Graduate Analysis Seminar: How Jean Bourgain Revolutionized the Study of Dispersive PDEs
Abstract: Around the time that he won the Fields medal ('94) until about 2000, Jean Bourgain made remarkable progress in the theory of well-posedenss and global well-posedness in dispersive PDEs. Paper after paper Professor Bourgain established (for anyone else) career defining results whose effects are still very much felt to this day. In this talk, I want to motivate and go over some of his results from this period with the hope of impressing just how amazing (compared to prior results) some of his work was. The talk will be introductory (in a sense), with the hope of gently kicking off the seminar. As always, there will be cookies. |
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Combinatorics Colloquium: Degenerate Turan problems for graphs
Abstract: In Turan type extremal problems, we want to determine how dense a graph or hypergraph is without containing a particular subgraph or family of subgraphs. Such problems are central to extremal graph theory, because solving them requires one to thoroughly investigate the interaction of global graph parameters with local structures. Efforts in solving these problems have spurred the developments of some powerful tools in extremal graph theory, such as the regularity method, probabilistic and algebraic methods. While Turan problems have satisfactory solutions for non-bipartite graphs, the problem is still generally wide-open for bipartite graphs with many intriguing conjectures and results. In this talk, we will discuss some conjectures on Turan problems for bipartite graphs and some recent progress on them. Time permitting, we will also discuss a colored variant of the Turan problem. There will also be a presentation on "Tao Jiang's favorite problems" (September 9, 11 - 11:50 a.m., Room 141 Altgeld Hall). |
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Graph Theory and Combinatorics Seminar: Weighted Turan Numbers and Maximum Crossing Numbers of Trees
Abstract: In extremal graph theory, the most natural question to consider involves finding the most edges in an n-vertex graph that does not contain any copy of some small forbidden graph F. We will explore a generalization of this to edge weighted graphs in which the edge weights are induced by a vertex weighting according to some rule. We will solve this problem for cliques when the rule involves weighting each edge by the product or the minimum of the weights of the endpoints. The main motivation for the study of such problems is in applications to other combinatorial problems. In particular, we will use the product weighting to solve an extremal problem in which the n-vertex host graph is not complete, and we will use the minimum edge weighting to solve a problem involving the maximum rectilinear crossing number of trees. This project was joint work with Patrick Bennett and Maria Talanda-Fisher. |
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Probability Seminar: Information Projections On Banach Spaces With Applications To KL Weighted Control And A Feynman-Kac Formula For ODEs
Abstract: In this talk, we discuss a portmanteau theorem establishing the equivalence between information projections on a Banach space, constrained Kullback-Leibler weighted control, finding the mode of a measure through Onsager-Machlup formalism and in the classical Wiener space case, an Euler-Lagrange equation. As one example of an application of our theorem, we discuss a Feynman-Kac type formula, showing that the solution to a second order linear ODE (or system of ODEs) is the mode of a particular diffusion. Our portmanteau theorem along with our Feynman-Kac result provides numerics and insight for solving these ODEs. Joint work with William Haskell and Harsha Honnappa. |
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Student Cluster Algebra Seminar: Generalized Snake Graphs from Triangulated Orbifolds
Abstract: We will continue the talk from last week. Cluster algebras, as originally defined by Fomin and Zelevinsky, are characterized by binomial exchange relations. A natural generalization of cluster algebras, due to Chekhov and Shapiro, allows the exchange relations to have arbitrarily many terms. For generalized cluster algebras that can be modeled by unpunctured triangulated orbifolds, we generalize the snake graph construction of Musiker, Schiffler, and Williams and obtain explicit combinatorial formulas for the Laurent expansion of any arc or closed curve. For ordinary arcs, this gives a combinatorial proof of positivity for the associated generalized cluster algebra. This talk is based on joint work with Esther Banaian. Email kelleye@illinois.edu for Zoom link |
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Combinatorics (Add-on Event): Tao Jiang's favorite problems
Please see Cominatorics Colloquium description from September 7. |
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Mathematical Biology Seminar: Understanding the role of phenotypic switching in cancer drug resistance
Abstract: Recent findings suggest that cancer cells can acquire transient resistant phenotypes via epigenetic modifications and other non-genetic mechanisms. Although these resistant phenotypes are eventually relinquished by individual cells, they can temporarily ’save’ the tumor from extinction and enable the emergence of more permanent resistance mechanisms. These observations have generated interest in the potential of epigenetic therapies for long-term tumor control or eradication. In this talk, I will discuss some mathematical models for exploring how phenotypic switching at the single-cell level affects resistance evolution in cancer. As an example, we will explore the role of MGMT promoter methylation in driving resistance to temozolomide in glioblastoma. |
