[et_pb_section fb_built=”1″ admin_label=”Section” _builder_version=”4.18.0″ _module_preset=”default” custom_padding=”0px|||||” box_shadow_style=”preset2″ locked=”off” global_colors_info=”{}”][et_pb_row _builder_version=”4.18.0″ _module_preset=”default” min_height=”75px” custom_margin=”|auto|-53px|auto||” global_colors_info=”{}”][et_pb_column type=”4_4″ _builder_version=”4.18.0″ _module_preset=”default” global_colors_info=”{}”][\/et_pb_column][\/et_pb_row][et_pb_row admin_label=”Algebra and Number Theory” _builder_version=”4.17.4″ _module_preset=”default” width=”100%” max_width=”850px” custom_margin=”5px||15px||false|false” custom_padding=”15px|15px|15px|15px|false|false” border_radii=”on|8px|8px|8px|8px” border_width_bottom=”1px” border_color_bottom=”#A9A9A9″ box_shadow_style=”preset1″ box_shadow_spread=”-3px” box_shadow_color=”#C0C0C0″ locked=”off” global_colors_info=”{}”][et_pb_column type=”4_4″ _builder_version=”4.17.4″ _module_preset=”default” global_colors_info=”{}”][dsm_text_divider header=”Algebra and Number Theory” color=”RGBA(255,255,255,0)” _builder_version=”4.18.0″ _module_preset=”default” background_color=”RGBA(255,255,255,0)” global_colors_info=”{}”][\/dsm_text_divider][et_pb_text _builder_version=”4.18.0″ _module_preset=”default” hover_enabled=”0″ global_colors_info=”{}” sticky_enabled=”0″]A wide array of mathematical research consists of topics in the area of \u201cAlgebra.\u201d Subjects such as algebraic geometry, number theory, and representation theory are ubiquitous in all of mathematics, and have innumerable applications in the natural sciences, social sciences, computer science, and more. Our research covers number theory, representation theory, and differential algebra. Some specializations within these research areas that we are more focused on are the theory of elliptic curves, automorphic forms, the Langlands program, and the algebraic theory of differential equations. [\/et_pb_text][et_pb_text _builder_version=”4.27.4″ _module_preset=”default” hover_enabled=”0″ global_colors_info=”{}” sticky_enabled=”0″]<\/p>\n
Moshe Adrian<\/b><\/a><\/span><\/span>\u00a0Langlands Program. Number theory and representation theory<\/i><\/p>\n Delaram Kahrobaei<\/b><\/a><\/span>\u00a0Combinatorial and Computational Group Theory, Algebraic Post-Quantum Cryptography, Mathematics of Artificial Intelligence<\/i><\/p>\n Krysztof Klosin<\/b><\/a><\/span><\/span>\u00a0Automorphic Forms, Paramodular Conjecture, Bloch-Kato Conjecture, <\/span>Galois representations, special L-values<\/span><\/i><\/p>\n Kenneth Kramer<\/b><\/span> Abelian Varieties, Galois Representations<\/span><\/i><\/p>\n Alexey Ovchinnikov<\/b><\/a><\/span><\/span><\/span> Differential Galois Theory, Symbolic Computation<\/span><\/i><\/p>\n Maria Sabitova<\/b><\/a><\/span><\/span><\/span>\u00a0Algebraic Number Theory, Abelian Varieties<\/span><\/i><\/p>\n [\/et_pb_text][\/et_pb_column][\/et_pb_row][et_pb_row admin_label=”Combinatorics” _builder_version=”4.17.4″ _module_preset=”default” width=”100%” max_width=”850px” custom_margin=”5px||15px||false|false” custom_padding=”15px|15px|15px|15px|false|false” border_radii=”on|8px|8px|8px|8px” border_width_bottom=”1px” border_color_bottom=”#A9A9A9″ box_shadow_style=”preset1″ box_shadow_spread=”-3px” box_shadow_color=”#C0C0C0″ locked=”off” global_colors_info=”{}”][et_pb_column type=”4_4″ _builder_version=”4.17.4″ _module_preset=”default” global_colors_info=”{}”][dsm_text_divider header=”Combinatorics” color=”RGBA(255,255,255,0)” _builder_version=”4.18.0″ _module_preset=”default” global_colors_info=”{}”][\/dsm_text_divider][et_pb_text _builder_version=”4.18.0″ _module_preset=”default” global_colors_info=”{}”]<\/p>\n Combinatorics is the study of discrete objects and counting techniques. Counting questions arise in all areas of mathematics, especially in algebra, representation theory, and number theory. Combinatorics is also intricately entwined with computer science \u2013 both because of the discrete nature of data structures and because computer exploration allows combinatorists to generate data that informs proof techniques. We do research in enumerative and algebraic combinatorics, especially the study of flow polytopes, Ehrhart Theory, and affine Coxeter groups.<\/p>\n [\/et_pb_text][et_pb_text _builder_version=”4.27.4″ _module_preset=”default” hover_enabled=”0″ global_colors_info=”{}” sticky_enabled=”0″]Christopher Hanusa<\/a><\/strong><\/span><\/span> Algebraic and Enumerative Combinatorics, Mathematical Art<\/i>[\/et_pb_text][\/et_pb_column][\/et_pb_row][et_pb_row admin_label=”Complex Analysis The classical theory of functions of a complex variable was a crowning achievement of mathematics in the 19th and early 20th century. The modern theory continues that rich tradition by introducing new powerful tools and establishing deep connections with other fields of mathematics such as PDE’s, differential geometry, low-dimensional topology, and dynamics. A notable example is Teichm\u00fcller theory, the study of deformations of complex structures on Riemann surfaces, which has become a fertile area of research largely due the pioneering works of Ahlfors, Bers, Sullivan, and Thurston. Our work covers Riemann surfaces and Kleinian groups, Teichm\u00fcller theory, holomorphic motions, and quasiconformal mappings and their generalizations.<\/p>\n [\/et_pb_text][et_pb_text _builder_version=”4.27.4″ _module_preset=”default” hover_enabled=”0″ global_colors_info=”{}” sticky_enabled=”0″]<\/p>\n Yunping Jiang<\/b><\/a><\/span>\u00a0Smooth Dynamics, Holomorphic Dynamics, Teichm\u00fcller Theory<\/i><\/span><\/p>\n Sudeb Mitra<\/b> Teichm\u00fcller Theory, Riemann surfaces, Quasiconformal Mappings<\/i><\/span><\/p>\n Dragomir Saric<\/b><\/a><\/span> Teichm\u00fcller Theory, Hyperbolic Geometry, Quasiconformal Mappings<\/i><\/span><\/p>\n Saeed Zakeri<\/b><\/a><\/span><\/span>\u00a0Holomorphic Dynamics, (Trans-)Quasiconformal Mappings, Geometric Complex Analysis<\/i><\/span><\/p>\n [\/et_pb_text][\/et_pb_column][\/et_pb_row][et_pb_row admin_label=”Data Science and Statistics” _builder_version=”4.17.4″ _module_preset=”default” width=”100%” max_width=”850px” custom_margin=”5px||15px||false|false” custom_padding=”15px|15px|15px|15px|false|false” border_radii=”on|8px|8px|8px|8px” border_width_bottom=”1px” border_color_bottom=”#A9A9A9″ box_shadow_style=”preset1″ box_shadow_spread=”-3px” box_shadow_color=”#C0C0C0″ locked=”off” global_colors_info=”{}”][et_pb_column type=”4_4″ _builder_version=”4.17.4″ _module_preset=”default” global_colors_info=”{}”][dsm_text_divider header=”Data Science and Statistics” color=”RGBA(255,255,255,0)” _builder_version=”4.18.0″ _module_preset=”default” global_colors_info=”{}”][\/dsm_text_divider][et_pb_text _builder_version=”4.18.0″ _module_preset=”default” global_colors_info=”{}”]<\/p>\n Data science is the interdisciplinary study of extracting knowledge from noisy structured or unstructured raw data. It is a \u201cscience\u201d in the