Difference between revisions of "Colloquia"

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__NOTOC__
 
__NOTOC__
  
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In 2022-2023, our colloquia will be in-person talks in B239 unless otherwise stated.
  
<b>UW Madison mathematics Colloquium is on Fridays at 4:00 pm. </b>
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==September 9 , 2022, Friday at 4pm  [https://math.ou.edu/~jing/ Jing Tao] (University of Oklahoma)==
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(host: Dymarz, Uyanik, WIMAW)
  
<!--- in Van Vleck B239, '''unless otherwise indicated'''. --->
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'''On surface homeomorphisms'''
  
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In the 1970s, Thurston generalized the classification of self-maps of the torus to surfaces of higher genus, thus completing the work initiated by Nielsen. This is known as the Nielsen-Thurston Classification Theorem. Over the years, many alternative proofs have been obtained, using different aspects of surface theory. In this talk, I will overview the classical theory and sketch the ideas of a new proof, one that offers new insights into the hyperbolic geometry of surfaces. This is joint work with Camille Horbez.
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==September 23, 2022, Friday at 4pm  [https://www.pabloshmerkin.org/ Pablo Shmerkin] (University of British Columbia) ==
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(host: Guo, Seeger)
  
== January 10, 2022, Monday at 4pm in B239 + [http://go.wisc.edu/wuas48 Live stream] + [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 Chat over Zoom], [https://www.stat.berkeley.edu/~gheissari/ Reza Gheissari] (UC Berkeley) ==
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'''Incidences and line counting: from the discrete to the fractal setting'''
  
(reserved by the hiring committee)
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How many lines are spanned by a set of planar points?. If the points are collinear, then the answer is clearly "one". If they are not collinear, however, several different answers exist when sets are finite and "how many" is measured by cardinality. I will discuss a bit of the history of this problem and present a recent extension to the continuum setting, obtained in collaboration with T. Orponen and H. Wang. No specialized background will be assumed.
  
'''Surface phenomena in the 2D and 3D Ising model'''
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==September 30, 2022, Friday at 4pm [https://alejandraquintos.com/ Alejandra Quintos] (University of Wisconsin-Madison, Statistics) ==
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(host: Stovall)
  
Since its introduction in 1920, the Ising model has been one of the most studied models of phase transitions in statistical physics. In its low-temperature regime, the model has two thermodynamically stable phases, which, when in contact with each other, form an interface: a random curve in 2D and a random surface in 3D. In this talk, I will survey the rich phenomenology of this interface in 2D and 3D, and describe recent progress in understanding its geometry in various parameter regimes where different surface phenomena and universality classes emerge.
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'''Dependent Stopping Times and an Application to Credit Risk Theory'''
  
== January 17, 2022, Monday at 4pm in B239 + [http://go.wisc.edu/wuas48 Live stream] + [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 Chat over Zoom], [https://sites.google.com/view/lovingmath/home Marissa Loving] (Georgia Tech) ==
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Stopping times are used in applications to model random arrivals. A standard assumption in many models is that the stopping times are conditionally independent, given an underlying filtration. This is a widely useful assumption, but there are circumstances where it seems to be unnecessarily strong. In the first part of the talk, we use a modified Cox construction, along with the bivariate exponential introduced by Marshall & Olkin (1967), to create a family of stopping times, which are not necessarily conditionally independent, allowing for a positive probability for them to be equal. We also present a series of results exploring the special properties of this construction.
  
(reserved by the hiring committee)
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In the second part of the talk, we present an application of our model to Credit Risk. We characterize the probability of a market failure which is defined as the default of two or more globally systemically important banks (G-SIBs) in a small interval of time. The default probabilities of the G-SIBs are correlated through the possible existence of a market-wide stress event. We derive various theorems related to market failure probabilities, such as the probability of a catastrophic market failure, the impact of increasing the number of G-SIBs in an economy, and the impact of changing the initial conditions of the economy's state variables. We also show that if there are too many G-SIBs, a market failure is inevitable, i.e., the probability of a market failure tends to one as the number of G-SIBs tends to infinity.
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==October 7, 2022, Friday at 4pm  [https://www.daniellitt.com/ Daniel Litt] (University of Toronto)==
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(host: Ananth Shankar)
  
'''Symmetries of surfaces: big and small'''
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'''The search for special symmetries'''
  
We will introduce both finite and infinite-type surfaces and study their collections of symmetries, known as mapping class groups. The study of the mapping class group of finite-type surfaces has played a central role in low-dimensional topology stretching back a hundred years to work of Max Dehn and Jakob Nielsen, and gaining momentum and significance through the celebrated work of Bill Thurston on the geometry of 3-manifolds. In comparison, the study of the mapping class group of infinite-type surfaces has exploded only within the past few years. Nevertheless, infinite-type surfaces appear quite regularly in the wilds of mathematics with connections to dynamics, the topology of 3-manifolds, and even descriptive set theory -- there is a great deal of rich mathematics to be gained in their study! In this talk, we will discuss the way that the study of surfaces intersects and interacts with geometry, algebra, and number theory, as well as some of my own contributions to this vibrant area of study.
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What are the canonical sets of symmetries of n-dimensional space? I'll describe the history of this question, going back to Schwarz, Fuchs, Painlevé, and others, and some new answers to it, obtained jointly with Aaron Landesman. While our results rely on low-dimensional topology, Hodge theory, and the Langlands program, and we'll get a peek into how these areas come into play, no knowledge of them will be assumed.
  