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Symplectic & Poisson Geometry Seminar: Around Viterbo's conjecture
Abstract: A conjecture of Claude Viterbo, from 1997, asserts that the symplectic capacity of a convex body should be bounded from above by its volume, when both are suitably normalized. This conjecture is of significant current interest, in part because its validity is now known to imply the Mahler conjecture in convex geometry. In this talk I will describe the proof of a weaker form of the inequality in which the role of the volume is played by a symplectic version of the mean-width. Time permitting, I will also describe some open problems suggested by this work. |
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Graph Theory and Combinatorics Seminar: Progress on pursuit-evasion games on graphs
Abstract: In pursuit-evasion games, a set of pursuers attempts to locate, eliminate, or contain an evader in a network. The rules, specified from the outset, greatly determine the difficulty of the questions posed above. For example, the evader may be visible, but the pursuers may have limited movement speed, only moving to nearby vertices adjacent to them. Central to pursuit-evasion games is the idea of optimizing certain parameters, whether they are the search number, burning number, or localization number, for example. We report on progress in several pursuit-evasion games on graphs and conjectures arising from their analysis. Finding the values, bounds, and algorithms to compute these graph parameters leads to topics intersecting graph theory, the probabilistic method, and geometry. |
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Algebra, Geometry & Combinatorics: Grothendieck-to-Lascoux Expansions
Abstract: We establish the conjecture of Reiner and Yong for an explicit combinatorial formula for the expansion of a Grothendieck polynomial into the basis of Lascoux polynomials. This expansion is a subtle refinement of its symmetric function version due to Buch, Kresch, Shimozono, Tamvakis, and Yong, which gives the expansion of stable Grothendieck polynomials indexed by permutations into Grassmannian stable Grothendieck polynomials. Our expansion is the K-theoretic analogue of that of a Schubert polynomial into Demazure characters, whose symmetric analogue is the expansion of a Stanley symmetric function into Schur functions. Our expansions extend to flagged Grothendieck polynomials. This is a joint work with Mark Shimozono. |
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Number Theory Seminar: Arithmetic properties of weakly holomorphic modular functions of arbitrary level
Abstract: The canonical basis of the space of modular functions on the modular group of genus zero form a Hecke system. From this fact, many important properties of modular functions were derived. Recently, we have proved that the Niebur-Poincare basis of the space of Harmonic Maass functions also forms a Hecke system. In this talk, we show its applications, including divisibility of Fourier coefficients of modular functions of arbitrary level, higher genus replicability, and values of modular functions on divisors of modular forms. |
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Graph Theory & Combinatorics: Linear Bounds for Cycle-free Saturation Games
Abstract: Given a family of graphs F, we define the F-saturation game as follows. Two players alternate adding edges to an initially empty graph on n vertices, with the only constraint being that neither player can add an edge that creates a subgraph in F. The game ends when no more edges can be added to the graph. One of the players wishes to end the game as quickly as possible, while the other wishes to prolong the game. We will consider the number of edges that are in the final graph when both players play optimally. In general there are very few non-trivial bounds on the order of magnitude of the number of edges in the final graph. In this talk, we discuss collections of infinite families of cycles such that the number of edges that are in the final graph has linear growth rate. This is joint work with Sean English, Tomáš Masařik, Erin Meger, Michael Ross, and Sam Spiro. |
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CAS/MillerComm: The Crochet Coral Reef: Intersections of Math, Science and Art through Hyperbolic Crochet
Abstract: Margaret Wertheim is a writer, artist and curator whose work intersects art, math and science as evidenced in her Crochet Coral Reef, an installation created through hyperbolic crochet. In this lecture Margaret will introduce the UIUC community to her work with the Institute for Figuring, a collaborative Los Angeles based practice started with Margaret's sister Christine in 2003. Devoted to the “aesthetic and poetic dimensions of science and mathematics,” the IFF has designed art & science exhibits for galleries and museums around the world. Margaret and Christine’s Crochet Coral Reef project is a global participatory art & science endeavor that has been seen by more than two million people and exhibited at the 2019 Venice Biennale. Margaret will discuss the interplay of art, science, and art as social practice in the Crochet Coral Reef project while promoting UIUC's very own locally made satellite reef, opening at the Siebel Center for Design September 23, 2021. |
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Undergraduate Friday Seminar: When are two knots the same?