sense that it is now the backbone of building models that explain and predict phenomena of interest. These models are many times complex and built through the modern methods of artificial intelligence or machine learning. It is \u201cinterdisciplinary\u201d in the sense that a data scientist must be well-versed in skills from a vast toolbox drawn from mathematics, statistics, optimization, computing, algorithms, etc. as well as domain-specific and context-specific knowledge. Statistics is arguably the theoretical core of data science. Its main focus is on inference: estimating quantities of interest, assessing the certainty in these estimates and understanding when these estimates apply. Sometimes these quantities represent causal effects measured from randomized experiments. This last setting is the focus of the research in our department.<\/p>\n [\/et_pb_text][et_pb_text _builder_version=”4.27.4″ _module_preset=”default” hover_enabled=”0″ global_colors_info=”{}” sticky_enabled=”0″]<\/p>\n Adam Kapelner<\/b><\/a><\/span><\/span>\u00a0Machine Learning, Experimental Design<\/i><\/p>\n [\/et_pb_text][\/et_pb_column][\/et_pb_row][et_pb_row admin_label=”Dynamical Systems” _builder_version=”4.17.4″ _module_preset=”default” width=”100%” max_width=”850px” custom_margin=”5px||15px||false|false” custom_padding=”15px|15px|15px|15px|false|false” border_radii=”on|8px|8px|8px|8px” border_width_bottom=”1px” border_color_bottom=”#A9A9A9″ box_shadow_style=”preset1″ box_shadow_spread=”-3px” box_shadow_color=”#C0C0C0″ locked=”off” global_colors_info=”{}”][et_pb_column type=”4_4″ _builder_version=”4.17.4″ _module_preset=”default” global_colors_info=”{}”][dsm_text_divider header=”Dynamical Systems” color=”RGBA(255,255,255,0)” _builder_version=”4.18.0″ _module_preset=”default” global_colors_info=”{}”][\/dsm_text_divider][et_pb_text _builder_version=”4.18.0″ _module_preset=”default” global_colors_info=”{}”]<\/p>\n The mathematical theory of dynamical systems studies the long term behavior of idealized systems that evolve with time and predicts how their future depends on various parameters involved. It is a branch of pure mathematics with deep connections to many other areas, especially analysis, geometry, topology, probability theory, and combinatorics. It also has historical links and applications to fields outside mathematics such as celestial mechanics, meteorology, and population dynamics. An area of particular interest is holomorphic dynamics, the study of (semi-)groups generated by analytic transformations, where the introduction of powerful tools from complex analysis and conformal geometry has inspired remarkable progress in the past couple of decades. Our research covers analytic, topological, measure-theoretic, and combinatorial aspects of dynamical systems and their moduli spaces. Our main areas of work include the dynamics of rational and transcendental maps, structure of Julia sets and connectedness loci of polynomials, and ergodic theory of circle endomorphisms.