== January 21, 2022, Friday at 4pm in B239 + [http://go.wisc.edu/wuas48 Live stream] + [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 Chat over Zoom], [https://web.math.princeton.edu/~nfm2/ Nicholas Marshall] (Princeton) ==
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==October 14, 2022, Friday at 4pm [https://math.sciences.ncsu.edu/people/asagema/ Andrew Sageman-Furnas] (North Carolina State)==
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(host: Mari-Beffa)
  
(reserved by the hiring committee)
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'''Constructing isometric tori with the same curvatures'''
  
'''Laplacian quadratic forms, function regularity, graphs, and optimal transport'''
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Which data determine an immersed surface in Euclidean three-space up to rigid motion? A generic surface is locally determined by only an intrinsic metric and extrinsic mean curvature function. However, there are exceptions. These may arise in a family like the isometric family of vanishing mean curvature surfaces transforming a catenoid into a helicoid.
  
In this talk, I will discuss two different applications of harmonic analysis to
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For compact surfaces, Lawson and Tribuzy proved in 1981 that a metric and non-constant mean curvature function determine at most one immersion with genus zero, but at most two compact immersions (compact Bonnet pairs) for higher genus. In this talk, we discuss our recent construction of the first examples of compact Bonnet pairs. It uses a local classification by Kamberov, Pedit, and Pinkall in terms of isothermic surfaces. Moreover, we describe how a structure-preserving discrete theory for isothermic surfaces and Bonnet pairs led to this discovery.
problems motivated by data science. Both problems involve using Laplacian
 
quadratic forms to measure the regularity of functions. In both cases the key
 
idea is to understand how to modify these quadratic forms to achieve a specific
 
goal. First, in the graph setting, we suppose that a collection of m graphs
 
G_1 = (V,E_1),...,G_m=(V,E_m) on a common set of vertices V is given,
 
and consider the problem of finding the 'smoothest' function f : V -> R with
 
respect to all graphs simultaneously, where the notion of smoothness is defined
 
using graph Laplacian quadratic forms. Second, on the unit square [0,1]^2, we
 
consider the problem of efficiently computing linearizations of 2-Wasserstein
 
distance; here, the solution involves quadratic forms of a Witten Laplacian.
 
  
== January 24, 2022, Monday at 4pm in B239 + [http://go.wisc.edu/wuas48 Live stream] + [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 Chat over Zoom], [https://sites.google.com/view/skippermath Rachel Skipper] (Ohio State) ==
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The smooth theory is joint work with Alexander Bobenko and Tim Hoffmann and the discrete theory is joint work with Tim Hoffmann and Max Wardetzky.
  
(reserved by the hiring committee)
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== October 20, 2022, Thursday at 4pm, VV911  [https://tavarelab.cancerdynamics.columbia.edu/ Simon Tavaré] (Columbia University) ==
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(host: Kurtz, Roch)
  
'''From simple groups to symmetries of surfaces'''
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''Note the unusual time and room!''
  
We will take a tour through some families of groups of historic importance in geometric group theory, including self-similar groups and Thompson’s groups. We will discuss the rich, continually developing theory of these groups which act as symmetries of the Cantor space, and how they can be used to understand the variety of infinite simple groups. Finally, we will discuss how these groups are serving an important role in the newly developing field of big mapping class groups which are used to describe symmetries of surfaces.
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'''An introduction to counts-of-counts data'''
  
== February 11, 2022, at 4pm in B239 + [http://go.wisc.edu/wuas48 Live stream] + [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 Chat over Zoom], [https://people.math.wisc.edu/~msoskova/ Mariya Soskova] (UW-Madison) ==
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Counts-of-counts data arise in many areas of biology and medicine, and have been studied by statisticians since the 1940s. One of the first examples, discussed by R. A. Fisher and collaborators in 1943 [1], concerns estimation of the number of unobserved species based on summary counts of the number of species observed once, twice, … in a sample of specimens. The data are summarized by the numbers ''C<sub>1</sub>, C<sub>2</sub>, …'' of species represented once, twice, … in a sample of size
  
'''The e-verse'''
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''N = C<sub>1</sub> + 2 C<sub>2</sub> + 3 C<sub>3</sub> + <sup>….</sup>''  containing ''S = C<sub>1</sub> + C<sub>2</sub> + <sup>…</sup>'' species; the vector ''C ='' ''(C<sub>1</sub>, C<sub>2</sub>, …)'' gives the counts-of-counts. Other examples include the frequencies of the distinct alleles in a human genetics sample, the counts of distinct variants of the SARS-CoV-2 S protein obtained from consensus sequencing experiments, counts of sizes of components in certain combinatorial structures [2], and counts of the numbers of SNVs arising in one cell, two cells, … in a cancer sequencing experiment.
  
Computability theory studies the relative algorithmic complexity of sets of natural numbers and other mathematical objects. Turing reducibility and the induced partial order of the Turing degrees serve as the well-established model of relative computability. Enumeration reducibility captures another natural relationship between sets of natural numbers in which positive information about the first set is used to produce positive information about the second set. The induced structure of the enumeration degrees can be viewed as an extension of the Turing degrees, as there is a natural way to embed the second partial order in the first. In certain cases, the enumeration degrees can be used to capture the algorithmic content of mathematical objects, while the Turing degrees fail. Certain open problems in degree theory present as more approachable in the extended context of the enumeration degrees, e.g. first order definability. We have been working to develop a richer “e-verse”: a system of classes of enumeration degrees with interesting properties and relationships, in order to better understand the enumeration degrees. I will outline several research directions in this context.
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In this talk I will outline some of the stochastic models used to model the distribution of ''C,'' and some of the inferential issues that come from estimating the parameters of these models. I will touch on the celebrated Ewens Sampling Formula [3] and Fisher’s multiple sampling problem concerning the variance expected between values of ''S'' in samples taken from the same population [3]. Variants of birth-death-immigration processes can be used, for example when different variants grow at different rates. Some of these models are mechanistic in spirit, others more statistical. For example, a non-mechanistic model is useful for describing the arrival of covid sequences at a database. Sequences arrive one at a time, and are either a new variant, or a copy of a variant that has appeared before. The classical Yule process with immigration provides a starting point to model this process, as I will illustrate.
  