Abstract: It is easy to tie knots, but hard to tell knots apart. As such, mathematicians have developed many tools and techniques to distinguish knots. In this talk, I discuss elementary knot invariants and introduce the Jones Polynomial as a powerful, combinatorial tool to differentiate many knots. |
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Graph Theory and Combinatorics Seminar: Large monochromatic components in expansive r-uniform hypergraphs
Abstract: An r-partite hole of size k in an r-uniform hypergraph H is a collection of pairwise disjoint vertex subsets V_1, ..., V_r, all of size k such that no edge touches each of V_1,..., V_r. Let a_r(H) be the largest size of an r-partite hole in H. We determine a relationship between a_r(H) and the order of the largest monochromatic component in an arbitrary edge coloring of H. We discuss some implications for random graphs and hypergraphs as well as random Steiner triple systems. Joint work with Louis DeBiasio. |
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Combinatorics Colloquium: Flag algebras and its applications
Abstract: Flag algebras is a method, developed by Razborov, to attack problems in extremal combinatorics. Razborov formulated the method in a very general way which made it applicable to various settings. The method was introduced in 2007 and since then its applications have led to solutions or significant improvements of best bounds on many long-standing open problems, including problems of Erd\H{o}s. The main contribution of the method was transferring problems from finite settings to limits settings. This allows for clean calculations ignoring lower order terms. The method can utilize semidefinite programming and computers to produce asymptotic results. This is often followed by stability type arguments with the goal of obtaining exact results. In this talk we will give a brief introduction of the basic notions used in flag algebras and demonstrate how the method works. Then we will discuss applications of the flag algebras in different settings. Additional event: Friday, October 1, 1:00-1:50 p.m. AH 447; the speakers talk with the students about their favorite problems. |
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Graduate Student Homotopy Theory Seminar: On the Lichtenbaum-Quillen conjectures in algebraic K theory
Abstract: Starting with some motivations and brief expositions on algebraic K theory, I’ll introduce some early important computations of algebraic K theory, including computations of K theory of finite fields and of rings of integers for which I will briefly outline the proofs. Then we’ll move on to K theory with finite coefficients of separably closed fields. With the motivation of recovering some information of K theory of an arbitrary field from its separable closure, we introduce a few versions of the Lichtenbaum-Quillen conjectures as descent spectrum sequences of etale Cohomology groups. If time permits, I’ll mention relation to motivic Cohomology that a key tool is some “motivic-to-K-theory” spectral sequence. |
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Symplectic and Poisson Geometry Seminar: Linearization of Poisson groupoids
Abstract: Motivated by the search for Lie group structures on groups of Poisson diffeomorphisms, we investigate linearizability of the Poisson structure of a Poisson groupoid around the unit section, and present some results in that direction. Our approach revolves around 'lifting' symplectic- and Poisson geometry to Lie algebroids: we will encounter an algebroid version of Weinstein's Lagrangian neighborhood theorem, and the integration of an algebroid-Poisson structure to an algebroid-symplectic groupoid. |
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Graph Theory and Combinatorics Seminar: Maximum determinant and permanent of sparse 0-1 matrices
Abstract: We prove that the maximum determinant of an $n \times n $ matrix, with entries in $\{0,1\}$ and at most $n+k$ non-zero entries, is at most $2^{k/3}$, which is best possible when $k$ is a multiple of 3. This result solves a conjecture of Bruhn and Rautenbach. We also obtain an upper bound on the number of perfect matchings in $C_4$-free bipartite graphs based on the number of edges, which, in the sparse case, improves on the classical Bregman's inequality for permanents. This bound is tight, as equality is achieved by the graph formed by vertex disjoint union of 6-vertex cycles. |
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Probability Seminar: Stein’s method for Conditional Central Limit Theorem
Abstract: It is common in probability theory and statistics to study distributional convergences of sums of random variables conditioned on another such sum. In this talk I will present a novel approach using Stein’s method for exchangeable pairs that allows to derive conditional central limit theorem of the form $(X_n | Y_n=k)$ with explicit rate of convergence as well as its extensions to multidimensional setting. We will apply these results to particular models including pattern count in a random binary sequence and subgraph count in Erdös-Rényi random graph. This talk is based on joint work with Partha S. Dey https://arxiv.org/abs/2109.09274. |
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Mathematics Colloquium/Combinatorics Colloquium: A problem on divisors of binomial coefficients, and a theorem on noncontractibility of coset posets
Abstract: Fix an integer n>1. It follows directly from a theorem of Kummer that the greatest common divisor of the members of the set BC(n) nontrivial binomial coefficients nC1,nC2,...nC(n-1) is one unless n is a prime power. With this in mind, we define b(n) to be the smallest size of a set P of primes such that every member of BC(n) is divisible by at least one member of P. In joint work with Russ Woodroofe, we ask whether b(n) is at most two for every n. The question remains open. I will discuss what we know about this question, and how we discovered it during our investigation of a problem raised by Ken Brown about certain topological spaces: Given a finite group G, let C(G) the set of all cosets of all proper subgroups of a finite group, partially ordered by containment. The order complex of C(G) is the simplicial complex whose k-dimensional faces are chains of size k+1 from C(G). We show that this order complex has nontrivial reduced homology in characteristic two, and is therefore not contractible. If time permits, I will discuss also related work on invariable generation of simple groups, joint with Bob Guralnick and Russ Woodroofe. |