<\/p>\n [\/et_pb_text][et_pb_text _builder_version=”4.27.4″ _module_preset=”default” hover_enabled=”0″ global_colors_info=”{}” sticky_enabled=”0″]<\/p>\n Yunping Jiang<\/b><\/a><\/span>\u00a0Smooth Dynamics, Holomorphic Dynamics, Teichm\u00fcller Theory<\/i><\/p>\n Saeed Zakeri<\/b><\/a><\/span><\/span>\u00a0Holomorphic Dynamics, (Trans-)Quasiconformal Mappings, Geometric Complex Analysis<\/i><\/p>\n [\/et_pb_text][\/et_pb_column][\/et_pb_row][et_pb_row admin_label=”Geometry and Topology Geometry and topology are the fundamental mathematical disciplines that study shapes and spaces. Geometry concerns quantitative aspects, such as length, area, and volume, as well as distance, angles, and curvature. Topology is a more qualitative study of space, in which geometric details, such as size, are de-emphasized so that other properties like dimension can be focused on. Both fields are organized around studying transformations and mappings that preserve certain structures and some of the deepest and most interesting results in geometry and topology reveal intricate algebraic objects working behind the scenes. The subjects are further enriched by connections with physics and other mathematical areas like real and complex analysis, number theory, and applied math.<\/p>\n [\/et_pb_text][et_pb_text _builder_version=”4.27.4″ _module_preset=”default” hover_enabled=”0″ global_colors_info=”{}” sticky_enabled=”0″]<\/p>\n Dan A. Lee<\/b><\/a><\/span>\u00a0Geometric Analysis<\/i><\/p>\n John Terilla<\/b><\/a><\/span> Algebraic Topology, Quantum Physics, Machine Learning<\/i><\/p>\n Yongwu Rong<\/b><\/a><\/span>\u00a0Low Dimensional Topology, Knot Theory, Quantum Invariants<\/i><\/p>\n Dragomir Saric<\/b><\/a><\/span>\u00a0Teichm\u00fcller Theory, Hyperbolic Geometry, Quasiconformal Mappings<\/i><\/p>\n Nicholas Vlamis<\/b><\/a><\/span>\u00a0Low Dimensional Topology, Hyperbolic Geometry, Teichm\u00fcller Theory<\/i><\/p>\n Scott Wilson<\/b><\/a><\/span><\/span>\u00a0Topology and Geometry of Manifolds<\/i><\/p>\n <\/p>\n [\/et_pb_text][\/et_pb_column][\/et_pb_row][et_pb_row admin_label=”Logic and Computability” _builder_version=”4.17.4″ _module_preset=”default” width=”100%” max_width=”850px” custom_margin=”5px||15px||false|false” custom_padding=”15px|15px|15px|15px|false|false” border_radii=”on|8px|8px|8px|8px” border_width_bottom=”1px” border_color_bottom=”#A9A9A9″ box_shadow_style=”preset1″ box_shadow_spread=”-3px” box_shadow_color=”#C0C0C0″ locked=”off” global_colors_info=”{}”][et_pb_column type=”4_4″ _builder_version=”4.17.4″ _module_preset=”default” global_colors_info=”{}”][dsm_text_divider header=”Logic and Computability” color=”RGBA(255,255,255,0)” _builder_version=”4.18.0″ _module_preset=”default” global_colors_info=”{}”][\/dsm_text_divider][et_pb_text _builder_version=”4.18.0″ _module_preset=”default” global_colors_info=”{}”]<\/p>\n Logic is the field that studies the axiomatic foundations of mathematics and the way mathematicians use the basic axioms to prove mathematical theorems. It divides into four main branches. Proof theory asks what we mean by \u201ca proof of a theorem.\u201d Set theory tries to determine the best collection of axioms to use for mathematics in general, making the common assumption that all of mathematics can be expressed in the language of sets. Model theory examines the relationship between syntax (the proofs that we write down) and semantics (actual mathematical structures such as groups or graphs or vector spaces). Computability theory examines the extent to which all of this can be done effectively, actually computing presentations of the objects in question rather than merely proving that they exist. Computability often intersects with theoretical computer science, while set theory and proof theory are of real interest to philosophers: thus logic often crosses over into computer science and philosophy, as well as into linguistics, probability, and many other academic disciplines.