== February 18, 2022, at 4pm in B239 + [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 Video over Zoom], [https://people.math.wisc.edu/~seeger/ Andreas Seeger] (UW-Madison) ==
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''References''
  
'''Spherical maximal functions and fractal dimensions of dilation sets'''
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[1] Fisher RA, Corbet AS & Williams CB. J Animal Ecology, 12, 1943
  
We survey old and new problems and results on spherical means, regarding pointwise convergence, $L^p$ improving and consequences for sparse domination.
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[2] Arratia R, Barbour AD & Tavaré S. ''Logarithmic Combinatorial Structures,'' EMS, 2002
  
== February 25, 2022, at 4pm in B239 + [http://go.wisc.edu/wuas48 Live Stream], [https://sites.google.com/view/rohini-ramadas/home Rohini Ramadas] (Warwick) ==
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[3] Ewens WJ. Theoret Popul Biol, 3, 1972
  
(hosted by WIMAW)
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[4] Da Silva P, Jamshidpey A, McCullagh P & Tavaré S. Bernoulli Journal, in press, 2022 (online)
  
'''Dynamics on the moduli space of point-configurations on the Riemann sphere'''
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==October 21, 2022, Friday at 4pm  [https://web.ma.utexas.edu/users/ntran/ Ngoc Mai Tran] (Texas)==
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(host: Rodriguez)
  
A degree-$d$ rational function $f(z)$ in one variable with complex coefficients defines a holomorphic self-map of the Riemann sphere. A rational function is called post-critically finite (PCF) if every critical point is (pre)-periodic. PCF rational functions have been central in complex dynamics, due to their special dynamical behavior, and their special distribution within the parameter space of all rational maps.  
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'''Forecast science, learn hidden networks and settle economics conjectures with combinatorics, geometry and probability.'''
  
By work of Koch building on a result of Thurston, every PCF map arises as an isolated fixed point of an algebraic dynamical system on the moduli space $M_{0,n}$ of point-configurations on the Riemann sphere. I will introduce PCF maps and $M_{0,n}$. I will then present results characterizing the ensuing dynamics on $M_{0,n}$.  
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In many problems, one observes noisy data coming from a hidden or complex combinatorial structure. My research aims to understand and exploit such structures to arrive at an efficient and optimal solution. I will showcase a few successes, achieved with different tools, from different different fields: networks forecasting, hydrology, and auction theory. Then I will outline some open questions in each field.  
  
This talk includes joint work with Nguyen-Bac Dang, Sarah Koch, David Speyer, and Rob Silversmith.
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==October 28, 2022, Friday at 4pm  [https://people.math.wisc.edu/~qinli/ Qin Li] (UW)==
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'''Multiscale inverse problem, from Schroedinger to Newton to Boltzmann'''
  
== March 1, 2 and 4, 2022 (Tuesday, Wednesday and Friday), [http://www.math.stonybrook.edu/~roblaz/ Robert Lazarsfeld] (Stony Brook) ==
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Inverse problems are ubiquitous. We probe the media with sources and measure the outputs, to infer the media information. At the scale of quantum, classical, statistical and fluid, we face inverse Schroedinger, inverse Newton’s second law, inverse Boltzmann problem, and inverse diffusion respectively. The universe, however, expects a universal mathematical description, as Hilbert proposed in 1900. In this talk, we discuss the connection between these problems. We will give arguments for justifying that these are the same problem merely represented at different scales. It is a light-hearted talk, and I will mostly focus on the story instead of the derivation. PDE background is appreciated but not necessary.
(''Departmental Distinguished Lecture series'')
 
  
'''Public Lecture: Pythagorean triples and parametrized curves'''
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== November 7, 2022, Monday at 4pm [https://ai.facebook.com/people/kristin-lauter/ Kristen Lauter] (Facebook) ==
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Distinguished lectures
  
''Tuesday, March 1, 4:00pm (Humanities 3650 + [http://go.wisc.edu/n6986j Live Stream]). Note unusual time and location!''
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(host: Yang).
  
In this lecture, aimed at advanced undergraduate and beginning graduate students, I will discuss the question of when a curve in the plane admits a parameterization by polynomials or rational functions.
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'''Private AI: Machine Learning on Encrypted Data'''
  
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As the world adopts Artificial Intelligence, the privacy risks are many. AI can improve our lives, but may leak our private data. Private AI is based on Homomorphic Encryption (HE), a new encryption paradigm which allows the cloud to operate on private data in encrypted form, without ever decrypting it, enabling private training and private prediction. Our 2016 ICML CryptoNets paper showed for the first time that it was possible to evaluate neural nets on homomorphically encrypted data, and opened new research directions combining machine learning and cryptography. The security of Homomorphic Encryption is based on hard problems in mathematics involving lattices, recently standardized by NIST for post-quantum cryptography. This talk will explain Homomorphic Encryption, Private AI, and explain HE in action.
  
'''Colloquium: How irrational is an irrational variety?'''
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== November 8, 2022, Tuesday at 4pm [https://ai.facebook.com/people/kristin-lauter/ Kristen Lauter] (Facebook) ==
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Distinguished lectures in VV911. ''Note: unusual room.''
  
''Wednesday, March 2, 4:00pm (VV B239 + [http://go.wisc.edu/wuas48 Live Stream]).''
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(host: Yang).
  
Recall that an algebraic variety is said to be rational if it has a Zariski open subset that is isomorphic to an open subset of projective space. There has been a great deal of recent activity and progress on questions of rationality, but most varieties aren't rational. I will survey a body of work concerned with measuring and controlling “how irrational” a given variety might be.
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'''Artificial Intelligence & Cryptography: Privacy and Security in the AI era'''
  
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How is Artificial Intelligence changing your life and the world?  How do you expect your data to be kept secure and private in the future?  Artificial intelligence (AI) refers to the science of utilizing data to formulate mathematical models that predict outcomes with high assurance. Such predictions can be used to make decisions automatically or give recommendations with high confidence. Cryptography is the science of protecting the privacy and security of data.  This talk will explain the dynamic relationship between cryptography and AI and how AI can be used to attack post-quantum cryptosystems.
  
