree zero on the curve.
Moduli space of vector bundles exhibit very rich and interesting properties, often reflecting the geometry of the underlying variety. For instance, moduli space of vector bundles on holomorphic symplectic surfaces (such as K3 and abelian surfaces) are higher-dimensional holomorphic symplectic manifolds.
In this talk I will review the development of the theory of moduli spaces of vector bundles, culminating in recent advances involving not just vector bundles, but so-called `stable complexes’ of vector bundles. - October 15, Richard Hain (Duke)
Completions of Mapping Class GroupsAbstract: Mapping class groups are groups of orientation preserving diffeomorphisms of a compact surface. They are a natural generalization of the modular group SL_2(Z), which is the mapping class group of a torus. These mysterious groups play an important role in topology and algebraic geometry, and an increasingly important role in number theory. Even though there is a vast literature on mapping class groups that begins with the seminal work of Dehn and Nielsen in the 1930s, there is a large (and increasing) number of important open
questions about them.One useful tool for studying discrete groups,such as mapping class groups, and also for applying results about them to questions in algebraic and arithmetic geometry, is “relative (unipotent) completion”. In this talk I will define relative completion, give explicit presentations of the relative completions of mapping class groups in genus 3 or more. I will also explain relative completion of the modular group SL_2(Z) and how it is related to modular forms and their periods. I will give some applications of relative completion of mapping class groups to algebraic and arithmetic geometry, and to topology and, if time, mention some open problems.
- October 29, Mario Bonk (UCLA)
The Quasiconformal Geometry of FractalsAbstract: Many questions in analysis and geometry lead to problems of quasiconformal geometry on non-smooth or fractal spaces. For example, there is a close relation of this subject to the problem of characterizing fundamental groups of hyperbolic 3-orbifolds or to Thurston’s characterization of rational functions with finite post-critical set.In recent years, the classical theory of quasiconformal maps between Euclidean spaces has been successfully extended to more general settings and powerful tools have become available. Fractal 2-spheres or Sierpin ́ski carpets are typical spaces for which this deeper understanding of their quasiconformal geometry is particularly relevant and interesting. In my talk I will give a survey on some recent developments in this area.
- November 9, Dan Christensen (Ontario)
An introduction to homotopy type theory
Abstract: Type theory is a formal system that was originally intended to describe set-like objects, and which is well-suited to formalizing proofs so that they can be verified by a computer. Recently, it was realized that type theory is intrinsically homotopical, and can be used to reason about spaces and other homotopical categories.
Voevodsky introduced an axiom that he calls Univalence, which says roughly that homotopy equivalence and equality agree. This axiom holds for spaces, and makes the theory truly homotopical. This talk will start with an introduction to type theory, will introduce Univalence and give examples of its consequences, and will briefly discuss some recent work on developing the theory of localization in homotopy type theory.Special Time : 4pm in 208 Deady
- November 26, Tom Roby (UConn)
Dynamical Algebraic Combinatorics: Actions, Orbits, and Averages
Abstract: Dynamical Algebraic Combinatorics explores actions on sets of discrete combinatorial objects, many of which can be built up by small local changes, e.g., Schützenberger’s promotion and evacuation, or the rowmotion map on order ideals. There are strong connections to the combinatorics of representation theory and with Coxeter groups. Birational liftings of these actions are related to the Y-systems of statistical mechanics, thereby to cluster algebras, in ways that are still relatively unexplored.The term “homomesy” (coined by Jim Propp and the speaker) describes the following widespread phenomenon: Given a group action on a set of combinatorial objects, a statistic on these objects is called “homomesic” if its average value is the same over all orbits. Along with its intrinsic interest as a kind of “hidden invariant”, homomesy can be used to prove certain properties of the action, e.g., facts about the orbit sizes. Proofs of homomesy often involve developing tools that further our understanding of the underlying dynamics, e.g., by finding an equivariant bijection.
This talk will be a introduction to these ideas, giving a number of examples of such actions and pointing out connections other areas.
- December 3, Maciej Zworski (Berkeley)
Microlocal methods in chaotic dynamicsAbstract: Microlocal analysis exploits mathematical manifestations of the classical/quantum (particle/wave) correspondence and has been a successful tool in spectral theory and partial differential equations. Recently, microlocal methods have been applied to the study of classical dynamical problems, in particular of chaotic (Anosov, Axiom A) flows. I will survey results obtained with Dyatlov and present some more recent results of, among others, Guillarmou, Dang–Riviere, Shen, Bonthonneau–Weich.