<\/p>\n [\/et_pb_text][et_pb_text _builder_version=”4.27.4″ _module_preset=”default” hover_enabled=”0″ global_colors_info=”{}” sticky_enabled=”0″]<\/p>\n Russell Miller<\/b><\/a><\/span><\/span>\u00a0Computability, Computable Structure Theory, Decidability in Number Theory<\/i><\/p>\n Alexey Ovchinnikov<\/b><\/a><\/span>\u00a0Differential Galois Theory, Symbolic Computation<\/i><\/p>\n [\/et_pb_text][\/et_pb_column][\/et_pb_row][et_pb_row admin_label=”Mathematical and Computational Biology” _builder_version=”4.17.4″ _module_preset=”default” width=”100%” max_width=”850px” custom_margin=”5px||15px||false|false” custom_padding=”15px|15px|15px|15px|false|false” border_radii=”on|8px|8px|8px|8px” border_width_bottom=”1px” border_color_bottom=”#A9A9A9″ box_shadow_style=”preset1″ box_shadow_spread=”-3px” box_shadow_color=”#C0C0C0″ locked=”off” global_colors_info=”{}”][et_pb_column type=”4_4″ _builder_version=”4.17.4″ _module_preset=”default” global_colors_info=”{}”][dsm_text_divider header=”Mathematical and Computational Biology” color=”RGBA(255,255,255,0)” _builder_version=”4.18.1″ _module_preset=”default” global_colors_info=”{}”][\/dsm_text_divider][et_pb_text _builder_version=”4.18.1″ _module_preset=”default” global_colors_info=”{}”]<\/p>\n Mathematical biology aims at analyzing mathematical representation and modeling of biological processes, using techniques and tools of pure and applied mathematics. This involves algebra, graph theory, numerical analysis, and probability and statistics, among many other areas of mathematics. Computational biology focuses on building and using computational models for problems in biology. Describing biological systems in a quantitative manner with precise models results in their behavior being better simulated, and hence properties can be predicted that might not be evident to the experimenter.<\/span><\/p>\n [\/et_pb_text][et_pb_text _builder_version=”4.27.4″ _module_preset=”default” hover_enabled=”0″ global_colors_info=”{}” sticky_enabled=”0″]<\/p>\n
\n” _builder_version=”4.17.4″ _module_preset=”default” width=”100%” max_width=”850px” custom_margin=”5px||15px||false|false” custom_padding=”15px|15px|15px|15px|false|false” border_radii=”on|8px|8px|8px|8px” border_width_bottom=”1px” border_color_bottom=”#A9A9A9″ box_shadow_style=”preset1″ box_shadow_spread=”-3px” box_shadow_color=”#C0C0C0″ locked=”off” global_colors_info=”{}”][et_pb_column type=”4_4″ _builder_version=”4.17.4″ _module_preset=”default” global_colors_info=”{}”][dsm_text_divider header=”Complex Analysis” color=”RGBA(255,255,255,0)” _builder_version=”4.18.0″ _module_preset=”default” global_colors_info=”{}”][\/dsm_text_divider][et_pb_text _builder_version=”4.18.0″ _module_preset=”default” global_colors_info=”{}”]<\/p>\n
\n” _builder_version=”4.17.4″ _module_preset=”default” width=”100%” max_width=”850px” custom_margin=”5px||15px||false|false” custom_padding=”15px|15px|15px|15px|false|false” border_radii=”on|8px|8px|8px|8px” border_width_bottom=”1px” border_color_bottom=”#A9A9A9″ box_shadow_style=”preset1″ box_shadow_spread=”-3px” box_shadow_color=”#C0C0C0″ locked=”off” global_colors_info=”{}”][et_pb_column type=”4_4″ _builder_version=”4.17.4″ _module_preset=”default” global_colors_info=”{}”][dsm_text_divider header=”Geometry and Topology” color=”RGBA(255,255,255,0)” _builder_version=”4.18.0″ _module_preset=”default” global_colors_info=”{}”][\/dsm_text_divider][et_pb_text _builder_version=”4.18.0″ _module_preset=”default” global_colors_info=”{}”]<\/p>\n