'''Seminar: Measures of association for algebraic varieties'''
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The first talk is based on my 2019 ICIAM Plenary Lecture and the second one is based on my 2022 SIAM Block Prize Lecture.
  
''Friday, March 4, 4:00pm (VV B239 )''
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== November 11, 2022, Friday at 4pm [http://users.cms.caltech.edu/~jtropp/ Joel Tropp] (Caltech)==
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This is the Annual LAA lecture. See [https://math.wisc.edu/laa-lecture/ this] for its history.
  
I will discuss some recent work with Olivier Martin that attempts to quantify how far two varieties are from being birationally isomorphic. Besides presenting a few results, I will discuss many open problems and avenues for further investigation.
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(host: Qin, Jordan)
  
== March 11, 2022, [https://people.math.wisc.edu/~anderson/ David Anderson] (UW-Madison) ==
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'''Rocket-propelled Cholesky: Addressing the challenges of large-scale kernel computations'''
  
'''Stochastic models of reaction networks and the Chemical Recurrence Conjecture'''
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Kernel methods are used for prediction and clustering in many data science and scientific computing applications, but applying kernel methods to a large number of data points N is expensive due to the high cost of manipulating the N x N kernel matrix. A basic approach for speeding up kernel computations is low-rank approximation, in which we replace the kernel matrix A with a factorized approximation that can be stored and manipulated more cheaply. When the kernel matrix A has rapidly decaying eigenvalues, mathematical existence proofs guarantee that A can be accurately approximated using a constant number of columns (without ever looking at the full matrix). Nevertheless, for a long time designing a practical and provably justified algorithm to select the appropriate columns proved challenging.
  
Cellular, chemical, and population processes are all often represented via networks that describe the interactions between the different population types (typically called the ''species'').  
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Recently, we introduced RPCholesky ("randomly pivoted" or "rocket-propelled" Cholesky), a natural algorithm for approximating an N x N positive semidefinite matrix using k adaptively sampled columns. RPCholesky can be implemented with just a few lines of code; it requires only (k+1)N entry evaluations and O(k^2 N) additional arithmetic operations. In experiments, RPCholesky matches or improves on the performance of alternative algorithms for low-rank psd approximation. Moreover, RPCholesky provably achieves near-optimal approximation guarantees. The simplicity, effectiveness, and robustness of this algorithm strongly support its use for large-scale kernel computations.
  
If the counts of the species are low, then these systems are most often modeled as continuous-time Markov chains on $Z^d$ (with d being the number of species), with rates determined by stochastic mass-action kinetics.  A natural (broad) question is:  how do the qualitative properties of the dynamical system relate to the properties of the network?  One specific conjecture, called the Chemical Recurrence Conjecture, and that has been open for decades, is the following: if each connected component of the network is strongly connected, then the associated stochastic model is positive recurrent (meaning the model is quite stable).
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Joint work with Yifan Chen, Ethan Epperly, and Rob Webber. Available at arXiv:2207.06503.
  
I will give a general introduction to this class of models and will present the latest work towards a proof of the Chemical Recurrence Conjecture.   I will make this talk accessible to graduate students, regardless of their field of study.   Some of the new results presented are joint with Daniele Cappelletti, Andrea Agazzi, and Jonathan Mattingly.
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==November 18, 2022, Friday at 4pm [http://homepages.math.uic.edu/~freitag/index.html Jim Freitag] (U of Illinois-Chicago) Zoom link: https://go.wisc.edu/jimfreitag<nowiki/>==
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'''Now available:''' [https://people.math.wisc.edu/logic/talks/221118-Freitag.mp4 Recording] and [https://people.math.wisc.edu/logic/talks/221118-Freitag.pdf Slides]
  
== March 25, 2022, Friday at 4pm on [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 Zoom]. [http://www.math.lsa.umich.edu/~canary/ Richard Canary] (Michigan) ==
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(hosts: Lempp, Andrews)
  
(hosted by Zimmer)
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'''When any three solutions are independent'''
  
'''Hitchin representations of Fuchsian groups'''
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In this talk, we'll talk about a surprising recent result about the algebraic relations between solutions of a differential equation. The result has applications to functional transcendence, diophantine geometry, and compact complex manifolds.
  
Abstract: The Teichm&uuml;ller space $\mathcal T(S)$ of all hyperbolic structures on a fixed closed surface $S$ is a central object in geometry, topology and dynamics. It may be viewed  as the orbifold universal cover of the moduli space of algebraic curves of fixed genus and also as a component of the space of (conjugacy classes of) representations of $\pi_1(S)$ into $\mathsf{PSL}(2,\mathbb R)$ which is topologically a cell.
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==November 21, 2022, <span style="color: red;">Monday</span> at 4pm [https://math.mit.edu/directory/profile.html?pid=1698 Andrei Negut] (MIT) Zoom link: [https://go.wisc.edu/andreinegut https://go.wisc.edu/andreinegut]==
Hitchin discovered a component $\mathcal H_d(S)$ of the space of (conjugacy classes of) representations of $\pi_1(S)$ into $\mathsf{PSL}(d,\mathbb R)$
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Hiring talk.
which is topologically a cell. Subsequently, many striking analogies between the Hitchin component $\mathcal H_d(S)$ and Teichm&uuml;ller space $\mathcal T(S)$ were found. For example, Labourie showed that all representations in $\mathcal H_d(S)$ are discrete, faithful quasi-isometric embeddings.
 