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Graduate Student Homotopy Theory Seminar: Simple Localizations and How to Find Them
Abstract: Abstractly defining ∞-categorical localization is easy, but explicitly constructing it is hard. Following a series of papers by Dwyer and Kan, I will describe the construction known as the hammock localization and use it to obtain a clearer picture of some important ∞-categories. |
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Undergraduate Friday Seminar: Distribution of Square Prime Numbers
Abstract: A number that is the product of a prime and square (square not 1) is defined as a square prime number. We study the asymptotic distribution, cardinality, and some other properties of these numbers, based on a recently published paper of mine. |
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Symplectic and Poisson Geometry Seminar: On the integration of transitive Lie algebroids
Abstract: We'll revisit the problem of integrating Lie algebroids A to Lie groupoids, for the special case that the Lie algebroid A is transitive. We obtain a geometric explanation of the Crainic-Fernandes obstructions for this situation, and an explicit construction of the integration whenever these obstructions vanish. Based on arXiv:2007.07120 |
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Graph Theory and Combinatorics Seminar: Clique packings in random graphs
Abstract: Let u(n,p,k) be the k-clique packing number of the random graph G(n,p), that is, the maximum number of edge-disjoint k-cliques in G(n,p). In 1992, Alon and Spencer conjectured that for p = 1/2 we have u(n,p,k) = Ω(n²/k²) when k = k(n,p) - 4, where k(n,p) is minimum t such that the expected number of t-cliques in G(n,p) is less than 1. This conjecture was disproved a few years ago by Acan and Kahn, who showed that in fact u(n,p,k) = O(n²/k³) with high probability, for any fixed p ∈ (0,1) and k = k(n,p) - O(1). In this talk, we show a lower bound which matches Acan and Kahn's upper bound on u(n,p,k) for all p ∈ (0,1) and k < k(n,p) - 3. To prove this result, we follow a random greedy process and use the differential equation method. This is a joint work with Simon Griffiths and Rob Morris. Contact Sean at SEnglish (at) illinois (dot) edu for Zoom information. |
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AWM Teaching & Diversity Seminar: Graduate Student Panel on Summer Opportunities
Abstract: Learn about various opportunities available for graduate students over the summer, and talk to people about their experiences! |
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Mathematical Biology Seminar: Flux in small noise dynamics: negative resistance and persistence
Abstract: Many biological and physical systems are well-modeled by Brownian particles subject to gradient dynamics plus noise. Important for many applications is the net steady-state particle current or "flux" enabled by the noise and an additional driving force, but this flux is rarely computable analytically. Motivated by this, I will describe joint work with Yuliy Baryshnikov investigating the steady-state flux of nondegenerate diffusion processes on compact manifolds. Such a flux is associated to each one-dimensional real cohomology class and is equivalent to an asymptotic winding rate of trajectories. When the deterministic part of the dynamics is "gradient-like" in a certain sense, I will describe a graph-theoretic formula for the small-noise asymptotics of the flux (in the sense of large deviations). When additionally the deterministic part is locally gradient and close to a generic global gradient, there is a natural flux for which the graph-theoretic formula becomes Morse-theoretic and admits a description in terms of persistent homology. As an application, I will explain the paradoxical "negative resistance" phenomenon in Brownian transport discovered by Cecchi and Magnasco (1996). |
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Algebra, Geometry & Combinatorics: The canonical bijection between pipe dreams and bumpless pipe dreams
Abstract: Pipe dreams and bumpless pipe dreams are two combinatorial objects that enumerate Schubert polynomials, and it has been an open problem to find a weight-preserving bijection between these two objects since bumpless pipe dreams were introduced by Lam, Lee and Shimozono. In this talk, we present such a bijection and establish its canonical nature by showing that it preserves Monk's rule. This is joint work with Daoji Huang. |
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Undergraduate Friday Seminar: Fixing the Cross Product: An Introduction to Geometric Algebra
Abstract: Although vectors are a fundamental object of study in mathematics and physics, it is surprisingly difficult to find a good way to multiply them. The dot and cross products commonly taught in schools work for many computational purposes, but they lack several key properties we might desire or expect from a vector product. Fortunately, there's a solution to these problems! In this talk we will explore geometric algebra, which draws numerous algebraic models of space together into a unified framework, elegantly providing vast generalizations of vector algebra and vector calculus while simultaneously offering deeper insight into the geometry of vectors, complex numbers, and more. Minimal background will be assumed beyond some prior exposure to vectors. |
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Department Picnic
Ambucs Park, Jean Driscoll Pavilion is off of University Avenue in east Urbana. The department will be providing food from Caribbean Grill. We encourage everyone to bring a dish to share. |