  
In this talk, we will begin by gently reviewing the parallel theories of Teichm&uuml;ller space and the Hitchin component. We will finish by reviewing a long term project to develop a geometric theory of the augmented Hitchin component which parallels the classical theory of the augmented Hitchin component (which one may view as the "orbifold universal cover" of the Deligne-Mumford compactification of Teichm&uuml;ller space). This program includes joint work with Harry Bray, Nyima Kao, Giuseppe Martone, Tengren Zhang and Andy Zimmer.
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(hosts: Arinkin, Caldararu)
  
== April 1, 2022, Friday at 4pm in B239 + [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 Zoom broadcast], [https://www.patelp.com/ Priyam Patel] (Utah) ==
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'''From gauge theory to geometric representation theory and back'''
  
(hosted by WIMAW)
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We start from the celebrated construction (due to Grojnowski and Nakajima) of a Heisenberg algebra action on the cohomology groups of Hilbert schemes of points on surfaces
  
'''Infinite-type surfaces'''
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# replacing Hilbert schemes with moduli spaces of higher rank sheaves yields a computation of Nekrasov partition functions in 5d supersymmetric gauge theory, and a proof of the deformed Alday-Gaiotto-Tachikawa conjecture.
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# replacing cohomology by Chow groups gives a proof of the Beauville conjecture in the hyperkahler geometry of Hilbert schemes of points on K3 surfaces (with Maulik)
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# working with derived categories allows us to construct a detailed framework realizing categorical knot invariants in terms of the geometry of Hilbert schemes of points on the affine plane (with Gorsky and Rasmussen)
  
Surfaces fall into two categories: finite-type and infinite-type. The theory of infinite-type surfaces has been historically less developed than that of finite-type surfaces, but in the last few years, there has been a surge of interest in surfaces of infinite type and their mapping class groups (informally thought of as the groups of topological symmetries of these surfaces). In this talk, I will survey some of the biggest open problems in this quickly growing subfield of geometric group theory and topology, and discuss some of my own recent joint work towards resolving them.
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==December 2, 2022, Friday at 4pm:  Promit Ghosal (MIT)==
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'''Fractal Geometry of the KPZ equation'''
  
== April 8, 2022, Friday at 4pm in B239 + [http://go.wisc.edu/wuas48 Live stream], [https://math.temple.edu/~tuf27009/index.html Matthew Stover] (Temple University) ==
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The Kardar-Parisi-Zhang (KPZ) equation is a fundamental stochastic PDE related to many important models like random growth processes, Burgers turbulence, interacting particles system, random polymers etc. In this talk, we focus on how the tall peaks and deep valleys of the KPZ height function grow as time increases. In particular, we will ask what is the appropriate scaling of the peaks and valleys of the (1+1)-d KPZ equation and whether they converge to any limit under those scaling. These questions will be answered via the law of iterated logarithms and fractal dimensions of the level sets. The talk will be based on joint works with Sayan Das and Jaeyun Yi. If time permits, I will also mention an interesting story about the  (2+1)-d and (3+1)-d case (work in progress with Jaeyun Yi).
  
(hosted by Zimmer)
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== December 5, 2022, Monday at 4pm:  Di Fang (Berkeley) ==
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(reserved by HC. contact: Stechmann)
  
'''A geometric characterization of arithmeticity'''
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== December 7, 2022, Wednesday at 4pm:  Benjamin Eichinger (Vienna University of Technology) ==
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(reserved by HC. contact: Stechmann)
  
An old, fundamental problem is classifying closed n-manifolds admitting a metric of constant curvature. The most mysterious case is constant curvature -1, that is, hyperbolic manifolds, and these divide further into "arithmetic" and "nonarithmetic" manifolds. However, it is not at all evident from the definitions that this distinction has anything to do with the differential geometry of the manifold. Uri Bader, David Fisher, Nicholas Miller and I gave a geometric characterization of arithmeticity in terms of properly immersed totally geodesic submanifolds, answering a question due independently to Alan Reid and Curtis McMullen. I will give an overview, assuming only basic differential topology, of how (non)arithmeticity and totally geodesic submanifolds are connected, then describe how this allows us to import tools from ergodic theory and homogeneous dynamics originating in groundbreaking work of Margulis to prove our characterization. Given time, I will mention some more recent developments and open questions.
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'''An approach to universality using Weyl m-functions'''
  
== April 15, 2022, Friday at 4pm in B239 + [http://go.wisc.edu/wuas48 Live stream], [https://www.qatar.tamu.edu/programs/science/faculty-and-staff/berhand-lamel Bernhard Lamel], (Texas A&M University at Qatar) ==
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In this talk I will present an approach to universality limits for orthogonal polynomials on the real line which is completely local and uses only the boundary behavior of the Weyl m-function at the point. We show that bulk universality of the Christoffel-Darboux kernel holds for any point where the imaginary part of the m-function has a positive finite nontangential limit. This approach is based on studying this problem in the more general setting of canonical systems and the realization that bulk universality for an associated matrix reproducing kernel at a point is equivalent to the fact that the corresponding m-function has normal limits at the same point. The talk is based on a joint work with Milivoje Lukic and Brian Simanek. If time permits, I will discuss some work in progress with Milivoje Lukic and Harald Woracek on rescaling limits for other
  
(hosted by Gong)
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universality classes.
  
'''Convergence and Divergence of Formal Power Series Maps'''
+
==December 9, 2022, Friday at 4pm:  Dallas Albritton (Princeton)==
 +
(reserved by HC. contact: Stechmann)
  
Consider two real-analytic hypersurfaces (i.e. defined by convergent power series) in complex spaces. A formal holomorphic map is said to take one into the other if the composition of the power series defining the target with the map (which is just another formal power series) is a (formal) multiple of the defining power series of the source. In this talk, we are going to be interested in conditions for formal holomorphic maps to necessarily be convergent. Now, a formal holomorphic map taking the real line to itself is just a formal power series with real coefficients; this example also gives rise to real hypersurfaces in higher dimensional complex spaces having divergent formal self-maps. On the other hand, a formal map taking the unit sphere in higher dimensional complex space to itself is necessarily a rational map with poles outside of the sphere, in particular, the formal power series defining it converges. The convergence theory for formal self-maps of real hypersurfaces has been developed in the late 1990s and early 2000s. For formal embeddings, “ideal" conditions had been long conjectured. I’m going to give an introduction to this problem and talk about some joint work from 2018 with Nordine Mir giving a basically complete answer to the question when a formal map taking a real-analytic hypersurface in complex space into another one is necessarily convergent.
+
== December 12, 2022, Monday at 4pm: Laurel Ohm (Princeton) ==
 
+
(reserved by HC. contact: Stechmann)
== April 25-26-27 (Monday [VV B239], Tuesday [Chamberlin 2241], Wednesday [VV B239]) 4 pm  [https://math.mit.edu/directory/profile.php?pid=1461 Larry Guth] (MIT) ==
 
 
 
(''Departmental Distinguished Lecture series'')
 
 
 
'''Reflections on decoupling and Vinogradov's mean value problem.'''
 
 
 
Decoupling is a recent development in Fourier analysis that has solved several longstanding problems.  The goal of the lectures is to describe this development to a general mathematical audience.
 
 
 
We will focus on one particular application of decoupling: Vinogradov's mean value problem from analytic number theory.  This problem is about the number of solutions of a certain system of diophantine equations.  It was raised in the 1930s and resolved in the last decade.
 
 
 
We will give some context about this problem, but the main goal of the lectures is to explore the ideas that go into the proof.  The method of decoupling came as a big surprise to me, and I think to other people working in the field.
 
The main idea in the proof of decoupling is to combine estimates from many different scales.  We will describe this process and reflect on why it is helpful.
 
 
 
 
 
 
 
 
 
'''Lecture 1:'''Introduction to decoupling and Vinogradov's mean value problem.
 
 
 
In this lecture, we introduce Vinogradov's problem and give an overview of the proof.
 
 
 
'''Lecture 2:''' Features of the proof of decoupling.
 
 
 
In this lecture, we look more closely at some features of the proof of decoupling.  The first feature we examine is the exact form of writing the inequality, which is especially suited for doing induction and connecting information from different scales.  The second feature we examine is called the wave packet decomposition.  This structure has roots in quantum physics and in information theory.
 
 
 
'''Lecture 3:''' Open problems.
 
 
 
In this lecture, we discuss some open problems in number theory that look superficially similar to Vinogradov mean value conjecture, such as Hardy and Littlewood's Hypothesis K*.  In this lecture, we probe the limitations of decoupling by exploring why the techniques from the first two lectures don't work on these open problems.  Hopefully this will give a sense of some of the issues and difficulties involved in these problems.
 
 
 
== May 10+12, 2022, Tuesday+Thursday, 12pm on Zoom. [http://www.ma.huji.ac.il/~kalai/ Gil Kalai] (Hebrew University) ==
 
 
 
(''Hilldale Lectures / Special colloquium'')
 
 
 
'''The argument against quantum computers'''
 
  
 
== Future Colloquia ==
 
== Future Colloquia ==
Line 183: Line 162:
  
 
== Past Colloquia ==
 
== Past Colloquia ==
 +
[[Spring 2022 Colloquiums|Spring 2022]]
 +
 
[[Colloquia/Fall2021|Fall 2021]]
 
[[Colloquia/Fall2021|Fall 2021]]
  

Latest revision as of 13:16, 30 November 2022


In 2022-2023, our colloquia will be in-person talks in B239 unless otherwise stated.

September 9 , 2022, Friday at 4pm Jing Tao (University of Oklahoma)

(host: Dymarz, Uyanik, WIMAW)

On surface homeomorphisms

In the 1970s, Thurston generalized the classification of self-maps of the torus to surfaces of higher genus, thus completing the work initiated by Nielsen. This is known as the Nielsen-Thurston Classification Theorem. Over the years, many alternative proofs have been obtained, using different aspects of surface theory. In this talk, I will overview the classical theory and sketch the ideas of a new proof, one that offers new insights into the hyperbolic geometry of surfaces. This is joint work with Camille Horbez.

September 23, 2022, Friday at 4pm Pablo Shmerkin (University of British Columbia)

(host: Guo, Seeger)

Incidences and line counting: from the discrete to the fractal setting

How many lines are spanned by a set of planar points?. If the points are collinear, then the answer is clearly "one". If they are not collinear, however, several different answers exist when sets are finite and "how many" is measured by cardinality. I will discuss a bit of the history of this problem and present a recent extension to the continuum setting, obtained in collaboration with T. Orponen and H. Wang. No specialized background will be assumed.

September 30, 2022, Friday at 4pm Alejandra Quintos (University of Wisconsin-Madison, Statistics)

(host: Stovall)

Dependent Stopping Times and an Application to Credit Risk Theory

Stopping times are used in applications to model random arrivals. A standard assumption in many models is that the stopping times are conditionally independent, given an underlying filtration. This is a widely useful assumption, but there are circumstances where it seems to be unnecessarily strong. In the first part of the talk, we use a modified Cox construction, along with the bivariate exponential introduced by Marshall & Olkin (1967), to create a family of stopping times, which are not necessarily conditionally independent, allowing for a positive probability for them to be equal. We also present a series of results exploring the special properties of this construction.

In the second part of the talk, we present an application of our model to Credit Risk. We characterize the probability of a market failure which is defined as the default of two or more globally systemically important banks (G-SIBs) in a small interval of time. The default probabilities of the G-SIBs are correlated through the possible existence of a market-wide stress event. We derive various theorems related to market failure probabilities, such as the probability of a catastrophic market failure, the impact of increasing the number of G-SIBs in an economy, and the impact of changing the initial conditions of the economy's state variables. We also show that if there are too many G-SIBs, a market failure is inevitable, i.e., the probability of a market failure tends to one as the number of G-SIBs tends to infinity.

October 7, 2022, Friday at 4pm Daniel Litt (University of Toronto)

(host: Ananth Shankar)

The search for special symmetries

What are the canonical sets of symmetries of n-dimensional space? I'll describe the history of this question, going back to Schwarz, Fuchs, Painlevé, and others, and some new answers to it, obtained jointly with Aaron Landesman. While our results rely on low-dimensional topology, Hodge theory, and the Langlands program, and we'll get a peek into how these areas come into play, no knowledge of them will be assumed.

October 14, 2022, Friday at 4pm Andrew Sageman-Furnas (North Carolina State)

(host: Mari-Beffa)

Constructing isometric tori with the same curvatures

Which data determine an immersed surface in Euclidean three-space up to rigid motion? A generic surface is locally determined by only an intrinsic metric and extrinsic mean curvature function. However, there are exceptions. These may arise in a family like the isometric family of vanishing mean curvature surfaces transforming a catenoid into a helicoid.

For compact surfaces, Lawson and Tribuzy proved in 1981 that a metric and non-constant mean curvature function determine at most one immersion with genus zero, but at most two compact immersions (compact Bonnet pairs) for higher genus. In this talk, we discuss our recent construction of the first examples of compact Bonnet pairs. It uses a local classification by Kamberov, Pedit, and Pinkall in terms of isothermic surfaces. Moreover, we describe how a structure-preserving discrete theory for isothermic surfaces and Bonnet pairs led to this discovery.

The smooth theory is joint work with Alexander Bobenko and Tim Hoffmann and the discrete theory is joint work with Tim Hoffmann and Max Wardetzky.

October 20, 2022, Thursday at 4pm, VV911 Simon Tavaré (Columbia University)

(host: Kurtz, Roch)

Note the unusual time and room!

An introduction to counts-of-counts data

Counts-of-counts data arise in many areas of biology and medicine, and have been studied by statisticians since the 1940s. One of the first examples, discussed by R. A. Fisher and collaborators in 1943 [1], concerns estimation of the number of unobserved species based on summary counts of the number of species observed once, twice, … in a sample of specimens. The data are summarized by the numbers C1, C2, … of species represented once, twice, … in a sample of size

N = C1 + 2 C2 + 3 C3 + ….  containing S = C1 + C2 + species; the vector C = (C1, C2, …) gives the counts-of-counts. Other examples include the frequencies of the distinct alleles in a human genetics sample, the counts of distinct variants of the SARS-CoV-2 S protein obtained from consensus sequencing experiments, counts of sizes of components in certain combinatorial structures [2], and counts of the numbers of SNVs arising in one cell, two cells, … in a cancer sequencing experiment.

In this talk I will outline some of the stochastic models used to model the distribution of C, and some of the inferential issues that come from estimating the parameters of these models. I will touch on the celebrated Ewens Sampling Formula [3] and Fisher’s multiple sampling problem concerning the variance expected between values of S in samples taken from the same population [3]. Variants of birth-death-immigration processes can be used, for example when different variants grow at different rates. Some of these models are mechanistic in spirit, others more statistical. For example, a non-mechanistic model is useful for describing the arrival of covid sequences at a database. Sequences arrive one at a time, and are either a new variant, or a copy of a variant that has appeared before. The classical Yule process with immigration provides a starting point to model this process, as I will illustrate.

References

[1] Fisher RA, Corbet AS & Williams CB. J Animal Ecology, 12, 1943

[2] Arratia R, Barbour AD & Tavaré S. Logarithmic Combinatorial Structures, EMS, 2002

[3] Ewens WJ. Theoret Popul Biol, 3, 1972

[4] Da Silva P, Jamshidpey A, McCullagh P & Tavaré S. Bernoulli Journal, in press, 2022 (online)

October 21, 2022, Friday at 4pm Ngoc Mai Tran (Texas)

(host: Rodriguez)

Forecast science, learn hidden networks and settle economics conjectures with combinatorics, geometry and probability.

In many problems, one observes noisy data coming from a hidden or complex combinatorial structure. My research aims to understand and exploit such structures to arrive at an efficient and optimal solution. I will showcase a few successes, achieved with different tools, from different different fields: networks forecasting, hydrology, and auction theory. Then I will outline some open questions in each field.

October 28, 2022, Friday at 4pm Qin Li (UW)

Multiscale inverse problem, from Schroedinger to Newton to Boltzmann

Inverse problems are ubiquitous. We probe the media with sources and measure the outputs, to infer the media information. At the scale of quantum, classical, statistical and fluid, we face inverse Schroedinger, inverse Newton’s second law, inverse Boltzmann problem, and inverse diffusion respectively. The universe, however, expects a universal mathematical description, as Hilbert proposed in 1900. In this talk, we discuss the connection between these problems. We will give arguments for justifying that these are the same problem merely represented at different scales. It is a light-hearted talk, and I will mostly focus on the story instead of the derivation. PDE background is appreciated but not necessary.

November 7, 2022, Monday at 4pm Kristen Lauter (Facebook)

Distinguished lectures

(host: Yang).

Private AI: Machine Learning on Encrypted Data

As the world adopts Artificial Intelligence, the privacy risks are many. AI can improve our lives, but may leak our private data. Private AI is based on Homomorphic Encryption (HE), a new encryption paradigm which allows the cloud to operate on private data in encrypted form, without ever decrypting it, enabling private training and private prediction. Our 2016 ICML CryptoNets paper showed for the first time that it was possible to evaluate neural nets on homomorphically encrypted data, and opened new research directions combining machine learning and cryptography. The security of Homomorphic Encryption is based on hard problems in mathematics involving lattices, recently standardized by NIST for post-quantum cryptography. This talk will explain Homomorphic Encryption, Private AI, and explain HE in action.

November 8, 2022, Tuesday at 4pm Kristen Lauter (Facebook)

Distinguished lectures in VV911. Note: unusual room.

(host: Yang).

Artificial Intelligence & Cryptography: Privacy and Security in the AI era

How is Artificial Intelligence changing your life and the world?  How do you expect your data to be kept secure and private in the future?  Artificial intelligence (AI) refers to the science of utilizing data to formulate mathematical models that predict outcomes with high assurance. Such predictions can be used to make decisions automatically or give recommendations with high confidence. Cryptography is the science of protecting the privacy and security of data.  This talk will explain the dynamic relationship between cryptography and AI and how AI can be used to attack post-quantum cryptosystems.

The first talk is based on my 2019 ICIAM Plenary Lecture and the second one is based on my 2022 SIAM Block Prize Lecture.

November 11, 2022, Friday at 4pm Joel Tropp (Caltech)

This is the Annual LAA lecture. See this for its history.

(host: Qin, Jordan)

Rocket-propelled Cholesky: Addressing the challenges of large-scale kernel computations

Kernel methods are used for prediction and clustering in many data science and scientific computing applications, but applying kernel methods to a large number of data points N is expensive due to the high cost of manipulating the N x N kernel matrix. A basic approach for speeding up kernel computations is low-rank approximation, in which we replace the kernel matrix A with a factorized approximation that can be stored and manipulated more cheaply. When the kernel matrix A has rapidly decaying eigenvalues, mathematical existence proofs guarantee that A can be accurately approximated using a constant number of columns (without ever looking at the full matrix). Nevertheless, for a long time designing a practical and provably justified algorithm to select the appropriate columns proved challenging.

Recently, we introduced RPCholesky ("randomly pivoted" or "rocket-propelled" Cholesky), a natural algorithm for approximating an N x N positive semidefinite matrix using k adaptively sampled columns. RPCholesky can be implemented with just a few lines of code; it requires only (k+1)N entry evaluations and O(k^2 N) additional arithmetic operations. In experiments, RPCholesky matches or improves on the performance of alternative algorithms for low-rank psd approximation. Moreover, RPCholesky provably achieves near-optimal approximation guarantees. The simplicity, effectiveness, and robustness of this algorithm strongly support its use for large-scale kernel computations.

Joint work with Yifan Chen, Ethan Epperly, and Rob Webber. Available at arXiv:2207.06503.

November 18, 2022, Friday at 4pm Jim Freitag (U of Illinois-Chicago) Zoom link: https://go.wisc.edu/jimfreitag

Now available: Recording and Slides

(hosts: Lempp, Andrews)

When any three solutions are independent

In this talk, we'll talk about a surprising recent result about the algebraic relations between solutions of a differential equation. The result has applications to functional transcendence, diophantine geometry, and compact complex manifolds.

November 21, 2022, Monday at 4pm Andrei Negut (MIT) Zoom link: https://go.wisc.edu/andreinegut

Hiring talk.

(hosts: Arinkin, Caldararu)

From gauge theory to geometric representation theory and back

We start from the celebrated construction (due to Grojnowski and Nakajima) of a Heisenberg algebra action on the cohomology groups of Hilbert schemes of points on surfaces

  1. replacing Hilbert schemes with moduli spaces of higher rank sheaves yields a computation of Nekrasov partition functions in 5d supersymmetric gauge theory, and a proof of the deformed Alday-Gaiotto-Tachikawa conjecture.
  2. replacing cohomology by Chow groups gives a proof of the Beauville conjecture in the hyperkahler geometry of Hilbert schemes of points on K3 surfaces (with Maulik)
  3. working with derived categories allows us to construct a detailed framework realizing categorical knot invariants in terms of the geometry of Hilbert schemes of points on the affine plane (with Gorsky and Rasmussen)

December 2, 2022, Friday at 4pm: Promit Ghosal (MIT)

Fractal Geometry of the KPZ equation

The Kardar-Parisi-Zhang (KPZ) equation is a fundamental stochastic PDE related to many important models like random growth processes, Burgers turbulence, interacting particles system, random polymers etc. In this talk, we focus on how the tall peaks and deep valleys of the KPZ height function grow as time increases. In particular, we will ask what is the appropriate scaling of the peaks and valleys of the (1+1)-d KPZ equation and whether they converge to any limit under those scaling. These questions will be answered via the law of iterated logarithms and fractal dimensions of the level sets. The talk will be based on joint works with Sayan Das and Jaeyun Yi. If time permits, I will also mention an interesting story about the  (2+1)-d and (3+1)-d case (work in progress with Jaeyun Yi).

December 5, 2022, Monday at 4pm: Di Fang (Berkeley)

(reserved by HC. contact: Stechmann)

December 7, 2022, Wednesday at 4pm: Benjamin Eichinger (Vienna University of Technology)

(reserved by HC. contact: Stechmann)

An approach to universality using Weyl m-functions

In this talk I will present an approach to universality limits for orthogonal polynomials on the real line which is completely local and uses only the boundary behavior of the Weyl m-function at the point. We show that bulk universality of the Christoffel-Darboux kernel holds for any point where the imaginary part of the m-function has a positive finite nontangential limit. This approach is based on studying this problem in the more general setting of canonical systems and the realization that bulk universality for an associated matrix reproducing kernel at a point is equivalent to the fact that the corresponding m-function has normal limits at the same point. The talk is based on a joint work with Milivoje Lukic and Brian Simanek. If time permits, I will discuss some work in progress with Milivoje Lukic and Harald Woracek on rescaling limits for other

universality classes.

December 9, 2022, Friday at 4pm: Dallas Albritton (Princeton)

(reserved by HC. contact: Stechmann)

December 12, 2022, Monday at 4pm: Laurel Ohm (Princeton)

(reserved by HC. contact: Stechmann)

Future Colloquia

Fall 2022

Spring 2023

Past Colloquia

Spring 2022

Fall 2021

Spring 2021

Fall 2020

Spring 2020

Fall 2019

Spring 2019

Fall 2018

Spring 2018

Fall 2017

Spring 2017

Fall 2016

Spring 2016

Fall 2015

Spring 2015

Fall 2014

Spring 2014

Fall 2013

Spring 2013

Fall 2012

WIMAW